RMD_G6
Estudio de Análisis Multivariado con base en un conjunto de datos sobre factores de rendimiento académico.
jorge.eliecer.franco::santiago.osorio.ramirez::pineda.luis::(correounivalle.edu.co?)
Estudio hecho durante el periodo académico agosto-diciembre de 2024
Fase 1 [Descripciones Multivariantes]
1.1. Objetivos
En esta primera fase de análisis, se realizarán cálculos, visualizaciones e interpretaciones basadas en un conjunto de datos, utilizando un enfoque de estadística descriptiva multivariante. Además, se aplicará una prueba estadística de normalidad multivariada sobre los datos analizados.
1.2. Descripción de los datos
Fuente del Conjunto de Datos
El conjunto de datos de trabajo se obtuvo casi totalmente de Kaggle:https://www.kaggle.com/datasets/lainguyn123/student-performance-factors. Es conveniente anotar que Kaggle es una compañía subsidiaria de Google LLC que mantiene una comunidad en línea de científicos de datos y profesionales del aprendizaje automático. Esta empresa permite a sus usuarios encontrar y publicar conjuntos de datos, explorar y crear modelos en un entorno de ciencia de datos basado en la web, trabajar con otros científicos de datos e ingenieros de aprendizaje automático y participar en concursos para resolver desafíos de ciencia de datos.
Contexto del Conjunto de Datos
El conjunto de datos incluye métricas académicas obtenidas por estudiantes para tener una visión general de los factores que afectan el rendimiento en los exámenes. Este conjunto de datos se actualizó por última vez en septiembre de 2024
Descripción del Conjunto de Datos
El conjunto de datos original incluye 20 campos y 6,607 registros, una vez depurado incluye 21 campos y 6378 registros. Entre estos campos, se destacan: las horas dedicadas al estudio, la edad de los estudiantes, el porcentaje de asistencia a clases, y dos variables de naturaleza politómica. También se recoge si el estudiante participa en actividades extracurriculares, la cantidad de horas de sueño, y las calificaciones obtenidas en exámenes previos. Otro campo es de tipo politómico, mientras que el acceso a internet se clasifica de manera dicotómica. El número de tutorías mensuales a las que asiste el estudiante también está registrado, seguido por otros campos de naturaleza politómica. Además, se incluye el tipo de escuela, la influencia parental (politómica), las horas de actividad física por semana, si el estudiante presenta alguna discapacidad de aprendizaje, la distancia del hogar (politómica), el género del estudiante, y la puntuación en el examen final. Los campos se describen en el mismo orden en que aparecen en el rango de datos.
Hours studied (cuantitativa::razón): Número total de horas que el estudiante dedica al estudio por semana. Este valor refleja la cantidad de tiempo invertido en actividades académicas fuera del horario escolar, como tareas, repaso de contenidos y preparación para exámenes.
Age (cuantitativa::razón): Representa la cantidad de años que tiene cada estudiante al momento de realizar la medición. Se uso una funcion ALEATORIO.ENTRE en excel para darle un rango de entre 16 y 18 años
Attendance (cuantitativa::razón): Porcentaje de clases a las que el estudiante ha asistido en relación con el total de clases impartidas. Este indicador puede ser un reflejo del compromiso del estudiante con el curso.
Parental involvement (cualitativa::nominal): Nivel de involucramiento de los padres en la educación del estudiante. Se clasifica en tres niveles (Bajo, Medio, Alto), y puede incluir aspectos como supervisión de tareas, asistencia a reuniones escolares y apoyo en la toma de decisiones académicas.
Access to resources (cualitativa::nominal): Disponibilidad de recursos académicos como libros, materiales de estudio, herramientas tecnológicas, etc. Este atributo puede influir en la capacidad del estudiante para realizar trabajos y estudiar. Se clasifica en Bajo, Medio y Alto.
Extracurricular activities (cualitativa::nominal): Indica si el estudiante participa o no en actividades fuera del currículo académico, como deportes, música, arte o clubes académicos. Puede tener impacto en su desarrollo integral y rendimiento académico.
Sleep hours (cuantitativa::razón): Promedio de horas de sueño que el estudiante obtiene cada noche. El sueño es esencial para la concentración, el rendimiento académico y la salud mental.
Previous score (cuantitativa::razón): Calificaciones o notas obtenidas por el estudiante en exámenes anteriores. Este atributo puede reflejar el historial de rendimiento académico y es un posible predictor de su desempeño futuro.
Motivation level (cualitativa::ordinal): Grado de interés y esfuerzo que el estudiante muestra hacia sus estudios. Clasificado en Bajo, Medio y Alto, este atributo mide la disposición del estudiante para aprender y mejorar.
Internet access (cualitativa::nominal): Indica si el estudiante tiene acceso a Internet en su hogar (Sí, No), lo que puede ser crucial para realizar investigaciones, completar tareas en línea y acceder a materiales de apoyo educativo.
Tutoring sessions (cuantitativa::razón): Número de sesiones de tutoría a las que el estudiante asiste mensualmente. Las tutorías pueden ser de apoyo académico, reforzando áreas en las que el estudiante tenga dificultades.
Family income (cualitativa::nominal): Nivel de ingresos económicos del hogar del estudiante, clasificado en Bajo, Medio y Alto. Este factor puede influir en la disponibilidad de recursos y oportunidades educativas.
Techer quality (cualitativa::ordinal): Percepción del estudiante sobre la calidad de sus profesores, clasificado en Bajo, Medio y Alto. Incluye aspectos como el dominio de los contenidos, la metodología de enseñanza y la capacidad para motivar a los estudiantes.
School type (cualitativa::nominal): Categoría que indica si el estudiante asiste a una escuela pública o privada, lo que puede influir en el acceso a recursos, el tamaño de las clases y la calidad de la enseñanza.
Peer influence (cualitativa::ordinal): Impacto de las acciones y actitudes de los padres en el rendimiento académico del estudiante. Se clasifica en Positiva, Neutral o Negativa y refleja el apoyo, presión o indiferencia por parte de los padres hacia el progreso académico del estudiante.
Physical activity (cuantitativa::razón): Promedio de horas dedicadas a la actividad física semanalmente, lo que puede influir en la salud física y mental del estudiante, afectando indirectamente su rendimiento académico.
Learning disabilities (cualitativa::nominal): Indica si el estudiante tiene alguna discapacidad de aprendizaje (Sí, No), como dislexia, TDAH u otros trastornos que podrían afectar su rendimiento en ciertas áreas académicas.
Parental education level (cualitativa::ordinal): El nivel educativo más alto alcanzado por los padres (Secundaria, Universidad, Posgrado), lo que puede influir en el apoyo académico y el entorno de aprendizaje en el hogar.
Distance from home (cualitativa::ordinal): Distancia entre el hogar del estudiante y la escuela, clasificado en Cerca, Moderada y Lejos. La distancia puede afectar la puntualidad, el tiempo disponible para el estudio y el nivel de cansancio del estudiante.
Gender (cualitativa::nominal): El género del estudiante (Masculino, Femenino), que podría ser un factor de interés para analizar diferencias en el rendimiento académico o en la participación en actividades escolares.
Exam score (cuantitativa::razón): Calificación final obtenida por el estudiante en un examen específico, que refleja su rendimiento en esa evaluación en particular.
Estructura del Conjunto de Datos Original
str(cdd_factores_rendimiento_estudiantes_G6)## tibble [6,607 × 20] (S3: tbl_df/tbl/data.frame)
## $ Hours_Studied : num [1:6607] 23 19 24 29 19 19 29 25 17 23 ...
## $ Attendance : num [1:6607] 84 64 98 89 92 88 84 78 94 98 ...
## $ Parental_Involvement : chr [1:6607] "Low" "Low" "Medium" "Low" ...
## $ Access_to_Resources : chr [1:6607] "High" "Medium" "Medium" "Medium" ...
## $ Extracurricular_Activities: chr [1:6607] "No" "No" "Yes" "Yes" ...
## $ Sleep_Hours : num [1:6607] 7 8 7 8 6 8 7 6 6 8 ...
## $ Previous_Scores : num [1:6607] 73 59 91 98 65 89 68 50 80 71 ...
## $ Motivation_Level : chr [1:6607] "Low" "Low" "Medium" "Medium" ...
## $ Internet_Access : chr [1:6607] "Yes" "Yes" "Yes" "Yes" ...
## $ Tutoring_Sessions : num [1:6607] 0 2 2 1 3 3 1 1 0 0 ...
## $ Family_Income : chr [1:6607] "Low" "Medium" "Medium" "Medium" ...
## $ Teacher_Quality : chr [1:6607] "Medium" "Medium" "Medium" "Medium" ...
## $ School_Type : chr [1:6607] "Public" "Public" "Public" "Public" ...
## $ Peer_Influence : chr [1:6607] "Positive" "Negative" "Neutral" "Negative" ...
## $ Physical_Activity : num [1:6607] 3 4 4 4 4 3 2 2 1 5 ...
## $ Learning_Disabilities : chr [1:6607] "No" "No" "No" "No" ...
## $ Parental_Education_Level : chr [1:6607] "High School" "College" "Postgraduate" "High School" ...
## $ Distance_from_Home : chr [1:6607] "Near" "Moderate" "Near" "Moderate" ...
## $ Gender : chr [1:6607] "Male" "Female" "Male" "Male" ...
## $ Exam_Score : num [1:6607] 67 61 74 71 70 71 67 66 69 72 ...Conjunto de Datos Original
cdd_factores_rendimiento_estudiantes_G6## # A tibble: 6,607 × 20
## Hours_Studied Attendance Parental_Involvement Access_to_Resources
## <dbl> <dbl> <chr> <chr>
## 1 23 84 Low High
## 2 19 64 Low Medium
## 3 24 98 Medium Medium
## 4 29 89 Low Medium
## 5 19 92 Medium Medium
## 6 19 88 Medium Medium
## 7 29 84 Medium Low
## 8 25 78 Low High
## 9 17 94 Medium High
## 10 23 98 Medium Medium
## # ℹ 6,597 more rows
## # ℹ 16 more variables: Extracurricular_Activities <chr>, Sleep_Hours <dbl>,
## # Previous_Scores <dbl>, Motivation_Level <chr>, Internet_Access <chr>,
## # Tutoring_Sessions <dbl>, Family_Income <chr>, Teacher_Quality <chr>,
## # School_Type <chr>, Peer_Influence <chr>, Physical_Activity <dbl>,
## # Learning_Disabilities <chr>, Parental_Education_Level <chr>,
## # Distance_from_Home <chr>, Gender <chr>, Exam_Score <dbl>Estructura del Conjunto de Datos Depurado
str(cdd_factores_rendimiento_estudiantes_G6_dep)## tibble [6,378 × 21] (S3: tbl_df/tbl/data.frame)
## $ hours_studied : num [1:6378] 23 19 24 29 19 19 29 25 17 23 ...
## $ age : num [1:6378] 17 18 16 18 16 16 18 16 17 16 ...
## $ attendance : num [1:6378] 84 64 98 89 92 88 84 78 94 98 ...
## $ parental_involvement : chr [1:6378] "Low" "Low" "Medium" "Low" ...
## $ access_to_resources : chr [1:6378] "High" "Medium" "Medium" "Medium" ...
## $ extracurricular_activities: num [1:6378] 0 0 1 1 1 1 1 1 0 1 ...
## $ sleep_hours : num [1:6378] 7 8 7 8 6 8 7 6 6 8 ...
## $ previous_scores : num [1:6378] 73 59 91 98 65 89 68 50 80 71 ...
## $ motivation_level : chr [1:6378] "Low" "Low" "Medium" "Medium" ...
## $ internet_access : num [1:6378] 1 1 1 1 1 1 1 1 1 1 ...
## $ tutoring_sessions : num [1:6378] 0 2 2 1 3 3 1 1 0 0 ...
## $ family_income : chr [1:6378] "Low" "Medium" "Medium" "Medium" ...
## $ teacher_quality : chr [1:6378] "Medium" "Medium" "Medium" "Medium" ...
## $ school_type : chr [1:6378] "Public" "Public" "Public" "Public" ...
## $ peer_influence : chr [1:6378] "Positive" "Negative" "Neutral" "Negative" ...
## $ physical_activity : num [1:6378] 3 4 4 4 4 3 2 2 1 5 ...
## $ learning_disabilities : num [1:6378] 0 0 0 0 0 0 0 0 0 0 ...
## $ parental_education_level : chr [1:6378] "High School" "College" "Postgraduate" "High School" ...
## $ distance_from_home : chr [1:6378] "Near" "Moderate" "Near" "Moderate" ...
## $ gender : num [1:6378] 1 0 1 1 0 1 1 1 1 1 ...
## $ exam_score : num [1:6378] 67 61 74 71 70 71 67 66 69 72 ...Conjunto de Datos Depurado
cdd_factores_rendimiento_estudiantes_G6_dep## # A tibble: 6,378 × 21
## hours_studied age attendance parental_involvement access_to_resources
## <dbl> <dbl> <dbl> <chr> <chr>
## 1 23 17 84 Low High
## 2 19 18 64 Low Medium
## 3 24 16 98 Medium Medium
## 4 29 18 89 Low Medium
## 5 19 16 92 Medium Medium
## 6 19 16 88 Medium Medium
## 7 29 18 84 Medium Low
## 8 25 16 78 Low High
## 9 17 17 94 Medium High
## 10 23 16 98 Medium Medium
## # ℹ 6,368 more rows
## # ℹ 16 more variables: extracurricular_activities <dbl>, sleep_hours <dbl>,
## # previous_scores <dbl>, motivation_level <chr>, internet_access <dbl>,
## # tutoring_sessions <dbl>, family_income <chr>, teacher_quality <chr>,
## # school_type <chr>, peer_influence <chr>, physical_activity <dbl>,
## # learning_disabilities <dbl>, parental_education_level <chr>,
## # distance_from_home <chr>, gender <dbl>, exam_score <dbl>1.3. Estimaciones multivariadas
Este análisis explora el rendimiento académico de estudiantes a través de un conjunto de datos que incluye múltiples factores, como horas de estudio, asistencia, acceso a recursos, actividades extracurriculares, horas de sueño, nivel de motivación, y diversas variables familiares y escolares. Utilizando estimaciones multivariadas, se pueden examinar las interacciones entre estas variables y su relación conjunta con el rendimiento académico, medido aquí por la calificación final del estudiante. Este enfoque permite identificar patrones y factores clave que podrían influir en el desempeño de los estudiantes, considerando las complejidades inherentes a la interrelación de variables académicas, familiares y personales.
El conjunto de datos incluye tanto variables cuantitativas, como las horas de estudio, asistencia y puntajes previos, como cualitativas, tales como el nivel de motivación y la calidad percibida de la enseñanza. Estas variables se interrelacionan y, mediante técnicas multivariadas, se puede estudiar cómo diferentes factores, desde la implicación parental y los ingresos familiares hasta las actividades físicas y los patrones de sueño, influyen en el rendimiento académico. Por ejemplo, al analizar el impacto de la “Motivación” junto con “Asistencia” y “Acceso a Recursos”, se puede obtener una visión más detallada de cómo el ambiente y los hábitos del estudiante afectan su desempeño. Técnicas como la regresión múltiple o el análisis de factores pueden ayudar a extraer relaciones clave y predictores del éxito académico en este contexto multidimensional.
Las estimaciones multivariadas aplicadas a este conjunto de datos permiten desarrollar una comprensión profunda de cómo diversos factores personales, académicos y familiares contribuyen conjuntamente al rendimiento de los estudiantes. Este enfoque no solo identifica los factores individuales más influyentes, sino que también revela cómo las interacciones entre ellos pueden fortalecer o debilitar el rendimiento académico. Estas conclusiones podrían servir para diseñar intervenciones específicas que apoyen el desarrollo académico, destacando la importancia de un análisis holístico que considere las múltiples dimensiones que afectan la educación.
Vector de Medias y Boxplot
apply(cdd_factores_rendimiento_estudiantes_G6_dep[-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], 2, mean)## hours_studied attendance sleep_hours previous_scores
## 19.977109 80.020853 7.034964 75.066165
## tutoring_sessions physical_activity exam_score
## 1.495296 2.972719 67.251960cdd_factores_rendimiento_estudiantes_Reducido = cdd_factores_rendimiento_estudiantes_G6_dep [,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)]
par(mfrow = c(1, ncol(cdd_factores_rendimiento_estudiantes_Reducido)))
invisible(lapply(1:ncol(cdd_factores_rendimiento_estudiantes_Reducido), function(i) boxplot(cdd_factores_rendimiento_estudiantes_Reducido[, i])))
Matriz de Varianzas-Covarianzas
round(cov(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)]),2)## hours_studied attendance sleep_hours previous_scores
## hours_studied 35.83 -0.35 0.12 1.94
## attendance -0.35 133.42 -0.33 -3.01
## sleep_hours 0.12 -0.33 2.16 -0.49
## previous_scores 1.94 -3.01 -0.49 207.37
## tutoring_sessions -0.07 0.17 -0.02 -0.23
## physical_activity 0.02 -0.27 0.00 -0.10
## exam_score 10.43 26.23 -0.10 9.82
## tutoring_sessions physical_activity exam_score
## hours_studied -0.07 0.02 10.43
## attendance 0.17 -0.27 26.23
## sleep_hours -0.02 0.00 -0.10
## previous_scores -0.23 -0.10 9.82
## tutoring_sessions 1.52 0.02 0.76
## physical_activity 0.02 1.06 0.10
## exam_score 0.76 0.10 15.31Matriz de Correlaciones
round(cor(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)]),3)## hours_studied attendance sleep_hours previous_scores
## hours_studied 1.000 -0.005 0.013 0.022
## attendance -0.005 1.000 -0.020 -0.018
## sleep_hours 0.013 -0.020 1.000 -0.023
## previous_scores 0.022 -0.018 -0.023 1.000
## tutoring_sessions -0.010 0.012 -0.011 -0.013
## physical_activity 0.003 -0.023 -0.003 -0.007
## exam_score 0.445 0.580 -0.017 0.174
## tutoring_sessions physical_activity exam_score
## hours_studied -0.010 0.003 0.445
## attendance 0.012 -0.023 0.580
## sleep_hours -0.011 -0.003 -0.017
## previous_scores -0.013 -0.007 0.174
## tutoring_sessions 1.000 0.016 0.157
## physical_activity 0.016 1.000 0.025
## exam_score 0.157 0.025 1.0001.4. Gráficas multivariadas
Los gráficos multivariados son herramientas analíticas que permiten visualizar y comprender relaciones complejas entre múltiples variables dentro de un conjunto de datos. Este tipo de gráficos es esencial para evaluar y analizar factores que influyen en fenómenos complejos, como el rendimiento académico. En este conjunto de datos específico, se registran diversas variables relacionadas con el rendimiento estudiantil, como las horas de estudio, la calidad del docente, el nivel de ingresos familiares, el acceso a recursos y otros factores sociales y personales. Utilizar gráficos multivariados en este contexto nos permite identificar patrones y correlaciones entre estos factores, proporcionando una comprensión más profunda de cómo interactúan para influir en el éxito académico de los estudiantes.
Este conjunto de datos contiene variables tanto cuantitativas como cualitativas que afectan el rendimiento estudiantil. Entre los factores cuantitativos se incluyen las horas de estudio, el nivel de asistencia, las sesiones de tutoría y el puntaje en exámenes. Estos datos permiten crear gráficos de dispersión, histogramas y diagramas de caja que ayudan a visualizar las distribuciones y tendencias de cada variable. Por otro lado, las variables cualitativas, como el nivel de participación parental, la calidad del docente y el tipo de escuela, permiten construir gráficos de barras apiladas o gráficos de mosaico para analizar la influencia de cada categoría. Al emplear gráficos de correlación y mapas de calor, se pueden identificar relaciones entre variables, por ejemplo, si un mayor número de horas de estudio o una alta calidad docente están relacionadas con mayores puntajes en exámenes.
En conclusión, los gráficos multivariados ofrecen una visión integrada de cómo diversos factores interactúan para influir en el rendimiento estudiantil. La visualización de estos datos permite a investigadores y educadores identificar patrones y áreas de mejora que pueden guiar intervenciones educativas efectivas. Por ejemplo, la relación entre el ingreso familiar y el acceso a recursos o entre la calidad del docente y el rendimiento académico puede esclarecer qué factores tienen un mayor impacto en los resultados de los estudiantes. Así, los gráficos multivariados no solo facilitan el análisis descriptivo, sino que también proporcionan una base sólida para el diseño de estrategias pedagógicas personalizadas y programas de apoyo, optimizando las oportunidades de éxito académico.
Diagrama Conjunto de Dispersión, Distribución y Correlaciones
ggpairs(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)])
Diagrama Conjunto de Dispersión, Distribución y Correlaciones [ES:GE]
ggpairs(cdd_factores_rendimiento_estudiantes_G6, columns = c(1,2,6,7,20), aes(color = Gender, alpha = 0.5), upper = list(continous = wrap("cor", size = 2.5)))
Diagrama Conjunto de Dispersión, Distribución y Correlaciones [ES:EA]
ggpairs(cdd_factores_rendimiento_estudiantes_G6, columns = c(1,2,6,7,20), aes(color = Extracurricular_Activities, alpha = 0.5), upper = list(continous = wrap("cor", size = 2.5)))
Diagrama Conjunto de Dispersión, Distribución y Correlaciones [ES:TQ]
ggpairs(cdd_factores_rendimiento_estudiantes_G6, columns = c(1,2,6,7,20), aes(color = Teacher_Quality, alpha = 0.5), upper = list(continous = wrap("cor", size = 2.5)))## Warning in ggally_statistic(data = data, mapping = mapping, na.rm = na.rm, :
## Removed 78 rows containing missing values
## Warning in ggally_statistic(data = data, mapping = mapping, na.rm = na.rm, :
## Removed 78 rows containing missing values
## Warning in ggally_statistic(data = data, mapping = mapping, na.rm = na.rm, :
## Removed 78 rows containing missing values
## Warning in ggally_statistic(data = data, mapping = mapping, na.rm = na.rm, :
## Removed 78 rows containing missing values
## Warning in ggally_statistic(data = data, mapping = mapping, na.rm = na.rm, :
## Removed 78 rows containing missing values
## Warning in ggally_statistic(data = data, mapping = mapping, na.rm = na.rm, :
## Removed 78 rows containing missing values
## Warning in ggally_statistic(data = data, mapping = mapping, na.rm = na.rm, :
## Removed 78 rows containing missing values
## Warning in ggally_statistic(data = data, mapping = mapping, na.rm = na.rm, :
## Removed 78 rows containing missing values
## Warning in ggally_statistic(data = data, mapping = mapping, na.rm = na.rm, :
## Removed 78 rows containing missing values
## Warning in ggally_statistic(data = data, mapping = mapping, na.rm = na.rm, :
## Removed 78 rows containing missing values
Diagrama de Estrellas
set.seed(780728)
cdd_factores_rendimiento_estudiantes_G6_muestreado = cdd_factores_rendimiento_estudiantes_G6_dep[sample(1:nrow(cdd_factores_rendimiento_estudiantes_G6_dep),23), -c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)]
stars(cdd_factores_rendimiento_estudiantes_G6_muestreado, len = 1, cex = 0.4, key.loc = c(21,20), draw.segments = TRUE)
Caras de Chernoff
set.seed(780728)
cdd_factores_rendimiento_estudiantes_G6_muestreado = cdd_factores_rendimiento_estudiantes_G6_dep[sample(1:nrow(cdd_factores_rendimiento_estudiantes_G6_dep),23), -c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)]
faces(cdd_factores_rendimiento_estudiantes_G6_muestreado)
## effect of variables:
## modified item Var
## "height of face " "hours_studied"
## "width of face " "attendance"
## "structure of face" "sleep_hours"
## "height of mouth " "previous_scores"
## "width of mouth " "tutoring_sessions"
## "smiling " "physical_activity"
## "height of eyes " "exam_score"
## "width of eyes " "hours_studied"
## "height of hair " "attendance"
## "width of hair " "sleep_hours"
## "style of hair " "previous_scores"
## "height of nose " "tutoring_sessions"
## "width of nose " "physical_activity"
## "width of ear " "exam_score"
## "height of ear " "hours_studied"1.5. Normalidad multivariada
La normalidad multivariada es un concepto esencial en el análisis estadístico multivariante, utilizado para evaluar patrones entre múltiples variables relacionadas. Este enfoque resulta especialmente relevante en estudios de rendimiento académico, en los cuales factores como horas de estudio, asistencia, apoyo parental y actividades extracurriculares influyen en el desempeño de los estudiantes. El presente informe examina la normalidad multivariada en estos datos de rendimiento académico, lo cual es crucial para aplicar modelos estadísticos avanzados, como el análisis de regresión multivariante y el análisis de componentes principales, que dependen de esta condición para producir resultados confiables y significativos.
Para determinar si los datos de rendimiento académico cumplen con la normalidad multivariada, se analizaron diversas variables mediante pruebas estadísticas como la de Mardia y representaciones gráficas de dispersión. Estas herramientas ayudan a visualizar la simetría y la forma de los datos, evaluando si variables como horas de sueño, actividades extracurriculares y participación parental siguen un comportamiento multivariante normal. Cumplir con esta condición permite aplicar modelos estadísticos con mayor precisión, como análisis de regresión y componentes principales, para identificar relaciones significativas.
Determinar la normalidad multivariada es fundamental para asegurar la validez de las inferencias estadísticas en este análisis de datos académicos. Si los datos cumplen con esta condición, los resultados obtenidos de los modelos estadísticos serán más confiables. En caso contrario, se puede optar por transformaciones de datos o modelos no paramétricos, lo que permite mantener la precisión en la interpretación de los factores.
La PNM de Mardia establece que si ambas pruebas (para asimetría y curtosis) indican una normalidad multivariante, los datos siguen una DNM con un nivel de significancia \(\alpha=0.05\); sin embargo, el caso tratado es contrario a esto. Obsérvese a través de la pestaña PNM Mardia que los \(p-value\) para la asimetría (Skewness) y curtoris (Kurtosis) son mayores que el nivel de significancia. Por lo tanto, las evidencias no apoyan una hipótesis de normalidad multivariada para el conjunto de datos restringido a sus variables numéricas.
La PNM de PNM Henze-Zirkler establece que el estadístico de prueba no se distribuye aproximadamente como lognormal dado que su \(p-value\) es menor que el nivel de significancia \(\alpha=0.05\), obsérvese esto a través de la pestaña PNM Henze-Zirkler. Así, por contrarrecíproco de la implicación formulada en la descripción de la prueba en la sección 5, el conjunto de datos no está apoyado por las evidencias para seguir una distribución normal multivariada.
La PNM de Doornik-Hansen establece que su estadístico de prueba no sigue una distribución aproximadamente \(\chi^2\) dado que su \(p-value\) es menor que el nivel de significancia \(\alpha=0.05\), obsérvese esto a través de la pestaña PNM Doornik-Hansen. Por lo tanto, las evidencias están lejos de apoyar que el conjunto de datos sigue una DNM.
La PNM de Royston establece que el conjunto de datos reducido a sus variables numéricas no sigue una DNM, dado que su \(p-value\) es menor que el nivel de significancia \(\alpha=0.05\). Obsérvese esto a través de la pestaña PNM Royston.
PNM Mardia
mvn(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], mvnTest="mardia")## $multivariateNormality
## Test Statistic p value Result
## 1 Mardia Skewness 47036.8836269216 0 NO
## 2 Mardia Kurtosis 229.692187509438 0 NO
## 3 MVN <NA> <NA> NO
##
## $univariateNormality
## Test Variable Statistic p value Normality
## 1 Anderson-Darling hours_studied 7.2322 <0.001 NO
## 2 Anderson-Darling attendance 72.1694 <0.001 NO
## 3 Anderson-Darling sleep_hours 120.6211 <0.001 NO
## 4 Anderson-Darling previous_scores 70.5347 <0.001 NO
## 5 Anderson-Darling tutoring_sessions 240.2729 <0.001 NO
## 6 Anderson-Darling physical_activity 246.7992 <0.001 NO
## 7 Anderson-Darling exam_score 50.7143 <0.001 NO
##
## $Descriptives
## n Mean Std.Dev Median Min Max 25th 75th
## hours_studied 6378 19.977109 5.985460 20 1 44 16 24
## attendance 6378 80.020853 11.550723 80 60 100 70 90
## sleep_hours 6378 7.034964 1.468033 7 4 10 6 8
## previous_scores 6378 75.066165 14.400389 75 50 100 63 88
## tutoring_sessions 6378 1.495296 1.233984 1 0 8 1 2
## physical_activity 6378 2.972719 1.028926 3 0 6 2 4
## exam_score 6378 67.251960 3.912884 67 55 100 65 69
## Skew Kurtosis
## hours_studied 0.016217000 0.004665859
## attendance 0.005660056 -1.196383298
## sleep_hours -0.026801717 -0.505112016
## previous_scores -0.001361042 -1.192770422
## tutoring_sessions 0.824872577 0.660788008
## physical_activity -0.037037401 -0.055396304
## exam_score 1.658189083 10.563143348PNM Henze-Zirkler
mvn(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], mvnTest="hz")## $multivariateNormality
## Test HZ p value MVN
## 1 Henze-Zirkler 2.728417 0 NO
##
## $univariateNormality
## Test Variable Statistic p value Normality
## 1 Anderson-Darling hours_studied 7.2322 <0.001 NO
## 2 Anderson-Darling attendance 72.1694 <0.001 NO
## 3 Anderson-Darling sleep_hours 120.6211 <0.001 NO
## 4 Anderson-Darling previous_scores 70.5347 <0.001 NO
## 5 Anderson-Darling tutoring_sessions 240.2729 <0.001 NO
## 6 Anderson-Darling physical_activity 246.7992 <0.001 NO
## 7 Anderson-Darling exam_score 50.7143 <0.001 NO
##
## $Descriptives
## n Mean Std.Dev Median Min Max 25th 75th
## hours_studied 6378 19.977109 5.985460 20 1 44 16 24
## attendance 6378 80.020853 11.550723 80 60 100 70 90
## sleep_hours 6378 7.034964 1.468033 7 4 10 6 8
## previous_scores 6378 75.066165 14.400389 75 50 100 63 88
## tutoring_sessions 6378 1.495296 1.233984 1 0 8 1 2
## physical_activity 6378 2.972719 1.028926 3 0 6 2 4
## exam_score 6378 67.251960 3.912884 67 55 100 65 69
## Skew Kurtosis
## hours_studied 0.016217000 0.004665859
## attendance 0.005660056 -1.196383298
## sleep_hours -0.026801717 -0.505112016
## previous_scores -0.001361042 -1.192770422
## tutoring_sessions 0.824872577 0.660788008
## physical_activity -0.037037401 -0.055396304
## exam_score 1.658189083 10.563143348PNM Doornik-Hansen
mvn(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], mvnTest="dh")## $multivariateNormality
## Test E df p value MVN
## 1 Doornik-Hansen 17198.72 14 0 NO
##
## $univariateNormality
## Test Variable Statistic p value Normality
## 1 Anderson-Darling hours_studied 7.2322 <0.001 NO
## 2 Anderson-Darling attendance 72.1694 <0.001 NO
## 3 Anderson-Darling sleep_hours 120.6211 <0.001 NO
## 4 Anderson-Darling previous_scores 70.5347 <0.001 NO
## 5 Anderson-Darling tutoring_sessions 240.2729 <0.001 NO
## 6 Anderson-Darling physical_activity 246.7992 <0.001 NO
## 7 Anderson-Darling exam_score 50.7143 <0.001 NO
##
## $Descriptives
## n Mean Std.Dev Median Min Max 25th 75th
## hours_studied 6378 19.977109 5.985460 20 1 44 16 24
## attendance 6378 80.020853 11.550723 80 60 100 70 90
## sleep_hours 6378 7.034964 1.468033 7 4 10 6 8
## previous_scores 6378 75.066165 14.400389 75 50 100 63 88
## tutoring_sessions 6378 1.495296 1.233984 1 0 8 1 2
## physical_activity 6378 2.972719 1.028926 3 0 6 2 4
## exam_score 6378 67.251960 3.912884 67 55 100 65 69
## Skew Kurtosis
## hours_studied 0.016217000 0.004665859
## attendance 0.005660056 -1.196383298
## sleep_hours -0.026801717 -0.505112016
## previous_scores -0.001361042 -1.192770422
## tutoring_sessions 0.824872577 0.660788008
## physical_activity -0.037037401 -0.055396304
## exam_score 1.658189083 10.563143348PNM Royston
# Seleccionar una muestra
set.seed(780729)
cdd_factores_rendimiento_estudiantes_G6_muestr = cdd_factores_rendimiento_estudiantes_G6_dep[sample(1:nrow(cdd_factores_rendimiento_estudiantes_G6_dep),250),-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)]
mvn(cdd_factores_rendimiento_estudiantes_G6_muestr, mvnTest="royston")## $multivariateNormality
## Test H p value MVN
## 1 Royston 215.7855 5.317622e-43 NO
##
## $univariateNormality
## Test Variable Statistic p value Normality
## 1 Anderson-Darling hours_studied 0.6507 0.0885 YES
## 2 Anderson-Darling attendance 3.1679 <0.001 NO
## 3 Anderson-Darling sleep_hours 5.1208 <0.001 NO
## 4 Anderson-Darling previous_scores 3.1934 <0.001 NO
## 5 Anderson-Darling tutoring_sessions 9.6882 <0.001 NO
## 6 Anderson-Darling physical_activity 10.0211 <0.001 NO
## 7 Anderson-Darling exam_score 1.7314 2e-04 NO
##
## $Descriptives
## n Mean Std.Dev Median Min Max 25th 75th Skew
## hours_studied 250 20.056 6.005090 20 4 34 16 24.75 -0.20604094
## attendance 250 78.928 11.581855 78 60 100 69 89.00 0.14855634
## sleep_hours 250 7.084 1.407433 7 4 10 6 8.00 0.01514575
## previous_scores 250 74.728 14.185164 73 50 99 63 87.00 0.02881172
## tutoring_sessions 250 1.488 1.326989 1 0 7 0 2.00 1.14526774
## physical_activity 250 2.984 1.029555 3 0 6 2 4.00 -0.05618942
## exam_score 250 67.028 3.708670 67 58 89 65 69.00 0.88416630
## Kurtosis
## hours_studied -0.3756403
## attendance -1.1864396
## sleep_hours -0.4284766
## previous_scores -1.2252237
## tutoring_sessions 1.9579467
## physical_activity -0.0998998
## exam_score 4.6683818Fase 2 [Componentes Principales]
2.1. Objetivos
El análisis de rendimiento académico basado en factores sociales, familiares y personales permite entender las dinámicas que influyen en el éxito estudiantil. Este conjunto de datos recopila información diversa, como horas de estudio, participación parental, calidad docente, y nivel de ingresos familiares, con el objetivo de identificar y analizar los elementos clave que afectan el desempeño en exámenes de estudiantes. A través de una exploración detallada de estos factores, se busca detectar patrones que puedan orientar intervenciones educativas y mejoras en el entorno de aprendizaje.
El conjunto de datos incluye tanto variables cuantitativas como cualitativas relacionadas con el contexto de los estudiantes y su rendimiento. Factores cuantitativos, como horas de estudio y puntajes en exámenes, y factores cualitativos, como el tipo de escuela y el nivel educativo de los padres, proporcionan una perspectiva amplia sobre el entorno de cada estudiante. Este enfoque permite analizar el impacto de elementos como el apoyo familiar, la motivación y las oportunidades de acceso a recursos. A partir de estos datos, se pueden aplicar técnicas analíticas, como el análisis de componentes principales, para identificar variables con mayor influencia y generar una interpretación global de los factores de éxito.
En conclusión, el objetivo principal de este conjunto de datos es ofrecer una base sólida para investigar cómo diversos factores interactúan y contribuyen al rendimiento académico. Este análisis no solo proporciona una comprensión más completa del entorno estudiantil, sino que también puede guiar la implementación de estrategias educativas enfocadas en el bienestar y desarrollo de los estudiantes. Al entender mejor los factores determinantes del rendimiento académico, los educadores y responsables pueden diseñar intervenciones personalizadas que promuevan mejores resultados en el ámbito educativo.
2.2. Selección de Componentes
Este análisis de componentes principales (ACP) aborda los factores que influyen en el rendimiento académico de los estudiantes. A partir de una variedad de datos, que incluyen factores de estudio, asistencia, actividades extracurriculares y características socioeconómicas, se utiliza el ACP para reducir la dimensionalidad del conjunto y destacar las variables más influyentes. Este proceso permite identificar patrones subyacentes entre las variables y descubrir cuáles contribuyentes de manera significativa al rendimiento en los exámenes, facilitando una interpretación más clara y estratégica de los datos.
En la etapa de análisis, se emplean múltiples herramientas para desentrañar las correlaciones y la importancia relativa de cada factor. Primero, la matriz de correlaciones permite observar las relaciones entre variables, mientras que la matriz ACP identifica aquellos factores con alta varianza explicada. Utilizando el gráfico de Cattell y el gráfico de Cattell-Kaiser, se determina el número adecuado de componentes a retener, lo cual simplifica el análisis sin perder información relevante. El círculo de correlaciones ofrece una representación visual de las variables, mostrando cómo se agrupan en torno a los principales componentes. Además, se evalúa la calidad de representación para medir la precisión con la que cada variable se representa en el espacio de componentes, junto con las coordenadas individuales, que proporcionan una visión detallada de cada estudiante respecto a los factores clave.
Este análisis confirma la utilidad del ACP en la identificación de los factores que más impactan el rendimiento estudiantil. Al simplificar la estructura de los datos, se logran resaltar las relaciones significativas y se puede enfocar en las variables que realmente contribuyen al desempeño académico. La comprensión de estos componentes principales no solo facilita una interpretación más precisa, sino que también proporciona una base sólida para futuras intervenciones educativas. Con esta información, se pueden diseñar estrategias enfocadas en mejorar los aspectos críticos identificados, beneficiando tanto a los estudiantes como al sistema educativo
Matriz ACP
get_eigenvalue(PCA(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], ncp = 7, scale.unit = TRUE, graph = F))## eigenvalue variance.percent cumulative.variance.percent
## Dim.1 1.7666725 25.238178 25.23818
## Dim.2 1.0425347 14.893353 40.13153
## Dim.3 1.0203547 14.576496 54.70803
## Dim.4 1.0193277 14.561825 69.26985
## Dim.5 0.9722169 13.888813 83.15866
## Dim.6 0.9502410 13.574872 96.73354
## Dim.7 0.2286525 3.266464 100.00000Matriz de Correlaciones
round(cor(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)]),3)## hours_studied attendance sleep_hours previous_scores
## hours_studied 1.000 -0.005 0.013 0.022
## attendance -0.005 1.000 -0.020 -0.018
## sleep_hours 0.013 -0.020 1.000 -0.023
## previous_scores 0.022 -0.018 -0.023 1.000
## tutoring_sessions -0.010 0.012 -0.011 -0.013
## physical_activity 0.003 -0.023 -0.003 -0.007
## exam_score 0.445 0.580 -0.017 0.174
## tutoring_sessions physical_activity exam_score
## hours_studied -0.010 0.003 0.445
## attendance 0.012 -0.023 0.580
## sleep_hours -0.011 -0.003 -0.017
## previous_scores -0.013 -0.007 0.174
## tutoring_sessions 1.000 0.016 0.157
## physical_activity 0.016 1.000 0.025
## exam_score 0.157 0.025 1.000Valores y Vectores Propios
princomp(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], cor = TRUE)$sdev^2## Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7
## 1.7666725 1.0425347 1.0203547 1.0193277 0.9722169 0.9502410 0.2286525princomp(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], cor = TRUE)$loadings[ ,1:7]## Comp.1 Comp.2 Comp.3 Comp.4 Comp.5
## hours_studied 0.40998846 0.543066055 0.2595324 0.088010346 0.029158535
## attendance 0.53223665 -0.447784870 -0.2169646 0.198094698 -0.262256510
## sleep_hours -0.02946127 0.206019969 0.4347771 0.663884450 0.255299855
## previous_scores 0.15870723 0.517923349 -0.3350981 -0.452593899 0.265257923
## tutoring_sessions 0.14592174 -0.439783239 0.3823780 -0.336130818 0.709009573
## physical_activity 0.01081249 0.021106701 0.6616205 -0.440928332 -0.540458473
## exam_score 0.70793192 0.004935298 0.0171087 0.005210456 -0.006448699
## Comp.6 Comp.7
## hours_studied 0.54293256 0.407749742
## attendance -0.27110227 0.536304742
## sleep_hours -0.51159815 -0.001844289
## previous_scores -0.54088589 0.162837780
## tutoring_sessions 0.05657156 0.142147005
## physical_activity -0.27185976 0.035706817
## exam_score -0.01815271 -0.705774099Correlaciones Comparadas
par(mfrow=c(1,2))
corrplot::corrplot(cor(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)]), method = "color", type = "upper", number.cex = 0.4)
corrplot::corrplot(cor(princomp(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], cor = TRUE)$scores), method = "color", type = "upper", number.cex = 0.4)
Gráfico de Cattell
fviz_eig(PCA(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], scale.unit = T, graph = F), addlabels = T, ylim=c(0,90), main = "")
Gráfico de Cattell-Kaiser
scree(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)],factors = FALSE, pc = TRUE, main ="")
2.3. Calidad de Representación
Al retomar el análisis basado en los datos del conjunto cdd_factores_rendimiento_estudiantes_G6_dep, se verifica que, una vez reducida la dimensionalidad del conjunto de datos mediante el análisis de componentes principales (ACP) y entendido que sus variables estandarizadas están representadas gráficamente como proyecciones en una hiperesfera de correlaciones, es necesario iniciar la interpretación de los componentes principales a partir de dichas correlaciones. Posteriormente, se debe evaluar la calidad de las representaciones de las variables dentro de este nuevo espacio reducido, considerando la información retenida tras la disminución dimensional del conjunto de datos.
El Círculo de Correlaciones expresa que se puede concebir una componente tipo tamaño en el sentido de que la dimensión 1 muestra en él una correlación positiva con las siete variables de interés, además cercanas a la frontera del círculo unitario y significativamente próximas al eje que la representa. Por otro lado, la dimensión 2 contrapone a las variables previous score y hours studied con las demás. Otro aspecto por resaltar es la correlación mostrada entre pares de variables, que en términos del fenómeno estudiado conservan su naturaleza correlacio
La Matriz de Representación, por otro lado, muestra valores significativamente cercanos a 1 del cociente de proyecciones coseno cuadrado en relación con la dimensión 1. En este sentido, los puntos proyectados están altamente asociados con este componente. Así, la estaña que muestra la Calidad de Representación indica en su escala un piso alto de \(0.88\) de esta manera las calidades de representación, en relación con la componente 1, están encabezadas por hours studied y cierran con physical activity.
Por último, las Coordenadas Individuales ayudan, aunque de manera menos digestiva, a identificar a nivel de observaciones a los perfiles de los registros
Círculo de Correlaciones
fviz_pca_var(PCA(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], scale.unit = T, graph = F),col.var="#3B83BD", repel = T, col.circle = "#CDCDCD", ggtheme = theme_bw())
Matriz de Representación
(get_pca_var(PCA(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], ncp = 7, scale.unit = TRUE, graph = F)))$cos2## Dim.1 Dim.2 Dim.3 Dim.4
## hours_studied 0.2969609159 3.074651e-01 0.0687281027 7.895530e-03
## attendance 0.5004556407 2.090400e-01 0.0480318251 3.999996e-02
## sleep_hours 0.0015334125 4.424958e-02 0.1928788342 4.492611e-01
## previous_scores 0.0444989201 2.796543e-01 0.1145763733 2.088004e-01
## tutoring_sessions 0.0376180301 2.016359e-01 0.1491890544 1.151676e-01
## physical_activity 0.0002065415 4.644417e-04 0.4466518350 1.981754e-01
## exam_score 0.8853989955 2.539319e-05 0.0002986654 2.767357e-05
## Dim.5 Dim.6 Dim.7
## hours_studied 8.265984e-04 0.2801080217 3.801573e-02
## attendance 6.686760e-02 0.0698393340 6.576567e-02
## sleep_hours 6.336717e-02 0.2487091124 7.777388e-07
## previous_scores 6.840690e-02 0.2780001779 6.062982e-03
## tutoring_sessions 4.887282e-01 0.0030410953 4.620100e-03
## physical_activity 2.839801e-01 0.0702301540 2.915266e-04
## exam_score 4.043034e-05 0.0003131244 1.138957e-01Calidad de Representación
fviz_pca_var(PCA(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], ncp = 7, scale.unit = TRUE, graph = F), col.var="cos2", gradient.cols=c("#00AFBB","#E7B800","#FC4E07"), repel = TRUE)
Coordenadas Individuales
head((PCA(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], ncp = 7, scale.unit = TRUE, graph = F))$ind$coord, n = 23L)## Dim.1 Dim.2 Dim.3 Dim.4 Dim.5 Dim.6
## 1 0.14623269 0.57399247 -0.35288265 -0.55714568 -0.992903368 -0.19602486
## 2 -2.06238350 -0.07667166 1.70798591 -0.06621650 -0.008430812 -0.33527551
## 3 2.57187063 0.08589251 0.30207005 0.71787876 -0.361947743 0.92303397
## 4 1.89565702 1.63327782 0.49822318 0.29805330 -0.400027977 0.90159693
## 5 1.08087837 -1.57218077 0.79925049 0.80709296 -0.321742960 -0.15379806
## 6 1.29136464 -0.29240801 0.26953959 0.29563674 1.082699808 1.09127606
## 7 0.60989233 0.56164078 -0.30951395 -0.95868563 0.044140830 -1.23822236
## 8 -0.29987133 -0.35773934 -0.25192369 -0.90913399 -0.342968623 -2.04549548
## 9 0.63411396 -0.28520850 -2.53740354 -0.82793909 -0.246853122 -0.02177835
## 10 1.67497041 0.14696215 1.03486191 -0.46191636 -2.232571687 0.95765543
## 11 0.36064134 -0.53938407 1.29621469 1.13580369 1.290439919 1.25183540
## 12 1.45842073 -0.83954113 -1.48505130 0.31136450 0.434605602 0.49340359
## 13 0.99744632 0.74634187 0.59281908 0.76091887 0.255077018 1.39843663
## 14 -0.88047565 -1.22852351 0.01247291 -0.25450085 0.290678168 1.24167011
## 15 -1.26294633 -0.53509398 0.41323479 0.01974464 -0.675052935 1.43763810
## 16 -1.35100839 -0.01378671 1.30324505 0.23634261 0.088692506 0.36716230
## 17 -2.99120398 0.81229284 0.75181008 -1.61208727 -0.105870176 0.76965230
## 18 -0.69176151 0.85045911 -0.33773670 0.70790617 -0.110097531 -0.49949914
## 19 -0.08358663 -0.15847567 -0.16455464 -0.32097060 1.987526266 1.40720979
## 20 -1.00300542 -1.66473742 1.25664889 0.53464370 0.081725856 0.14919766
## 21 0.94832435 1.96702315 -2.45482749 0.31948593 0.351006371 -1.12073961
## 22 1.72770922 -0.46531376 -1.02853126 0.30817466 -0.757897436 0.56095236
## 23 -0.43699854 1.02712821 -0.70883401 -0.44569003 0.789781129 0.47620713
## Dim.7
## 1 0.241526569
## 2 0.228178515
## 3 0.165698774
## 4 0.592296203
## 5 0.090426014
## 6 -0.041504412
## 7 0.674251978
## 8 0.101217496
## 9 -0.052670992
## 10 0.035210891
## 11 0.130523281
## 12 0.070131484
## 13 0.052941980
## 14 -0.406917587
## 15 -0.402018435
## 16 -0.139104295
## 17 -0.317586494
## 18 0.102331011
## 19 0.022707617
## 20 -0.534727646
## 21 0.238051791
## 22 0.004447883
## 23 0.0133665752.4. Contribuciones
Este trabajo presenta un análisis estadístico multivariante detallado, aplicando metodologías clave para explorar y modelar relaciones complejas entre variables. El objetivo principal es demostrar cómo estas técnicas pueden proporcionar insights significativos en la interpretación de datos reales, promoviendo una comprensión profunda de los fenómenos analizados.
El análisis se llevó a cabo utilizando herramientas estadísticas como análisis de componentes principales (PCA), análisis discriminante, y clustering jerárquico. Cada técnica fue aplicada de manera rigurosa a un conjunto de datos específicos, seleccionados por su relevancia en contextos industriales y académicos. Los resultados obtenidos destacan patrones ocultos en los datos, identificando agrupaciones naturales y estableciendo relaciones lineales que permiten reducir la dimensionalidad de los mismos sin perder información clave. Además, las visualizaciones generadas proporcionan una interpretación clara y accesible, lo que facilita la comunicación de hallazgos incluso a audiencias no especializadas.
En conjunto, este trabajo contribuye significativamente al campo del análisis multivariante al ofrecer un caso práctico que combina técnicas robustas con una presentación clara de resultados. Los métodos y conclusiones expuestos no solo demuestran su aplicabilidad en situaciones concretas, sino que también establecen una base para futuras investigaciones. Esto refuerza la utilidad de estas herramientas para resolver problemas complejos en ingeniería y otras disciplinas, promoviendo su integración en prácticas cotidianas de análisis de datos.
Matriz de Contribuciones
(get_pca_var(PCA(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], ncp = 7, scale.unit = TRUE, graph = F)))$contrib## Dim.1 Dim.2 Dim.3 Dim.4
## hours_studied 16.80905336 29.492073965 6.73570704 0.774582093
## attendance 28.32758494 20.051129013 4.70736555 3.924150945
## sleep_hours 0.08679665 4.244422750 18.90311634 44.074256243
## previous_scores 2.51879855 26.824459582 11.22907303 20.484123741
## tutoring_sessions 2.12931548 19.340929689 14.62129354 11.298392662
## physical_activity 0.01169099 0.044549284 43.77417376 19.441779432
## exam_score 50.11676003 0.002435716 0.02927074 0.002714885
## Dim.5 Dim.6 Dim.7
## hours_studied 0.085022016 29.4775763 1.662599e+01
## attendance 6.877847718 7.3496442 2.876228e+01
## sleep_hours 6.517801591 26.1732663 3.401401e-04
## previous_scores 7.036176549 29.2557543 2.651614e+00
## tutoring_sessions 50.269457415 0.3200341 2.020577e+00
## physical_activity 29.209536139 7.3907727 1.274977e-01
## exam_score 0.004158572 0.0329521 4.981171e+01Contribuciones a D1
fviz_contrib(PCA(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], ncp = 7, scale.unit = TRUE, graph = F), choice = "var", axes = 1, top = 10)
Contribuciones a D2
fviz_contrib(PCA(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], ncp = 7, scale.unit = TRUE, graph = F), choice = "var", axes = 2, top = 10)
Contribuciones a D3
fviz_contrib(PCA(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], ncp = 7, scale.unit = TRUE, graph = F), choice = "var", axes = 3, top = 10)
Contribuciones a D4
fviz_contrib(PCA(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], ncp = 7, scale.unit = TRUE, graph = F), choice = "var", axes = 4, top = 10)
Contribuciones a D5
fviz_contrib(PCA(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], ncp = 7, scale.unit = TRUE, graph = F), choice = "var", axes = 5, top = 10)
Contribuciones a D6
fviz_contrib(PCA(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], ncp = 7, scale.unit = TRUE, graph = F), choice = "var", axes = 6, top = 10)
Contribuciones a D7
fviz_contrib(PCA(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], ncp = 7, scale.unit = TRUE, graph = F), choice = "var", axes = 7, top = 10)
2.5 Interpretación
Según lo señalado, las coordenadas obtenidas de registros tras una reducción dimensional permiten ubicarlos en un plano de factores, facilitando el análisis e interpretación. En este plano, las variables reducidas corresponden a los componentes principales, que se representan como ejes, mientras que los valores asociados a estos componentes son los puntajes obtenidos. Este enfoque subraya que las distancias entre los puntos, definidas por los puntajes de los componentes, poseen un significado importante, ya que ayudan a identificar similitudes entre los perfiles observados. Sin embargo, dichas similitudes pueden no abarcar todas las variables, sino solo algunas. En consecuencia, se espera que las distancias originales en el espacio multidimensional sean adecuadamente reflejadas en el espacio reducido formado por los componentes principales.
Con base en las variables cuantitativas del conjunto de datos se requiere definir e interpretar sus componentes principales.
Biplot de Variables y Registros [Total GN]
# Seleccionar una muestra
set.seed(780729)
cdd_factores_rendimiento_estudiantes_G6_muestr = cdd_factores_rendimiento_estudiantes_G6_dep[sample(1:nrow(cdd_factores_rendimiento_estudiantes_G6_dep),100),-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19)]
# Convertir en campo categorico
cdd_factores_rendimiento_estudiantes_G6_muestr$gender <- as.factor(cdd_factores_rendimiento_estudiantes_G6_muestr$gender)
#Graficar el biplot PCA
fviz_pca_biplot(PCA(cdd_factores_rendimiento_estudiantes_G6_muestr[,], ncp = 7, scale.unit = TRUE, graph = F, quali.sup = "gender"), axes = c(1, 2), repel = TRUE, habillage = "gender")
Biplot de Variables y Registros [Total SH]
# Seleccionar una muestra
set.seed(780729)
cdd_factores_rendimiento_estudiantes_G6_muestr = cdd_factores_rendimiento_estudiantes_G6_dep[sample(1:nrow(cdd_factores_rendimiento_estudiantes_G6_dep),100),-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)]
# Convertir en campo categorico
cdd_factores_rendimiento_estudiantes_G6_muestr$sleep_hours <- as.factor(cdd_factores_rendimiento_estudiantes_G6_muestr$sleep_hours)
#Graficar el biplot PCA
fviz_pca_biplot(PCA(cdd_factores_rendimiento_estudiantes_G6_muestr[,], ncp = 7, scale.unit = TRUE, graph = F, quali.sup = "sleep_hours"), axes = c(1, 2), repel = TRUE, habillage = "sleep_hours")
Biplot de Variables y Registros [Total TQ]
# Seleccionar una muestra
set.seed(780729)
cdd_factores_rendimiento_estudiantes_G6_muestr = cdd_factores_rendimiento_estudiantes_G6_dep[sample(1:nrow(cdd_factores_rendimiento_estudiantes_G6_dep),100),-c(2, 4, 5, 6, 9, 10, 12, 14, 15, 17, 18, 19, 20)]
# Convertir en campo categorico
cdd_factores_rendimiento_estudiantes_G6_muestr$teacher_quality <- as.factor(cdd_factores_rendimiento_estudiantes_G6_muestr$teacher_quality)
#Graficar el biplot PCA
fviz_pca_biplot(PCA(cdd_factores_rendimiento_estudiantes_G6_muestr[,], ncp = 7, scale.unit = TRUE, graph = F, quali.sup = "teacher_quality"), axes = c(1, 2), repel = TRUE, habillage = "teacher_quality")
Coordenadas individuales [Subconjunto SH]
set.seed(780729)
data_61_SH <- cdd_factores_rendimiento_estudiantes_G6_dep[sample(1:nrow(cdd_factores_rendimiento_estudiantes_G6_dep),61),-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)]
set.seed(780728)
data_61 <- cbind(cdd_factores_rendimiento_estudiantes_G6_dep[sample(1:nrow(cdd_factores_rendimiento_estudiantes_G6_dep),61),-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], data_61_SH$sleep_hours)
head(PCA(data_61, ncp = 7, scale.unit = T, graph = F, quali.sup = "sleep_hours")$ind$coord, n = 61L)## Dim.1 Dim.2 Dim.3 Dim.4 Dim.5 Dim.6
## 1 -0.200074968 -0.28406878 -0.71194293 0.61666797 0.81815639 -0.270410758
## 2 -0.234848761 -0.88320028 0.33810499 -0.45540530 -0.83421861 -1.154912802
## 3 1.170799324 -0.28697158 1.71586040 -1.24988371 1.15834798 0.342681781
## 4 1.585861407 -0.36837261 -0.47560291 -0.09944727 1.00003076 0.827129079
## 5 -0.162401411 0.23780415 -1.29363767 -0.22239135 -1.83024030 -1.512164291
## 6 -0.557841906 -0.97580681 -0.28644232 -1.41290850 -0.83510642 0.984018909
## 7 0.141486211 0.16739772 -2.36199291 0.64335267 0.32474285 1.229850378
## 8 -2.555306891 -0.95095881 0.25056166 -0.42506314 -0.70766744 -0.195968371
## 9 -0.599921794 -0.51052946 0.73565496 0.09260679 -1.80428790 -0.518462107
## 10 -3.772799164 -0.41796867 -0.24036053 -0.71148174 0.35956552 0.607823712
## 11 3.063583114 -1.02859072 -0.92184415 -1.99242543 0.64840845 -0.939090946
## 12 -2.227477029 0.34460730 0.52313023 -1.36020052 1.29071715 0.655589585
## 13 -1.578488460 -1.15657323 -0.79638893 0.45294445 0.88058822 -0.776841689
## 14 1.982922300 -1.14074699 0.95391747 -0.26523204 0.19954159 -1.210287453
## 15 0.318579024 0.31142833 -0.06244942 2.86858175 0.47124413 -0.272566485
## 16 -0.441984801 0.74764806 -0.83648891 -0.45985673 -0.57695288 -0.939684422
## 17 1.150101092 -2.08906166 1.12110716 -0.06315172 -0.18879437 0.279552380
## 18 1.649415771 0.07914589 0.47863142 -0.62458958 -0.78405792 -0.208834668
## 19 0.957680163 0.89201632 0.04420352 -0.93660573 0.14136476 -0.187210374
## 20 1.558140640 -0.61405396 1.14126265 -1.30270493 0.35595909 0.293868790
## 21 0.825274686 -1.33687274 -2.23281427 1.52800793 -0.18022932 1.612057847
## 22 -2.072725056 -2.30671535 -2.23059916 -0.12826072 -0.04230148 -1.083721222
## 23 -0.862700741 -0.24584708 -1.50191152 -1.07248818 1.26356334 0.541108816
## 24 2.030417484 -1.20994500 -0.69924412 2.31918390 0.20773575 -0.225538238
## 25 -0.802724126 -0.54243564 1.38801344 1.95842162 0.12914754 0.301141619
## 26 0.528600355 -1.29684970 1.31069208 0.37328445 -0.73400076 0.734846132
## 27 1.598329744 -1.29283821 -0.99288653 -0.60381566 0.13927160 -0.095031119
## 28 -0.444831326 -0.09316904 -0.99143211 -0.49101590 -0.48480123 -2.195670475
## 29 0.176795384 -1.12832784 0.19040985 0.08853733 1.35921760 -0.365453252
## 30 -1.248750183 1.63839990 -0.51660197 -0.72597602 0.80977058 -0.688633163
## 31 0.658561383 -0.44597212 0.11059891 -0.65467109 0.26421521 1.834437291
## 32 -1.351435020 2.46429585 0.73146699 0.77950952 -0.25278211 0.062532896
## 33 -0.009177087 1.30733925 -1.21644813 -0.56220875 -1.54495525 0.317529608
## 34 -0.102030601 2.49660790 0.05141452 -0.72064235 -0.18563296 -0.399518569
## 35 1.358341760 -0.05360175 0.48302224 0.28023664 -1.01664310 1.117676918
## 36 0.067907304 0.24889478 0.81072544 -0.19780833 0.28248820 -0.055180779
## 37 -1.689962399 -1.53288770 0.92538103 0.12375510 -0.21617543 -0.122660480
## 38 1.386343231 0.09629107 0.84361148 0.90824146 -0.16714548 0.219540850
## 39 3.252826089 2.01080580 -0.85647886 0.02028472 -0.23912794 0.655481579
## 40 -2.324780409 -0.14586689 1.22730327 0.71681088 0.87485328 0.140008452
## 41 -2.671310709 0.10437883 1.81958803 1.54474529 -0.75417076 0.001606055
## 42 -0.395708562 -0.37670340 -0.23365143 -0.65937891 -1.31146641 0.971404549
## 43 0.097252885 0.04904795 -0.44433856 -2.02249813 -0.30313957 0.479727626
## 44 1.700104687 2.00537680 1.14531109 -0.94649559 0.13052093 0.545138958
## 45 2.848497757 -0.47839839 0.39302668 1.53237789 0.52760477 -0.015315972
## 46 -0.027566735 -0.72571818 0.52292403 0.39655087 0.73641194 0.882384758
## 47 -0.803705812 -0.73299764 0.53537028 -0.20058030 -1.02305767 1.079154098
## 48 0.171634897 -0.24002474 0.39911413 -0.64419230 1.22205909 -0.607369859
## 49 0.152318533 0.13479140 0.51428694 -0.72656349 -0.79570189 0.372546263
## 50 1.472228277 -0.68614979 1.91666863 0.73159935 -0.40188288 -1.073173964
## 51 -2.385303733 -0.61575479 -0.72712101 -0.14255123 1.01978776 -0.177974172
## 52 -1.637300168 0.22946990 0.65306869 0.34936710 0.03825140 -0.626554556
## 53 0.702287144 1.11950605 1.46352543 0.22511263 -1.31439930 -0.836866263
## 54 0.116038891 0.46321702 -0.64458535 -0.47277202 -0.87192352 -0.793454749
## 55 2.836981421 1.11963160 -1.18219123 0.99310919 1.58125103 -1.105069603
## 56 -0.220292922 0.45388441 -0.96765336 0.90659174 -0.90154342 -0.435101301
## 57 -0.329072095 1.65190732 -0.71299104 0.32308930 0.88412531 -0.136314327
## 58 -1.646863292 1.57726746 -1.53909852 1.52840732 -0.10444512 0.283655068
## 59 -0.108058895 2.03645435 0.22086623 0.38629057 -0.80660399 0.725219431
## 60 -1.585584439 1.69167672 1.63378672 -0.25093434 2.38666740 -0.377429353
## 61 -0.508281461 0.51468744 -0.91541278 0.11653257 -0.29215419 1.504752354
## Dim.7
## 1 -0.11048035
## 2 -0.09743806
## 3 0.13212857
## 4 0.45174617
## 5 -0.28932128
## 6 0.18850828
## 7 -0.41104550
## 8 -0.17547609
## 9 -0.11212964
## 10 0.23147818
## 11 0.35410793
## 12 -0.04135688
## 13 0.11699938
## 14 -2.29451854
## 15 0.16193980
## 16 -0.25893652
## 17 0.49899505
## 18 0.40099074
## 19 0.01017492
## 20 0.14339675
## 21 -0.06904387
## 22 -0.23606784
## 23 -0.34779686
## 24 0.13495102
## 25 -0.14482281
## 26 -0.09971296
## 27 0.60694399
## 28 0.47970241
## 29 -0.22084182
## 30 0.23413592
## 31 -0.20815753
## 32 0.09296399
## 33 -0.32148762
## 34 0.23796478
## 35 0.36999998
## 36 -0.32922459
## 37 0.28580613
## 38 0.06273839
## 39 -0.28504526
## 40 -0.43434122
## 41 -0.05429974
## 42 0.22732258
## 43 0.03302980
## 44 -0.02655752
## 45 -0.04459825
## 46 -0.13380805
## 47 -0.04004259
## 48 -0.21711783
## 49 -0.21780986
## 50 0.71319736
## 51 0.36277754
## 52 0.43308699
## 53 -0.07856649
## 54 -0.06009823
## 55 0.29326909
## 56 0.44426711
## 57 -0.07905496
## 58 -0.06692887
## 59 -0.12645478
## 60 0.07569368
## 61 -0.14573412Biplot de variables y registros [Subconjunto SH]
set.seed(780729)
data_61_SH <- cdd_factores_rendimiento_estudiantes_G6_dep[sample(1:nrow(cdd_factores_rendimiento_estudiantes_G6_dep),61),-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)]
set.seed(780728)
data_61 <- cbind(cdd_factores_rendimiento_estudiantes_G6_dep[sample(1:nrow(cdd_factores_rendimiento_estudiantes_G6_dep),61),-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], data_61_SH$sleep_hours)
fviz_pca_biplot(PCA(data_61, ncp = 7, scale.unit = T, graph = F, quali.sup = "sleep_hours"), axes = c(1, 2), repel = T, habillage = "sleep_hours")
Fase 3 [Correspondencias]
3.1. Objetivos
En términos generales, los objetivos del análisis de correspondencias simples y múltiples se centran en explorar y visualizar relaciones entre variables cualitativas de un conjunto de datos. Esta metodología permite identificar patrones, similitudes y diferencias significativas mediante herramientas como tablas de contingencia y disyuntivas completas. Además, se busca evaluar las calidades de representación y las contribuciones de cada variable o categoría, facilitando interpretaciones que aporten un entendimiento más profundo de las estructuras y asociaciones presentes en los datos analizados.
3.2. Correspondencias Simples
El análisis de correspondencias simples (ACS) es una técnica estadística que permite visualizar relaciones entre categorías de variables categóricas en un espacio reducido y multidimensional. En este trabajo, se implementa el ACS para explorar las asociaciones entre pares de variables categóricas provenientes del conjunto de datos analizado. Este método no solo facilita la interpretación gráfica de tablas de contingencia, sino que también ofrece una comprensión intuitiva de las relaciones entre los niveles de las variables.
El ACS se aplicó a pares seleccionados de variables categóricas, destacando aquellas con relevancia en contextos multivariantes. A través del cálculo de perfiles fila y columna, y su representación gráfica en dos dimensiones, se identificaron proximidades entre categorías que sugieren patrones de asociación significativos. El número de dimensiones consideradas se determinó en función del menor número de categorías entre las variables analizadas, siguiendo la metodología estándar. Los resultados muestran cómo las categorías agrupadas tienden a compartir características comunes o a reflejar asociaciones que serían menos evidentes en el análisis directo de tablas de frecuencias. Además, se generaron gráficos bidimensionales que ilustran estas relaciones, aportando claridad visual a los hallazgos.
El análisis de correspondencias simples realizado en este trabajo aporta una perspectiva única sobre las relaciones entre las categorías de las variables del conjunto de datos. Al permitir la reducción de dimensiones y la visualización de patrones de proximidad, el ACS demuestra ser una herramienta poderosa para identificar asociaciones clave en datos categóricos. Este análisis sienta las bases para extender la metodología hacia análisis de correspondencias múltiples (ACM), ofreciendo la posibilidad de examinar conjuntos de datos más complejos en futuras investigaciones.
Análisis de Correspondencias
Tablas de contingencia
addmargins(table(cdd_factores_rendimiento_estudiantes_G6_dep$gender,cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours))##
## 4 5 6 7 8 9 10 Sum
## 0 126 278 519 743 579 321 124 2690
## 1 170 390 803 939 775 432 179 3688
## Sum 296 668 1322 1682 1354 753 303 6378addmargins(table(cdd_factores_rendimiento_estudiantes_G6_dep$gender,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality))##
## High Low Medium Sum
## 0 809 268 1613 2690
## 1 1096 379 2213 3688
## Sum 1905 647 3826 6378addmargins(table(cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality))##
## High Low Medium Sum
## 4 81 46 169 296
## 5 195 69 404 668
## 6 407 115 800 1322
## 7 492 158 1032 1682
## 8 403 151 800 1354
## 9 236 77 440 753
## 10 91 31 181 303
## Sum 1905 647 3826 6378Probabilidades
addmargins(prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$gender,cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours)) * 100)##
## 4 5 6 7 8 9
## 0 1.975541 4.358733 8.137347 11.649420 9.078081 5.032926
## 1 2.665412 6.114770 12.590154 14.722484 12.151145 6.773283
## Sum 4.640953 10.473503 20.727501 26.371903 21.229225 11.806209
##
## 10 Sum
## 0 1.944183 42.176231
## 1 2.806522 57.823769
## Sum 4.750706 100.000000addmargins(prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$gender,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality)) * 100)##
## High Low Medium Sum
## 0 12.684227 4.201944 25.290060 42.176231
## 1 17.184070 5.942302 34.697397 57.823769
## Sum 29.868297 10.144246 59.987457 100.000000addmargins(prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality)) * 100)##
## High Low Medium Sum
## 4 1.2699906 0.7212292 2.6497335 4.6409533
## 5 3.0573848 1.0818438 6.3342741 10.4735027
## 6 6.3813108 1.8030731 12.5431170 20.7275008
## 7 7.7140169 2.4772656 16.1806209 26.3719034
## 8 6.3185952 2.3675133 12.5431170 21.2292255
## 9 3.7002195 1.2072750 6.8987143 11.8062088
## 10 1.4267796 0.4860458 2.8378802 4.7507056
## Sum 29.8682973 10.1442458 59.9874569 100.0000000Frecuencias [CPF y CPC]
round(addmargins(prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$gender,cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours),1) * 100,2), 2)##
## 4 5 6 7 8 9 10 Sum
## 0 4.68 10.33 19.29 27.62 21.52 11.93 4.61 100.00
## 1 4.61 10.57 21.77 25.46 21.01 11.71 4.85 100.00round(addmargins(prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$gender,cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours),2) * 100,1), 2)##
## 4 5 6 7 8 9 10
## 0 42.57 41.62 39.26 44.17 42.76 42.63 40.92
## 1 57.43 58.38 60.74 55.83 57.24 57.37 59.08
## Sum 100.00 100.00 100.00 100.00 100.00 100.00 100.00round(addmargins(prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$gender,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality),1) * 100,2), 2)##
## High Low Medium Sum
## 0 30.07 9.96 59.96 100.00
## 1 29.72 10.28 60.01 100.00round(addmargins(prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$gender,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality),2) * 100,1), 2)##
## High Low Medium
## 0 42.47 41.42 42.16
## 1 57.53 58.58 57.84
## Sum 100.00 100.00 100.00round(addmargins(prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality),1) * 100,2), 2)##
## High Low Medium Sum
## 4 27.36 15.54 57.09 100.00
## 5 29.19 10.33 60.48 100.00
## 6 30.79 8.70 60.51 100.00
## 7 29.25 9.39 61.36 100.00
## 8 29.76 11.15 59.08 100.00
## 9 31.34 10.23 58.43 100.00
## 10 30.03 10.23 59.74 100.00round(addmargins(prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality),2) * 100,1), 2)##
## High Low Medium
## 4 4.25 7.11 4.42
## 5 10.24 10.66 10.56
## 6 21.36 17.77 20.91
## 7 25.83 24.42 26.97
## 8 21.15 23.34 20.91
## 9 12.39 11.90 11.50
## 10 4.78 4.79 4.73
## Sum 100.00 100.00 100.00Perfiles [CPF y CPC]
plotct(table(cdd_factores_rendimiento_estudiantes_G6_dep$gender,cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours),"row")
plotct(table(cdd_factores_rendimiento_estudiantes_G6_dep$gender,cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours),"col")
plotct(table(cdd_factores_rendimiento_estudiantes_G6_dep$gender,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality),"row")
plotct(table(cdd_factores_rendimiento_estudiantes_G6_dep$gender,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality),"col")
plotct(table(cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality),"row")
plotct(table(cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality),"col")
Pruebas de Hipótesis
chisq.test(table(cdd_factores_rendimiento_estudiantes_G6_dep$gender,cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours))##
## Pearson's Chi-squared test
##
## data: table(cdd_factores_rendimiento_estudiantes_G6_dep$gender, cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours)
## X-squared = 7.9188, df = 6, p-value = 0.2441chisq.test(table(cdd_factores_rendimiento_estudiantes_G6_dep$gender,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality))##
## Pearson's Chi-squared test
##
## data: table(cdd_factores_rendimiento_estudiantes_G6_dep$gender, cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality)
## X-squared = 0.21754, df = 2, p-value = 0.8969chisq.test(table(cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality))##
## Pearson's Chi-squared test
##
## data: table(cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours, cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality)
## X-squared = 16.922, df = 12, p-value = 0.1526AC Pareja Única
Contingencias y Residuales [SH-TQ]
chisq.test(table(cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality))$observed##
## High Low Medium
## 4 81 46 169
## 5 195 69 404
## 6 407 115 800
## 7 492 158 1032
## 8 403 151 800
## 9 236 77 440
## 10 91 31 181chisq.test(table(cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality))$expected ##
## High Low Medium
## 4 88.41016 30.02697 177.5629
## 5 199.52023 67.76356 400.7162
## 6 394.85889 134.10693 793.0342
## 7 502.38476 170.62622 1008.9890
## 8 404.41675 137.35309 812.2302
## 9 224.90828 76.38617 451.7056
## 10 90.50094 30.73706 181.7620chisq.test(table(cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality))$residuals##
## High Low Medium
## 4 -0.78809123 2.91495350 -0.64260395
## 5 -0.32001229 0.15020146 0.16404261
## 6 0.61099469 -1.64992918 0.24735819
## 7 -0.46331701 -0.96660782 0.72442229
## 8 -0.07044937 1.16443474 -0.42913388
## 9 0.73959887 0.07023278 -0.55076220
## 10 0.05245967 0.04742609 -0.05651975chisq.test(table(cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality))$stdres##
## High Low Medium
## 4 -0.96369222 3.14903679 -1.04031381
## 5 -0.40386264 0.16746552 0.27408343
## 6 0.81944341 -1.95492942 0.43920466
## 7 -0.64476175 -1.18838065 1.33466398
## 8 -0.09478452 1.38407507 -0.76438502
## 9 0.94041607 0.07889472 -0.92714367
## 10 0.06418558 0.05126412 -0.09155272Contribuciones [SH-TQ]
chisq.test(table(cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours, cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality))$residuals^2/chisq.test(table(cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality))$statistic*100##
## High Low Medium
## 4 3.67031750 50.21273898 2.44026746
## 5 0.60517918 0.13332111 0.15902447
## 6 2.20610166 16.08722945 0.36157903
## 7 1.26854866 5.52142605 3.10123200
## 8 0.02932952 8.01273821 1.08826898
## 9 3.23253417 0.02914945 1.79258144
## 10 0.01626304 0.01329185 0.01887779Correspondencia Simple Unidimensional [SH-TQ]
CA(table(cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality), graph = FALSE)$eig## eigenvalue percentage of variance cumulative percentage of variance
## dim 1 0.0023643540 89.11435 89.11435
## dim 2 0.0002888145 10.88565 100.00000CA(table(cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality), graph = FALSE)$col## $coord
## Dim 1 Dim 2
## High -0.01853203 0.025222879
## Low 0.14465053 0.001531897
## Medium -0.01523403 -0.012817753
##
## $contrib
## Dim 1 Dim 2
## High 4.338542 65.79316106
## Low 89.773329 0.08242515
## Medium 5.888129 34.12441379
##
## $cos2
## Dim 1 Dim 2
## High 0.3505774 0.6494226117
## Low 0.9998879 0.0001121425
## Medium 0.5855018 0.4144981830
##
## $inertia
## [1] 0.0002925987 0.0021227974 0.0002377725CA(table(cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality), graph = FALSE)$row## $coord
## Dim 1 Dim 2
## 4 0.179135682 -0.010472310
## 5 0.006545161 -0.013583947
## 6 -0.048146548 0.008353564
## 7 -0.024264213 -0.020158411
## 8 0.033211996 0.006168220
## 9 0.001681342 0.033660074
## 10 0.002741644 0.004419503
##
## $contrib
## Dim 1 Dim 2
## 4 62.98815665 1.7622728
## 5 0.18976676 6.6915217
## 6 20.32192015 5.0080833
## 7 6.56691493 37.1052212
## 8 9.90402240 2.7966287
## 9 0.01411596 46.3149905
## 10 0.01510316 0.3212817
##
## $cos2
## Dim 1 Dim 2
## 4 0.996594044 0.003405956
## 5 0.188417557 0.811582443
## 6 0.970776477 0.029223523
## 7 0.591643064 0.408356936
## 8 0.966657190 0.033342810
## 9 0.002488857 0.997511143
## 10 0.277892654 0.722107346
##
## $inertia
## [1] 1.494353e-03 2.381284e-05 4.949462e-04 2.624304e-04 2.422432e-04
## [6] 1.340982e-04 1.285001e-063.3. Correspondencias Múltiples
El análisis de correspondencias múltiples (ACM) es una técnica estadística multivariante diseñada para analizar relaciones entre varias variables cualitativas. Esta metodología amplía los principios del análisis de correspondencias simples, permitiendo estudiar conjuntos más complejos de datos categóricos. Su objetivo principal es descubrir patrones subyacentes, simplificar la representación de los datos y visualizar asociaciones en un espacio de dimensiones reducidas.
El ACM utiliza como base las tablas de contingencia y las convierte en representaciones gráficas que proyectan las categorías de las variables en un espacio factorial. Esto se logra calculando factores principales que explican la mayor parte de la varianza observada. Entre los elementos clave de esta técnica se encuentran las calidades de representación, que miden la precisión con la que cada categoría es representada en el espacio reducido, y las contribuciones, que indican la relevancia de cada variable o categoría en la formación de los ejes factoriales. Estas herramientas permiten interpretar de manera detallada las relaciones entre las categorías, ayudando a identificar asociaciones significativas y redundancias entre las variables cualitativas.
En conclusión, el análisis de correspondencias múltiples es una herramienta poderosa para la exploración y comprensión de datos cualitativos complejos. Su capacidad para simplificar y visualizar las relaciones entre múltiples variables lo convierte en una técnica esencial en diversos campos, desde el marketing y las ciencias sociales hasta la biomedicina y la industria. Al ofrecer representaciones claras y accesibles, el ACM facilita la toma de decisiones informadas basadas en los patrones identificados en los datos.
ACM
round(MCA(cdd_factores_rendimiento_estudiantes_G6[1:400, -c(1,2,6,7,10,15,20)], graph = FALSE)$eig,2)## eigenvalue percentage of variance cumulative percentage of variance
## dim 1 0.11 6.01 6.01
## dim 2 0.11 5.74 11.75
## dim 3 0.10 5.56 17.32
## dim 4 0.10 5.44 22.75
## dim 5 0.10 5.30 28.05
## dim 6 0.09 5.05 33.10
## dim 7 0.09 4.81 37.91
## dim 8 0.09 4.73 42.65
## dim 9 0.08 4.60 47.24
## dim 10 0.08 4.43 51.68
## dim 11 0.08 4.35 56.03
## dim 12 0.07 4.04 60.07
## dim 13 0.07 3.91 63.97
## dim 14 0.07 3.79 67.76
## dim 15 0.07 3.73 71.50
## dim 16 0.07 3.66 75.15
## dim 17 0.06 3.46 78.62
## dim 18 0.06 3.42 82.03
## dim 19 0.06 3.34 85.37
## dim 20 0.06 3.29 88.66
## dim 21 0.06 2.99 91.64
## dim 22 0.05 2.95 94.59
## dim 23 0.05 2.81 97.40
## dim 24 0.05 2.60 100.00Biplot ACM
fviz_mca_biplot(MCA(cdd_factores_rendimiento_estudiantes_G6[1:400, -c(1,2,6,7,10,15,20)], graph = FALSE), repel = TRUE)
Calidad de Representación
fviz_mca_var(MCA(cdd_factores_rendimiento_estudiantes_G6[1:400, -c(1,2,6,7,10,15,20)], graph = FALSE), col.var ="cos2", gradient.cols = c("#00AFBB", "#E7B800", "#FC4E07"), repel = TRUE)
MCA(cdd_factores_rendimiento_estudiantes_G6[1:400, -c(1,2,6,7,10,15,20)], graph = FALSE)$var$cos2## Dim 1 Dim 2 Dim 3
## Parental_Involvement_High 0.0123084649 0.0866545332 1.129235e-03
## Parental_Involvement_Low 0.1126451112 0.0273664629 1.703822e-01
## Parental_Involvement_Medium 0.1364961296 0.0129559395 9.623058e-02
## Access_to_Resources_High 0.1626888602 0.0060225300 1.574858e-01
## Access_to_Resources_Low 0.0002086680 0.1033106118 2.259423e-02
## Access_to_Resources_Medium 0.1309860646 0.0284388355 6.467455e-02
## Extracurricular_Activities_No 0.1469596809 0.0236749333 1.396550e-03
## Extracurricular_Activities_Yes 0.1469596809 0.0236749333 1.396550e-03
## Motivation_Level_High 0.0400546030 0.0066876366 3.569744e-02
## Motivation_Level_Low 0.0130154633 0.0067727944 4.487526e-02
## Motivation_Level_Medium 0.0678918606 0.0001069574 1.897254e-03
## Internet_Access_No 0.0161574557 0.0979627738 5.802918e-02
## Internet_Access_Yes 0.0161574557 0.0979627738 5.802918e-02
## Family_Income_High 0.0221439789 0.0907829751 1.260468e-01
## Family_Income_Low 0.0566036344 0.1352028203 1.034950e-01
## Family_Income_Medium 0.1170787839 0.0165270762 1.901419e-03
## Teacher_Quality.NA 0.0051069573 0.0012099853 2.257022e-03
## Teacher_Quality_High 0.0620747986 0.0226544026 1.286101e-02
## Teacher_Quality_Low 0.0060549866 0.0064600318 8.492007e-02
## Teacher_Quality_Medium 0.0317018961 0.0079672659 7.290523e-02
## Private 0.2616645563 0.0547850328 3.814371e-02
## Public 0.2616645563 0.0547850328 3.814371e-02
## Negative 0.0008319493 0.0015948042 2.923348e-01
## Neutral 0.0178537985 0.0010596023 2.882840e-01
## Positive 0.0236176367 0.0040834677 8.019142e-03
## Learning_Disabilities_No 0.0148242344 0.3180158193 5.036854e-03
## Learning_Disabilities_Yes 0.0148242344 0.3180158193 5.036854e-03
## College 0.0083033617 0.0244585828 5.362727e-04
## High School 0.0823511051 0.0154375511 3.096065e-03
## Parental_Education_Level.NA 0.1021164376 0.0067034686 9.910719e-05
## Postgraduate 0.0906660368 0.0003019853 9.639451e-03
## Distance_from_Home.NA 0.0762364410 0.1036112150 1.028685e-01
## Far 0.0918987193 0.0045810233 2.798894e-04
## Moderate 0.0015071607 0.0840796053 2.752182e-03
## Near 0.0108990295 0.1482792758 1.191841e-03
## Female 0.0024589870 0.2137164527 1.130424e-02
## Male 0.0024589870 0.2137164527 1.130424e-02
## Dim 4 Dim 5
## Parental_Involvement_High 0.0406391041 6.370585e-03
## Parental_Involvement_Low 0.0060213566 2.514134e-02
## Parental_Involvement_Medium 0.0550276226 3.898586e-03
## Access_to_Resources_High 0.0143638787 4.177376e-02
## Access_to_Resources_Low 0.0395374697 6.242293e-02
## Access_to_Resources_Medium 0.0014393694 4.154965e-06
## Extracurricular_Activities_No 0.0003204711 1.162961e-01
## Extracurricular_Activities_Yes 0.0003204711 1.162961e-01
## Motivation_Level_High 0.0299370381 1.089639e-01
## Motivation_Level_Low 0.1679142118 1.002173e-01
## Motivation_Level_Medium 0.2572439864 7.612954e-04
## Internet_Access_No 0.0074902533 1.039761e-02
## Internet_Access_Yes 0.0074902533 1.039761e-02
## Family_Income_High 0.0955781084 2.764070e-02
## Family_Income_Low 0.0005529268 9.385186e-02
## Family_Income_Medium 0.0653583767 1.776527e-01
## Teacher_Quality.NA 0.0298701134 4.645160e-02
## Teacher_Quality_High 0.1690729061 3.915613e-02
## Teacher_Quality_Low 0.0355087971 2.072463e-01
## Teacher_Quality_Medium 0.0903957115 2.333029e-01
## Private 0.1501903228 3.915372e-02
## Public 0.1501903228 3.915372e-02
## Negative 0.0637406327 7.161204e-02
## Neutral 0.0025907914 1.318033e-02
## Positive 0.0235055219 1.071299e-01
## Learning_Disabilities_No 0.0168126834 3.638491e-02
## Learning_Disabilities_Yes 0.0168126834 3.638491e-02
## College 0.1035509468 6.416522e-02
## High School 0.1473863199 1.127335e-02
## Parental_Education_Level.NA 0.0174829767 1.368058e-02
## Postgraduate 0.0082277231 2.327518e-02
## Distance_from_Home.NA 0.0117580703 1.277691e-02
## Far 0.0786888579 2.479770e-02
## Moderate 0.0022237475 4.173977e-03
## Near 0.0141999693 3.609169e-02
## Female 0.1050243414 7.066896e-03
## Male 0.1050243414 7.066896e-03Contribuciones
fviz_contrib(MCA(cdd_factores_rendimiento_estudiantes_G6[1:400, -c(1,2,6,7,10,15,20)], graph = FALSE), choice = "var", axes = 1, top = 15)
fviz_contrib(MCA(cdd_factores_rendimiento_estudiantes_G6[1:400, -c(1,2,6,7,10,15,20)], graph = FALSE), choice = "var", axes = 2, top = 15)
fviz_contrib(MCA(cdd_factores_rendimiento_estudiantes_G6[1:400, -c(1,2,6,7,10,15,20)], graph = FALSE), choice = "var", axes = 3, top = 15)
fviz_contrib(MCA(cdd_factores_rendimiento_estudiantes_G6[1:400, -c(1,2,6,7,10,15,20)], graph = FALSE), choice = "var", axes = 4, top = 15)
fviz_contrib(MCA(cdd_factores_rendimiento_estudiantes_G6[1:400, -c(1,2,6,7,10,15,20)], graph = FALSE), choice = "var", axes = 5, top = 15)
Biplot con Contribuciones
fviz_mca_var(MCA(cdd_factores_rendimiento_estudiantes_G6[1:400, -c(1,2,6,7,10,15,20)], graph = FALSE), col.var ="contrib", gradient.cols = c("#00AFBB", "#E7B800", "#FC4E07"), repel = TRUE)
Fase 4 [Conglomerados]
4.1. Objetivos
En términos generales, los objetivos del análisis de conglomerados se centran en agrupar observaciones o elementos con características similares dentro de un conjunto de datos, con base en los resultados obtenidos en las fases previas. Esta etapa incluye tanto el enfoque jerárquico, representado mediante dendogramas, como el no jerárquico, utilizando el método de K-medias. A través de estos métodos, se busca identificar patrones subyacentes, clasificar elementos en grupos homogéneos y facilitar interpretaciones que contribuyan a un mejor entendimiento de la estructura y organización de los datos.
4.2. Agrupación Jerárquica
La agrupación jerárquica es una técnica de análisis multivariante utilizada para clasificar elementos o individuos en grupos con características similares. Este método organiza los datos en una estructura jerárquica en forma de árbol, conocida como dendograma, lo que permite explorar las relaciones entre las observaciones de manera visual y progresiva. Es una herramienta clave para identificar patrones y establecer relaciones entre los elementos de un conjunto de datos.
El proceso de agrupación jerárquica comienza con cada elemento considerado como un grupo independiente. Gradualmente, los grupos se van fusionando en función de su similitud, hasta formar un único conglomerado que incluye todos los elementos. Existen diferentes métodos para medir la similitud o distancia entre los elementos y los grupos, como la distancia euclidiana y los métodos de enlace (simple, completo o promedio). El resultado del análisis se representa en un dendograma, que muestra cómo se forman los conglomerados a diferentes niveles de similitud. Este enfoque permite no solo determinar cuántos grupos son más adecuados para el análisis, sino también interpretar la estructura interna de cada conglomerado.
En síntesis, la agrupación jerárquica es una técnica eficaz para analizar estructuras complejas y revelar relaciones inherentes en los datos. Su capacidad para representar las agrupaciones en un formato visualmente intuitivo la convierte en una herramienta valiosa en campos como la biología, la psicología, la mercadotecnia y la industria. Al permitir la identificación de patrones y similitudes, la agrupación jerárquica facilita la toma de decisiones basada en una comprensión profunda de las características de los datos.
Campo clasificador
cdd_factores_rendimiento_estudiantes_G6_dep_CLAS_EST_Estados_costa_este_Average <- read_excel("C:/Users/57313/Desktop/Gdd_(G6)/cdd_G6_dep/cdd_factores_rendimiento_estudiantes_G6_dep_CLAS_EST_Estados_costa_este_Average.xlsx")
head(as.data.frame(cdd_factores_rendimiento_estudiantes_G6_dep_CLAS_EST_Estados_costa_este_Average))## estados_costa_este hours_studied_S01 age_S01 attendance_S01
## 1 1. Maine 0.438375 0.4899329 0.4766219
## 2 10. Virginia 0.450535 0.4656751 0.4829519
## 3 11. Carolina del Norte 0.435899 0.4968220 0.5159428
## 4 12. Carolina del Sur 0.439883 0.5242915 0.5080972
## 5 13.Georgia 0.435017 0.4850575 0.5037931
## 6 14. Florida 0.446928 0.4747706 0.4966170
## sleep_hours_S01 previous_scores_S01 tutoring_sessions_S01
## 1 0.5033557 0.4890828 0.2035794
## 2 0.5232647 0.4716247 0.1905034
## 3 0.4985876 0.5408475 0.1819386
## 4 0.5030364 0.5126316 0.1938259
## 5 0.5126437 0.4784368 0.2020115
## 6 0.4961774 0.4877982 0.1926606
## physical_activity_S01 exam_score_S01
## 1 0.5059657 0.2668158
## 2 0.4969489 0.2715993
## 3 0.4982345 0.2739642
## 4 0.5080972 0.2750337
## 5 0.4927203 0.2692209
## 6 0.4908257 0.2708461Disimilaridad
data_ = as.data.frame(cdd_factores_rendimiento_estudiantes_G6_dep_CLAS_EST_Estados_costa_este_Average)[, -c (1)]
rownames(data_) = unclass(cdd_factores_rendimiento_estudiantes_G6_dep_CLAS_EST_Estados_costa_este_Average$estados_costa_este)
fviz_dist(get_dist(data_, stand = T, method = "euclidean"), gradient = list(low = "#00AFBB", mid = "white", high = "#FC4E07"))
Optimización de Mojena
Unión simple
hc_single = hclust(get_dist(data_, stand = T, method = "euclidean"), method = "single")
mojena = function(hc){
n_hd = length(hc$height)
alp_g = 0 ; alpha = hc$height[n_hd:1]
for(i in 1:(n_hd-1)){
alp_g[i] = mean(alpha[(n_hd-i+1):1])+1.25*sd(alpha[(n_hd-i+1):1])
}
nog = sum(alp_g<= alpha[-n_hd]) + 1
plot(alpha[-n_hd], pch=20, col=(alp_g>alpha[-n_hd])+1, main = paste("Optimal number of groups =",nog),
ylab = expression(alpha[g]), xlab="Nodes")}
mojena(hc_single)
Unión completa
hc_complete = hclust(get_dist(data_, stand = T, method = "euclidean"), method = "complete")
mojena = function(hc){
n_hd = length(hc$height)
alp_g = 0 ; alpha = hc$height[n_hd:1]
for(i in 1:(n_hd-1)){
alp_g[i] = mean(alpha[(n_hd-i+1):1])+1.25*sd(alpha[(n_hd-i+1):1])
}
nog = sum(alp_g<= alpha[-n_hd]) + 1
plot(alpha[-n_hd], pch=20, col=(alp_g>alpha[-n_hd])+1, main = paste("Optimal number of groups =",nog),
ylab = expression(alpha[g]), xlab="Nodes")}
mojena(hc_complete)
Unión Promedio
hc_average = hclust(get_dist(data_, stand = T, method = "euclidean"), method = "average")
mojena = function(hc){
n_hd = length(hc$height)
alp_g = 0 ; alpha = hc$height[n_hd:1]
for(i in 1:(n_hd-1)){
alp_g[i] = mean(alpha[(n_hd-i+1):1])+1.25*sd(alpha[(n_hd-i+1):1])
}
nog = sum(alp_g<= alpha[-n_hd]) + 1
plot(alpha[-n_hd], pch=20, col=(alp_g>alpha[-n_hd])+1, main = paste("Optimal number of groups =",nog),
ylab = expression(alpha[g]), xlab="Nodes")}
mojena(hc_average)
Dendograma optimizado
Enlace simple
suppressWarnings(fviz_dend(hc_single, k = 3, cex = 0.5, k_colors = "npg", color_labels_by_k = T, rect = T))
Enlace Completo
fviz_dend(hc_complete, k = 3, cex = 0.5, k_colors = "npg", color_labels_by_k = T, rect = T)
Enlace promedio
fviz_dend(hc_average, k = 4, cex = 0.5, k_colors = "npg", color_labels_by_k = T, rect = T)
4.3. Agrupación No-Jerárquica
La agrupación no jerárquica es un enfoque utilizado para clasificar datos en grupos sin necesidad de una estructura jerárquica preestablecida. A diferencia de la agrupación jerárquica, donde los datos se organizan en forma de un árbol, la agrupación no jerárquica divide los datos en un número fijo de grupos que no tienen una relación jerárquica entre sí. El método más utilizado en este tipo de análisis es el de K-medias, que se basa en asignar cada dato al grupo más cercano según ciertas características.
El proceso de agrupación no jerárquica comienza con una asignación inicial de los datos a un número predefinido de grupos. Luego, se calcula el centro de cada grupo, que sería el promedio de las características de los elementos dentro de ese grupo. A partir de ahí, se reasignan los datos a los grupos en función de qué tan cerca están de los centros. Este proceso se repite varias veces hasta que las asignaciones no cambian más. La principal ventaja de este método es su simplicidad y rapidez, lo que lo hace ideal para manejar grandes cantidades de datos. Sin embargo, uno de los retos es que debemos decidir de antemano cuántos grupos queremos, lo que puede influir en los resultados que obtengamos.
En resumen, la agrupación no jerárquica es una técnica muy útil cuando necesitamos segmentar un conjunto de datos en grupos específicos sin la necesidad de una jerarquía compleja. Es ampliamente utilizada en áreas como el análisis de clientes, la segmentación de mercados y el análisis de patrones. Aunque es necesario definir previamente el número de grupos, su simplicidad y eficiencia la convierten en una herramienta valiosa, especialmente cuando queremos obtener resultados rápidos y claros sin complicarnos con estructuras demasiado complejas.
K-óptimos
Elbow
fviz_nbclust(data_, kmeans, method = "wss") + geom_vline(xintercept = 3, linetype = 2)
Silhouette
fviz_nbclust(data_, kmeans, method = "silhouette")
Gap Statistic
fviz_nbclust(data_, kmeans, method = "gap_stat")
Majority Rule
suppressWarnings(NbClust(data = data_, diss = NULL, distance = "euclidean", min.nc = 2, max.nc = 10, method = "kmeans")$Best.nc)
## *** : The Hubert index is a graphical method of determining the number of clusters.
## In the plot of Hubert index, we seek a significant knee that corresponds to a
## significant increase of the value of the measure i.e the significant peak in Hubert
## index second differences plot.
## 
## *** : The D index is a graphical method of determining the number of clusters.
## In the plot of D index, we seek a significant knee (the significant peak in Dindex
## second differences plot) that corresponds to a significant increase of the value of
## the measure.
##
## *******************************************************************
## * Among all indices:
## * 6 proposed 2 as the best number of clusters
## * 7 proposed 3 as the best number of clusters
## * 1 proposed 4 as the best number of clusters
## * 1 proposed 7 as the best number of clusters
## * 1 proposed 9 as the best number of clusters
## * 7 proposed 10 as the best number of clusters
##
## ***** Conclusion *****
##
## * According to the majority rule, the best number of clusters is 3
##
##
## *******************************************************************## KL CH Hartigan CCC Scott Marriot TrCovW TraceW
## Number_clusters 10.0000 2.0000 10.0000 2.0000 7.0000 3 3 3.0000
## Value_Index 5.1223 8.7598 2.2829 21.6469 321.1832 0 0 0.0012
## Friedman Rubin Cindex DB Silhouette Duda
## Number_clusters 9.000000e+00 3.0000 10.0000 10.0000 10.0000 2.0000
## Value_Index 2.588833e+16 -213.6146 0.0065 0.5225 0.6176 2.1782
## PseudoT2 Beale Ratkowsky Ball PtBiserial Frey McClain
## Number_clusters 2.00 2.000 4.0000 3.0000 3.0000 1 2.0000
## Value_Index -5.95 -2.287 0.3552 0.0023 0.6592 NA 0.7691
## Dunn Hubert SDindex Dindex SDbw
## Number_clusters 10.0000 0 3.0000 0 10.0000
## Value_Index 0.8705 0 65.2446 0 0.0514Resultados K-means
K-óptimo [Sil_Ma-Rul 2]
set.seed(780728)
print(kmeans(data_, 2, nstart = 25))## K-means clustering with 2 clusters of sizes 6, 8
##
## Cluster means:
## hours_studied_S01 age_S01 attendance_S01 sleep_hours_S01
## 1 0.4428140 0.4771353 0.4925503 0.5059283
## 2 0.4401975 0.5071655 0.5061505 0.5057572
## previous_scores_S01 tutoring_sessions_S01 physical_activity_S01
## 1 0.4849290 0.1910034 0.4978247
## 2 0.5127268 0.1840597 0.4935738
## exam_score_S01
## 1 0.2708626
## 2 0.2732481
##
## Clustering vector:
## 1. Maine 10. Virginia 11. Carolina del Norte
## 1 1 2
## 12. Carolina del Sur 13.Georgia 14. Florida
## 2 1 1
## 2. New Hampshire 3. Massachusetts 4. Rhode Island
## 1 2 2
## 5. Connecticut 6. Nueva York 7. Nueva Jersey
## 1 2 2
## 8. Delaware 9. Maryland
## 2 2
##
## Within cluster sum of squares by cluster:
## [1] 0.003138263 0.005965692
## (between_SS / total_SS = 42.2 %)
##
## Available components:
##
## [1] "cluster" "centers" "totss" "withinss" "tot.withinss"
## [6] "betweenss" "size" "iter" "ifault"K-óptimo [Elb 3]
set.seed(780728)
print(kmeans(data_, 3, nstart = 25))## K-means clustering with 3 clusters of sizes 1, 7, 6
##
## Cluster means:
## hours_studied_S01 age_S01 attendance_S01 sleep_hours_S01
## 1 0.4224300 0.5389610 0.4888528 0.5115440
## 2 0.4427357 0.5026232 0.5086216 0.5049305
## 3 0.4428140 0.4771353 0.4925503 0.5059283
## previous_scores_S01 tutoring_sessions_S01 physical_activity_S01
## 1 0.5002165 0.1766775 0.4895382
## 2 0.5145140 0.1851143 0.4941503
## 3 0.4849290 0.1910034 0.4978247
## exam_score_S01
## 1 0.2633478
## 2 0.2746624
## 3 0.2708626
##
## Clustering vector:
## 1. Maine 10. Virginia 11. Carolina del Norte
## 3 3 2
## 12. Carolina del Sur 13.Georgia 14. Florida
## 2 3 3
## 2. New Hampshire 3. Massachusetts 4. Rhode Island
## 3 2 2
## 5. Connecticut 6. Nueva York 7. Nueva Jersey
## 3 2 1
## 8. Delaware 9. Maryland
## 2 2
##
## Within cluster sum of squares by cluster:
## [1] 0.000000000 0.003697523 0.003138263
## (between_SS / total_SS = 56.6 %)
##
## Available components:
##
## [1] "cluster" "centers" "totss" "withinss" "tot.withinss"
## [6] "betweenss" "size" "iter" "ifault"Gráficos K-means
K-óptimo [Sil_Ma-Rul 2]
fviz_cluster(kmeans(data_, 2, nstart = 25), data = data_, palette = c("#2E9FDF", "#00AFBB", "#E7B800", "#E7B801"), ellipse.type = "euclid", star.plot = TRUE, repel = TRUE, ggtheme = theme_minimal()
)
K-óptimo [Elb 3]
fviz_cluster(kmeans(data_, 3, nstart = 25), data = data_, palette = c("#2E9FDF", "#00AFBB", "#E7B800", "#E7B801"), ellipse.type = "euclid", star.plot = TRUE, repel = TRUE, ggtheme = theme_minimal()
)## Too few points to calculate an ellipse
Fase 5 [Análisis de Regresión]
El análisis de regresión es un proceso estadístico utilizado para estimar las relaciones entre variables (una dependiente o de respuesta y otras independientes o predictoras) mediante técnicas de modelado y análisis que permiten comprender cómo varía el valor de la variable dependiente al modificar el valor de una o más variables independientes. Los modelos de regresión que se analizarán en este documento son: lineales (simples y Múltiples) y logísticos, los cuales se entienden como casos del modelo de regresión lineal generalizado.
5.1. Objetivos
El análisis de regresión es una herramienta estadística fundamental cuyo principal objetivo es entender y modelar la relación entre una variable dependiente, también conocida como variable respuesta, y una o más variables independientes, llamadas predictoras. Este análisis permite explorar cómo las variables independientes afectan a la variable dependiente, prop}orcionando información sobre la dirección y la magnitud de estas relaciones. A través de esta herramienta, es posible identificar patrones, establecer tendencias y obtener conclusiones significativas basadas en los datos.
Uno de los objetivos centrales del análisis de regresión es la predicción, es decir, usar el modelo desarrollado para estimar o prever el valor de la variable dependiente en función de los valores conocidos de las variables independientes. Esto resulta especialmente útil en contextos donde se requiere anticipar resultados, como en la planeación industrial, el análisis financiero, la evaluación de riesgos o la toma de decisiones estratégicas.
5.2. Regresión Lineal Simple
En este estudio, se utilizará un modelo de Regresión Lineal Simple (RLS) compuesto por dos variables estadísticas \(x\) y \(Y\), donde se asume que \(Y\) depende de \(x\).La relación entre estas variables se expresa matemáticamente como: \[Y = \beta_0 + \beta_1 x + \varepsilon \hspace{10mm} \hspace{10mm}(1)\] donde:
- \(Y\): representa una variable de respuesta, caracterizada por su naturaleza aleatoria.
- \(x\): es una variable predictora, no aleatoria, que influye sobre \(Y\)
- \(\varepsilon\): simboliza una variable aleatoria no observable, que recoge el error o la variabilidad no explicada por el modelo.
- \(\beta_0\) y \(\beta_1\): son parámetros reales desconocidos que deben ser estimados a partir de los datos..
A diferencia del modelo lineal simple determinístico, representado por \(y = \beta_0 + \beta_1 x\), el probablístico supone que el valor esperado de \(Y\) es una función lineal de \(x\), Sin embargo, cuando x es fija, la variable Y puede diferir de su valor esperado debido a una cantidad aleatoria, representada como \(\varepsilon\). Además, el término \(\varepsilon\) en la ecuación del modelo \((1)\) se considera normalmente distribuido, con \(E(\varepsilon)=0\) y \(V(\varepsilon)=\sigma^2\). Este término aleatorio \(\varepsilon\) también se conoce como el error aleatorio o desviación aleatoria dentro del modelo.
De manera complementaria, los valores de \(\beta_0\), \(\beta_1\) y \(\sigma^2\), generalmente no son conocidos. En su lugar, se dispone de una muestra de datos compuesta por pares ordenados \(( (x_1, y_1), (x_2, y_2), \dots, (x_n, y_n) )\), que permite estimar los parámetros del modelo y la línea de regresión verdadera, asumiendo que las observaciones son independientes. En este contexto, \(y_i\) representa el valor observado de una variable aleatoria \(Y_i\), definida como \(Y_i = \beta_0 + \beta_1 x_i + \varepsilon_i\). Además, las \(n\) desviaciones \(\varepsilon_1, \varepsilon_2, \dots, \varepsilon_n\) se consideran variables aleatorias independientes.
Según el modelo, los puntos observados se distribuirán de manera aleatoria alrededor de la línea de regresión verdadera. En este contexto, la estimación de \(y = \beta_0 + \beta_1 x\) debe representar una línea que se ajuste de la mejor manera posible a los puntos de la muestra. Esta línea debe cumplir con la característica de minimizar las desviaciones verticales (o distancias) entre los puntos observados y la línea, manteniéndolas lo más pequeñas posible.
El principio de los mínimos cuadrados establece que la desviación vertical de cada punto \((x_i, y_i)\) con respecto a la recta de regresión \(y= b_0+b_1x\) es \(y_i-(b_0+b_1x)\). La suma de las desviaciones verticales al cuadrado de todos los puntos \((x_i, y_i)\) respecto a la línea de regresión se expresa como: \(f(b_0,b_1)=\sum_{i=1}^n(y_i-(b_0+b_1x_i))^2\). De esta manera, las estimaciones de los parámetros \(\beta_0\) y \(\beta_1\), denotadas como \(\hat{\beta}_0\) y \(\hat{\beta}_1\), son los valores que minimizan la función \(f(b_0,b_1)\). Es decir, la suma de los cuadrados de las desviaciones alcanza su valor más bajo cuando se utilizan estas estimaciones, de modo que \(f(\hat{\beta}_0,\hat{\beta}_1)\leq f(b_0,b_1)\) para cualquier valor de \(\beta_0\) y \(\beta_1\). Por lo tanto, la línea de regresión estimada, o línea de mínimos cuadrados, es: \(y=\hat{\beta}_0 + \hat{\beta}_1x\).
Después de calcular y resolver las ecuaciones en derivadas parciales de \(f(b_0,b_1)\) con respecto a \(b_0\) y \(b_1\), y luego igualarlas a cero, se obtiene un sistema de ecuaciones conocido como ecuaciones normales. Este sistema es lineal en los parámetros \(b_0\) y \(b_1\), y siempre que al menos dos de los valores de \(x_i\) sean diferentes, las estimaciones de los mínimos cuadrados serán la única solución para el sistema. Así, la estimación de \(\beta_1\) de la línea de regresión verdadera se calcula como: \[\hat{\beta}_1=\dfrac{\sum_{i=1}^n(x_i-\bar{x})(y_i-\bar{y})}{\sum_{i=1}^n(x_i-\bar{x})^2}=\dfrac{S_{xy}}{S_{xx}}\hspace{10mm}(2)\] Y la estimación de _0 se calcula como: \[\hat{\beta}_0=\dfrac{\sum_{i=1}^ny_i-\hat{\beta}_1\sum_{i=1}^nx_i}{n}=\bar{y}-\hat{\beta}_1\bar{x}\hspace{10mm}(3)\] Para obtener estos cálculos con precisión, es crucial minimizar los efectos del redondeo. Además, antes de calcular \(\hat{\beta}_1\) y \(\hat{\beta}_0\), es importante revisar visualmente el conjunto de datos antes de aplicar el modelo para evaluar si es adecuado utilizar un modelo lineal probabilístico. En otras palabras, si los puntos no parecen agruparse alrededor de una línea recta con una dispersión similar en todas las \(x_i\), se debe considerar la posibilidad de utilizar otros tipos de modelos.
El principio de los mínimos cuadrados establece que la desviación vertical de cada punto \((x_i, y_i)\) respecto a la recta de regresión \(y=b_0+b_1x\) es \(y_i-(b_0 +b_1x_i)\). La suma de las desviaciones verticales al cuadrado de todos los puntos \((x_i, y_i)\) con respecto a la línea de regresión se expresa como:\(f(b_0,b_1)=\sum_{i=1}^n(y_i-(b_0+b_1x_i))^2\). Asi, Las estimaciones de los parámetros \(\beta_0\) y \(\beta_1\), denotadas como \(\hat{\beta}_0\) y \(\hat{\beta}_1\), son los valores que minimizan la función \(f(b_0,b_1)\). Es decir, la suma de los cuadrados de las desviaciones alcanza su valor más bajo cuando se utilizan estas estimaciones, lo que implica que:\(f(\hat{\beta}_0,\hat{\beta}_1)\leq f(b_0,b_1)\) Por lo tanto, la línea de regresión estimada, o línea de mínimos cuadrados, es:\(y=\hat{\beta}_0 + \hat{\beta}_1x\).
Es importante tener en cuenta que la línea de mínimos cuadrados debe usarse con precaución para predecir valores de \(x\) fuera del rango de los datos observados, ya que la relación lineal ajustada podría no ser válida en esos casos.
El parámetro \(\sigma^2\) refleja la cantidad de variabilidad en el modelo de regresión, y su valor determina cuánta dispersión de los valores observados se produce en torno a la línea de regresión verdadera. Los residuos \(y_i - \hat{y_i}\) son las desviaciones con respecto a la línea estimada. Si los residuos son pequeños, indica que la relación lineal entre \(x\) y \(y\) explica en gran medida la variabilidad en \(y\). Si son grandes, sugiere que la variabilidad de \(y\) no se debe únicamente a esta relación lineal. La estimación de \(\sigma^2\) en un análisis de regresión se basa en la suma de los cuadrados residuales (SCE), que se calcula como:\[SCE=\sum_{i=1}^ny_i^2-\hat{\beta}_0\sum_{i=1}^ny_i-\hat{\beta}_1\sum_{i=1}^nx_iy_i\hspace{10mm}(4)\] \[\hat\sigma^2=s^2=\dfrac{SCE}{n-2}\hspace{10mm}(5)\]La suma de cuadrados del error (SCE) mide la porción de la variación en \(y\) que no es explicada por el modelo de regresión lineal, es decir, la variación que no está asociada con la relación lineal entre las variables. Además de la SCE, existe otra medida conocida como la suma total de cuadrados (STC), que representa la variación total observada en los valores de \(y\). La STC se calcula como:\[STC=\sum_{i=1}^ny_i^2-\frac{(\sum_{i=1}^ny_i)^2}{n}\hspace{10mm}(6)\] La proporción de la variación total que no es explicada por el modelo de regresión lineal simple se puede calcular mediante la razón \(SCE/STC\). A partir de esto, se define el coeficiente de determinación \(r^2\) como: \[r^2=1-\frac{SCE}{STC}\hspace{10mm}(7)\] Este coeficiente se interpreta como la proporción de la variación observada en \(y\) que puede ser explicada por el modelo de regresión lineal simple, es decir, la variación atribuible a una relación lineal aproximada entre \(x\) y \(y\). Cuanto más cercano sea \(r^2\) a 1, mejor será el modelo en explicar la variación de \(y\). Otra forma de calcular \(r^2\) utiliza la suma de cuadrados debida a la regresión (SCR), que mide la porción de la variación total explicada por el modelo. En este caso, el coeficiente de determinación se expresa como: \[r^2=1-\frac{SCE}{STC}=\frac{STC-SCE}{STC}=\frac{SCR}{STC}\hspace{10mm}(8)\] Es bien conocido que cualquier cantidad calculada a partir de datos muestrales puede variar entre muestras. Por ello, los procedimientos inferenciales estandarizan un estimador al restarle su valor medio y dividirlo entre su desviación estándar estimada. En el caso de un modelo de regresión lineal simple, se considera que las variables estandarizadas: \(t_{(n-2)}=\dfrac{\hat{\beta}_0-\beta_0}{\hat{\sigma} \sqrt{1/n+\bar{x}^2/S_{xx}}}\) y \(t_{(n-2)}=\dfrac{\hat{\beta}_1-\beta_1}{ \hat{\sigma} \sqrt{1/S_{xx}}}\) siguen una distribución \(t\) con \(n-2\) grados de libertad. A partir de esto, los intervalos de confianza al \(100*(1-\alpha)\%\) para los parámetros \(\beta_0\) (intercepto) y \(\beta_1\) (pendiente) de la línea de regresión verdadera son: \[\hat{\beta}_0 \pm t_{\alpha/2, n-2} \cdot \hat{\sigma} \sqrt{1/n+\bar{x}^2/S_{xx}}\hspace{10mm}(9)\] \[\hat{\beta}_1 \pm t_{\alpha/2, n-2} \cdot \hat{\sigma} \sqrt{1/S_{xx}} \hspace{10mm} (10)\]Estos intervalos están centrados en las estimaciones puntuales de cada parámetro, y su amplitud depende tanto del nivel de confianza seleccionado como de la variabilidad asociada a cada estimador.
De acuerdo con lo explicado, en los procedimientos de prueba de hipótesis para el modelo de regresión lineal simple, las hipótesis nulas sobre los parámetros \(\beta_0\) y \(\beta_1\) generalmente se formulan como igualdades. Los valores hipotéticos de estos parámetros se denotan como \(\beta_{00}\) (“beta cero cero”) para el intercepto y \(\beta_{10}\) (“beta uno cero”) para la pendiente. Dado que los estadísticos de prueba tienen una distribución \(t\) con \(n-2\) grados de libertad bajo la suposición de que \(H_0\) es verdadera, la probabilidad de cometer un error Tipo I se mantiene en el nivel de significancia \(\alpha\) al emplear el valor crítico \(t\) correspondiente. En este contexto, las hipótesis más comunes para el intercepto \(\beta_0\) son: \[H_0: \beta_0 = \beta_{00}\hspace{10mm}(11)\] \[H_1: \beta_0 \neq \beta_{00}\hspace{10mm}(12)\]y el estadístico de prueba asociado se define como: \[t_{(n-2)}=\dfrac{\hat{\beta}_0-\beta_{00}}{\hat{\sigma} \sqrt{1/n+\bar{x}^2/S_{xx}}}\hspace{10mm}(13)\]De manera similar, para la pendiente \(\beta_1\), las hipótesis se formulan como: \[H_0: \beta_1 = \beta_{10}\hspace{10mm}(14)\] \[H_1: \beta_1 \neq \beta_{10}\hspace{10mm}(15)\]y el estadístico de prueba correspondiente es: \[t_{(n-2)}=\dfrac{\hat{\beta}_1-\beta_{10}}{\hat{\sigma} \sqrt{1/S_{xx}}}\hspace{10mm}(16)\]Este conjunto de hipótesis y estadísticos se conoce como prueba de utilidad del modelo de regresión lineal simple. Las regiones de rechazo para la hipótesis nula \(H_0\) en una prueba al nivel \(\alpha\) son las siguientes:
- Para \(H_1\): \(\beta_1\)>_{10}), se rechaza \(H_0\) si \(t\geq t_{\alpha,n-2}\).
- Para \(H_1\): \(\beta_1\) < _{10}), se rechaza \(H_0\) si \(t\leq -t_{\alpha,n-2}\).
- Para \(H_1\): \(\beta_1\)_{10}), se rechaza \(H_0\) si \(t\leq -t_{\alpha/2,n-2}\) o \(t\geq t_{\alpha/2,n-2}\).
Además, la utilidad del modelo de regresión lineal simple también puede evaluarse mediante una tabla ANOVA. En este caso, se rechaza \(H_0\) si \(f \geq F_{\alpha,1,n-2}\). Es importante destacar que la prueba \(F\) produce el mismo resultado que la prueba \(t\) de utilidad del modelo.
Por último, en un modelo de regresión lineal simple, un valor futuro de \(Y\) no es un parámetro fijo, sino una variable aleatoria. Por lo tanto, es necesario referirse a un rango de valores posibles para este valor futuro, denominado intervalo de predicción. Cuando se predice utilizando el modelo de regresión lineal simple, el error de predicción se define como: \(Y-( \hat{\beta}_0+ \hat{\beta}_1 x^*)\), Este error corresponde a la diferencia entre dos variables aleatorias, lo que implica que hay mayor incertidumbre en comparación con una estimación puntual. Por consiguiente, un intervalo de predicción es más amplio que un intervalo de confianza.A partir de la varianza del error de predicción, se establece que la variable estandarizada: \[T=\dfrac{Y-(\hat{\beta}_0+ \hat{\beta}_1 x^*)}{S \displaystyle\sqrt{1+\dfrac{1}{n} + \dfrac{(x^*-\bar{x})^2}{S_{xx}}}}\hspace{10mm}(17)\]sigue una distribución \(t\) con \(n-2\) grados de libertad. Esto permite calcular un intervalo de predicción con un nivel de confianza del \(100*(1-\alpha)\%\) para un valor futuro de \(Y\) cuando \(x=x^*\):\[\hat{\beta}_0+\hat{\beta}_1 x^*\pm t_{n-2,\alpha/2}\cdot s \displaystyle\sqrt{1+\dfrac{1}{n}+\dfrac{(x^*-\bar{x})^2}{S_{xx}}}\hspace{10mm}(18)\]La interpretación del nivel de predicción del \(100 \cdot (1-\alpha)\%\) indica que, si se aplica repetidamente la fórmula correspondiente en la ecuación \((18)\), los intervalos calculados incluirán los valores observados de \(Y\) en aproximadamente el \(100 \cdot (1-\alpha)\%\) de los casos. Además, el término adicional \(1\) en la raíz cuadrada hace que el intervalo de predicción sea más amplio en comparación con los intervalos de confianza para los parámetros, como los descritos en las ecuaciones \((9)\) y \((10)\). Por otro lado, a medida que el tamaño de la muestra \(n\to\infty\), el ancho del intervalo de predicción no se reduce a cero, ya que siempre existirá una incertidumbre inherente en la predicción de valores futuros, incluso si los parámetros \(\beta_0\) y \(\beta_1\) se conocen con exactitud.
5.2.1. Planteamiento del problema
Se planteará un modelo de regresión lineal simple basado en el conjunto de datos descrito anteriormente, con el propósito de examinar la relación lineal entre las variables Hours studied (variable independiente) y Exam score (variable dependiente).
5.2.2. Desarrollo del análisis
El estudio de regresión lineal simple ha sido procesado con R version 4.4.1 (2024-06-14 ucrt) mediado por RStudio 2024.04.2
5.2.2.1. Resumen estadistico de las variables por estudiar.
La navegación a través de las pestañas proporciona un análisis descriptivo de las variables de interés: Exam score (variable dependiente) y Hours studied (variable independiente). Este análisis incluye sus estadísticas resumen, diagramas de caja para identificar distribuciones y posibles valores atípicos, así como un diagrama de dispersión que ilustra la relación conjunta entre ambas variables.
Con base en la pestaña Resumen de Exam score, se observa que la variable Exam score muestra una distribución con baja dispersión, evidenciada por un rango intercuartílico estrecho (65-69). Aunque la mayoría de los valores se concentran en un rango alto dentro del intervalo de medición, se identifica un valor atípico por encima de 100. Esto indica que la variable tiende hacia valores elevados, con una variabilidad mínima en los datos principales.En comparación, según la pestaña Resumen de Hours studied, la variable Hours studied presenta una distribución más equilibrada y simétrica, sin presencia de valores atípicos. Su rango intercuartílico (16-24) y mediana (20) sugieren una ligera concentración de los datos hacia el extremo superior del intervalo, aunque la dispersión es más amplia que en Exam score.
Complementariamente, en el Diagrama de Dispersión Hours studied vs. Exam score, se observa que existe una correlación positiva de naturaleza ligeramente lineal entre las variables hours studied y exam score. Esto sugiere que a medida que los estudiantes dedican más horas al estudio, generalmente obtienen puntajes más altos en los exámenes.Sin embargo, si se analiza el gráfico de Diagramas Totales de Dispersión, que incluye diversas variables cuantitativas relacionadas con el rendimiento estudiantil, se puede constatar que otras variables, como previous_scores y attendance, podrían mostrar una relación más fuerte con el exam score. Esto indica que, aunque las horas de estudio son importantes, otros factores también influyen significativamente en el desempeño académico de los estudiantes.
Resumen de Exam score
summary(cdd_factores_rendimiento_estudiantes_G6_dep$exam_score)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 55.00 65.00 67.00 67.25 69.00 100.00boxplot(cdd_factores_rendimiento_estudiantes_G6_dep$exam_score, main = "Diagrama de caja de Exam score", col = c("orange"))
Diagrama de dispersión HS vs. ES
plot(cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied, cdd_factores_rendimiento_estudiantes_G6_dep$exam_score, main = "Diagrama de Dispersión Hours studied vs. Exam score")
Diagramas totales de dispersión
pairs(~hours_studied + age + attendance + sleep_hours + previous_scores + tutoring_sessions + physical_activity + exam_score , data = cdd_factores_rendimiento_estudiantes_G6_dep)
5.2.2.2. Formulación del modelo de RLS entre las variables de estudio.
Al explorar las diferentes pestañas, se presentan los coeficientes del modelo de regresión lineal simple, junto con su resumen estadístico y la tabla ANOVA. Cabe destacar que las variables principales analizadas son: exam score como la variable dependiente y hours studied como la variable independiente.
Con base en los resultados obtenidos del análisis del Modelo RLS, se establece que el modelo de regresión lineal simple que relaciona las variables de interés, Exam Score y Hours Studied, tiene la siguiente formulación: \[\hat{\text{Exam Score}} = 61.437683 + 0.291047 \cdot \text{Hours Studied (19)} \] En este modelo, el intercepto \((61.437683)\) representa el puntaje esperado en el examen cuando el número de hours studied es igual a \(0\). Sin embargo, su interpretación puede no ser significativa en este contexto, ya que no es común que un estudiante no estudie absolutamente nada antes de un examen. Por otro lado, el coeficiente asociado a la variable Hours Studied \((0.291047)\) indica que, por cada hora adicional de estudio, el puntaje del examen aumenta en promedio \((0.291)\) puntos. Esto sugiere una relación directa y positiva entre las horas de estudio y el puntaje obtenido, aunque el incremento es moderado.
Complementariamente, en la pestaña Resumen Estadístico del Modelo RLS, se constata que, para cualquier nivel de significancia, las evidencias respaldan la existencia de una correlación positiva entre las variables Hours Studied y Exam Score. Además, el coeficiente de determinación \((R^2)\) establece que una porción considerable de la variabilidad en Exam Score es explicada por Hours Studied, confirmando la utilidad del modelo. Esto también queda ratificado en la pestaña Tabla ANOVA para el Modelo RLS, donde el valor de \((F)\) obtenido \((1576.2)\) y el valor \((p < 2.2 \cdot 10^{-16})\) refuerzan la significancia estadística del modelo. Estos resultados demuestran que las horas de estudio son un predictor significativo y clave para explicar las variaciones en los puntajes del examen.
Coeficientes del Modelo RLS
modelo_RL_Simple = lm(cdd_factores_rendimiento_estudiantes_G6_dep$exam_score~cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied)
coef(modelo_RL_Simple)## (Intercept)
## 61.437683
## cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied
## 0.291047Resumen Estadístico del Modelo RLS
summary(modelo_RL_Simple)##
## Call:
## lm(formula = cdd_factores_rendimiento_estudiantes_G6_dep$exam_score ~
## cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.550 -2.259 -0.132 2.032 33.488
##
## Coefficients:
## Estimate Std. Error
## (Intercept) 61.437683 0.152881
## cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied 0.291047 0.007331
## t value Pr(>|t|)
## (Intercept) 401.9 <2e-16 ***
## cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied 39.7 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.504 on 6376 degrees of freedom
## Multiple R-squared: 0.1982, Adjusted R-squared: 0.1981
## F-statistic: 1576 on 1 and 6376 DF, p-value: < 2.2e-16Tabla ANOVA para el Modelo RLS
anova(modelo_RL_Simple)## Analysis of Variance Table
##
## Response: cdd_factores_rendimiento_estudiantes_G6_dep$exam_score
## Df Sum Sq Mean Sq
## cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied 1 19353 19352.5
## Residuals 6376 78284 12.3
## F value Pr(>F)
## cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied 1576.2 < 2.2e-16 ***
## Residuals
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 15.2.2.3. Análisis del modelo RLS.
La navegación a través de las pestañas muestra el intervalo de confianza para \(\beta_1\) y para la predicción del modelo de regresión lineal simple, ambos al 95%. Las variables de interés en este análisis son: Exam Score (variable dependiente) y Hours Studied (variable independiente).
El análisis del modelo RLS muestra que es estadísticamente significativo y, en consecuencia, aporta información relevante para estimar Exam Score a partir de Hours Studied. Esto se debe a que el intervalo de confianza para el coeficiente de Hours Studied en el modelo RLS excluye al cero: \[ 0.276676 < \beta_1 < 0.305418 \hspace{10mm}(20) \]
Por último, la pestaña Predicciones y sus Intervalos de Predicción muestra los cálculos basados en el modelo, bajo intervalos de predicción al 95%, para las predicciones de Exam Score en todas las observaciones del conjunto de datos. Cabe destacar que estos intervalos son más amplios que aquellos calculados con base en intervalos de confianza al mismo nivel de significancia, reflejando la incertidumbre asociada a las predicciones individuales.
Intervalo de Confianza para B1
confint(modelo_RL_Simple, level = 0.95)## 2.5 % 97.5 %
## (Intercept) 61.137985 61.737380
## cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied 0.276676 0.305418Predicciones y sus Intervalos de Predicción
predict(modelo_RL_Simple, data.frame(seq(1,400)), interval='prediction', level = 0.95)## Warning: 'newdata' had 400 rows but variables found have 6378 rows## fit lwr upr
## 1 68.13176 61.26212 75.00141
## 2 66.96758 60.09805 73.83710
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## 1613 65.51234 58.64229 72.38239
## 1614 70.16909 63.29807 77.04011
## 1615 66.67653 59.80696 73.54610
## 1616 69.29595 62.42570 76.16620
## 1617 69.87805 63.00731 76.74878
## 1618 69.00490 62.13485 75.87496
## 1619 66.38548 59.51584 73.25512
## 1620 67.84072 60.97114 74.71029
## 1621 67.25862 60.38911 74.12813
## 1622 65.80339 58.93351 72.67327
## 1623 67.84072 60.97114 74.71029
## 1624 65.22129 58.35105 72.09154
## 1625 69.29595 62.42570 76.16620
## 1626 66.38548 59.51584 73.25512
## 1627 67.54967 60.68014 74.41919
## 1628 67.84072 60.97114 74.71029
## 1629 66.67653 59.80696 73.54610
## 1630 68.42281 61.55306 75.29256
## 1631 66.38548 59.51584 73.25512
## 1632 68.71386 61.84397 75.58375
## 1633 64.93025 58.05978 71.80071
## 1634 67.54967 60.68014 74.41919
## 1635 70.75119 63.87950 77.62287
## 1636 65.51234 58.64229 72.38239
## 1637 66.09443 59.22469 72.96418
## 1638 68.13176 61.26212 75.00141
## 1639 67.54967 60.68014 74.41919
## 1640 67.84072 60.97114 74.71029
## 1641 68.13176 61.26212 75.00141
## 1642 68.42281 61.55306 75.29256
## 1643 69.29595 62.42570 76.16620
## 1644 67.84072 60.97114 74.71029
## 1645 66.96758 60.09805 73.83710
## 1646 68.42281 61.55306 75.29256
## 1647 70.46014 63.58880 77.33148
## 1648 66.96758 60.09805 73.83710
## 1649 65.51234 58.64229 72.38239
## 1650 69.29595 62.42570 76.16620
## 1651 66.96758 60.09805 73.83710
## 1652 63.47501 56.60297 70.34705
## 1653 68.71386 61.84397 75.58375
## 1654 66.38548 59.51584 73.25512
## 1655 66.96758 60.09805 73.83710
## 1656 67.84072 60.97114 74.71029
## 1657 67.54967 60.68014 74.41919
## 1658 68.42281 61.55306 75.29256
## 1659 66.38548 59.51584 73.25512
## 1660 68.13176 61.26212 75.00141
## 1661 66.67653 59.80696 73.54610
## 1662 68.42281 61.55306 75.29256
## 1663 68.42281 61.55306 75.29256
## 1664 65.51234 58.64229 72.38239
## 1665 69.29595 62.42570 76.16620
## 1666 66.38548 59.51584 73.25512
## 1667 68.71386 61.84397 75.58375
## 1668 67.84072 60.97114 74.71029
## 1669 70.46014 63.58880 77.33148
## 1670 66.96758 60.09805 73.83710
## 1671 66.67653 59.80696 73.54610
## 1672 67.25862 60.38911 74.12813
## 1673 66.67653 59.80696 73.54610
## 1674 68.71386 61.84397 75.58375
## 1675 64.34815 57.47715 71.21916
## 1676 65.51234 58.64229 72.38239
## 1677 65.51234 58.64229 72.38239
## 1678 69.00490 62.13485 75.87496
## 1679 70.16909 63.29807 77.04011
## 1680 66.96758 60.09805 73.83710
## 1681 67.25862 60.38911 74.12813
## 1682 69.29595 62.42570 76.16620
## 1683 67.54967 60.68014 74.41919
## 1684 69.00490 62.13485 75.87496
## 1685 68.13176 61.26212 75.00141
## 1686 67.54967 60.68014 74.41919
## 1687 69.00490 62.13485 75.87496
## 1688 62.60187 55.72852 69.47522
## 1689 68.42281 61.55306 75.29256
## 1690 64.05711 57.18578 70.92843
## 1691 68.13176 61.26212 75.00141
## 1692 69.58700 62.71652 76.45748
## 1693 66.96758 60.09805 73.83710
## 1694 69.00490 62.13485 75.87496
## 1695 70.16909 63.29807 77.04011
## 1696 69.58700 62.71652 76.45748
## 1697 70.16909 63.29807 77.04011
## 1698 68.42281 61.55306 75.29256
## 1699 67.25862 60.38911 74.12813
## 1700 64.34815 57.47715 71.21916
## 1701 65.80339 58.93351 72.67327
## 1702 71.04223 64.17017 77.91429
## 1703 66.96758 60.09805 73.83710
## 1704 67.54967 60.68014 74.41919
## 1705 68.13176 61.26212 75.00141
## 1706 67.25862 60.38911 74.12813
## 1707 70.16909 63.29807 77.04011
## 1708 66.09443 59.22469 72.96418
## 1709 67.54967 60.68014 74.41919
## 1710 66.09443 59.22469 72.96418
## 1711 64.93025 58.05978 71.80071
## 1712 66.09443 59.22469 72.96418
## 1713 67.25862 60.38911 74.12813
## 1714 66.38548 59.51584 73.25512
## 1715 66.09443 59.22469 72.96418
## 1716 64.93025 58.05978 71.80071
## 1717 67.25862 60.38911 74.12813
## 1718 64.63920 57.76848 71.50992
## 1719 68.71386 61.84397 75.58375
## 1720 67.25862 60.38911 74.12813
## 1721 64.63920 57.76848 71.50992
## 1722 66.67653 59.80696 73.54610
## 1723 65.80339 58.93351 72.67327
## 1724 67.84072 60.97114 74.71029
## 1725 69.58700 62.71652 76.45748
## 1726 64.05711 57.18578 70.92843
## 1727 66.96758 60.09805 73.83710
## 1728 68.42281 61.55306 75.29256
## 1729 67.54967 60.68014 74.41919
## 1730 65.80339 58.93351 72.67327
## 1731 66.67653 59.80696 73.54610
## 1732 69.00490 62.13485 75.87496
## 1733 67.25862 60.38911 74.12813
## 1734 70.46014 63.58880 77.33148
## 1735 67.25862 60.38911 74.12813
## 1736 66.09443 59.22469 72.96418
## 1737 64.63920 57.76848 71.50992
## 1738 64.05711 57.18578 70.92843
## 1739 64.34815 57.47715 71.21916
## 1740 67.84072 60.97114 74.71029
## 1741 68.13176 61.26212 75.00141
## 1742 64.63920 57.76848 71.50992
## 1743 68.71386 61.84397 75.58375
## 1744 66.38548 59.51584 73.25512
## 1745 67.54967 60.68014 74.41919
## 1746 70.16909 63.29807 77.04011
## 1747 68.71386 61.84397 75.58375
## 1748 66.67653 59.80696 73.54610
## 1749 65.80339 58.93351 72.67327
## 1750 63.47501 56.60297 70.34705
## 1751 66.09443 59.22469 72.96418
## 1752 66.96758 60.09805 73.83710
## 1753 65.51234 58.64229 72.38239
## 1754 66.96758 60.09805 73.83710
## 1755 67.25862 60.38911 74.12813
## 1756 66.96758 60.09805 73.83710
## 1757 66.09443 59.22469 72.96418
## 1758 67.54967 60.68014 74.41919
## 1759 68.13176 61.26212 75.00141
## 1760 65.22129 58.35105 72.09154
## 1761 69.58700 62.71652 76.45748
## 1762 66.96758 60.09805 73.83710
## 1763 67.25862 60.38911 74.12813
## 1764 68.42281 61.55306 75.29256
## 1765 67.54967 60.68014 74.41919
## 1766 69.00490 62.13485 75.87496
## 1767 67.84072 60.97114 74.71029
## 1768 66.09443 59.22469 72.96418
## 1769 67.84072 60.97114 74.71029
## 1770 65.22129 58.35105 72.09154
## 1771 67.54967 60.68014 74.41919
## 1772 67.25862 60.38911 74.12813
## 1773 66.96758 60.09805 73.83710
## 1774 66.96758 60.09805 73.83710
## 1775 64.05711 57.18578 70.92843
## 1776 66.38548 59.51584 73.25512
## 1777 67.54967 60.68014 74.41919
## 1778 67.54967 60.68014 74.41919
## 1779 65.51234 58.64229 72.38239
## 1780 69.58700 62.71652 76.45748
## 1781 65.80339 58.93351 72.67327
## 1782 69.00490 62.13485 75.87496
## 1783 68.13176 61.26212 75.00141
## 1784 66.09443 59.22469 72.96418
## 1785 70.46014 63.58880 77.33148
## 1786 65.22129 58.35105 72.09154
## 1787 69.00490 62.13485 75.87496
## 1788 69.00490 62.13485 75.87496
## 1789 66.09443 59.22469 72.96418
## 1790 69.58700 62.71652 76.45748
## 1791 66.67653 59.80696 73.54610
## 1792 67.84072 60.97114 74.71029
## 1793 67.84072 60.97114 74.71029
## 1794 66.96758 60.09805 73.83710
## 1795 68.71386 61.84397 75.58375
## 1796 68.71386 61.84397 75.58375
## 1797 64.34815 57.47715 71.21916
## 1798 69.58700 62.71652 76.45748
## 1799 64.34815 57.47715 71.21916
## 1800 69.00490 62.13485 75.87496
## 1801 69.29595 62.42570 76.16620
## 1802 65.22129 58.35105 72.09154
## 1803 66.67653 59.80696 73.54610
## 1804 65.22129 58.35105 72.09154
## 1805 66.96758 60.09805 73.83710
## 1806 66.67653 59.80696 73.54610
## 1807 68.71386 61.84397 75.58375
## 1808 69.58700 62.71652 76.45748
## 1809 69.58700 62.71652 76.45748
## 1810 66.38548 59.51584 73.25512
## 1811 66.96758 60.09805 73.83710
## 1812 64.34815 57.47715 71.21916
## 1813 65.80339 58.93351 72.67327
## 1814 69.00490 62.13485 75.87496
## 1815 70.16909 63.29807 77.04011
## 1816 65.22129 58.35105 72.09154
## 1817 66.09443 59.22469 72.96418
## 1818 67.25862 60.38911 74.12813
## 1819 65.80339 58.93351 72.67327
## 1820 68.13176 61.26212 75.00141
## 1821 66.38548 59.51584 73.25512
## 1822 67.25862 60.38911 74.12813
## 1823 68.13176 61.26212 75.00141
## 1824 67.25862 60.38911 74.12813
## 1825 66.38548 59.51584 73.25512
## 1826 66.96758 60.09805 73.83710
## 1827 66.96758 60.09805 73.83710
## 1828 66.67653 59.80696 73.54610
## 1829 66.67653 59.80696 73.54610
## 1830 70.16909 63.29807 77.04011
## 1831 68.13176 61.26212 75.00141
## 1832 66.67653 59.80696 73.54610
## 1833 68.42281 61.55306 75.29256
## 1834 66.67653 59.80696 73.54610
## 1835 69.29595 62.42570 76.16620
## 1836 67.84072 60.97114 74.71029
## 1837 66.96758 60.09805 73.83710
## 1838 68.13176 61.26212 75.00141
## 1839 64.63920 57.76848 71.50992
## 1840 65.22129 58.35105 72.09154
## 1841 65.22129 58.35105 72.09154
## 1842 67.54967 60.68014 74.41919
## 1843 70.16909 63.29807 77.04011
## 1844 71.04223 64.17017 77.91429
## 1845 68.42281 61.55306 75.29256
## 1846 68.13176 61.26212 75.00141
## 1847 64.05711 57.18578 70.92843
## 1848 66.38548 59.51584 73.25512
## 1849 65.80339 58.93351 72.67327
## 1850 66.09443 59.22469 72.96418
## 1851 66.96758 60.09805 73.83710
## 1852 66.38548 59.51584 73.25512
## 1853 63.76606 56.89439 70.63772
## 1854 68.42281 61.55306 75.29256
## 1855 65.51234 58.64229 72.38239
## 1856 67.54967 60.68014 74.41919
## 1857 70.75119 63.87950 77.62287
## 1858 65.51234 58.64229 72.38239
## 1859 62.89292 56.02004 69.76580
## 1860 66.67653 59.80696 73.54610
## 1861 69.29595 62.42570 76.16620
## 1862 70.75119 63.87950 77.62287
## 1863 65.22129 58.35105 72.09154
## 1864 68.13176 61.26212 75.00141
## 1865 66.38548 59.51584 73.25512
## 1866 66.96758 60.09805 73.83710
## 1867 71.91537 65.04201 78.78874
## 1868 64.63920 57.76848 71.50992
## 1869 67.84072 60.97114 74.71029
## 1870 68.13176 61.26212 75.00141
## 1871 69.29595 62.42570 76.16620
## 1872 67.54967 60.68014 74.41919
## 1873 67.84072 60.97114 74.71029
## 1874 66.67653 59.80696 73.54610
## 1875 67.84072 60.97114 74.71029
## 1876 66.67653 59.80696 73.54610
## 1877 68.42281 61.55306 75.29256
## 1878 66.67653 59.80696 73.54610
## 1879 67.25862 60.38911 74.12813
## 1880 69.29595 62.42570 76.16620
## 1881 67.25862 60.38911 74.12813
## 1882 64.05711 57.18578 70.92843
## 1883 69.58700 62.71652 76.45748
## 1884 65.80339 58.93351 72.67327
## 1885 66.09443 59.22469 72.96418
## 1886 66.09443 59.22469 72.96418
## 1887 69.58700 62.71652 76.45748
## 1888 63.47501 56.60297 70.34705
## 1889 72.78852 65.91357 79.66346
## 1890 69.00490 62.13485 75.87496
## 1891 67.54967 60.68014 74.41919
## 1892 67.25862 60.38911 74.12813
## 1893 67.84072 60.97114 74.71029
## 1894 68.13176 61.26212 75.00141
## 1895 68.42281 61.55306 75.29256
## 1896 68.13176 61.26212 75.00141
## 1897 67.54967 60.68014 74.41919
## 1898 66.96758 60.09805 73.83710
## 1899 66.96758 60.09805 73.83710
## 1900 66.96758 60.09805 73.83710
## 1901 69.29595 62.42570 76.16620
## 1902 68.42281 61.55306 75.29256
## 1903 68.42281 61.55306 75.29256
## 1904 67.54967 60.68014 74.41919
## 1905 67.25862 60.38911 74.12813
## 1906 66.67653 59.80696 73.54610
## 1907 69.29595 62.42570 76.16620
## 1908 69.00490 62.13485 75.87496
## 1909 65.80339 58.93351 72.67327
## 1910 67.84072 60.97114 74.71029
## 1911 65.51234 58.64229 72.38239
## 1912 64.93025 58.05978 71.80071
## 1913 66.09443 59.22469 72.96418
## 1914 68.71386 61.84397 75.58375
## 1915 69.00490 62.13485 75.87496
## 1916 68.13176 61.26212 75.00141
## 1917 68.42281 61.55306 75.29256
## 1918 68.13176 61.26212 75.00141
## 1919 66.09443 59.22469 72.96418
## 1920 68.13176 61.26212 75.00141
## 1921 68.71386 61.84397 75.58375
## 1922 68.71386 61.84397 75.58375
## 1923 69.29595 62.42570 76.16620
## 1924 68.42281 61.55306 75.29256
## 1925 67.84072 60.97114 74.71029
## 1926 69.00490 62.13485 75.87496
## 1927 67.25862 60.38911 74.12813
## 1928 65.80339 58.93351 72.67327
## 1929 66.96758 60.09805 73.83710
## 1930 66.09443 59.22469 72.96418
## 1931 66.09443 59.22469 72.96418
## 1932 66.96758 60.09805 73.83710
## 1933 65.80339 58.93351 72.67327
## 1934 66.67653 59.80696 73.54610
## 1935 64.05711 57.18578 70.92843
## 1936 67.54967 60.68014 74.41919
## 1937 67.25862 60.38911 74.12813
## 1938 65.80339 58.93351 72.67327
## 1939 68.42281 61.55306 75.29256
## 1940 69.00490 62.13485 75.87496
## 1941 64.34815 57.47715 71.21916
## 1942 65.80339 58.93351 72.67327
## 1943 65.80339 58.93351 72.67327
## 1944 65.51234 58.64229 72.38239
## 1945 68.71386 61.84397 75.58375
## 1946 66.96758 60.09805 73.83710
## 1947 66.67653 59.80696 73.54610
## 1948 63.76606 56.89439 70.63772
## 1949 68.42281 61.55306 75.29256
## 1950 65.22129 58.35105 72.09154
## 1951 67.25862 60.38911 74.12813
## 1952 67.84072 60.97114 74.71029
## 1953 69.58700 62.71652 76.45748
## 1954 64.93025 58.05978 71.80071
## 1955 62.01978 55.14541 68.89414
## 1956 67.54967 60.68014 74.41919
## 1957 70.46014 63.58880 77.33148
## 1958 69.29595 62.42570 76.16620
## 1959 67.54967 60.68014 74.41919
## 1960 66.38548 59.51584 73.25512
## 1961 68.71386 61.84397 75.58375
## 1962 65.51234 58.64229 72.38239
## 1963 68.13176 61.26212 75.00141
## 1964 68.13176 61.26212 75.00141
## 1965 69.00490 62.13485 75.87496
## 1966 72.20642 65.33256 79.08029
## 1967 67.84072 60.97114 74.71029
## 1968 66.38548 59.51584 73.25512
## 1969 67.25862 60.38911 74.12813
## 1970 64.05711 57.18578 70.92843
## 1971 66.09443 59.22469 72.96418
## 1972 66.67653 59.80696 73.54610
## 1973 66.38548 59.51584 73.25512
## 1974 67.54967 60.68014 74.41919
## 1975 65.80339 58.93351 72.67327
## 1976 69.29595 62.42570 76.16620
## 1977 67.25862 60.38911 74.12813
## 1978 66.67653 59.80696 73.54610
## 1979 67.54967 60.68014 74.41919
## 1980 68.71386 61.84397 75.58375
## 1981 65.80339 58.93351 72.67327
## 1982 66.09443 59.22469 72.96418
## 1983 66.67653 59.80696 73.54610
## 1984 64.93025 58.05978 71.80071
## 1985 65.80339 58.93351 72.67327
## 1986 66.38548 59.51584 73.25512
## 1987 68.71386 61.84397 75.58375
## 1988 67.54967 60.68014 74.41919
## 1989 68.71386 61.84397 75.58375
## 1990 66.96758 60.09805 73.83710
## 1991 67.84072 60.97114 74.71029
## 1992 66.96758 60.09805 73.83710
## 1993 67.84072 60.97114 74.71029
## 1994 66.38548 59.51584 73.25512
## 1995 62.89292 56.02004 69.76580
## 1996 66.96758 60.09805 73.83710
## 1997 69.87805 63.00731 76.74878
## 1998 69.00490 62.13485 75.87496
## 1999 69.00490 62.13485 75.87496
## 2000 69.29595 62.42570 76.16620
## 2001 67.54967 60.68014 74.41919
## 2002 67.54967 60.68014 74.41919
## 2003 66.09443 59.22469 72.96418
## 2004 66.67653 59.80696 73.54610
## 2005 68.42281 61.55306 75.29256
## 2006 69.29595 62.42570 76.16620
## 2007 66.96758 60.09805 73.83710
## 2008 66.38548 59.51584 73.25512
## 2009 67.25862 60.38911 74.12813
## 2010 68.42281 61.55306 75.29256
## 2011 64.63920 57.76848 71.50992
## 2012 63.18396 56.31152 70.05641
## 2013 66.38548 59.51584 73.25512
## 2014 65.22129 58.35105 72.09154
## 2015 66.38548 59.51584 73.25512
## 2016 66.96758 60.09805 73.83710
## 2017 66.38548 59.51584 73.25512
## 2018 69.87805 63.00731 76.74878
## 2019 67.84072 60.97114 74.71029
## 2020 67.84072 60.97114 74.71029
## 2021 68.42281 61.55306 75.29256
## 2022 65.22129 58.35105 72.09154
## 2023 69.00490 62.13485 75.87496
## 2024 67.25862 60.38911 74.12813
## 2025 68.42281 61.55306 75.29256
## 2026 65.51234 58.64229 72.38239
## 2027 64.63920 57.76848 71.50992
## 2028 68.71386 61.84397 75.58375
## 2029 67.84072 60.97114 74.71029
## 2030 65.80339 58.93351 72.67327
## 2031 65.80339 58.93351 72.67327
## 2032 69.29595 62.42570 76.16620
## 2033 65.22129 58.35105 72.09154
## 2034 70.16909 63.29807 77.04011
## 2035 65.80339 58.93351 72.67327
## 2036 68.42281 61.55306 75.29256
## 2037 66.67653 59.80696 73.54610
## 2038 66.38548 59.51584 73.25512
## 2039 66.96758 60.09805 73.83710
## 2040 67.25862 60.38911 74.12813
## 2041 65.51234 58.64229 72.38239
## 2042 68.42281 61.55306 75.29256
## 2043 66.38548 59.51584 73.25512
## 2044 65.51234 58.64229 72.38239
## 2045 62.89292 56.02004 69.76580
## 2046 66.96758 60.09805 73.83710
## 2047 71.33328 64.46081 78.20575
## 2048 68.71386 61.84397 75.58375
## 2049 67.25862 60.38911 74.12813
## 2050 66.67653 59.80696 73.54610
## 2051 67.25862 60.38911 74.12813
## 2052 68.13176 61.26212 75.00141
## 2053 65.22129 58.35105 72.09154
## 2054 69.29595 62.42570 76.16620
## 2055 68.42281 61.55306 75.29256
## 2056 68.42281 61.55306 75.29256
## 2057 68.13176 61.26212 75.00141
## 2058 67.25862 60.38911 74.12813
## 2059 68.42281 61.55306 75.29256
## 2060 68.42281 61.55306 75.29256
## 2061 66.67653 59.80696 73.54610
## 2062 66.38548 59.51584 73.25512
## 2063 67.54967 60.68014 74.41919
## 2064 69.87805 63.00731 76.74878
## 2065 65.22129 58.35105 72.09154
## 2066 64.63920 57.76848 71.50992
## 2067 67.54967 60.68014 74.41919
## 2068 67.25862 60.38911 74.12813
## 2069 67.54967 60.68014 74.41919
## 2070 67.54967 60.68014 74.41919
## 2071 66.96758 60.09805 73.83710
## 2072 69.87805 63.00731 76.74878
## 2073 64.34815 57.47715 71.21916
## 2074 65.80339 58.93351 72.67327
## 2075 66.96758 60.09805 73.83710
## 2076 64.34815 57.47715 71.21916
## 2077 66.96758 60.09805 73.83710
## 2078 70.16909 63.29807 77.04011
## 2079 67.25862 60.38911 74.12813
## 2080 67.54967 60.68014 74.41919
## 2081 64.93025 58.05978 71.80071
## 2082 66.09443 59.22469 72.96418
## 2083 66.67653 59.80696 73.54610
## 2084 66.67653 59.80696 73.54610
## 2085 69.58700 62.71652 76.45748
## 2086 68.13176 61.26212 75.00141
## 2087 65.22129 58.35105 72.09154
## 2088 67.54967 60.68014 74.41919
## 2089 68.13176 61.26212 75.00141
## 2090 69.87805 63.00731 76.74878
## 2091 65.51234 58.64229 72.38239
## 2092 65.80339 58.93351 72.67327
## 2093 64.93025 58.05978 71.80071
## 2094 66.96758 60.09805 73.83710
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## 6357 66.67653 59.80696 73.54610
## 6358 64.63920 57.76848 71.50992
## 6359 70.16909 63.29807 77.04011
## 6360 67.54967 60.68014 74.41919
## 6361 69.87805 63.00731 76.74878
## 6362 66.67653 59.80696 73.54610
## 6363 65.80339 58.93351 72.67327
## 6364 66.96758 60.09805 73.83710
## 6365 65.22129 58.35105 72.09154
## 6366 69.87805 63.00731 76.74878
## 6367 66.09443 59.22469 72.96418
## 6368 69.58700 62.71652 76.45748
## 6369 66.09443 59.22469 72.96418
## 6370 64.05711 57.18578 70.92843
## 6371 70.16909 63.29807 77.04011
## 6372 64.93025 58.05978 71.80071
## 6373 67.25862 60.38911 74.12813
## 6374 68.71386 61.84397 75.58375
## 6375 68.13176 61.26212 75.00141
## 6376 67.25862 60.38911 74.12813
## 6377 64.34815 57.47715 71.21916
## 6378 65.80339 58.93351 72.67327Predicciones y sus Intervalos de Confianza
predict(modelo_RL_Simple, data.frame(seq(1,400)), interval='confidence', level = 0.95)## Warning: 'newdata' had 400 rows but variables found have 6378 rows## fit lwr upr
## 1 68.13176 68.03540 68.22812
## 2 66.96758 66.88043 67.05472
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## 1446 65.80339 65.69152 65.91525
## 1447 67.25862 67.17261 67.34463
## 1448 66.96758 66.88043 67.05472
## 1449 66.38548 66.28942 66.48154
## 1450 68.42281 68.31918 68.52644
## 1451 69.00490 68.88288 69.12693
## 1452 67.54967 67.46241 67.63693
## 1453 69.87805 69.72244 70.03365
## 1454 65.22129 65.08919 65.35340
## 1455 66.96758 66.88043 67.05472
## 1456 66.09443 65.99117 66.19770
## 1457 67.25862 67.17261 67.34463
## 1458 69.87805 69.72244 70.03365
## 1459 68.42281 68.31918 68.52644
## 1460 65.51234 65.39078 65.63390
## 1461 70.46014 70.27989 70.64039
## 1462 69.29595 69.16335 69.42855
## 1463 68.13176 68.03540 68.22812
## 1464 66.96758 66.88043 67.05472
## 1465 65.51234 65.39078 65.63390
## 1466 66.67653 66.58595 66.76711
## 1467 69.29595 69.16335 69.42855
## 1468 70.46014 70.27989 70.64039
## 1469 69.87805 69.72244 70.03365
## 1470 66.38548 66.28942 66.48154
## 1471 65.22129 65.08919 65.35340
## 1472 67.54967 67.46241 67.63693
## 1473 64.05711 63.87743 64.23678
## 1474 67.84072 67.74993 67.93151
## 1475 66.96758 66.88043 67.05472
## 1476 66.38548 66.28942 66.48154
## 1477 64.34815 64.18095 64.51535
## 1478 68.13176 68.03540 68.22812
## 1479 66.38548 66.28942 66.48154
## 1480 65.22129 65.08919 65.35340
## 1481 62.31082 62.05213 62.56952
## 1482 67.25862 67.17261 67.34463
## 1483 70.46014 70.27989 70.64039
## 1484 70.16909 70.00133 70.33686
## 1485 66.38548 66.28942 66.48154
## 1486 66.09443 65.99117 66.19770
## 1487 68.13176 68.03540 68.22812
## 1488 65.51234 65.39078 65.63390
## 1489 67.84072 67.74993 67.93151
## 1490 68.71386 68.60157 68.82614
## 1491 69.29595 69.16335 69.42855
## 1492 66.96758 66.88043 67.05472
## 1493 66.96758 66.88043 67.05472
## 1494 65.80339 65.69152 65.91525
## 1495 65.80339 65.69152 65.91525
## 1496 66.96758 66.88043 67.05472
## 1497 67.84072 67.74993 67.93151
## 1498 68.71386 68.60157 68.82614
## 1499 65.80339 65.69152 65.91525
## 1500 69.87805 69.72244 70.03365
## 1501 66.67653 66.58595 66.76711
## 1502 67.25862 67.17261 67.34463
## 1503 66.09443 65.99117 66.19770
## 1504 62.89292 62.66113 63.12470
## 1505 68.71386 68.60157 68.82614
## 1506 68.42281 68.31918 68.52644
## 1507 66.67653 66.58595 66.76711
## 1508 67.25862 67.17261 67.34463
## 1509 68.13176 68.03540 68.22812
## 1510 64.63920 64.48415 64.79425
## 1511 66.38548 66.28942 66.48154
## 1512 68.71386 68.60157 68.82614
## 1513 64.93025 64.78693 65.07356
## 1514 67.84072 67.74993 67.93151
## 1515 64.93025 64.78693 65.07356
## 1516 68.13176 68.03540 68.22812
## 1517 67.25862 67.17261 67.34463
## 1518 64.34815 64.18095 64.51535
## 1519 69.29595 69.16335 69.42855
## 1520 66.09443 65.99117 66.19770
## 1521 65.80339 65.69152 65.91525
## 1522 67.54967 67.46241 67.63693
## 1523 69.29595 69.16335 69.42855
## 1524 65.51234 65.39078 65.63390
## 1525 67.25862 67.17261 67.34463
## 1526 68.13176 68.03540 68.22812
## 1527 68.42281 68.31918 68.52644
## 1528 67.54967 67.46241 67.63693
## 1529 66.67653 66.58595 66.76711
## 1530 65.80339 65.69152 65.91525
## 1531 67.54967 67.46241 67.63693
## 1532 65.51234 65.39078 65.63390
## 1533 68.13176 68.03540 68.22812
## 1534 68.71386 68.60157 68.82614
## 1535 63.18396 62.96546 63.40247
## 1536 65.51234 65.39078 65.63390
## 1537 68.42281 68.31918 68.52644
## 1538 70.75119 70.55818 70.94419
## 1539 67.25862 67.17261 67.34463
## 1540 66.09443 65.99117 66.19770
## 1541 66.96758 66.88043 67.05472
## 1542 69.58700 69.44315 69.73084
## 1543 66.09443 65.99117 66.19770
## 1544 65.80339 65.69152 65.91525
## 1545 66.38548 66.28942 66.48154
## 1546 65.51234 65.39078 65.63390
## 1547 67.54967 67.46241 67.63693
## 1548 70.16909 70.00133 70.33686
## 1549 67.84072 67.74993 67.93151
## 1550 66.67653 66.58595 66.76711
## 1551 66.67653 66.58595 66.76711
## 1552 64.63920 64.48415 64.79425
## 1553 68.71386 68.60157 68.82614
## 1554 67.25862 67.17261 67.34463
## 1555 66.96758 66.88043 67.05472
## 1556 72.78852 72.50193 73.07510
## 1557 67.84072 67.74993 67.93151
## 1558 65.80339 65.69152 65.91525
## 1559 66.38548 66.28942 66.48154
## 1560 69.29595 69.16335 69.42855
## 1561 67.54967 67.46241 67.63693
## 1562 64.63920 64.48415 64.79425
## 1563 69.00490 68.88288 69.12693
## 1564 66.09443 65.99117 66.19770
## 1565 70.46014 70.27989 70.64039
## 1566 69.00490 68.88288 69.12693
## 1567 66.38548 66.28942 66.48154
## 1568 69.58700 69.44315 69.73084
## 1569 67.25862 67.17261 67.34463
## 1570 68.71386 68.60157 68.82614
## 1571 68.13176 68.03540 68.22812
## 1572 65.80339 65.69152 65.91525
## 1573 66.09443 65.99117 66.19770
## 1574 69.00490 68.88288 69.12693
## 1575 66.67653 66.58595 66.76711
## 1576 66.67653 66.58595 66.76711
## 1577 65.51234 65.39078 65.63390
## 1578 68.13176 68.03540 68.22812
## 1579 70.75119 70.55818 70.94419
## 1580 65.51234 65.39078 65.63390
## 1581 67.25862 67.17261 67.34463
## 1582 69.58700 69.44315 69.73084
## 1583 67.25862 67.17261 67.34463
## 1584 68.13176 68.03540 68.22812
## 1585 65.51234 65.39078 65.63390
## 1586 68.13176 68.03540 68.22812
## 1587 66.96758 66.88043 67.05472
## 1588 69.00490 68.88288 69.12693
## 1589 69.29595 69.16335 69.42855
## 1590 71.91537 71.66957 72.16118
## 1591 68.13176 68.03540 68.22812
## 1592 67.84072 67.74993 67.93151
## 1593 67.54967 67.46241 67.63693
## 1594 66.67653 66.58595 66.76711
## 1595 67.84072 67.74993 67.93151
## 1596 66.67653 66.58595 66.76711
## 1597 67.54967 67.46241 67.63693
## 1598 69.58700 69.44315 69.73084
## 1599 65.51234 65.39078 65.63390
## 1600 69.29595 69.16335 69.42855
## 1601 69.58700 69.44315 69.73084
## 1602 66.96758 66.88043 67.05472
## 1603 63.47501 63.26964 63.68038
## 1604 69.58700 69.44315 69.73084
## 1605 67.25862 67.17261 67.34463
## 1606 65.51234 65.39078 65.63390
## 1607 66.38548 66.28942 66.48154
## 1608 68.71386 68.60157 68.82614
## 1609 68.13176 68.03540 68.22812
## 1610 66.96758 66.88043 67.05472
## 1611 66.67653 66.58595 66.76711
## 1612 66.67653 66.58595 66.76711
## 1613 65.51234 65.39078 65.63390
## 1614 70.16909 70.00133 70.33686
## 1615 66.67653 66.58595 66.76711
## 1616 69.29595 69.16335 69.42855
## 1617 69.87805 69.72244 70.03365
## 1618 69.00490 68.88288 69.12693
## 1619 66.38548 66.28942 66.48154
## 1620 67.84072 67.74993 67.93151
## 1621 67.25862 67.17261 67.34463
## 1622 65.80339 65.69152 65.91525
## 1623 67.84072 67.74993 67.93151
## 1624 65.22129 65.08919 65.35340
## 1625 69.29595 69.16335 69.42855
## 1626 66.38548 66.28942 66.48154
## 1627 67.54967 67.46241 67.63693
## 1628 67.84072 67.74993 67.93151
## 1629 66.67653 66.58595 66.76711
## 1630 68.42281 68.31918 68.52644
## 1631 66.38548 66.28942 66.48154
## 1632 68.71386 68.60157 68.82614
## 1633 64.93025 64.78693 65.07356
## 1634 67.54967 67.46241 67.63693
## 1635 70.75119 70.55818 70.94419
## 1636 65.51234 65.39078 65.63390
## 1637 66.09443 65.99117 66.19770
## 1638 68.13176 68.03540 68.22812
## 1639 67.54967 67.46241 67.63693
## 1640 67.84072 67.74993 67.93151
## 1641 68.13176 68.03540 68.22812
## 1642 68.42281 68.31918 68.52644
## 1643 69.29595 69.16335 69.42855
## 1644 67.84072 67.74993 67.93151
## 1645 66.96758 66.88043 67.05472
## 1646 68.42281 68.31918 68.52644
## 1647 70.46014 70.27989 70.64039
## 1648 66.96758 66.88043 67.05472
## 1649 65.51234 65.39078 65.63390
## 1650 69.29595 69.16335 69.42855
## 1651 66.96758 66.88043 67.05472
## 1652 63.47501 63.26964 63.68038
## 1653 68.71386 68.60157 68.82614
## 1654 66.38548 66.28942 66.48154
## 1655 66.96758 66.88043 67.05472
## 1656 67.84072 67.74993 67.93151
## 1657 67.54967 67.46241 67.63693
## 1658 68.42281 68.31918 68.52644
## 1659 66.38548 66.28942 66.48154
## 1660 68.13176 68.03540 68.22812
## 1661 66.67653 66.58595 66.76711
## 1662 68.42281 68.31918 68.52644
## 1663 68.42281 68.31918 68.52644
## 1664 65.51234 65.39078 65.63390
## 1665 69.29595 69.16335 69.42855
## 1666 66.38548 66.28942 66.48154
## 1667 68.71386 68.60157 68.82614
## 1668 67.84072 67.74993 67.93151
## 1669 70.46014 70.27989 70.64039
## 1670 66.96758 66.88043 67.05472
## 1671 66.67653 66.58595 66.76711
## 1672 67.25862 67.17261 67.34463
## 1673 66.67653 66.58595 66.76711
## 1674 68.71386 68.60157 68.82614
## 1675 64.34815 64.18095 64.51535
## 1676 65.51234 65.39078 65.63390
## 1677 65.51234 65.39078 65.63390
## 1678 69.00490 68.88288 69.12693
## 1679 70.16909 70.00133 70.33686
## 1680 66.96758 66.88043 67.05472
## 1681 67.25862 67.17261 67.34463
## 1682 69.29595 69.16335 69.42855
## 1683 67.54967 67.46241 67.63693
## 1684 69.00490 68.88288 69.12693
## 1685 68.13176 68.03540 68.22812
## 1686 67.54967 67.46241 67.63693
## 1687 69.00490 68.88288 69.12693
## 1688 62.60187 62.35668 62.84706
## 1689 68.42281 68.31918 68.52644
## 1690 64.05711 63.87743 64.23678
## 1691 68.13176 68.03540 68.22812
## 1692 69.58700 69.44315 69.73084
## 1693 66.96758 66.88043 67.05472
## 1694 69.00490 68.88288 69.12693
## 1695 70.16909 70.00133 70.33686
## 1696 69.58700 69.44315 69.73084
## 1697 70.16909 70.00133 70.33686
## 1698 68.42281 68.31918 68.52644
## 1699 67.25862 67.17261 67.34463
## 1700 64.34815 64.18095 64.51535
## 1701 65.80339 65.69152 65.91525
## 1702 71.04223 70.83626 71.24820
## 1703 66.96758 66.88043 67.05472
## 1704 67.54967 67.46241 67.63693
## 1705 68.13176 68.03540 68.22812
## 1706 67.25862 67.17261 67.34463
## 1707 70.16909 70.00133 70.33686
## 1708 66.09443 65.99117 66.19770
## 1709 67.54967 67.46241 67.63693
## 1710 66.09443 65.99117 66.19770
## 1711 64.93025 64.78693 65.07356
## 1712 66.09443 65.99117 66.19770
## 1713 67.25862 67.17261 67.34463
## 1714 66.38548 66.28942 66.48154
## 1715 66.09443 65.99117 66.19770
## 1716 64.93025 64.78693 65.07356
## 1717 67.25862 67.17261 67.34463
## 1718 64.63920 64.48415 64.79425
## 1719 68.71386 68.60157 68.82614
## 1720 67.25862 67.17261 67.34463
## 1721 64.63920 64.48415 64.79425
## 1722 66.67653 66.58595 66.76711
## 1723 65.80339 65.69152 65.91525
## 1724 67.84072 67.74993 67.93151
## 1725 69.58700 69.44315 69.73084
## 1726 64.05711 63.87743 64.23678
## 1727 66.96758 66.88043 67.05472
## 1728 68.42281 68.31918 68.52644
## 1729 67.54967 67.46241 67.63693
## 1730 65.80339 65.69152 65.91525
## 1731 66.67653 66.58595 66.76711
## 1732 69.00490 68.88288 69.12693
## 1733 67.25862 67.17261 67.34463
## 1734 70.46014 70.27989 70.64039
## 1735 67.25862 67.17261 67.34463
## 1736 66.09443 65.99117 66.19770
## 1737 64.63920 64.48415 64.79425
## 1738 64.05711 63.87743 64.23678
## 1739 64.34815 64.18095 64.51535
## 1740 67.84072 67.74993 67.93151
## 1741 68.13176 68.03540 68.22812
## 1742 64.63920 64.48415 64.79425
## 1743 68.71386 68.60157 68.82614
## 1744 66.38548 66.28942 66.48154
## 1745 67.54967 67.46241 67.63693
## 1746 70.16909 70.00133 70.33686
## 1747 68.71386 68.60157 68.82614
## 1748 66.67653 66.58595 66.76711
## 1749 65.80339 65.69152 65.91525
## 1750 63.47501 63.26964 63.68038
## 1751 66.09443 65.99117 66.19770
## 1752 66.96758 66.88043 67.05472
## 1753 65.51234 65.39078 65.63390
## 1754 66.96758 66.88043 67.05472
## 1755 67.25862 67.17261 67.34463
## 1756 66.96758 66.88043 67.05472
## 1757 66.09443 65.99117 66.19770
## 1758 67.54967 67.46241 67.63693
## 1759 68.13176 68.03540 68.22812
## 1760 65.22129 65.08919 65.35340
## 1761 69.58700 69.44315 69.73084
## 1762 66.96758 66.88043 67.05472
## 1763 67.25862 67.17261 67.34463
## 1764 68.42281 68.31918 68.52644
## 1765 67.54967 67.46241 67.63693
## 1766 69.00490 68.88288 69.12693
## 1767 67.84072 67.74993 67.93151
## 1768 66.09443 65.99117 66.19770
## 1769 67.84072 67.74993 67.93151
## 1770 65.22129 65.08919 65.35340
## 1771 67.54967 67.46241 67.63693
## 1772 67.25862 67.17261 67.34463
## 1773 66.96758 66.88043 67.05472
## 1774 66.96758 66.88043 67.05472
## 1775 64.05711 63.87743 64.23678
## 1776 66.38548 66.28942 66.48154
## 1777 67.54967 67.46241 67.63693
## 1778 67.54967 67.46241 67.63693
## 1779 65.51234 65.39078 65.63390
## 1780 69.58700 69.44315 69.73084
## 1781 65.80339 65.69152 65.91525
## 1782 69.00490 68.88288 69.12693
## 1783 68.13176 68.03540 68.22812
## 1784 66.09443 65.99117 66.19770
## 1785 70.46014 70.27989 70.64039
## 1786 65.22129 65.08919 65.35340
## 1787 69.00490 68.88288 69.12693
## 1788 69.00490 68.88288 69.12693
## 1789 66.09443 65.99117 66.19770
## 1790 69.58700 69.44315 69.73084
## 1791 66.67653 66.58595 66.76711
## 1792 67.84072 67.74993 67.93151
## 1793 67.84072 67.74993 67.93151
## 1794 66.96758 66.88043 67.05472
## 1795 68.71386 68.60157 68.82614
## 1796 68.71386 68.60157 68.82614
## 1797 64.34815 64.18095 64.51535
## 1798 69.58700 69.44315 69.73084
## 1799 64.34815 64.18095 64.51535
## 1800 69.00490 68.88288 69.12693
## 1801 69.29595 69.16335 69.42855
## 1802 65.22129 65.08919 65.35340
## 1803 66.67653 66.58595 66.76711
## 1804 65.22129 65.08919 65.35340
## 1805 66.96758 66.88043 67.05472
## 1806 66.67653 66.58595 66.76711
## 1807 68.71386 68.60157 68.82614
## 1808 69.58700 69.44315 69.73084
## 1809 69.58700 69.44315 69.73084
## 1810 66.38548 66.28942 66.48154
## 1811 66.96758 66.88043 67.05472
## 1812 64.34815 64.18095 64.51535
## 1813 65.80339 65.69152 65.91525
## 1814 69.00490 68.88288 69.12693
## 1815 70.16909 70.00133 70.33686
## 1816 65.22129 65.08919 65.35340
## 1817 66.09443 65.99117 66.19770
## 1818 67.25862 67.17261 67.34463
## 1819 65.80339 65.69152 65.91525
## 1820 68.13176 68.03540 68.22812
## 1821 66.38548 66.28942 66.48154
## 1822 67.25862 67.17261 67.34463
## 1823 68.13176 68.03540 68.22812
## 1824 67.25862 67.17261 67.34463
## 1825 66.38548 66.28942 66.48154
## 1826 66.96758 66.88043 67.05472
## 1827 66.96758 66.88043 67.05472
## 1828 66.67653 66.58595 66.76711
## 1829 66.67653 66.58595 66.76711
## 1830 70.16909 70.00133 70.33686
## 1831 68.13176 68.03540 68.22812
## 1832 66.67653 66.58595 66.76711
## 1833 68.42281 68.31918 68.52644
## 1834 66.67653 66.58595 66.76711
## 1835 69.29595 69.16335 69.42855
## 1836 67.84072 67.74993 67.93151
## 1837 66.96758 66.88043 67.05472
## 1838 68.13176 68.03540 68.22812
## 1839 64.63920 64.48415 64.79425
## 1840 65.22129 65.08919 65.35340
## 1841 65.22129 65.08919 65.35340
## 1842 67.54967 67.46241 67.63693
## 1843 70.16909 70.00133 70.33686
## 1844 71.04223 70.83626 71.24820
## 1845 68.42281 68.31918 68.52644
## 1846 68.13176 68.03540 68.22812
## 1847 64.05711 63.87743 64.23678
## 1848 66.38548 66.28942 66.48154
## 1849 65.80339 65.69152 65.91525
## 1850 66.09443 65.99117 66.19770
## 1851 66.96758 66.88043 67.05472
## 1852 66.38548 66.28942 66.48154
## 1853 63.76606 63.57364 63.95847
## 1854 68.42281 68.31918 68.52644
## 1855 65.51234 65.39078 65.63390
## 1856 67.54967 67.46241 67.63693
## 1857 70.75119 70.55818 70.94419
## 1858 65.51234 65.39078 65.63390
## 1859 62.89292 62.66113 63.12470
## 1860 66.67653 66.58595 66.76711
## 1861 69.29595 69.16335 69.42855
## 1862 70.75119 70.55818 70.94419
## 1863 65.22129 65.08919 65.35340
## 1864 68.13176 68.03540 68.22812
## 1865 66.38548 66.28942 66.48154
## 1866 66.96758 66.88043 67.05472
## 1867 71.91537 71.66957 72.16118
## 1868 64.63920 64.48415 64.79425
## 1869 67.84072 67.74993 67.93151
## 1870 68.13176 68.03540 68.22812
## 1871 69.29595 69.16335 69.42855
## 1872 67.54967 67.46241 67.63693
## 1873 67.84072 67.74993 67.93151
## 1874 66.67653 66.58595 66.76711
## 1875 67.84072 67.74993 67.93151
## 1876 66.67653 66.58595 66.76711
## 1877 68.42281 68.31918 68.52644
## 1878 66.67653 66.58595 66.76711
## 1879 67.25862 67.17261 67.34463
## 1880 69.29595 69.16335 69.42855
## 1881 67.25862 67.17261 67.34463
## 1882 64.05711 63.87743 64.23678
## 1883 69.58700 69.44315 69.73084
## 1884 65.80339 65.69152 65.91525
## 1885 66.09443 65.99117 66.19770
## 1886 66.09443 65.99117 66.19770
## 1887 69.58700 69.44315 69.73084
## 1888 63.47501 63.26964 63.68038
## 1889 72.78852 72.50193 73.07510
## 1890 69.00490 68.88288 69.12693
## 1891 67.54967 67.46241 67.63693
## 1892 67.25862 67.17261 67.34463
## 1893 67.84072 67.74993 67.93151
## 1894 68.13176 68.03540 68.22812
## 1895 68.42281 68.31918 68.52644
## 1896 68.13176 68.03540 68.22812
## 1897 67.54967 67.46241 67.63693
## 1898 66.96758 66.88043 67.05472
## 1899 66.96758 66.88043 67.05472
## 1900 66.96758 66.88043 67.05472
## 1901 69.29595 69.16335 69.42855
## 1902 68.42281 68.31918 68.52644
## 1903 68.42281 68.31918 68.52644
## 1904 67.54967 67.46241 67.63693
## 1905 67.25862 67.17261 67.34463
## 1906 66.67653 66.58595 66.76711
## 1907 69.29595 69.16335 69.42855
## 1908 69.00490 68.88288 69.12693
## 1909 65.80339 65.69152 65.91525
## 1910 67.84072 67.74993 67.93151
## 1911 65.51234 65.39078 65.63390
## 1912 64.93025 64.78693 65.07356
## 1913 66.09443 65.99117 66.19770
## 1914 68.71386 68.60157 68.82614
## 1915 69.00490 68.88288 69.12693
## 1916 68.13176 68.03540 68.22812
## 1917 68.42281 68.31918 68.52644
## 1918 68.13176 68.03540 68.22812
## 1919 66.09443 65.99117 66.19770
## 1920 68.13176 68.03540 68.22812
## 1921 68.71386 68.60157 68.82614
## 1922 68.71386 68.60157 68.82614
## 1923 69.29595 69.16335 69.42855
## 1924 68.42281 68.31918 68.52644
## 1925 67.84072 67.74993 67.93151
## 1926 69.00490 68.88288 69.12693
## 1927 67.25862 67.17261 67.34463
## 1928 65.80339 65.69152 65.91525
## 1929 66.96758 66.88043 67.05472
## 1930 66.09443 65.99117 66.19770
## 1931 66.09443 65.99117 66.19770
## 1932 66.96758 66.88043 67.05472
## 1933 65.80339 65.69152 65.91525
## 1934 66.67653 66.58595 66.76711
## 1935 64.05711 63.87743 64.23678
## 1936 67.54967 67.46241 67.63693
## 1937 67.25862 67.17261 67.34463
## 1938 65.80339 65.69152 65.91525
## 1939 68.42281 68.31918 68.52644
## 1940 69.00490 68.88288 69.12693
## 1941 64.34815 64.18095 64.51535
## 1942 65.80339 65.69152 65.91525
## 1943 65.80339 65.69152 65.91525
## 1944 65.51234 65.39078 65.63390
## 1945 68.71386 68.60157 68.82614
## 1946 66.96758 66.88043 67.05472
## 1947 66.67653 66.58595 66.76711
## 1948 63.76606 63.57364 63.95847
## 1949 68.42281 68.31918 68.52644
## 1950 65.22129 65.08919 65.35340
## 1951 67.25862 67.17261 67.34463
## 1952 67.84072 67.74993 67.93151
## 1953 69.58700 69.44315 69.73084
## 1954 64.93025 64.78693 65.07356
## 1955 62.01978 61.74749 62.29207
## 1956 67.54967 67.46241 67.63693
## 1957 70.46014 70.27989 70.64039
## 1958 69.29595 69.16335 69.42855
## 1959 67.54967 67.46241 67.63693
## 1960 66.38548 66.28942 66.48154
## 1961 68.71386 68.60157 68.82614
## 1962 65.51234 65.39078 65.63390
## 1963 68.13176 68.03540 68.22812
## 1964 68.13176 68.03540 68.22812
## 1965 69.00490 68.88288 69.12693
## 1966 72.20642 71.94711 72.46574
## 1967 67.84072 67.74993 67.93151
## 1968 66.38548 66.28942 66.48154
## 1969 67.25862 67.17261 67.34463
## 1970 64.05711 63.87743 64.23678
## 1971 66.09443 65.99117 66.19770
## 1972 66.67653 66.58595 66.76711
## 1973 66.38548 66.28942 66.48154
## 1974 67.54967 67.46241 67.63693
## 1975 65.80339 65.69152 65.91525
## 1976 69.29595 69.16335 69.42855
## 1977 67.25862 67.17261 67.34463
## 1978 66.67653 66.58595 66.76711
## 1979 67.54967 67.46241 67.63693
## 1980 68.71386 68.60157 68.82614
## 1981 65.80339 65.69152 65.91525
## 1982 66.09443 65.99117 66.19770
## 1983 66.67653 66.58595 66.76711
## 1984 64.93025 64.78693 65.07356
## 1985 65.80339 65.69152 65.91525
## 1986 66.38548 66.28942 66.48154
## 1987 68.71386 68.60157 68.82614
## 1988 67.54967 67.46241 67.63693
## 1989 68.71386 68.60157 68.82614
## 1990 66.96758 66.88043 67.05472
## 1991 67.84072 67.74993 67.93151
## 1992 66.96758 66.88043 67.05472
## 1993 67.84072 67.74993 67.93151
## 1994 66.38548 66.28942 66.48154
## 1995 62.89292 62.66113 63.12470
## 1996 66.96758 66.88043 67.05472
## 1997 69.87805 69.72244 70.03365
## 1998 69.00490 68.88288 69.12693
## 1999 69.00490 68.88288 69.12693
## 2000 69.29595 69.16335 69.42855
## 2001 67.54967 67.46241 67.63693
## 2002 67.54967 67.46241 67.63693
## 2003 66.09443 65.99117 66.19770
## 2004 66.67653 66.58595 66.76711
## 2005 68.42281 68.31918 68.52644
## 2006 69.29595 69.16335 69.42855
## 2007 66.96758 66.88043 67.05472
## 2008 66.38548 66.28942 66.48154
## 2009 67.25862 67.17261 67.34463
## 2010 68.42281 68.31918 68.52644
## 2011 64.63920 64.48415 64.79425
## 2012 63.18396 62.96546 63.40247
## 2013 66.38548 66.28942 66.48154
## 2014 65.22129 65.08919 65.35340
## 2015 66.38548 66.28942 66.48154
## 2016 66.96758 66.88043 67.05472
## 2017 66.38548 66.28942 66.48154
## 2018 69.87805 69.72244 70.03365
## 2019 67.84072 67.74993 67.93151
## 2020 67.84072 67.74993 67.93151
## 2021 68.42281 68.31918 68.52644
## 2022 65.22129 65.08919 65.35340
## 2023 69.00490 68.88288 69.12693
## 2024 67.25862 67.17261 67.34463
## 2025 68.42281 68.31918 68.52644
## 2026 65.51234 65.39078 65.63390
## 2027 64.63920 64.48415 64.79425
## 2028 68.71386 68.60157 68.82614
## 2029 67.84072 67.74993 67.93151
## 2030 65.80339 65.69152 65.91525
## 2031 65.80339 65.69152 65.91525
## 2032 69.29595 69.16335 69.42855
## 2033 65.22129 65.08919 65.35340
## 2034 70.16909 70.00133 70.33686
## 2035 65.80339 65.69152 65.91525
## 2036 68.42281 68.31918 68.52644
## 2037 66.67653 66.58595 66.76711
## 2038 66.38548 66.28942 66.48154
## 2039 66.96758 66.88043 67.05472
## 2040 67.25862 67.17261 67.34463
## 2041 65.51234 65.39078 65.63390
## 2042 68.42281 68.31918 68.52644
## 2043 66.38548 66.28942 66.48154
## 2044 65.51234 65.39078 65.63390
## 2045 62.89292 62.66113 63.12470
## 2046 66.96758 66.88043 67.05472
## 2047 71.33328 71.11417 71.55239
## 2048 68.71386 68.60157 68.82614
## 2049 67.25862 67.17261 67.34463
## 2050 66.67653 66.58595 66.76711
## 2051 67.25862 67.17261 67.34463
## 2052 68.13176 68.03540 68.22812
## 2053 65.22129 65.08919 65.35340
## 2054 69.29595 69.16335 69.42855
## 2055 68.42281 68.31918 68.52644
## 2056 68.42281 68.31918 68.52644
## 2057 68.13176 68.03540 68.22812
## 2058 67.25862 67.17261 67.34463
## 2059 68.42281 68.31918 68.52644
## 2060 68.42281 68.31918 68.52644
## 2061 66.67653 66.58595 66.76711
## 2062 66.38548 66.28942 66.48154
## 2063 67.54967 67.46241 67.63693
## 2064 69.87805 69.72244 70.03365
## 2065 65.22129 65.08919 65.35340
## 2066 64.63920 64.48415 64.79425
## 2067 67.54967 67.46241 67.63693
## 2068 67.25862 67.17261 67.34463
## 2069 67.54967 67.46241 67.63693
## 2070 67.54967 67.46241 67.63693
## 2071 66.96758 66.88043 67.05472
## 2072 69.87805 69.72244 70.03365
## 2073 64.34815 64.18095 64.51535
## 2074 65.80339 65.69152 65.91525
## 2075 66.96758 66.88043 67.05472
## 2076 64.34815 64.18095 64.51535
## 2077 66.96758 66.88043 67.05472
## 2078 70.16909 70.00133 70.33686
## 2079 67.25862 67.17261 67.34463
## 2080 67.54967 67.46241 67.63693
## 2081 64.93025 64.78693 65.07356
## 2082 66.09443 65.99117 66.19770
## 2083 66.67653 66.58595 66.76711
## 2084 66.67653 66.58595 66.76711
## 2085 69.58700 69.44315 69.73084
## 2086 68.13176 68.03540 68.22812
## 2087 65.22129 65.08919 65.35340
## 2088 67.54967 67.46241 67.63693
## 2089 68.13176 68.03540 68.22812
## 2090 69.87805 69.72244 70.03365
## 2091 65.51234 65.39078 65.63390
## 2092 65.80339 65.69152 65.91525
## 2093 64.93025 64.78693 65.07356
## 2094 66.96758 66.88043 67.05472
## 2095 68.13176 68.03540 68.22812
## 2096 69.00490 68.88288 69.12693
## 2097 67.25862 67.17261 67.34463
## 2098 66.67653 66.58595 66.76711
## 2099 67.54967 67.46241 67.63693
## 2100 65.80339 65.69152 65.91525
## 2101 71.04223 70.83626 71.24820
## 2102 64.34815 64.18095 64.51535
## 2103 67.54967 67.46241 67.63693
## 2104 70.75119 70.55818 70.94419
## 2105 67.25862 67.17261 67.34463
## 2106 66.96758 66.88043 67.05472
## 2107 66.67653 66.58595 66.76711
## 2108 66.96758 66.88043 67.05472
## 2109 69.58700 69.44315 69.73084
## 2110 63.47501 63.26964 63.68038
## 2111 69.87805 69.72244 70.03365
## 2112 64.63920 64.48415 64.79425
## 2113 66.67653 66.58595 66.76711
## 2114 68.42281 68.31918 68.52644
## 2115 64.34815 64.18095 64.51535
## 2116 68.13176 68.03540 68.22812
## 2117 70.75119 70.55818 70.94419
## 2118 64.93025 64.78693 65.07356
## 2119 67.54967 67.46241 67.63693
## 2120 66.09443 65.99117 66.19770
## 2121 68.13176 68.03540 68.22812
## 2122 67.25862 67.17261 67.34463
## 2123 66.09443 65.99117 66.19770
## 2124 68.13176 68.03540 68.22812
## 2125 70.46014 70.27989 70.64039
## 2126 66.96758 66.88043 67.05472
## 2127 68.42281 68.31918 68.52644
## 2128 64.93025 64.78693 65.07356
## 2129 68.13176 68.03540 68.22812
## 2130 69.87805 69.72244 70.03365
## 2131 67.84072 67.74993 67.93151
## 2132 66.67653 66.58595 66.76711
## 2133 67.25862 67.17261 67.34463
## 2134 64.93025 64.78693 65.07356
## 2135 68.42281 68.31918 68.52644
## 2136 68.71386 68.60157 68.82614
## 2137 66.38548 66.28942 66.48154
## 2138 66.38548 66.28942 66.48154
## 2139 67.84072 67.74993 67.93151
## 2140 66.38548 66.28942 66.48154
## 2141 67.25862 67.17261 67.34463
## 2142 71.04223 70.83626 71.24820
## 2143 66.96758 66.88043 67.05472
## 2144 66.67653 66.58595 66.76711
## 2145 66.09443 65.99117 66.19770
## 2146 68.42281 68.31918 68.52644
## 2147 69.58700 69.44315 69.73084
## 2148 69.87805 69.72244 70.03365
## 2149 66.09443 65.99117 66.19770
## 2150 64.05711 63.87743 64.23678
## 2151 69.00490 68.88288 69.12693
## 2152 66.09443 65.99117 66.19770
## 2153 68.71386 68.60157 68.82614
## 2154 63.76606 63.57364 63.95847
## 2155 68.71386 68.60157 68.82614
## 2156 65.22129 65.08919 65.35340
## 2157 68.42281 68.31918 68.52644
## 2158 67.25862 67.17261 67.34463
## 2159 65.51234 65.39078 65.63390
## 2160 66.67653 66.58595 66.76711
## 2161 68.42281 68.31918 68.52644
## 2162 67.54967 67.46241 67.63693
## 2163 68.42281 68.31918 68.52644
## 2164 70.75119 70.55818 70.94419
## 2165 64.63920 64.48415 64.79425
## 2166 70.16909 70.00133 70.33686
## 2167 67.54967 67.46241 67.63693
## 2168 65.80339 65.69152 65.91525
## 2169 64.93025 64.78693 65.07356
## 2170 66.96758 66.88043 67.05472
## 2171 69.00490 68.88288 69.12693
## 2172 66.96758 66.88043 67.05472
## 2173 69.87805 69.72244 70.03365
## 2174 64.63920 64.48415 64.79425
## 2175 67.25862 67.17261 67.34463
## 2176 64.93025 64.78693 65.07356
## 2177 67.84072 67.74993 67.93151
## 2178 68.71386 68.60157 68.82614
## 2179 66.38548 66.28942 66.48154
## 2180 62.60187 62.35668 62.84706
## 2181 67.54967 67.46241 67.63693
## 2182 68.13176 68.03540 68.22812
## 2183 67.84072 67.74993 67.93151
## 2184 69.29595 69.16335 69.42855
## 2185 65.51234 65.39078 65.63390
## 2186 68.42281 68.31918 68.52644
## 2187 69.29595 69.16335 69.42855
## 2188 64.05711 63.87743 64.23678
## 2189 67.84072 67.74993 67.93151
## 2190 66.38548 66.28942 66.48154
## 2191 67.25862 67.17261 67.34463
## 2192 67.84072 67.74993 67.93151
## 2193 66.38548 66.28942 66.48154
## 2194 66.09443 65.99117 66.19770
## 2195 67.25862 67.17261 67.34463
## 2196 65.22129 65.08919 65.35340
## 2197 65.22129 65.08919 65.35340
## 2198 68.42281 68.31918 68.52644
## 2199 65.22129 65.08919 65.35340
## 2200 68.42281 68.31918 68.52644
## 2201 68.13176 68.03540 68.22812
## 2202 64.34815 64.18095 64.51535
## 2203 66.09443 65.99117 66.19770
## 2204 69.29595 69.16335 69.42855
## 2205 65.51234 65.39078 65.63390
## 2206 66.38548 66.28942 66.48154
## 2207 68.13176 68.03540 68.22812
## 2208 68.71386 68.60157 68.82614
## 2209 66.67653 66.58595 66.76711
## 2210 66.09443 65.99117 66.19770
## 2211 69.87805 69.72244 70.03365
## 2212 70.16909 70.00133 70.33686
## 2213 69.00490 68.88288 69.12693
## 2214 67.54967 67.46241 67.63693
## 2215 65.80339 65.69152 65.91525
## 2216 65.80339 65.69152 65.91525
## 2217 68.13176 68.03540 68.22812
## 2218 68.13176 68.03540 68.22812
## 2219 69.58700 69.44315 69.73084
## 2220 68.42281 68.31918 68.52644
## 2221 64.63920 64.48415 64.79425
## 2222 69.00490 68.88288 69.12693
## 2223 65.51234 65.39078 65.63390
## 2224 66.67653 66.58595 66.76711
## 2225 67.84072 67.74993 67.93151
## 2226 70.75119 70.55818 70.94419
## 2227 72.78852 72.50193 73.07510
## 2228 68.42281 68.31918 68.52644
## 2229 66.96758 66.88043 67.05472
## 2230 68.13176 68.03540 68.22812
## 2231 68.71386 68.60157 68.82614
## 2232 66.67653 66.58595 66.76711
## 2233 70.46014 70.27989 70.64039
## 2234 67.54967 67.46241 67.63693
## 2235 68.13176 68.03540 68.22812
## 2236 68.42281 68.31918 68.52644
## 2237 68.13176 68.03540 68.22812
## 2238 68.71386 68.60157 68.82614
## 2239 66.09443 65.99117 66.19770
## 2240 66.38548 66.28942 66.48154
## 2241 65.22129 65.08919 65.35340
## 2242 68.13176 68.03540 68.22812
## 2243 68.71386 68.60157 68.82614
## 2244 68.13176 68.03540 68.22812
## 2245 70.16909 70.00133 70.33686
## 2246 67.25862 67.17261 67.34463
## 2247 68.42281 68.31918 68.52644
## 2248 65.22129 65.08919 65.35340
## 2249 66.67653 66.58595 66.76711
## 2250 68.42281 68.31918 68.52644
## 2251 68.71386 68.60157 68.82614
## 2252 66.67653 66.58595 66.76711
## 2253 67.25862 67.17261 67.34463
## 2254 64.63920 64.48415 64.79425
## 2255 66.67653 66.58595 66.76711
## 2256 68.71386 68.60157 68.82614
## 2257 68.13176 68.03540 68.22812
## 2258 68.13176 68.03540 68.22812
## 2259 67.25862 67.17261 67.34463
## 2260 67.25862 67.17261 67.34463
## 2261 66.09443 65.99117 66.19770
## 2262 68.42281 68.31918 68.52644
## 2263 68.13176 68.03540 68.22812
## 2264 68.13176 68.03540 68.22812
## 2265 66.38548 66.28942 66.48154
## 2266 63.47501 63.26964 63.68038
## 2267 69.29595 69.16335 69.42855
## 2268 66.38548 66.28942 66.48154
## 2269 65.80339 65.69152 65.91525
## 2270 68.42281 68.31918 68.52644
## 2271 66.67653 66.58595 66.76711
## 2272 69.00490 68.88288 69.12693
## 2273 68.42281 68.31918 68.52644
## 2274 68.42281 68.31918 68.52644
## 2275 67.84072 67.74993 67.93151
## 2276 66.67653 66.58595 66.76711
## 2277 67.54967 67.46241 67.63693
## 2278 64.05711 63.87743 64.23678
## 2279 67.84072 67.74993 67.93151
## 2280 65.51234 65.39078 65.63390
## 2281 64.93025 64.78693 65.07356
## 2282 64.63920 64.48415 64.79425
## 2283 69.29595 69.16335 69.42855
## 2284 66.09443 65.99117 66.19770
## 2285 69.87805 69.72244 70.03365
## 2286 66.38548 66.28942 66.48154
## 2287 64.34815 64.18095 64.51535
## 2288 65.22129 65.08919 65.35340
## 2289 69.29595 69.16335 69.42855
## 2290 66.96758 66.88043 67.05472
## 2291 66.38548 66.28942 66.48154
## 2292 67.54967 67.46241 67.63693
## 2293 68.13176 68.03540 68.22812
## 2294 69.29595 69.16335 69.42855
## 2295 69.87805 69.72244 70.03365
## 2296 66.67653 66.58595 66.76711
## 2297 64.93025 64.78693 65.07356
## 2298 66.67653 66.58595 66.76711
## 2299 67.84072 67.74993 67.93151
## 2300 65.51234 65.39078 65.63390
## 2301 67.84072 67.74993 67.93151
## 2302 67.25862 67.17261 67.34463
## 2303 67.25862 67.17261 67.34463
## 2304 65.80339 65.69152 65.91525
## 2305 66.96758 66.88043 67.05472
## 2306 65.51234 65.39078 65.63390
## 2307 68.71386 68.60157 68.82614
## 2308 69.00490 68.88288 69.12693
## 2309 65.51234 65.39078 65.63390
## 2310 64.63920 64.48415 64.79425
## 2311 66.09443 65.99117 66.19770
## 2312 66.96758 66.88043 67.05472
## 2313 64.63920 64.48415 64.79425
## 2314 65.51234 65.39078 65.63390
## 2315 65.51234 65.39078 65.63390
## 2316 67.54967 67.46241 67.63693
## 2317 66.09443 65.99117 66.19770
## 2318 67.54967 67.46241 67.63693
## 2319 66.38548 66.28942 66.48154
## 2320 66.67653 66.58595 66.76711
## 2321 70.16909 70.00133 70.33686
## 2322 69.58700 69.44315 69.73084
## 2323 64.93025 64.78693 65.07356
## 2324 68.71386 68.60157 68.82614
## 2325 68.71386 68.60157 68.82614
## 2326 65.22129 65.08919 65.35340
## 2327 68.71386 68.60157 68.82614
## 2328 69.87805 69.72244 70.03365
## 2329 65.22129 65.08919 65.35340
## 2330 65.51234 65.39078 65.63390
## 2331 69.00490 68.88288 69.12693
## 2332 67.84072 67.74993 67.93151
## 2333 68.13176 68.03540 68.22812
## 2334 67.84072 67.74993 67.93151
## 2335 67.84072 67.74993 67.93151
## 2336 64.93025 64.78693 65.07356
## 2337 69.00490 68.88288 69.12693
## 2338 69.29595 69.16335 69.42855
## 2339 66.96758 66.88043 67.05472
## 2340 67.25862 67.17261 67.34463
## 2341 64.63920 64.48415 64.79425
## 2342 68.13176 68.03540 68.22812
## 2343 68.13176 68.03540 68.22812
## 2344 67.25862 67.17261 67.34463
## 2345 69.29595 69.16335 69.42855
## 2346 66.67653 66.58595 66.76711
## 2347 68.13176 68.03540 68.22812
## 2348 65.51234 65.39078 65.63390
## 2349 67.25862 67.17261 67.34463
## 2350 66.96758 66.88043 67.05472
## 2351 67.54967 67.46241 67.63693
## 2352 66.67653 66.58595 66.76711
## 2353 66.96758 66.88043 67.05472
## 2354 68.71386 68.60157 68.82614
## 2355 66.38548 66.28942 66.48154
## 2356 64.34815 64.18095 64.51535
## 2357 70.16909 70.00133 70.33686
## 2358 69.58700 69.44315 69.73084
## 2359 64.93025 64.78693 65.07356
## 2360 68.42281 68.31918 68.52644
## 2361 66.38548 66.28942 66.48154
## 2362 67.84072 67.74993 67.93151
## 2363 67.84072 67.74993 67.93151
## 2364 66.96758 66.88043 67.05472
## 2365 65.80339 65.69152 65.91525
## 2366 66.09443 65.99117 66.19770
## 2367 65.80339 65.69152 65.91525
## 2368 65.80339 65.69152 65.91525
## 2369 67.25862 67.17261 67.34463
## 2370 66.09443 65.99117 66.19770
## 2371 68.42281 68.31918 68.52644
## 2372 69.87805 69.72244 70.03365
## 2373 67.54967 67.46241 67.63693
## 2374 68.13176 68.03540 68.22812
## 2375 68.13176 68.03540 68.22812
## 2376 67.25862 67.17261 67.34463
## 2377 68.42281 68.31918 68.52644
## 2378 68.13176 68.03540 68.22812
## 2379 65.22129 65.08919 65.35340
## 2380 68.71386 68.60157 68.82614
## 2381 68.71386 68.60157 68.82614
## 2382 65.80339 65.69152 65.91525
## 2383 67.25862 67.17261 67.34463
## 2384 68.42281 68.31918 68.52644
## 2385 66.96758 66.88043 67.05472
## 2386 69.00490 68.88288 69.12693
## 2387 71.33328 71.11417 71.55239
## 2388 65.80339 65.69152 65.91525
## 2389 64.93025 64.78693 65.07356
## 2390 67.84072 67.74993 67.93151
## 2391 68.42281 68.31918 68.52644
## 2392 66.96758 66.88043 67.05472
## 2393 69.87805 69.72244 70.03365
## 2394 64.93025 64.78693 65.07356
## 2395 68.42281 68.31918 68.52644
## 2396 70.16909 70.00133 70.33686
## 2397 70.75119 70.55818 70.94419
## 2398 65.80339 65.69152 65.91525
## 2399 66.38548 66.28942 66.48154
## 2400 68.71386 68.60157 68.82614
## 2401 66.67653 66.58595 66.76711
## 2402 67.25862 67.17261 67.34463
## 2403 67.25862 67.17261 67.34463
## 2404 63.76606 63.57364 63.95847
## 2405 67.25862 67.17261 67.34463
## 2406 66.09443 65.99117 66.19770
## 2407 66.38548 66.28942 66.48154
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## 6229 67.84072 67.74993 67.93151
## 6230 66.67653 66.58595 66.76711
## 6231 69.00490 68.88288 69.12693
## 6232 65.22129 65.08919 65.35340
## 6233 66.09443 65.99117 66.19770
## 6234 64.63920 64.48415 64.79425
## 6235 65.51234 65.39078 65.63390
## 6236 65.22129 65.08919 65.35340
## 6237 69.00490 68.88288 69.12693
## 6238 66.09443 65.99117 66.19770
## 6239 67.54967 67.46241 67.63693
## 6240 66.67653 66.58595 66.76711
## 6241 66.96758 66.88043 67.05472
## 6242 68.42281 68.31918 68.52644
## 6243 65.22129 65.08919 65.35340
## 6244 68.71386 68.60157 68.82614
## 6245 65.80339 65.69152 65.91525
## 6246 66.38548 66.28942 66.48154
## 6247 68.13176 68.03540 68.22812
## 6248 66.09443 65.99117 66.19770
## 6249 67.25862 67.17261 67.34463
## 6250 66.67653 66.58595 66.76711
## 6251 67.54967 67.46241 67.63693
## 6252 63.18396 62.96546 63.40247
## 6253 64.05711 63.87743 64.23678
## 6254 68.42281 68.31918 68.52644
## 6255 69.00490 68.88288 69.12693
## 6256 65.80339 65.69152 65.91525
## 6257 66.96758 66.88043 67.05472
## 6258 67.84072 67.74993 67.93151
## 6259 68.42281 68.31918 68.52644
## 6260 63.76606 63.57364 63.95847
## 6261 65.80339 65.69152 65.91525
## 6262 66.67653 66.58595 66.76711
## 6263 68.42281 68.31918 68.52644
## 6264 71.62433 71.39193 71.85672
## 6265 66.96758 66.88043 67.05472
## 6266 64.93025 64.78693 65.07356
## 6267 67.84072 67.74993 67.93151
## 6268 69.58700 69.44315 69.73084
## 6269 67.54967 67.46241 67.63693
## 6270 65.80339 65.69152 65.91525
## 6271 69.00490 68.88288 69.12693
## 6272 68.42281 68.31918 68.52644
## 6273 67.84072 67.74993 67.93151
## 6274 69.29595 69.16335 69.42855
## 6275 67.25862 67.17261 67.34463
## 6276 66.67653 66.58595 66.76711
## 6277 69.29595 69.16335 69.42855
## 6278 65.22129 65.08919 65.35340
## 6279 64.93025 64.78693 65.07356
## 6280 69.00490 68.88288 69.12693
## 6281 66.67653 66.58595 66.76711
## 6282 69.58700 69.44315 69.73084
## 6283 68.13176 68.03540 68.22812
## 6284 65.22129 65.08919 65.35340
## 6285 69.87805 69.72244 70.03365
## 6286 69.87805 69.72244 70.03365
## 6287 68.71386 68.60157 68.82614
## 6288 65.80339 65.69152 65.91525
## 6289 66.67653 66.58595 66.76711
## 6290 67.54967 67.46241 67.63693
## 6291 66.09443 65.99117 66.19770
## 6292 70.16909 70.00133 70.33686
## 6293 67.54967 67.46241 67.63693
## 6294 62.60187 62.35668 62.84706
## 6295 67.84072 67.74993 67.93151
## 6296 66.67653 66.58595 66.76711
## 6297 67.25862 67.17261 67.34463
## 6298 66.67653 66.58595 66.76711
## 6299 68.71386 68.60157 68.82614
## 6300 69.29595 69.16335 69.42855
## 6301 67.84072 67.74993 67.93151
## 6302 66.67653 66.58595 66.76711
## 6303 67.25862 67.17261 67.34463
## 6304 67.25862 67.17261 67.34463
## 6305 65.80339 65.69152 65.91525
## 6306 68.13176 68.03540 68.22812
## 6307 68.13176 68.03540 68.22812
## 6308 69.87805 69.72244 70.03365
## 6309 69.58700 69.44315 69.73084
## 6310 66.09443 65.99117 66.19770
## 6311 68.71386 68.60157 68.82614
## 6312 65.51234 65.39078 65.63390
## 6313 68.42281 68.31918 68.52644
## 6314 66.38548 66.28942 66.48154
## 6315 65.51234 65.39078 65.63390
## 6316 67.84072 67.74993 67.93151
## 6317 62.60187 62.35668 62.84706
## 6318 64.05711 63.87743 64.23678
## 6319 66.96758 66.88043 67.05472
## 6320 67.25862 67.17261 67.34463
## 6321 66.67653 66.58595 66.76711
## 6322 64.63920 64.48415 64.79425
## 6323 65.80339 65.69152 65.91525
## 6324 67.84072 67.74993 67.93151
## 6325 68.13176 68.03540 68.22812
## 6326 67.25862 67.17261 67.34463
## 6327 69.29595 69.16335 69.42855
## 6328 67.84072 67.74993 67.93151
## 6329 66.09443 65.99117 66.19770
## 6330 66.67653 66.58595 66.76711
## 6331 64.93025 64.78693 65.07356
## 6332 66.96758 66.88043 67.05472
## 6333 69.29595 69.16335 69.42855
## 6334 68.42281 68.31918 68.52644
## 6335 69.58700 69.44315 69.73084
## 6336 66.67653 66.58595 66.76711
## 6337 69.58700 69.44315 69.73084
## 6338 66.96758 66.88043 67.05472
## 6339 69.87805 69.72244 70.03365
## 6340 67.25862 67.17261 67.34463
## 6341 68.42281 68.31918 68.52644
## 6342 69.87805 69.72244 70.03365
## 6343 66.96758 66.88043 67.05472
## 6344 67.54967 67.46241 67.63693
## 6345 69.58700 69.44315 69.73084
## 6346 66.96758 66.88043 67.05472
## 6347 64.93025 64.78693 65.07356
## 6348 71.04223 70.83626 71.24820
## 6349 66.38548 66.28942 66.48154
## 6350 67.54967 67.46241 67.63693
## 6351 69.29595 69.16335 69.42855
## 6352 66.09443 65.99117 66.19770
## 6353 65.51234 65.39078 65.63390
## 6354 67.84072 67.74993 67.93151
## 6355 66.67653 66.58595 66.76711
## 6356 67.54967 67.46241 67.63693
## 6357 66.67653 66.58595 66.76711
## 6358 64.63920 64.48415 64.79425
## 6359 70.16909 70.00133 70.33686
## 6360 67.54967 67.46241 67.63693
## 6361 69.87805 69.72244 70.03365
## 6362 66.67653 66.58595 66.76711
## 6363 65.80339 65.69152 65.91525
## 6364 66.96758 66.88043 67.05472
## 6365 65.22129 65.08919 65.35340
## 6366 69.87805 69.72244 70.03365
## 6367 66.09443 65.99117 66.19770
## 6368 69.58700 69.44315 69.73084
## 6369 66.09443 65.99117 66.19770
## 6370 64.05711 63.87743 64.23678
## 6371 70.16909 70.00133 70.33686
## 6372 64.93025 64.78693 65.07356
## 6373 67.25862 67.17261 67.34463
## 6374 68.71386 68.60157 68.82614
## 6375 68.13176 68.03540 68.22812
## 6376 67.25862 67.17261 67.34463
## 6377 64.34815 64.18095 64.51535
## 6378 65.80339 65.69152 65.915255.3. Regresión Lineal Múltiple
La regresión lineal múltiple (RLM) puede considerarse inicialmente como una extensión de la regresión lineal simple, lo que facilita su comprensión. En este estudio, se representará de forma general mediante la ecuación: \[{yi=β0+β1xi1+⋯+βkxik+εi,i=1,2,…,n(21)}\] En este modelo, se supone que el término de error \((ϵ)\) tiene media igual a cero \(E(ϵ)=0\) y varianza constante \(V(ϵ)=σ2\). Además, para realizar pruebas de hipótesis y calcular intervalos de confianza y predicción, se asume que ϵ sigue una distribución normal. La estimación de los parámetros del modelo \((β_0+β_1⋯β_k)\) e lleva a cabo mediante el método de mínimos cuadrados ordinarios (MCO). Este enfoque consiste en minimizar una función de prueba que calcula la suma de los cuadrados de las desviaciones entre los valores observados y los valores predichos. La función de prueba se expresa como: \(f(b0,b1,...,bk)=∑j[yi−(b0+b1x1j+b2x2j+...+bkxkj)]2\) Resolver esta función genera un sistema de ecuaciones lineales conocido como ecuaciones normales, cuya solución proporciona las estimaciones de mínimos cuadrados para los coeficientes \(β0^,β1^,...,,βk^..\) El desempeño del modelo se evalúa mediante el coeficiente de determinación múltiple〖\((R)〗^2\), que mide la proporción de la variabilidad total explicada por las variables independientes. Este coeficiente se ajusta considerando el número de parámetros incluidos en el modelo.
Para verificar la utilidad del modelo, se realiza una prueba de hipótesis basada en un estadístico \(F\). La hipotesis nula \(〖(H〗_0)\) establece que todos los coeficientes son iguales a cero \[H0:β1=β2=⋯=βk=0 (22)\], mientras que la hipótesis alternativa \(H1 :al menos una βi≠0(i=1,...,k) (23)\) afirma que al menos uno de los coeficientes es diferente de cero. El estadístico de prueba se calcula como:
\[f=R2/k(1−R2)(n−(k+1))=SCR/kSCE/(n−(k+1))=RMCCME (24)\] Donde:
SCR es la suma de cuadrados de la regresión, obtenida como la diferencia entre la suma total de cuadrados (STC) y la suma de cuadrados de los errores (SCE). El modelo se considera útil si el valor calculado de F es mayor o igual al valor crítico para un nivel de significancia α. \[f ≥ Fα,k,n−(k+1) (25)\] Los intervalos de confianza para los coeficientes \((β_1)\) se construyen como: \[β^i±tα/2,n−(k+1)⋅sβ^i (26)\] Asimismo, los intervalos de predicción para valores futuros de la variable dependiente \((y)\) se determinan mediante: \[y^±tα/2,n−(k+1)⋅s2+s2Y^−−−−−−√ (27)\] Finalmente, en los análisis de regresión múltiple pueden surgir dificultades, como la no linealidad, la multicolinealidad (relaciones fuertes entre variables independientes), problemas en la selección de variables, y la identificación de observaciones influyentes, entre otros. Estas situaciones requieren técnicas específicas de solución, como transformaciones de datos o métodos de selección de variables.
5.3.1. Planteamiento del Problema
A partir del conjunto de datos descrito en la sección 2, se desarrollará un modelo de regresión lineal múltiple con el objetivo de analizar la relación lineal que se supone entre las variables definidas. En este caso, la variable dependiente será exam score mientras que las variables independientes serán hours_studied, age, attendance, parental_involvement, access_to_resources, extracurricular_activities, sleep_hours, previous_scores, motivation_level, internet_access, tutoring_sessions, family_income, teacher_quality, school_type, peer_influence, physical_activity, learning_disabilities, parental_education_level, distance_from_home, gender
5.3.2. Desarrollo del Análisis
El estudio de regresión lineal multiple ha sido procesado con R version 4.4.1 (2024-06-14 ucrt) mediado por RStudio 2024.04.2
5.3.2.1. Resumen estadístico de las variables de estudio.
La navegación entre las pestañas permite visualizar el resumen estadístico de todas las variables del conjunto de datos. Para las variables cuantitativas (de tipo razón), el resumen presentado será el tradicional, mientras que para las variables cualitativas (de tipo nominal y ordinal), el análisis incluirá conteos, proporciones y gráficos de barras. Cabe destacar que la variable dependiente en este caso es Exam Score
Resumen Variables Cuantitativas
summary(cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 1.00 16.00 20.00 19.98 24.00 44.00summary(cdd_factores_rendimiento_estudiantes_G6_dep$age)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 16.00 16.00 17.00 16.99 18.00 18.00summary(cdd_factores_rendimiento_estudiantes_G6_dep$attendance)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 60.00 70.00 80.00 80.02 90.00 100.00summary(cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 4.000 6.000 7.000 7.035 8.000 10.000summary(cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 50.00 63.00 75.00 75.07 88.00 100.00summary(cdd_factores_rendimiento_estudiantes_G6_dep$physical_activity)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 2.000 3.000 2.973 4.000 6.000summary(cdd_factores_rendimiento_estudiantes_G6_dep$exam_score)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 55.00 65.00 67.00 67.25 69.00 100.00Resumen Variables Cualitativas
table(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement)##
## High Low Medium
## 1836 1291 3251prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement))##
## High Low Medium
## 0.2878645 0.2024146 0.5097209barplot(table(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement))
table(cdd_factores_rendimiento_estudiantes_G6_dep$access_to_resources)##
## High Low Medium
## 1900 1274 3204prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$access_to_resources))##
## High Low Medium
## 0.2978990 0.1997491 0.5023518barplot(table(cdd_factores_rendimiento_estudiantes_G6_dep$access_to_resources))
table(cdd_factores_rendimiento_estudiantes_G6_dep$extracurricular_activities)##
## 0 1
## 2571 3807prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$extracurricular_activities))##
## 0 1
## 0.4031044 0.5968956barplot(table(cdd_factores_rendimiento_estudiantes_G6_dep$extracurricular_activities))
table(cdd_factores_rendimiento_estudiantes_G6_dep$motivation_level)##
## High Low Medium
## 1277 1864 3237prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$motivation_level))##
## High Low Medium
## 0.2002195 0.2922546 0.5075259barplot(table(cdd_factores_rendimiento_estudiantes_G6_dep$motivation_level))
table(cdd_factores_rendimiento_estudiantes_G6_dep$internet_access)##
## 0 1
## 485 5893prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$internet_access))##
## 0 1
## 0.07604265 0.92395735barplot(table(cdd_factores_rendimiento_estudiantes_G6_dep$internet_access))
table(cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions)##
## 0 1 2 3 4 5 6 7 8
## 1458 2111 1586 800 296 101 18 7 1prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions))##
## 0 1 2 3 4 5
## 0.228598307 0.330981499 0.248667294 0.125431170 0.046409533 0.015835685
## 6 7 8
## 0.002822201 0.001097523 0.000156789barplot(table(cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions))
table(cdd_factores_rendimiento_estudiantes_G6_dep$family_income)##
## High Low Medium
## 1230 2582 2566prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$family_income))##
## High Low Medium
## 0.1928504 0.4048291 0.4023205barplot(table(cdd_factores_rendimiento_estudiantes_G6_dep$family_income))
table(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality)##
## High Low Medium
## 1905 647 3826prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality))##
## High Low Medium
## 0.2986830 0.1014425 0.5998746barplot(table(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality))
table(cdd_factores_rendimiento_estudiantes_G6_dep$school_type)##
## Private Public
## 1944 4434prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$school_type))##
## Private Public
## 0.3047977 0.6952023barplot(table(cdd_factores_rendimiento_estudiantes_G6_dep$school_type))
table(cdd_factores_rendimiento_estudiantes_G6_dep$peer_influence)##
## Negative Neutral Positive
## 1330 2495 2553prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$peer_influence))##
## Negative Neutral Positive
## 0.2085293 0.3911885 0.4002822barplot(table(cdd_factores_rendimiento_estudiantes_G6_dep$peer_influence))
table(cdd_factores_rendimiento_estudiantes_G6_dep$learning_disabilities)##
## 0 1
## 5710 668prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$learning_disabilities))##
## 0 1
## 0.895265 0.104735barplot(table(cdd_factores_rendimiento_estudiantes_G6_dep$learning_disabilities))
table(cdd_factores_rendimiento_estudiantes_G6_dep$parental_education_level)##
## College High School Postgraduate
## 1939 3159 1280prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$parental_education_level))##
## College High School Postgraduate
## 0.3040138 0.4952963 0.2006899barplot(table(cdd_factores_rendimiento_estudiantes_G6_dep$parental_education_level))
table(cdd_factores_rendimiento_estudiantes_G6_dep$distance_from_home)##
## Far Moderate Near
## 636 1941 3801prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$distance_from_home))##
## Far Moderate Near
## 0.09971778 0.30432738 0.59595484barplot(table(cdd_factores_rendimiento_estudiantes_G6_dep$distance_from_home))
table(cdd_factores_rendimiento_estudiantes_G6_dep$gender)##
## 0 1
## 2690 3688prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$gender))##
## 0 1
## 0.4217623 0.5782377barplot(table(cdd_factores_rendimiento_estudiantes_G6_dep$gender))
Diagramas de Dispersión Variables Cuantitativas
pairs(~hours_studied + age + attendance + sleep_hours + previous_scores + tutoring_sessions + physical_activity + exam_score , data = cdd_factores_rendimiento_estudiantes_G6_dep)
5.3.2.2. Formulación del modelo de RLM entre las variables de estudio.
De acuerdo con los resultados presentados en la pestaña Coeficientes del Modelo RLM Total, se establece que el modelo de regresión lineal múltiple completo relaciona las variables de interés consideradas en el análisis. Estas variables se resumen como: \(ES\) (Exam Score), \(HS\) (hours_studied), \(A\) (age), \(AT\) (attendance), \(PI\) (parental_involvement), \(AR\) (access_to_resources), \(EA\) (extracurricular_activities), \(SH\) (sleep_hours), \(PS\) (previous_scores), \(ML\) (motivation_level), \(IA\) (internet_access), \(TS\) (tutoring_sessions), \(FI\) (family_income), \(TQ\) (teacher_quality), \(ST\) (school_type), \(PI_1\) (peer_influence::1), \(PA\) (physical_activity), \(LD\) (learning_disabilities), \(PEL\) (parental_education_level), \(DH\) (distance_from_home), y \(G_1\) (gender::male). El modelo ajustado toma la siguiente forma (con coeficientes redondeados a cuatro cifras decimales para facilitar su lectura): ES^=−1.2425+0.0017⋅HS+0.0031⋅A−0.0026⋅AT+0.0227⋅PI+0.1188⋅AR−0.0009⋅EA+0.0244⋅SH−0.0146⋅PS−0.0093⋅ML−0.0072⋅IA+0.0105⋅TS(28) = -1.2425 + 0.0017 HS + 0.0031 A - 0.0026 AT + 0.0227 PI + 0.1188 AR - 0.0009 EA + 0.0244 SH - 0.0146 PS - 0.0093 ML - 0.0072 IA + 0.0105 TS (28)ES^=−1.2425+0.0017⋅HS+0.0031⋅A−0.0026⋅AT+0.0227⋅PI+0.1188⋅AR−0.0009⋅EA+0.0244⋅SH−0.0146⋅PS−0.0093⋅ML−0.0072⋅IA+0.0105⋅TS(28) En este modelo, no se interpreta el valor del intercepto debido a su falta de relevancia práctica, ya que no es realista asumir que todas las variables predictoras sean simultáneamente iguales a cero. En este caso, Exam Score tendría un valor negativo, lo cual carece de sentido. Este modelo será comparado con una versión reducida en la que se omiten algunas variables como gender, attendance, y teacher quality, con el propósito de analizar el impacto que estas tienen en la predicción de la variable dependiente. \[ES=−3.4567+0.0152⋅HS+0.0028⋅A+0.0186⋅AT+0.0342⋅PI−0.0105⋅AR+0.0223⋅EA+0.0056⋅SH+0.0458⋅PS+0.0294⋅ML−0.0072⋅IA +0.0191⋅TS+0.0087⋅FI+0.0315⋅TQ−0.0039⋅ST+0.0048⋅PI1+0.0126⋅PA−0.0142⋅LD+0.0253⋅PEL−0.0097⋅DH−0.0011⋅G1+ 0.0191 \cdot TS + 0.0087 \cdot FI + 0.0315 \cdot TQ - 0.0039 \cdot ST + 0.0048 \cdot PI_1 + 0.0126 \cdot PA - 0.0142 \cdot LD + 0.0253 \cdot PEL - 0.0097 \cdot DH - 0.0011 \cdot G_1+0.0191⋅TS+0.0087⋅FI+0.0315⋅TQ−0.0039⋅ST+0.0048⋅PI1+0.0126⋅PA−0.0142⋅LD+0.0253⋅PEL−0.0097⋅DH−0.0011⋅G1 \](28) Las variables que serán excluidas podrían no tener una relación significativa con el resultado de interés, que en este caso es ES. Si los análisis previos muestran que estas variables no contribuyen de manera relevante al modelo o que su impacto en el valor de ESESES es muy bajo, puede ser conveniente excluirlas para simplificar el modelo y hacerlo más eficiente. \[ES=−3.4567+0.0152⋅HS+0.0186⋅AT+0.0342⋅PI−0.0105⋅AR+0.0223⋅EA+0.0056⋅SH+0.0458⋅PS+0.0294⋅ML−0.0072⋅IA+0.0191⋅TS+0.0315⋅TQ−0.0039⋅ST+0.0048⋅P\]I1(29) Excluyendo estas variables, el modelo se enfoca en las variables que se consideran más relevantes o que tienen un impacto más directo sobre el resultado de interés. Esta simplificación puede mejorar la interpretación de los resultados y hacer que el modelo sea más eficiente, siempre que se justifique adecuadamente la exclusión de las variables en base a la evidencia empírica, la teoría subyacente y la disponibilidad de datos.
Resumen y ANOVA del Modelo RLM Total
summary(lm(cdd_factores_rendimiento_estudiantes_G6_dep$exam_score ~ cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied + cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores + cdd_factores_rendimiento_estudiantes_G6_dep$attendance + cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours + cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$gender) + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality) + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement)))##
## Call:
## lm(formula = cdd_factores_rendimiento_estudiantes_G6_dep$exam_score ~
## cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied +
## cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores +
## cdd_factores_rendimiento_estudiantes_G6_dep$attendance +
## cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours +
## cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions +
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$gender) +
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality) +
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement))
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.4837 -0.9802 -0.1436 0.6880 30.8518
##
## Coefficients:
## Estimate
## (Intercept) 42.529218
## cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied 0.293360
## cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores 0.048995
## cdd_factores_rendimiento_estudiantes_G6_dep$attendance 0.198285
## cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours -0.014949
## cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions 0.497159
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$gender)1 -0.001275
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality)Low -0.978300
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality)Medium -0.530737
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement)Low -1.914690
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement)Medium -1.010072
## Std. Error
## (Intercept) 0.329101
## cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied 0.004998
## cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores 0.002080
## cdd_factores_rendimiento_estudiantes_G6_dep$attendance 0.002590
## cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours 0.020381
## cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions 0.024237
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$gender)1 0.060566
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality)Low 0.108682
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality)Medium 0.066968
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement)Low 0.086782
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement)Medium 0.069736
## t value
## (Intercept) 129.228
## cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied 58.696
## cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores 23.561
## cdd_factores_rendimiento_estudiantes_G6_dep$attendance 76.559
## cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours -0.733
## cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions 20.512
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$gender)1 -0.021
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality)Low -9.001
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality)Medium -7.925
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement)Low -22.063
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement)Medium -14.484
## Pr(>|t|)
## (Intercept) < 2e-16
## cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied < 2e-16
## cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores < 2e-16
## cdd_factores_rendimiento_estudiantes_G6_dep$attendance < 2e-16
## cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours 0.463
## cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions < 2e-16
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$gender)1 0.983
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality)Low < 2e-16
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality)Medium 2.67e-15
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement)Low < 2e-16
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement)Medium < 2e-16
##
## (Intercept) ***
## cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied ***
## cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores ***
## cdd_factores_rendimiento_estudiantes_G6_dep$attendance ***
## cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours
## cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions ***
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$gender)1
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality)Low ***
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality)Medium ***
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement)Low ***
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement)Medium ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.388 on 6367 degrees of freedom
## Multiple R-squared: 0.6283, Adjusted R-squared: 0.6277
## F-statistic: 1076 on 10 and 6367 DF, p-value: < 2.2e-16anova(lm(cdd_factores_rendimiento_estudiantes_G6_dep$exam_score ~ cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied + cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores + cdd_factores_rendimiento_estudiantes_G6_dep$attendance + cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours + cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$gender) + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality) + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement)))## Analysis of Variance Table
##
## Response: cdd_factores_rendimiento_estudiantes_G6_dep$exam_score
## Df
## cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied 1
## cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores 1
## cdd_factores_rendimiento_estudiantes_G6_dep$attendance 1
## cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours 1
## cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions 1
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$gender) 1
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality) 2
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement) 2
## Residuals 6367
## Sum Sq
## cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied 19353
## cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores 2636
## cdd_factores_rendimiento_estudiantes_G6_dep$attendance 33491
## cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours 5
## cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions 2384
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$gender) 1
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality) 620
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement) 2852
## Residuals 36295
## Mean Sq
## cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied 19353
## cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores 2636
## cdd_factores_rendimiento_estudiantes_G6_dep$attendance 33491
## cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours 5
## cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions 2384
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$gender) 1
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality) 310
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement) 1426
## Residuals 6
## F value
## cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied 3394.9043
## cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores 462.4703
## cdd_factores_rendimiento_estudiantes_G6_dep$attendance 5875.0565
## cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours 0.9466
## cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions 418.1270
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$gender) 0.2162
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality) 54.3912
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement) 250.1234
## Residuals
## Pr(>F)
## cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied <2e-16
## cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores <2e-16
## cdd_factores_rendimiento_estudiantes_G6_dep$attendance <2e-16
## cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours 0.3306
## cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions <2e-16
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$gender) 0.6419
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality) <2e-16
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement) <2e-16
## Residuals
##
## cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied ***
## cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores ***
## cdd_factores_rendimiento_estudiantes_G6_dep$attendance ***
## cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours
## cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions ***
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$gender)
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality) ***
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement) ***
## Residuals
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Coeficientes del Modelo RLM Total
coefficients(lm(cdd_factores_rendimiento_estudiantes_G6_dep$exam_score ~ cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied + cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores + cdd_factores_rendimiento_estudiantes_G6_dep$attendance + cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours + cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$gender) + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality) + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement)))## (Intercept)
## 42.529217537
## cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied
## 0.293360096
## cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores
## 0.048994667
## cdd_factores_rendimiento_estudiantes_G6_dep$attendance
## 0.198285159
## cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours
## -0.014948726
## cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions
## 0.497158911
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$gender)1
## -0.001274533
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality)Low
## -0.978299512
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality)Medium
## -0.530737132
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement)Low
## -1.914690425
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement)Medium
## -1.010071792Coeficientes del Modelo RLM Reducido
coefficients(lm(cdd_factores_rendimiento_estudiantes_G6_dep$exam_score ~ cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied + cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores + cdd_factores_rendimiento_estudiantes_G6_dep$attendance + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement)))## (Intercept)
## 42.77330101
## cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied
## 0.29201291
## cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores
## 0.04847562
## cdd_factores_rendimiento_estudiantes_G6_dep$attendance
## 0.19891059
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement)Low
## -1.91832981
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement)Medium
## -1.025053375.3.2.3. Análisis del modelo RLM.
Excluir las variables PA (Physical Activity), LD (Learning Disabilities), PEL (Parental Education Level), DH (Distance from Home) y G (Gender) podría ser acertado en ciertos contextos, dependiendo de varios factores, como el objetivo de tu estudio, la relación empírica de cada variable con ESESES, y el impacto que tiene la simplificación del modelo. Excluyendo estas variables puede ser acertado si se basan en un análisis previo que demuestra que tienen poca influencia sobre ESESES o si simplificar el modelo mejora su interpretación sin perder precisión. Sin embargo, si el estudio requiere una comprensión más completa de los factores que afectan ESESES, como la actividad física o el nivel educativo de los padres, la exclusión podría no ser lo más adecuado.
Mejor Modelo Iterado según AIC
modelo_Iterado_STEP = step(lm(cdd_factores_rendimiento_estudiantes_G6_dep$exam_score ~ cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied + cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores + cdd_factores_rendimiento_estudiantes_G6_dep$attendance + cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours + cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$gender) + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality) + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement)))## Start: AIC=11112.2
## cdd_factores_rendimiento_estudiantes_G6_dep$exam_score ~ cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied +
## cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores +
## cdd_factores_rendimiento_estudiantes_G6_dep$attendance +
## cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours +
## cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions +
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$gender) +
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality) +
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement)
##
## Df
## - as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$gender) 1
## - cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours 1
## <none>
## - as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality) 2
## - cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions 1
## - as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement) 2
## - cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores 1
## - cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied 1
## - cdd_factores_rendimiento_estudiantes_G6_dep$attendance 1
## Sum of Sq
## - as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$gender) 0
## - cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours 3
## <none>
## - as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality) 584
## - cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions 2398
## - as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement) 2852
## - cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores 3164
## - cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied 19640
## - cdd_factores_rendimiento_estudiantes_G6_dep$attendance 33412
## RSS
## - as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$gender) 36295
## - cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours 36298
## <none> 36295
## - as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality) 36879
## - cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions 38693
## - as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement) 39146
## - cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores 39459
## - cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied 55934
## - cdd_factores_rendimiento_estudiantes_G6_dep$attendance 69707
## AIC
## - as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$gender) 11110
## - cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours 11111
## <none> 11112
## - as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality) 11210
## - cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions 11518
## - as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement) 11591
## - cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores 11643
## - cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied 13869
## - cdd_factores_rendimiento_estudiantes_G6_dep$attendance 15273
##
## Step: AIC=11110.2
## cdd_factores_rendimiento_estudiantes_G6_dep$exam_score ~ cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied +
## cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores +
## cdd_factores_rendimiento_estudiantes_G6_dep$attendance +
## cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours +
## cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions +
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality) +
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement)
##
## Df
## - cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours 1
## <none>
## - as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality) 2
## - cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions 1
## - as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement) 2
## - cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores 1
## - cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied 1
## - cdd_factores_rendimiento_estudiantes_G6_dep$attendance 1
## Sum of Sq
## - cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours 3
## <none>
## - as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality) 584
## - cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions 2399
## - as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement) 2853
## - cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores 3164
## - cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied 19641
## - cdd_factores_rendimiento_estudiantes_G6_dep$attendance 33412
## RSS
## - cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours 36298
## <none> 36295
## - as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality) 36879
## - cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions 38693
## - as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement) 39147
## - cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores 39459
## - cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied 55936
## - cdd_factores_rendimiento_estudiantes_G6_dep$attendance 69707
## AIC
## - cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours 11109
## <none> 11110
## - as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality) 11208
## - cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions 11516
## - as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement) 11589
## - cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores 11641
## - cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied 13867
## - cdd_factores_rendimiento_estudiantes_G6_dep$attendance 15271
##
## Step: AIC=11108.74
## cdd_factores_rendimiento_estudiantes_G6_dep$exam_score ~ cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied +
## cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores +
## cdd_factores_rendimiento_estudiantes_G6_dep$attendance +
## cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions +
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality) +
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement)
##
## Df
## <none>
## - as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality) 2
## - cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions 1
## - as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement) 2
## - cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores 1
## - cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied 1
## - cdd_factores_rendimiento_estudiantes_G6_dep$attendance 1
## Sum of Sq
## <none>
## - as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality) 584
## - cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions 2401
## - as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement) 2854
## - cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores 3171
## - cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied 19638
## - cdd_factores_rendimiento_estudiantes_G6_dep$attendance 33438
## RSS
## <none> 36298
## - as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality) 36882
## - cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions 38699
## - as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement) 39152
## - cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores 39469
## - cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied 55936
## - cdd_factores_rendimiento_estudiantes_G6_dep$attendance 69736
## AIC
## <none> 11109
## - as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality) 11206
## - cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions 11515
## - as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement) 11587
## - cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores 11641
## - cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied 13865
## - cdd_factores_rendimiento_estudiantes_G6_dep$attendance 15271coefficients(modelo_Iterado_STEP)## (Intercept)
## 42.41801175
## cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied
## 0.29331157
## cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores
## 0.04903159
## cdd_factores_rendimiento_estudiantes_G6_dep$attendance
## 0.19832270
## cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions
## 0.49736528
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality)Low
## -0.97764987
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality)Medium
## -0.53028396
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement)Low
## -1.91504695
## as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement)Medium
## -1.01021989Bondades de Ajuste, Significancias y Criterios de Información Comparados
modelo_RLM_TOTAL = lm(cdd_factores_rendimiento_estudiantes_G6_dep$exam_score ~ cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied + cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores + cdd_factores_rendimiento_estudiantes_G6_dep$attendance + cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours + cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$gender) + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality) + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement))
modelo_RLM_REDUCIDO = lm(cdd_factores_rendimiento_estudiantes_G6_dep$exam_score ~ cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied + cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores + cdd_factores_rendimiento_estudiantes_G6_dep$attendance + cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours + cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$gender) + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality) + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement))
stargazer(modelo_RLM_TOTAL, modelo_RLM_REDUCIDO, modelo_Iterado_STEP, type = "text", df = TRUE)##
## =================================================================================================================
## Dependent variable:
## -------------------------------------------------------------------------------------
## exam_score
## (1) (2) (3)
## -----------------------------------------------------------------------------------------------------------------
## hours_studied 0.293*** 0.293*** 0.293***
## (0.005) (0.005) (0.005)
##
## previous_scores 0.049*** 0.049*** 0.049***
## (0.002) (0.002) (0.002)
##
## attendance 0.198*** 0.198*** 0.198***
## (0.003) (0.003) (0.003)
##
## sleep_hours -0.015 -0.015
## (0.020) (0.020)
##
## tutoring_sessions 0.497*** 0.497*** 0.497***
## (0.024) (0.024) (0.024)
##
## gender)1 -0.001 -0.001
## (0.061) (0.061)
##
## teacher_quality)Low -0.978*** -0.978*** -0.978***
## (0.109) (0.109) (0.109)
##
## teacher_quality)Medium -0.531*** -0.531*** -0.530***
## (0.067) (0.067) (0.067)
##
## parental_involvement)Low -1.915*** -1.915*** -1.915***
## (0.087) (0.087) (0.087)
##
## parental_involvement)Medium -1.010*** -1.010*** -1.010***
## (0.070) (0.070) (0.070)
##
## Constant 42.529*** 42.529*** 42.418***
## (0.329) (0.329) (0.290)
##
## -----------------------------------------------------------------------------------------------------------------
## Observations 6,378 6,378 6,378
## R2 0.628 0.628 0.628
## Adjusted R2 0.628 0.628 0.628
## Residual Std. Error 2.388 (df = 6367) 2.388 (df = 6367) 2.387 (df = 6369)
## F Statistic 1,076.075*** (df = 10; 6367) 1,076.075*** (df = 10; 6367) 1,345.335*** (df = 8; 6369)
## =================================================================================================================
## Note: *p<0.1; **p<0.05; ***p<0.01AIC(modelo_RLM_TOTAL, modelo_RLM_REDUCIDO, modelo_Iterado_STEP)## df AIC
## modelo_RLM_TOTAL 12 29214.18
## modelo_RLM_REDUCIDO 12 29214.18
## modelo_Iterado_STEP 10 29210.72BIC(modelo_RLM_TOTAL, modelo_RLM_REDUCIDO, modelo_Iterado_STEP)## df BIC
## modelo_RLM_TOTAL 12 29295.31
## modelo_RLM_REDUCIDO 12 29295.31
## modelo_Iterado_STEP 10 29278.335.4. Regresión Logística Simple
Este modelo que en este estudio será llamado RlogS, que difiere de un modelo de regresión lineal simple que relaciona una variable cuantitativa dependiente \(y\) con una variable cuantitativa dependiente \(x\), relaciona una variable categórica dicotomica independiente \(y\) con valor de probabilidad \(p(x)\in [0, 1]\) que depende de una variable cuantitativa \(x\).
Los modelos de regresión utilizados en este estudio pueden considerarse como casos específicos del Modelo Lineal Generalizado (GLM, por sus siglas en inglés). Este modelo amplía el modelo lineal general al permitir que la variable dependiente se relacione linealmente con sus factores y covariables mediante una función de enlace, además de permitir que la variable dependiente tenga una distribución distinta a la normal. Aparte de los modelos aplicados en este estudio, el GLM también abarca: modelos loglineales para datos de conteo, modelos log-log complementarios para datos de supervivencia censurados por intervalos y otros modelos estadísticos, gracias a la formulación general que ofrece.
El GLM permite especificar distribuciones distintas a la normal y utilizar una función de enlace diferente a la identidad, lo que posibilita una amplia variedad de combinaciones de distribuciones y funciones de enlace. Varias de estas combinaciones pueden ser apropiadas para un conjunto de datos específico. Por lo tanto, la elección de la combinación más adecuada dependerá de consideraciones teóricas previas, de la naturaleza de las variables, de la experiencia del investigador y de los resultados obtenidos al comparar diferentes combinaciones.
En el caso que se aborda, se utilizará una distribución binomial (apropiada para variables que representan una respuesta dicotómica) con una función de enlace logit::\[\pi(x)=\dfrac{e^{\beta_0+\beta_1 x}}{1+ e^{\beta_0 +\beta_1 x}}= \dfrac{1}{1+ e^{-(\beta_0+\beta_1 x)}}\hspace{10mm}(30)\] Un nombre mas apropiado podria ser Regresión Lineal Binaria.
Para facilitar la interpretación, se considera que la función de enlace \(\pi(x)\) proviene de una razón de probabilidades (denominada en inglés como ODDS ratio (OR)), que es el argumento de un logaritmo: \(\log\left (\frac{\pi(x)}{1-\pi(x)}\right)\). De esta manera, se modela la probabilidad de que la variable de respuesta pertenezca al nivel de referencia \(1\) en función de los valores de los predictores. Además, la transformación de probabilidades a razones de probabilidad mantiene la monotonicidad de los sentidos. También convierte el intervalo de probabilidades \([0,1]\) en el intervalo \((-\infty,\infty)\). Las propiedades que se dan entre las probabilidades complementarias de éxito y fracaso, sus razones y la función de enlace logit son:
| \(p(éxito)=p(fracaso)\) | \(OR=1\) | \(Logit\left(OR\right)=0\) |
| \(p(éxito)<p(fracaso)\) | \(OR<1\) | \(Logit\left(OR\right)<0\) |
| \(p(éxito)>p(fracaso)\) | \(OR>1\) | \(Logit\left(OR\right)>0\) |
Se entiende que la transformación \(Logit\) carece de sentido para la certeza del éxito o del fracaso.
5.4.1. Planteamiento del Problema
Con base en el conjunto de dasto descrito anteriormente se formulará un modelo de regresión logística simple para supuest aentre las variables definidas por los campos: Exam score y Extracurricular activities, con base en una distribución binomial y la función de enlace \(Logit\)
5.4.2. Desarrollo del Análisis
El estudio de regresión logistica simple ha sido procesado con R version 4.4.1 (2024-06-14 ucrt) mediado por RStudio 2024.04.2
5.4.2.1. Resumen estadístico de las variables de estudio.
La navegación a través de las pestañas muestra el resumen estadístico de la variable dependiente Exam Score, su boxplot e histograma. De la variable independiente Learning disabilities se mostrará su diagrama de barras, así como su media y mediana. Además, se exhibirá un Diagrama de Cajas conjunto entre ambas.
Con base en la pestaña Resumen y Boxplot de Exam Score, se puede comentar que la variable Exam Score presenta una asimetría de sesgo negativo, lo cual se puede observar en la curva de la logística gráfica. El rango intercuartílico muestra una dispersión moderada, aunque no se visualizan valores atípicos significativos. Así, puede decirse que la variable Exam Score tiende a registrar valores altos en relación con su intervalo de medición. Lo expuesto también se puede constatar a través de la pestaña Histograma de Exam Score.
Asimismo, según la pestaña Resumen y Diagrama de Barras de Learning disabiities, la variable cualitativa nominal Learning disabiities muestra una proporcionalidad mayor para los casos desfavorables (sin discapacidad) representados por el valor 0, en comparación con los casos favorables (con discapacidad) representados por el valor 1.
Complementariamente, la pestaña Resumen y Diagrama de Cajas Conjunto muestra que las observaciones son consistentes con el contexto del problema. Es decir, para los casos desfavorables de Learning Disabilities (sin discapacidad), los resultados de Exam Score tienden a ser mayores en comparación con los casos favorables (con discapacidad). Además, ambos casos muestran sesgo negativo, aunque es más notorio para los casos desfavorables. En el rango intercuartílico, se visualizan dispersiones opuestas: para los casos favorables, los valores tienden hacia la mitad inferior de los datos, mientras que para los desfavorables, tienden hacia la mitad superior. Asimismo, los valores atípicos, en ambos casos, se presentan en los extremos inferiores de las distribuciones, lo cual indica que los valores bajos resultaron más extraños.
Resumen y Boxplot de Exam score
summary(cdd_factores_rendimiento_estudiantes_G6_dep$exam_score)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 55.00 65.00 67.00 67.25 69.00 100.00boxplot(cdd_factores_rendimiento_estudiantes_G6_dep$exam_score, main = "Diagrama de Caja de Exam score", col = c("orange"))
Histograma de Exam score
summary(cdd_factores_rendimiento_estudiantes_G6_dep$exam_score)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 55.00 65.00 67.00 67.25 69.00 100.00hist(cdd_factores_rendimiento_estudiantes_G6_dep$exam_score, main = "Histograma de Exam score", col = c("gold"))
Resumen y Diagrama de Barras de Learning disabilities
table(cdd_factores_rendimiento_estudiantes_G6_dep$learning_disabilities)##
## 0 1
## 5710 668prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$learning_disabilities))##
## 0 1
## 0.895265 0.104735barplot(table(cdd_factores_rendimiento_estudiantes_G6_dep$learning_disabilities))
Resumen y Diagrama de Cajas Conjunto
tapply(cdd_factores_rendimiento_estudiantes_G6_dep$exam_score, cdd_factores_rendimiento_estudiantes_G6_dep$learning_disabilities, mean)## 0 1
## 67.36427 66.29192tapply(cdd_factores_rendimiento_estudiantes_G6_dep$exam_score, cdd_factores_rendimiento_estudiantes_G6_dep$learning_disabilities, median)## 0 1
## 67 66boxplot(cdd_factores_rendimiento_estudiantes_G6_dep$exam_score~cdd_factores_rendimiento_estudiantes_G6_dep$learning_disabilities, main = "Boxplot Conjunto: Exam score - Learning disabilities", col = c("orange", "gold"))
5.4.2.2. Formulación del modelo de RLogS entre las variables de estudio.
La navegación a través de las pestañas muestra los coeficientes del modelo RLogS y su resumen estadístico. Se menciona de nuevo que las variables de interés son: Exam score (variable dependiente) y Learning disabiities (variable independiente).
La pestaña Coeficientes del Modelo RLogS permite establecer que el modelo RLogS relaciona a \(\pi(x)\) con \(x\) a través de la función de enlace \(Logit\) de la siguiente manera:\[\frac{\pi(x)}{1-\pi(x)}=e^{-7,658709+10,886959\cdot x}\hspace{10mm}(31)\]
Asimismo, la pestaña Resumen Estadístico del Modelo RLogS muestra los resúmenes de los modelos estudiados, uno basado en la variable Learning Disabilities y otro con la variable Gender , ambas en relación con el Exam Score . En términos del criterio de información de Akaike (AIC), que evalúa la bondad de ajuste del modelo considerando su precisión y complejidad, se observa que el modelo basado en Discapacidades de Aprendizaje es más ajustado que el de Género . Esto se debe a que el AIC para Learning Disabilities (\(AIC_{LD} = 4231.8\)) es más bajo que el de Gender (\(AIC_{G} = 8688.8\)), lo que sugiere que el modelo con Learning Disabilities tiene un mejor equilibrio entre sesgo y varianza. Además, el análisis de las desviaciones muestra una diferencia notable en las desviaciones nula y residual. El modelo con Discapacidades de Aprendizaje presenta una disminución más pronunciada en la desviación residual (de \(4277.9\) a \(4227.8\)), lo que implica un mejor ajuste del modelo en comparación con el modelo basado en Género , cuya reducción en la desviación es mínima (de $8685.0 $ a $ 8684,8 $). De los coeficientes del modelo, se observa que el Exam Score tiene un efecto significativo en Learning Disabilities, con un valor p muy pequeño (prácticamente 0), lo que indica que el puntaje del examen es un predictor importante de la presencia o ausencia de discapacidades. de aprendizaje. En contraste, el modelo con Gender muestra un valor p más alto (0.266), lo que sugiere que el Exam Score no es un predictor significativo de género en este contexto.
Coeficientes del Modelo RLogS
modelo_RLog_Simple = glm(cdd_factores_rendimiento_estudiantes_G6_dep$learning_disabilities~cdd_factores_rendimiento_estudiantes_G6_dep$exam_score, family = "binomial", data = data.frame(cdd_factores_rendimiento_estudiantes_G6_dep$learning_disabilities, cdd_factores_rendimiento_estudiantes_G6_dep$exam_score))
coef(modelo_RLog_Simple)## (Intercept)
## 3.32351525
## cdd_factores_rendimiento_estudiantes_G6_dep$exam_score
## -0.08186994Resumen Estadístico del Modelo RLogS
summary(modelo_RLog_Simple)##
## Call:
## glm(formula = cdd_factores_rendimiento_estudiantes_G6_dep$learning_disabilities ~
## cdd_factores_rendimiento_estudiantes_G6_dep$exam_score, family = "binomial",
## data = data.frame(cdd_factores_rendimiento_estudiantes_G6_dep$learning_disabilities,
## cdd_factores_rendimiento_estudiantes_G6_dep$exam_score))
##
## Coefficients:
## Estimate Std. Error
## (Intercept) 3.32352 0.79571
## cdd_factores_rendimiento_estudiantes_G6_dep$exam_score -0.08187 0.01197
## z value Pr(>|z|)
## (Intercept) 4.177 2.96e-05 ***
## cdd_factores_rendimiento_estudiantes_G6_dep$exam_score -6.840 7.89e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 4277.9 on 6377 degrees of freedom
## Residual deviance: 4227.8 on 6376 degrees of freedom
## AIC: 4231.8
##
## Number of Fisher Scoring iterations: 5modelo_RLog_Simple_S = glm(cdd_factores_rendimiento_estudiantes_G6_dep$gender~cdd_factores_rendimiento_estudiantes_G6_dep$exam_score, family = "binomial", data = data.frame(cdd_factores_rendimiento_estudiantes_G6_dep$gender, cdd_factores_rendimiento_estudiantes_G6_dep$exam_score))
summary(modelo_RLog_Simple_S)##
## Call:
## glm(formula = cdd_factores_rendimiento_estudiantes_G6_dep$gender ~
## cdd_factores_rendimiento_estudiantes_G6_dep$exam_score, family = "binomial",
## data = data.frame(cdd_factores_rendimiento_estudiantes_G6_dep$gender,
## cdd_factores_rendimiento_estudiantes_G6_dep$exam_score))
##
## Coefficients:
## Estimate Std. Error
## (Intercept) 0.485456 0.436112
## cdd_factores_rendimiento_estudiantes_G6_dep$exam_score -0.002526 0.006473
## z value Pr(>|z|)
## (Intercept) 1.113 0.266
## cdd_factores_rendimiento_estudiantes_G6_dep$exam_score -0.390 0.696
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 8685.0 on 6377 degrees of freedom
## Residual deviance: 8684.8 on 6376 degrees of freedom
## AIC: 8688.8
##
## Number of Fisher Scoring iterations: 45.4.2.3. Análisis del modelo RLogS.
Se presentarán, mediante las pestañas, los resultados de algunas predicciones generadas por el modelo RLogS, con el fin de identificar en sus respuestas la relación de sentido en las razones de probabilidades (odds) a favor o en contra del evento en cuestión: \(\frac{\pi}{1-\pi}\) y \(\frac{1-\pi}{\pi}\), respectivamente. Se menciona de nuevo que las variables de interés son: Exam score (variable dependiente) y Learning disabilities (variable independiente).
La pestaña Variable Predictora igual a Cero plantea dos situaciones interpretativas. La primera permite comprender que el coeficiente del factor en el cual está presente la variable predictora, estima un valor base para la puntuación en el examen. En la primera, el intercepto tiene un valor de 3.32, lo que indica que, cuando la variable predictora no tiene influencia (es cero), la puntuación inicial estimada en el examen sería de aproximadamente 3.32 puntos. Sin embargo, el coeficiente asociado a la variable exam_score es negativo (-0.0819), lo que sugiere que un aumento en una unidad de esta variable se asocia con una leve disminución en el puntaje del examen.
Por otro lado, en la segunda, el intercepto tiene un valor de 27.76, lo que indica una puntuación inicial significativamente mayor en ausencia de influencia de la variable predictora. En este modelo, el coeficiente de exam_score es positivo (0.9214), lo que implica que cada incremento de una unidad en la variable predictora se traduce en un aumento promedio de 0.92 puntos en el resultado del examen.
Con base en los valores calculados en la pestaña de Probabilidades Estimadas, se puede observar entre los registros \(79\) y \(80\) un cambio absoluto de \(0.050139\) en las probabilidades (lo que representa un incremento relativo aproximado del \(41.45%\)). Este incremento ocurre al aumentar la variable predictora en una unidad, en línea con las definiciones establecidas previamente en el análisis.
El gráfico muestra que a medida que aumenta el puntaje en el examen, la probabilidad de tener una discapacidad de aprendizaje disminuye, con una relación negativa típica de un modelo logístico. Las observaciones se concentran en puntajes bajos para quienes tienen discapacidad (1) y en puntajes altos para quienes no la tienen (0). Esto sugiere que las discapacidades de aprendizaje pueden estar asociadas a un menor rendimiento académico.
Variable Predictora igual a Cero
coef(modelo_RLog_Simple)## (Intercept)
## 3.32351525
## cdd_factores_rendimiento_estudiantes_G6_dep$exam_score
## -0.08186994round(exp(coef(modelo_RLog_Simple)),6)## (Intercept)
## 27.757755
## cdd_factores_rendimiento_estudiantes_G6_dep$exam_score
## 0.921392Probabilidades Estimadas
predict(modelo_RLog_Simple, data.frame(seq(1, 400)), type = "response")## Warning: 'newdata' had 400 rows but variables found have 6378 rows## 1 2 3 4 5 6
## 0.103236410 0.158350725 0.060947403 0.076615209 0.082611626 0.076615209
## 7 8 9 10 11 12
## 0.103236410 0.111065767 0.089032111 0.071020351 0.095899426 0.076615209
## 13 14 15 16 17 18
## 0.082611626 0.111065767 0.119409831 0.128290293 0.169569538 0.119409831
## 19 20 21 22 23 24
## 0.103236410 0.111065767 0.089032111 0.071020351 0.111065767 0.111065767
## 25 26 27 28 29 30
## 0.137727896 0.128290293 0.119409831 0.076615209 0.111065767 0.128290293
## 31 32 33 34 35 36
## 0.137727896 0.089032111 0.103236410 0.095899426 0.158350725 0.128290293
## 37 38 39 40 41 42
## 0.103236410 0.137727896 0.128290293 0.103236410 0.119409831 0.128290293
## 43 44 45 46 47 48
## 0.082611626 0.095899426 0.119409831 0.095899426 0.128290293 0.095899426
## 49 50 51 52 53 54
## 0.137727896 0.128290293 0.128290293 0.103236410 0.103236410 0.111065767
## 55 56 57 58 59 60
## 0.071020351 0.119409831 0.076615209 0.111065767 0.095899426 0.082611626
## 61 62 63 64 65 66
## 0.147742102 0.095899426 0.103236410 0.076615209 0.095899426 0.089032111
## 67 68 69 70 71 72
## 0.082611626 0.089032111 0.119409831 0.082611626 0.076615209 0.119409831
## 73 74 75 76 77 78
## 0.071020351 0.089032111 0.111065767 0.095899426 0.119409831 0.128290293
## 79 80 81 82 83 84
## 0.169569538 0.119409831 0.111065767 0.060947403 0.147742102 0.082611626
## 85 86 87 88 89 90
## 0.089032111 0.103236410 0.128290293 0.076615209 0.095899426 0.128290293
## 91 92 93 94 95 96
## 0.137727896 0.076615209 0.089032111 0.007664373 0.095899426 0.147742102
## 97 98 99 100 101 102
## 0.095899426 0.111065767 0.128290293 0.103236410 0.119409831 0.119409831
## 103 104 105 106 107 108
## 0.089032111 0.137727896 0.071020351 0.052222807 0.089032111 0.103236410
## 109 110 111 112 113 114
## 0.089032111 0.137727896 0.103236410 0.111065767 0.041320030 0.128290293
## 115 116 117 118 119 120
## 0.065804943 0.137727896 0.119409831 0.065804943 0.082611626 0.095899426
## 121 122 123 124 125 126
## 0.119409831 0.076615209 0.128290293 0.095899426 0.089032111 0.089032111
## 127 128 129 130 131 132
## 0.103236410 0.095899426 0.128290293 0.111065767 0.137727896 0.111065767
## 133 134 135 136 137 138
## 0.111065767 0.103236410 0.119409831 0.095899426 0.095899426 0.119409831
## 139 140 141 142 143 144
## 0.076615209 0.119409831 0.128290293 0.071020351 0.119409831 0.111065767
## 145 146 147 148 149 150
## 0.158350725 0.095899426 0.089032111 0.071020351 0.076615209 0.082611626
## 151 152 153 154 155 156
## 0.095899426 0.103236410 0.095899426 0.095899426 0.071020351 0.119409831
## 157 158 159 160 161 162
## 0.111065767 0.119409831 0.103236410 0.111065767 0.089032111 0.111065767
## 163 164 165 166 167 168
## 0.089032111 0.103236410 0.111065767 0.076615209 0.082611626 0.076615209
## 169 170 171 172 173 174
## 0.111065767 0.103236410 0.111065767 0.071020351 0.095899426 0.089032111
## 175 176 177 178 179 180
## 0.076615209 0.082611626 0.103236410 0.089032111 0.119409831 0.119409831
## 181 182 183 184 185 186
## 0.128290293 0.137727896 0.089032111 0.128290293 0.071020351 0.137727896
## 187 188 189 190 191 192
## 0.128290293 0.128290293 0.103236410 0.076615209 0.082611626 0.158350725
## 193 194 195 196 197 198
## 0.103236410 0.071020351 0.111065767 0.082611626 0.103236410 0.103236410
## 199 200 201 202 203 204
## 0.076615209 0.111065767 0.095899426 0.060947403 0.128290293 0.119409831
## 205 206 207 208 209 210
## 0.111065767 0.111065767 0.095899426 0.044687700 0.103236410 0.060947403
## 211 212 213 214 215 216
## 0.060947403 0.147742102 0.111065767 0.089032111 0.095899426 0.018653179
## 217 218 219 220 221 222
## 0.111065767 0.095899426 0.089032111 0.103236410 0.056426775 0.169569538
## 223 224 225 226 227 228
## 0.111065767 0.065804943 0.095899426 0.147742102 0.147742102 0.119409831
## 229 230 231 232 233 234
## 0.089032111 0.128290293 0.103236410 0.082611626 0.082611626 0.089032111
## 235 236 237 238 239 240
## 0.158350725 0.089032111 0.137727896 0.103236410 0.111065767 0.137727896
## 241 242 243 244 245 246
## 0.095899426 0.119409831 0.103236410 0.095899426 0.128290293 0.076615209
## 247 248 249 250 251 252
## 0.089032111 0.103236410 0.119409831 0.095899426 0.103236410 0.103236410
## 253 254 255 256 257 258
## 0.111065767 0.082611626 0.119409831 0.089032111 0.119409831 0.137727896
## 259 260 261 262 263 264
## 0.103236410 0.158350725 0.119409831 0.111065767 0.111065767 0.089032111
## 265 266 267 268 269 270
## 0.137727896 0.119409831 0.103236410 0.089032111 0.119409831 0.119409831
## 271 272 273 274 275 276
## 0.119409831 0.103236410 0.071020351 0.111065767 0.119409831 0.089032111
## 277 278 279 280 281 282
## 0.095899426 0.128290293 0.089032111 0.181411860 0.095899426 0.169569538
## 283 284 285 286 287 288
## 0.147742102 0.103236410 0.095899426 0.128290293 0.119409831 0.119409831
## 289 290 291 292 293 294
## 0.137727896 0.082611626 0.111065767 0.095899426 0.111065767 0.137727896
## 295 296 297 298 299 300
## 0.111065767 0.119409831 0.111065767 0.103236410 0.111065767 0.137727896
## 301 302 303 304 305 306
## 0.111065767 0.103236410 0.076615209 0.103236410 0.089032111 0.103236410
## 307 308 309 310 311 312
## 0.071020351 0.082611626 0.103236410 0.128290293 0.103236410 0.089032111
## 313 314 315 316 317 318
## 0.137727896 0.095899426 0.082611626 0.111065767 0.095899426 0.103236410
## 319 320 321 322 323 324
## 0.089032111 0.111065767 0.095899426 0.089032111 0.082611626 0.060947403
## 325 326 327 328 329 330
## 0.082611626 0.111065767 0.089032111 0.128290293 0.147742102 0.082611626
## 331 332 333 334 335 336
## 0.095899426 0.119409831 0.119409831 0.076615209 0.111065767 0.158350725
## 337 338 339 340 341 342
## 0.103236410 0.119409831 0.119409831 0.119409831 0.065804943 0.147742102
## 343 344 345 346 347 348
## 0.169569538 0.158350725 0.103236410 0.082611626 0.089032111 0.111065767
## 349 350 351 352 353 354
## 0.128290293 0.095899426 0.103236410 0.128290293 0.119409831 0.111065767
## 355 356 357 358 359 360
## 0.095899426 0.111065767 0.137727896 0.137727896 0.060947403 0.082611626
## 361 362 363 364 365 366
## 0.095899426 0.111065767 0.137727896 0.103236410 0.147742102 0.076615209
## 367 368 369 370 371 372
## 0.071020351 0.071020351 0.095899426 0.111065767 0.082611626 0.111065767
## 373 374 375 376 377 378
## 0.071020351 0.128290293 0.158350725 0.147742102 0.095899426 0.095899426
## 379 380 381 382 383 384
## 0.137727896 0.082611626 0.095899426 0.071020351 0.103236410 0.158350725
## 385 386 387 388 389 390
## 0.089032111 0.158350725 0.056426775 0.095899426 0.119409831 0.056426775
## 391 392 393 394 395 396
## 0.128290293 0.082611626 0.103236410 0.137727896 0.103236410 0.023723042
## 397 398 399 400 401 402
## 0.103236410 0.137727896 0.111065767 0.103236410 0.089032111 0.111065767
## 403 404 405 406 407 408
## 0.128290293 0.065804943 0.111065767 0.095899426 0.082611626 0.103236410
## 409 410 411 412 413 414
## 0.147742102 0.095899426 0.071020351 0.089032111 0.111065767 0.082611626
## 415 416 417 418 419 420
## 0.076615209 0.119409831 0.082611626 0.119409831 0.147742102 0.147742102
## 421 422 423 424 425 426
## 0.082611626 0.111065767 0.103236410 0.071020351 0.111065767 0.095899426
## 427 428 429 430 431 432
## 0.147742102 0.128290293 0.119409831 0.111065767 0.128290293 0.103236410
## 433 434 435 436 437 438
## 0.089032111 0.095899426 0.128290293 0.095899426 0.128290293 0.076615209
## 439 440 441 442 443 444
## 0.158350725 0.095899426 0.128290293 0.071020351 0.119409831 0.095899426
## 445 446 447 448 449 450
## 0.076615209 0.111065767 0.137727896 0.060947403 0.111065767 0.137727896
## 451 452 453 454 455 456
## 0.071020351 0.089032111 0.103236410 0.111065767 0.128290293 0.103236410
## 457 458 459 460 461 462
## 0.089032111 0.128290293 0.119409831 0.071020351 0.111065767 0.111065767
## 463 464 465 466 467 468
## 0.103236410 0.089032111 0.065804943 0.089032111 0.095899426 0.158350725
## 469 470 471 472 473 474
## 0.137727896 0.137727896 0.095899426 0.137727896 0.111065767 0.119409831
## 475 476 477 478 479 480
## 0.095899426 0.111065767 0.111065767 0.082611626 0.158350725 0.158350725
## 481 482 483 484 485 486
## 0.076615209 0.103236410 0.082611626 0.089032111 0.119409831 0.158350725
## 487 488 489 490 491 492
## 0.095899426 0.060947403 0.103236410 0.082611626 0.181411860 0.128290293
## 493 494 495 496 497 498
## 0.111065767 0.137727896 0.158350725 0.095899426 0.119409831 0.089032111
## 499 500 501 502 503 504
## 0.095899426 0.089032111 0.103236410 0.082611626 0.095899426 0.103236410
## 505 506 507 508 509 510
## 0.103236410 0.056426775 0.137727896 0.082611626 0.082611626 0.128290293
## 511 512 513 514 515 516
## 0.181411860 0.137727896 0.103236410 0.095899426 0.095899426 0.009777277
## 517 518 519 520 521 522
## 0.111065767 0.158350725 0.082611626 0.119409831 0.128290293 0.089032111
## 523 524 525 526 527 528
## 0.128290293 0.158350725 0.128290293 0.095899426 0.103236410 0.147742102
## 529 530 531 532 533 534
## 0.111065767 0.071020351 0.095899426 0.076615209 0.119409831 0.095899426
## 535 536 537 538 539 540
## 0.119409831 0.128290293 0.119409831 0.119409831 0.065804943 0.071020351
## 541 542 543 544 545 546
## 0.128290293 0.089032111 0.089032111 0.082611626 0.030128568 0.095899426
## 547 548 549 550 551 552
## 0.027826117 0.056426775 0.103236410 0.089032111 0.076615209 0.111065767
## 553 554 555 556 557 558
## 0.119409831 0.082611626 0.111065767 0.060947403 0.095899426 0.095899426
## 559 560 561 562 563 564
## 0.147742102 0.076615209 0.137727896 0.089032111 0.076615209 0.111065767
## 565 566 567 568 569 570
## 0.128290293 0.119409831 0.089032111 0.111065767 0.103236410 0.119409831
## 571 572 573 574 575 576
## 0.119409831 0.103236410 0.111065767 0.089032111 0.082611626 0.095899426
## 577 578 579 580 581 582
## 0.095899426 0.082611626 0.103236410 0.071020351 0.089032111 0.089032111
## 583 584 585 586 587 588
## 0.128290293 0.137727896 0.082611626 0.082611626 0.111065767 0.082611626
## 589 590 591 592 593 594
## 0.111065767 0.095899426 0.158350725 0.071020351 0.089032111 0.071020351
## 595 596 597 598 599 600
## 0.119409831 0.060947403 0.147742102 0.119409831 0.095899426 0.076615209
## 601 602 603 604 605 606
## 0.147742102 0.119409831 0.095899426 0.082611626 0.137727896 0.137727896
## 607 608 609 610 611 612
## 0.089032111 0.111065767 0.082611626 0.137727896 0.071020351 0.119409831
## 613 614 615 616 617 618
## 0.103236410 0.137727896 0.119409831 0.103236410 0.038196000 0.095899426
## 619 620 621 622 623 624
## 0.137727896 0.128290293 0.065804943 0.111065767 0.111065767 0.111065767
## 625 626 627 628 629 630
## 0.089032111 0.181411860 0.095899426 0.103236410 0.128290293 0.119409831
## 631 632 633 634 635 636
## 0.095899426 0.082611626 0.147742102 0.071020351 0.147742102 0.095899426
## 637 638 639 640 641 642
## 0.089032111 0.065804943 0.089032111 0.103236410 0.119409831 0.137727896
## 643 644 645 646 647 648
## 0.111065767 0.111065767 0.082611626 0.089032111 0.082611626 0.193888125
## 649 650 651 652 653 654
## 0.103236410 0.111065767 0.082611626 0.103236410 0.169569538 0.119409831
## 655 656 657 658 659 660
## 0.137727896 0.071020351 0.147742102 0.082611626 0.082611626 0.128290293
## 661 662 663 664 665 666
## 0.128290293 0.071020351 0.111065767 0.111065767 0.076615209 0.111065767
## 667 668 669 670 671 672
## 0.128290293 0.111065767 0.147742102 0.082611626 0.089032111 0.119409831
## 673 674 675 676 677 678
## 0.082611626 0.082611626 0.103236410 0.089032111 0.147742102 0.103236410
## 679 680 681 682 683 684
## 0.119409831 0.119409831 0.089032111 0.137727896 0.111065767 0.103236410
## 685 686 687 688 689 690
## 0.089032111 0.103236410 0.082611626 0.103236410 0.082611626 0.119409831
## 691 692 693 694 695 696
## 0.137727896 0.128290293 0.089032111 0.169569538 0.089032111 0.071020351
## 697 698 699 700 701 702
## 0.137727896 0.137727896 0.082611626 0.095899426 0.089032111 0.071020351
## 703 704 705 706 707 708
## 0.103236410 0.095899426 0.082611626 0.095899426 0.089032111 0.095899426
## 709 710 711 712 713 714
## 0.111065767 0.071020351 0.103236410 0.082611626 0.065804943 0.065804943
## 715 716 717 718 719 720
## 0.169569538 0.103236410 0.089032111 0.119409831 0.111065767 0.119409831
## 721 722 723 724 725 726
## 0.128290293 0.103236410 0.119409831 0.128290293 0.103236410 0.082611626
## 727 728 729 730 731 732
## 0.137727896 0.076615209 0.095899426 0.103236410 0.103236410 0.095899426
## 733 734 735 736 737 738
## 0.111065767 0.095899426 0.082611626 0.103236410 0.103236410 0.147742102
## 739 740 741 742 743 744
## 0.103236410 0.082611626 0.071020351 0.082611626 0.012465347 0.111065767
## 745 746 747 748 749 750
## 0.089032111 0.095899426 0.076615209 0.089032111 0.103236410 0.128290293
## 751 752 753 754 755 756
## 0.137727896 0.111065767 0.111065767 0.089032111 0.128290293 0.147742102
## 757 758 759 760 761 762
## 0.082611626 0.103236410 0.095899426 0.147742102 0.103236410 0.095899426
## 763 764 765 766 767 768
## 0.128290293 0.128290293 0.111065767 0.071020351 0.071020351 0.111065767
## 769 770 771 772 773 774
## 0.071020351 0.089032111 0.071020351 0.169569538 0.089032111 0.095899426
## 775 776 777 778 779 780
## 0.089032111 0.147742102 0.082611626 0.095899426 0.095899426 0.082611626
## 781 782 783 784 785 786
## 0.119409831 0.065804943 0.111065767 0.076615209 0.147742102 0.095899426
## 787 788 789 790 791 792
## 0.082611626 0.095899426 0.076615209 0.082611626 0.071020351 0.089032111
## 793 794 795 796 797 798
## 0.137727896 0.147742102 0.095899426 0.082611626 0.089032111 0.095899426
## 799 800 801 802 803 804
## 0.119409831 0.181411860 0.111065767 0.128290293 0.137727896 0.071020351
## 805 806 807 808 809 810
## 0.103236410 0.012465347 0.082611626 0.111065767 0.089032111 0.128290293
## 811 812 813 814 815 816
## 0.137727896 0.076615209 0.119409831 0.095899426 0.128290293 0.128290293
## 817 818 819 820 821 822
## 0.111065767 0.095899426 0.103236410 0.111065767 0.095899426 0.111065767
## 823 824 825 826 827 828
## 0.076615209 0.076615209 0.095899426 0.128290293 0.103236410 0.128290293
## 829 830 831 832 833 834
## 0.095899426 0.147742102 0.128290293 0.082611626 0.095899426 0.089032111
## 835 836 837 838 839 840
## 0.119409831 0.103236410 0.076615209 0.119409831 0.111065767 0.119409831
## 841 842 843 844 845 846
## 0.119409831 0.095899426 0.095899426 0.103236410 0.119409831 0.111065767
## 847 848 849 850 851 852
## 0.137727896 0.071020351 0.103236410 0.103236410 0.111065767 0.111065767
## 853 854 855 856 857 858
## 0.089032111 0.147742102 0.111065767 0.128290293 0.103236410 0.076615209
## 859 860 861 862 863 864
## 0.095899426 0.111065767 0.056426775 0.071020351 0.095899426 0.095899426
## 865 866 867 868 869 870
## 0.089032111 0.137727896 0.082611626 0.111065767 0.111065767 0.071020351
## 871 872 873 874 875 876
## 0.076615209 0.071020351 0.089032111 0.076615209 0.089032111 0.095899426
## 877 878 879 880 881 882
## 0.158350725 0.095899426 0.071020351 0.082611626 0.065804943 0.103236410
## 883 884 885 886 887 888
## 0.119409831 0.095899426 0.119409831 0.009777277 0.111065767 0.103236410
## 889 890 891 892 893 894
## 0.082611626 0.111065767 0.089032111 0.147742102 0.095899426 0.103236410
## 895 896 897 898 899 900
## 0.095899426 0.089032111 0.147742102 0.056426775 0.119409831 0.137727896
## 901 902 903 904 905 906
## 0.111065767 0.076615209 0.111065767 0.111065767 0.111065767 0.095899426
## 907 908 909 910 911 912
## 0.076615209 0.095899426 0.095899426 0.095899426 0.147742102 0.119409831
## 913 914 915 916 917 918
## 0.076615209 0.137727896 0.095899426 0.089032111 0.089032111 0.076615209
## 919 920 921 922 923 924
## 0.076615209 0.147742102 0.082611626 0.082611626 0.065804943 0.076615209
## 925 926 927 928 929 930
## 0.089032111 0.119409831 0.076615209 0.095899426 0.128290293 0.076615209
## 931 932 933 934 935 936
## 0.103236410 0.111065767 0.076615209 0.147742102 0.103236410 0.103236410
## 937 938 939 940 941 942
## 0.111065767 0.128290293 0.147742102 0.119409831 0.137727896 0.128290293
## 943 944 945 946 947 948
## 0.119409831 0.095899426 0.071020351 0.128290293 0.147742102 0.089032111
## 949 950 951 952 953 954
## 0.128290293 0.158350725 0.071020351 0.111065767 0.089032111 0.119409831
## 955 956 957 958 959 960
## 0.128290293 0.147742102 0.111065767 0.147742102 0.111065767 0.082611626
## 961 962 963 964 965 966
## 0.056426775 0.103236410 0.111065767 0.076615209 0.056426775 0.071020351
## 967 968 969 970 971 972
## 0.119409831 0.103236410 0.137727896 0.147742102 0.089032111 0.082611626
## 973 974 975 976 977 978
## 0.137727896 0.076615209 0.137727896 0.076615209 0.137727896 0.119409831
## 979 980 981 982 983 984
## 0.111065767 0.111065767 0.111065767 0.111065767 0.103236410 0.095899426
## 985 986 987 988 989 990
## 0.095899426 0.089032111 0.128290293 0.103236410 0.137727896 0.128290293
## 991 992 993 994 995 996
## 0.111065767 0.119409831 0.095899426 0.111065767 0.076615209 0.060947403
## 997 998 999 1000 1001 1002
## 0.119409831 0.095899426 0.095899426 0.071020351 0.060947403 0.158350725
## 1003 1004 1005 1006 1007 1008
## 0.119409831 0.137727896 0.111065767 0.111065767 0.065804943 0.137727896
## 1009 1010 1011 1012 1013 1014
## 0.065804943 0.137727896 0.119409831 0.119409831 0.111065767 0.119409831
## 1015 1016 1017 1018 1019 1020
## 0.082611626 0.082611626 0.095899426 0.128290293 0.095899426 0.147742102
## 1021 1022 1023 1024 1025 1026
## 0.158350725 0.089032111 0.119409831 0.119409831 0.071020351 0.082611626
## 1027 1028 1029 1030 1031 1032
## 0.119409831 0.089032111 0.119409831 0.089032111 0.119409831 0.071020351
## 1033 1034 1035 1036 1037 1038
## 0.082611626 0.128290293 0.103236410 0.089032111 0.082611626 0.119409831
## 1039 1040 1041 1042 1043 1044
## 0.089032111 0.060947403 0.111065767 0.071020351 0.056426775 0.082611626
## 1045 1046 1047 1048 1049 1050
## 0.137727896 0.119409831 0.128290293 0.119409831 0.082611626 0.103236410
## 1051 1052 1053 1054 1055 1056
## 0.103236410 0.095899426 0.119409831 0.158350725 0.137727896 0.038196000
## 1057 1058 1059 1060 1061 1062
## 0.089032111 0.235172433 0.082611626 0.119409831 0.095899426 0.128290293
## 1063 1064 1065 1066 1067 1068
## 0.089032111 0.018653179 0.103236410 0.014650540 0.103236410 0.082611626
## 1069 1070 1071 1072 1073 1074
## 0.095899426 0.082611626 0.103236410 0.089032111 0.103236410 0.089032111
## 1075 1076 1077 1078 1079 1080
## 0.137727896 0.111065767 0.095899426 0.089032111 0.111065767 0.071020351
## 1081 1082 1083 1084 1085 1086
## 0.147742102 0.103236410 0.103236410 0.103236410 0.119409831 0.103236410
## 1087 1088 1089 1090 1091 1092
## 0.111065767 0.082611626 0.076615209 0.128290293 0.089032111 0.071020351
## 1093 1094 1095 1096 1097 1098
## 0.169569538 0.076615209 0.076615209 0.111065767 0.060947403 0.128290293
## 1099 1100 1101 1102 1103 1104
## 0.095899426 0.111065767 0.119409831 0.128290293 0.119409831 0.089032111
## 1105 1106 1107 1108 1109 1110
## 0.147742102 0.137727896 0.119409831 0.147742102 0.103236410 0.128290293
## 1111 1112 1113 1114 1115 1116
## 0.158350725 0.119409831 0.103236410 0.082611626 0.181411860 0.056426775
## 1117 1118 1119 1120 1121 1122
## 0.103236410 0.128290293 0.071020351 0.052222807 0.089032111 0.119409831
## 1123 1124 1125 1126 1127 1128
## 0.071020351 0.089032111 0.076615209 0.128290293 0.060947403 0.095899426
## 1129 1130 1131 1132 1133 1134
## 0.111065767 0.119409831 0.103236410 0.089032111 0.111065767 0.089032111
## 1135 1136 1137 1138 1139 1140
## 0.103236410 0.128290293 0.103236410 0.158350725 0.158350725 0.095899426
## 1141 1142 1143 1144 1145 1146
## 0.111065767 0.111065767 0.137727896 0.111065767 0.111065767 0.111065767
## 1147 1148 1149 1150 1151 1152
## 0.119409831 0.089032111 0.076615209 0.111065767 0.111065767 0.119409831
## 1153 1154 1155 1156 1157 1158
## 0.076615209 0.089032111 0.158350725 0.082611626 0.082611626 0.119409831
## 1159 1160 1161 1162 1163 1164
## 0.128290293 0.095899426 0.137727896 0.137727896 0.076615209 0.193888125
## 1165 1166 1167 1168 1169 1170
## 0.095899426 0.119409831 0.119409831 0.095899426 0.089032111 0.076615209
## 1171 1172 1173 1174 1175 1176
## 0.119409831 0.128290293 0.056426775 0.103236410 0.095899426 0.071020351
## 1177 1178 1179 1180 1181 1182
## 0.103236410 0.071020351 0.089032111 0.095899426 0.082611626 0.089032111
## 1183 1184 1185 1186 1187 1188
## 0.103236410 0.137727896 0.158350725 0.111065767 0.065804943 0.103236410
## 1189 1190 1191 1192 1193 1194
## 0.071020351 0.089032111 0.089032111 0.137727896 0.103236410 0.137727896
## 1195 1196 1197 1198 1199 1200
## 0.111065767 0.147742102 0.095899426 0.119409831 0.181411860 0.071020351
## 1201 1202 1203 1204 1205 1206
## 0.137727896 0.095899426 0.158350725 0.119409831 0.095899426 0.158350725
## 1207 1208 1209 1210 1211 1212
## 0.103236410 0.111065767 0.056426775 0.095899426 0.119409831 0.128290293
## 1213 1214 1215 1216 1217 1218
## 0.128290293 0.103236410 0.137727896 0.103236410 0.119409831 0.082611626
## 1219 1220 1221 1222 1223 1224
## 0.147742102 0.089032111 0.082611626 0.095899426 0.095899426 0.128290293
## 1225 1226 1227 1228 1229 1230
## 0.137727896 0.095899426 0.103236410 0.095899426 0.111065767 0.103236410
## 1231 1232 1233 1234 1235 1236
## 0.089032111 0.082611626 0.095899426 0.111065767 0.103236410 0.119409831
## 1237 1238 1239 1240 1241 1242
## 0.103236410 0.137727896 0.128290293 0.128290293 0.111065767 0.095899426
## 1243 1244 1245 1246 1247 1248
## 0.103236410 0.071020351 0.103236410 0.111065767 0.111065767 0.103236410
## 1249 1250 1251 1252 1253 1254
## 0.119409831 0.082611626 0.089032111 0.089032111 0.111065767 0.111065767
## 1255 1256 1257 1258 1259 1260
## 0.103236410 0.103236410 0.103236410 0.071020351 0.095899426 0.082611626
## 1261 1262 1263 1264 1265 1266
## 0.089032111 0.119409831 0.071020351 0.089032111 0.095899426 0.147742102
## 1267 1268 1269 1270 1271 1272
## 0.169569538 0.103236410 0.103236410 0.103236410 0.137727896 0.076615209
## 1273 1274 1275 1276 1277 1278
## 0.111065767 0.119409831 0.089032111 0.137727896 0.128290293 0.103236410
## 1279 1280 1281 1282 1283 1284
## 0.147742102 0.095899426 0.111065767 0.089032111 0.076615209 0.111065767
## 1285 1286 1287 1288 1289 1290
## 0.071020351 0.082611626 0.119409831 0.128290293 0.137727896 0.137727896
## 1291 1292 1293 1294 1295 1296
## 0.111065767 0.095899426 0.060947403 0.076615209 0.103236410 0.111065767
## 1297 1298 1299 1300 1301 1302
## 0.103236410 0.147742102 0.095899426 0.147742102 0.032615142 0.082611626
## 1303 1304 1305 1306 1307 1308
## 0.071020351 0.128290293 0.111065767 0.095899426 0.128290293 0.158350725
## 1309 1310 1311 1312 1313 1314
## 0.089032111 0.095899426 0.128290293 0.103236410 0.103236410 0.119409831
## 1315 1316 1317 1318 1319 1320
## 0.095899426 0.103236410 0.128290293 0.082611626 0.089032111 0.111065767
## 1321 1322 1323 1324 1325 1326
## 0.089032111 0.111065767 0.065804943 0.111065767 0.089032111 0.082611626
## 1327 1328 1329 1330 1331 1332
## 0.076615209 0.082611626 0.089032111 0.137727896 0.076615209 0.103236410
## 1333 1334 1335 1336 1337 1338
## 0.065804943 0.095899426 0.089032111 0.052222807 0.095899426 0.119409831
## 1339 1340 1341 1342 1343 1344
## 0.111065767 0.119409831 0.089032111 0.111065767 0.089032111 0.181411860
## 1345 1346 1347 1348 1349 1350
## 0.119409831 0.071020351 0.128290293 0.089032111 0.082611626 0.103236410
## 1351 1352 1353 1354 1355 1356
## 0.103236410 0.128290293 0.095899426 0.071020351 0.119409831 0.082611626
## 1357 1358 1359 1360 1361 1362
## 0.056426775 0.119409831 0.137727896 0.076615209 0.111065767 0.089032111
## 1363 1364 1365 1366 1367 1368
## 0.095899426 0.076615209 0.048316010 0.082611626 0.111065767 0.065804943
## 1369 1370 1371 1372 1373 1374
## 0.147742102 0.119409831 0.076615209 0.111065767 0.071020351 0.128290293
## 1375 1376 1377 1378 1379 1380
## 0.137727896 0.137727896 0.071020351 0.076615209 0.071020351 0.119409831
## 1381 1382 1383 1384 1385 1386
## 0.103236410 0.095899426 0.111065767 0.071020351 0.076615209 0.082611626
## 1387 1388 1389 1390 1391 1392
## 0.095899426 0.071020351 0.111065767 0.089032111 0.119409831 0.147742102
## 1393 1394 1395 1396 1397 1398
## 0.089032111 0.119409831 0.089032111 0.089032111 0.056426775 0.082611626
## 1399 1400 1401 1402 1403 1404
## 0.060947403 0.119409831 0.137727896 0.060947403 0.103236410 0.076615209
## 1405 1406 1407 1408 1409 1410
## 0.095899426 0.089032111 0.076615209 0.111065767 0.119409831 0.111065767
## 1411 1412 1413 1414 1415 1416
## 0.119409831 0.103236410 0.082611626 0.137727896 0.128290293 0.076615209
## 1417 1418 1419 1420 1421 1422
## 0.095899426 0.071020351 0.076615209 0.128290293 0.128290293 0.111065767
## 1423 1424 1425 1426 1427 1428
## 0.095899426 0.119409831 0.103236410 0.128290293 0.082611626 0.119409831
## 1429 1430 1431 1432 1433 1434
## 0.119409831 0.095899426 0.158350725 0.119409831 0.089032111 0.128290293
## 1435 1436 1437 1438 1439 1440
## 0.089032111 0.103236410 0.060947403 0.111065767 0.071020351 0.065804943
## 1441 1442 1443 1444 1445 1446
## 0.065804943 0.137727896 0.089032111 0.076615209 0.137727896 0.089032111
## 1447 1448 1449 1450 1451 1452
## 0.071020351 0.095899426 0.147742102 0.095899426 0.076615209 0.103236410
## 1453 1454 1455 1456 1457 1458
## 0.119409831 0.169569538 0.095899426 0.181411860 0.076615209 0.082611626
## 1459 1460 1461 1462 1463 1464
## 0.147742102 0.147742102 0.095899426 0.095899426 0.060947403 0.158350725
## 1465 1466 1467 1468 1469 1470
## 0.103236410 0.119409831 0.007664373 0.119409831 0.103236410 0.158350725
## 1471 1472 1473 1474 1475 1476
## 0.128290293 0.128290293 0.103236410 0.089032111 0.089032111 0.103236410
## 1477 1478 1479 1480 1481 1482
## 0.147742102 0.111065767 0.071020351 0.103236410 0.111065767 0.103236410
## 1483 1484 1485 1486 1487 1488
## 0.103236410 0.095899426 0.137727896 0.147742102 0.119409831 0.119409831
## 1489 1490 1491 1492 1493 1494
## 0.082611626 0.065804943 0.065804943 0.095899426 0.137727896 0.147742102
## 1495 1496 1497 1498 1499 1500
## 0.089032111 0.111065767 0.095899426 0.103236410 0.119409831 0.082611626
## 1501 1502 1503 1504 1505 1506
## 0.128290293 0.119409831 0.103236410 0.147742102 0.111065767 0.095899426
## 1507 1508 1509 1510 1511 1512
## 0.128290293 0.089032111 0.089032111 0.103236410 0.158350725 0.082611626
## 1513 1514 1515 1516 1517 1518
## 0.119409831 0.111065767 0.111065767 0.089032111 0.103236410 0.076615209
## 1519 1520 1521 1522 1523 1524
## 0.095899426 0.103236410 0.137727896 0.128290293 0.065804943 0.089032111
## 1525 1526 1527 1528 1529 1530
## 0.119409831 0.082611626 0.103236410 0.103236410 0.082611626 0.089032111
## 1531 1532 1533 1534 1535 1536
## 0.095899426 0.111065767 0.103236410 0.076615209 0.095899426 0.111065767
## 1537 1538 1539 1540 1541 1542
## 0.128290293 0.076615209 0.158350725 0.103236410 0.119409831 0.111065767
## 1543 1544 1545 1546 1547 1548
## 0.095899426 0.111065767 0.137727896 0.111065767 0.065804943 0.020212403
## 1549 1550 1551 1552 1553 1554
## 0.103236410 0.089032111 0.103236410 0.128290293 0.128290293 0.119409831
## 1555 1556 1557 1558 1559 1560
## 0.095899426 0.048316010 0.137727896 0.089032111 0.089032111 0.082611626
## 1561 1562 1563 1564 1565 1566
## 0.065804943 0.111065767 0.076615209 0.111065767 0.076615209 0.111065767
## 1567 1568 1569 1570 1571 1572
## 0.137727896 0.082611626 0.111065767 0.111065767 0.111065767 0.147742102
## 1573 1574 1575 1576 1577 1578
## 0.089032111 0.137727896 0.095899426 0.128290293 0.089032111 0.076615209
## 1579 1580 1581 1582 1583 1584
## 0.076615209 0.111065767 0.137727896 0.071020351 0.082611626 0.076615209
## 1585 1586 1587 1588 1589 1590
## 0.119409831 0.119409831 0.103236410 0.095899426 0.082611626 0.082611626
## 1591 1592 1593 1594 1595 1596
## 0.071020351 0.119409831 0.119409831 0.076615209 0.060947403 0.128290293
## 1597 1598 1599 1600 1601 1602
## 0.128290293 0.119409831 0.103236410 0.065804943 0.071020351 0.119409831
## 1603 1604 1605 1606 1607 1608
## 0.103236410 0.082611626 0.111065767 0.111065767 0.095899426 0.095899426
## 1609 1610 1611 1612 1613 1614
## 0.103236410 0.119409831 0.128290293 0.082611626 0.137727896 0.119409831
## 1615 1616 1617 1618 1619 1620
## 0.103236410 0.076615209 0.065804943 0.103236410 0.071020351 0.076615209
## 1621 1622 1623 1624 1625 1626
## 0.071020351 0.095899426 0.119409831 0.095899426 0.103236410 0.111065767
## 1627 1628 1629 1630 1631 1632
## 0.128290293 0.119409831 0.089032111 0.095899426 0.089032111 0.082611626
## 1633 1634 1635 1636 1637 1638
## 0.119409831 0.137727896 0.076615209 0.095899426 0.082611626 0.071020351
## 1639 1640 1641 1642 1643 1644
## 0.128290293 0.128290293 0.111065767 0.089032111 0.076615209 0.089032111
## 1645 1646 1647 1648 1649 1650
## 0.103236410 0.111065767 0.076615209 0.137727896 0.147742102 0.065804943
## 1651 1652 1653 1654 1655 1656
## 0.111065767 0.071020351 0.103236410 0.119409831 0.111065767 0.095899426
## 1657 1658 1659 1660 1661 1662
## 0.089032111 0.095899426 0.158350725 0.095899426 0.128290293 0.103236410
## 1663 1664 1665 1666 1667 1668
## 0.095899426 0.158350725 0.111065767 0.128290293 0.103236410 0.128290293
## 1669 1670 1671 1672 1673 1674
## 0.071020351 0.147742102 0.137727896 0.082611626 0.089032111 0.128290293
## 1675 1676 1677 1678 1679 1680
## 0.111065767 0.103236410 0.076615209 0.082611626 0.089032111 0.128290293
## 1681 1682 1683 1684 1685 1686
## 0.082611626 0.089032111 0.169569538 0.119409831 0.103236410 0.111065767
## 1687 1688 1689 1690 1691 1692
## 0.095899426 0.193888125 0.103236410 0.147742102 0.089032111 0.065804943
## 1693 1694 1695 1696 1697 1698
## 0.111065767 0.103236410 0.076615209 0.111065767 0.060947403 0.103236410
## 1699 1700 1701 1702 1703 1704
## 0.095899426 0.158350725 0.119409831 0.076615209 0.158350725 0.082611626
## 1705 1706 1707 1708 1709 1710
## 0.082611626 0.089032111 0.082611626 0.103236410 0.137727896 0.082611626
## 1711 1712 1713 1714 1715 1716
## 0.158350725 0.128290293 0.119409831 0.119409831 0.111065767 0.111065767
## 1717 1718 1719 1720 1721 1722
## 0.137727896 0.111065767 0.128290293 0.119409831 0.147742102 0.076615209
## 1723 1724 1725 1726 1727 1728
## 0.137727896 0.082611626 0.082611626 0.119409831 0.111065767 0.095899426
## 1729 1730 1731 1732 1733 1734
## 0.095899426 0.089032111 0.076615209 0.095899426 0.103236410 0.071020351
## 1735 1736 1737 1738 1739 1740
## 0.095899426 0.169569538 0.095899426 0.119409831 0.103236410 0.082611626
## 1741 1742 1743 1744 1745 1746
## 0.082611626 0.158350725 0.095899426 0.103236410 0.147742102 0.111065767
## 1747 1748 1749 1750 1751 1752
## 0.089032111 0.111065767 0.128290293 0.111065767 0.137727896 0.147742102
## 1753 1754 1755 1756 1757 1758
## 0.119409831 0.095899426 0.119409831 0.137727896 0.095899426 0.071020351
## 1759 1760 1761 1762 1763 1764
## 0.089032111 0.111065767 0.095899426 0.128290293 0.158350725 0.103236410
## 1765 1766 1767 1768 1769 1770
## 0.095899426 0.060947403 0.103236410 0.128290293 0.089032111 0.137727896
## 1771 1772 1773 1774 1775 1776
## 0.103236410 0.119409831 0.103236410 0.095899426 0.147742102 0.082611626
## 1777 1778 1779 1780 1781 1782
## 0.103236410 0.018653179 0.119409831 0.089032111 0.128290293 0.060947403
## 1783 1784 1785 1786 1787 1788
## 0.128290293 0.128290293 0.103236410 0.137727896 0.095899426 0.111065767
## 1789 1790 1791 1792 1793 1794
## 0.095899426 0.095899426 0.089032111 0.076615209 0.119409831 0.103236410
## 1795 1796 1797 1798 1799 1800
## 0.119409831 0.119409831 0.038196000 0.060947403 0.181411860 0.089032111
## 1801 1802 1803 1804 1805 1806
## 0.111065767 0.111065767 0.119409831 0.103236410 0.089032111 0.095899426
## 1807 1808 1809 1810 1811 1812
## 0.065804943 0.076615209 0.103236410 0.103236410 0.111065767 0.137727896
## 1813 1814 1815 1816 1817 1818
## 0.119409831 0.103236410 0.089032111 0.103236410 0.111065767 0.060947403
## 1819 1820 1821 1822 1823 1824
## 0.111065767 0.089032111 0.137727896 0.119409831 0.128290293 0.128290293
## 1825 1826 1827 1828 1829 1830
## 0.089032111 0.065804943 0.147742102 0.147742102 0.095899426 0.111065767
## 1831 1832 1833 1834 1835 1836
## 0.095899426 0.111065767 0.089032111 0.147742102 0.095899426 0.103236410
## 1837 1838 1839 1840 1841 1842
## 0.119409831 0.089032111 0.103236410 0.128290293 0.147742102 0.076615209
## 1843 1844 1845 1846 1847 1848
## 0.071020351 0.060947403 0.111065767 0.071020351 0.119409831 0.103236410
## 1849 1850 1851 1852 1853 1854
## 0.082611626 0.147742102 0.111065767 0.119409831 0.082611626 0.089032111
## 1855 1856 1857 1858 1859 1860
## 0.082611626 0.111065767 0.065804943 0.089032111 0.111065767 0.119409831
## 1861 1862 1863 1864 1865 1866
## 0.082611626 0.082611626 0.103236410 0.095899426 0.095899426 0.158350725
## 1867 1868 1869 1870 1871 1872
## 0.082611626 0.119409831 0.111065767 0.103236410 0.089032111 0.089032111
## 1873 1874 1875 1876 1877 1878
## 0.103236410 0.111065767 0.137727896 0.103236410 0.128290293 0.095899426
## 1879 1880 1881 1882 1883 1884
## 0.119409831 0.119409831 0.082611626 0.147742102 0.095899426 0.147742102
## 1885 1886 1887 1888 1889 1890
## 0.095899426 0.071020351 0.082611626 0.147742102 0.041320030 0.065804943
## 1891 1892 1893 1894 1895 1896
## 0.095899426 0.095899426 0.089032111 0.137727896 0.060947403 0.089032111
## 1897 1898 1899 1900 1901 1902
## 0.158350725 0.089032111 0.128290293 0.137727896 0.095899426 0.071020351
## 1903 1904 1905 1906 1907 1908
## 0.089032111 0.082611626 0.137727896 0.147742102 0.071020351 0.089032111
## 1909 1910 1911 1912 1913 1914
## 0.111065767 0.076615209 0.147742102 0.095899426 0.137727896 0.111065767
## 1915 1916 1917 1918 1919 1920
## 0.056426775 0.089032111 0.095899426 0.089032111 0.103236410 0.065804943
## 1921 1922 1923 1924 1925 1926
## 0.076615209 0.076615209 0.095899426 0.082611626 0.076615209 0.089032111
## 1927 1928 1929 1930 1931 1932
## 0.137727896 0.076615209 0.089032111 0.076615209 0.095899426 0.095899426
## 1933 1934 1935 1936 1937 1938
## 0.111065767 0.103236410 0.181411860 0.111065767 0.128290293 0.128290293
## 1939 1940 1941 1942 1943 1944
## 0.089032111 0.089032111 0.158350725 0.147742102 0.158350725 0.137727896
## 1945 1946 1947 1948 1949 1950
## 0.076615209 0.119409831 0.095899426 0.103236410 0.082611626 0.128290293
## 1951 1952 1953 1954 1955 1956
## 0.076615209 0.076615209 0.103236410 0.137727896 0.111065767 0.082611626
## 1957 1958 1959 1960 1961 1962
## 0.082611626 0.103236410 0.089032111 0.111065767 0.103236410 0.119409831
## 1963 1964 1965 1966 1967 1968
## 0.071020351 0.095899426 0.071020351 0.052222807 0.119409831 0.095899426
## 1969 1970 1971 1972 1973 1974
## 0.119409831 0.111065767 0.137727896 0.137727896 0.111065767 0.089032111
## 1975 1976 1977 1978 1979 1980
## 0.071020351 0.111065767 0.128290293 0.103236410 0.111065767 0.095899426
## 1981 1982 1983 1984 1985 1986
## 0.119409831 0.076615209 0.076615209 0.147742102 0.089032111 0.119409831
## 1987 1988 1989 1990 1991 1992
## 0.082611626 0.082611626 0.082611626 0.089032111 0.076615209 0.089032111
## 1993 1994 1995 1996 1997 1998
## 0.065804943 0.119409831 0.147742102 0.103236410 0.119409831 0.119409831
## 1999 2000 2001 2002 2003 2004
## 0.082611626 0.082611626 0.076615209 0.076615209 0.147742102 0.119409831
## 2005 2006 2007 2008 2009 2010
## 0.027826117 0.095899426 0.082611626 0.095899426 0.103236410 0.089032111
## 2011 2012 2013 2014 2015 2016
## 0.095899426 0.181411860 0.103236410 0.137727896 0.119409831 0.137727896
## 2017 2018 2019 2020 2021 2022
## 0.128290293 0.071020351 0.137727896 0.111065767 0.103236410 0.111065767
## 2023 2024 2025 2026 2027 2028
## 0.111065767 0.095899426 0.111065767 0.095899426 0.119409831 0.103236410
## 2029 2030 2031 2032 2033 2034
## 0.128290293 0.181411860 0.169569538 0.119409831 0.128290293 0.111065767
## 2035 2036 2037 2038 2039 2040
## 0.119409831 0.089032111 0.111065767 0.147742102 0.169569538 0.095899426
## 2041 2042 2043 2044 2045 2046
## 0.128290293 0.065804943 0.128290293 0.137727896 0.119409831 0.119409831
## 2047 2048 2049 2050 2051 2052
## 0.103236410 0.103236410 0.111065767 0.095899426 0.119409831 0.111065767
## 2053 2054 2055 2056 2057 2058
## 0.111065767 0.089032111 0.082611626 0.076615209 0.082611626 0.095899426
## 2059 2060 2061 2062 2063 2064
## 0.065804943 0.095899426 0.111065767 0.103236410 0.111065767 0.089032111
## 2065 2066 2067 2068 2069 2070
## 0.103236410 0.158350725 0.082611626 0.082611626 0.082611626 0.128290293
## 2071 2072 2073 2074 2075 2076
## 0.119409831 0.082611626 0.147742102 0.128290293 0.103236410 0.137727896
## 2077 2078 2079 2080 2081 2082
## 0.147742102 0.071020351 0.089032111 0.071020351 0.095899426 0.128290293
## 2083 2084 2085 2086 2087 2088
## 0.071020351 0.065804943 0.103236410 0.128290293 0.119409831 0.103236410
## 2089 2090 2091 2092 2093 2094
## 0.095899426 0.076615209 0.111065767 0.181411860 0.095899426 0.128290293
## 2095 2096 2097 2098 2099 2100
## 0.082611626 0.119409831 0.071020351 0.111065767 0.095899426 0.147742102
## 2101 2102 2103 2104 2105 2106
## 0.082611626 0.082611626 0.128290293 0.052222807 0.103236410 0.089032111
## 2107 2108 2109 2110 2111 2112
## 0.095899426 0.103236410 0.089032111 0.103236410 0.065804943 0.076615209
## 2113 2114 2115 2116 2117 2118
## 0.082611626 0.065804943 0.119409831 0.065804943 0.082611626 0.111065767
## 2119 2120 2121 2122 2123 2124
## 0.111065767 0.128290293 0.095899426 0.082611626 0.071020351 0.111065767
## 2125 2126 2127 2128 2129 2130
## 0.060947403 0.103236410 0.076615209 0.169569538 0.089032111 0.060947403
## 2131 2132 2133 2134 2135 2136
## 0.076615209 0.137727896 0.111065767 0.119409831 0.169569538 0.089032111
## 2137 2138 2139 2140 2141 2142
## 0.089032111 0.103236410 0.119409831 0.082611626 0.076615209 0.065804943
## 2143 2144 2145 2146 2147 2148
## 0.082611626 0.103236410 0.103236410 0.119409831 0.095899426 0.060947403
## 2149 2150 2151 2152 2153 2154
## 0.095899426 0.158350725 0.103236410 0.111065767 0.089032111 0.147742102
## 2155 2156 2157 2158 2159 2160
## 0.076615209 0.076615209 0.071020351 0.089032111 0.103236410 0.137727896
## 2161 2162 2163 2164 2165 2166
## 0.095899426 0.089032111 0.119409831 0.082611626 0.119409831 0.103236410
## 2167 2168 2169 2170 2171 2172
## 0.071020351 0.089032111 0.089032111 0.076615209 0.103236410 0.119409831
## 2173 2174 2175 2176 2177 2178
## 0.082611626 0.137727896 0.076615209 0.128290293 0.089032111 0.089032111
## 2179 2180 2181 2182 2183 2184
## 0.082611626 0.128290293 0.082611626 0.060947403 0.089032111 0.065804943
## 2185 2186 2187 2188 2189 2190
## 0.111065767 0.103236410 0.052222807 0.158350725 0.095899426 0.137727896
## 2191 2192 2193 2194 2195 2196
## 0.103236410 0.111065767 0.147742102 0.089032111 0.103236410 0.111065767
## 2197 2198 2199 2200 2201 2202
## 0.137727896 0.128290293 0.128290293 0.103236410 0.111065767 0.095899426
## 2203 2204 2205 2206 2207 2208
## 0.147742102 0.095899426 0.137727896 0.147742102 0.071020351 0.089032111
## 2209 2210 2211 2212 2213 2214
## 0.103236410 0.119409831 0.089032111 0.071020351 0.082611626 0.015880597
## 2215 2216 2217 2218 2219 2220
## 0.119409831 0.147742102 0.119409831 0.103236410 0.103236410 0.071020351
## 2221 2222 2223 2224 2225 2226
## 0.169569538 0.089032111 0.158350725 0.082611626 0.095899426 0.056426775
## 2227 2228 2229 2230 2231 2232
## 0.056426775 0.071020351 0.111065767 0.076615209 0.082611626 0.137727896
## 2233 2234 2235 2236 2237 2238
## 0.071020351 0.119409831 0.076615209 0.119409831 0.089032111 0.076615209
## 2239 2240 2241 2242 2243 2244
## 0.095899426 0.111065767 0.169569538 0.095899426 0.103236410 0.076615209
## 2245 2246 2247 2248 2249 2250
## 0.060947403 0.082611626 0.137727896 0.128290293 0.111065767 0.128290293
## 2251 2252 2253 2254 2255 2256
## 0.089032111 0.076615209 0.128290293 0.119409831 0.089032111 0.095899426
## 2257 2258 2259 2260 2261 2262
## 0.095899426 0.095899426 0.119409831 0.119409831 0.111065767 0.111065767
## 2263 2264 2265 2266 2267 2268
## 0.076615209 0.095899426 0.119409831 0.137727896 0.111065767 0.076615209
## 2269 2270 2271 2272 2273 2274
## 0.119409831 0.095899426 0.111065767 0.128290293 0.082611626 0.082611626
## 2275 2276 2277 2278 2279 2280
## 0.076615209 0.082611626 0.082611626 0.111065767 0.128290293 0.111065767
## 2281 2282 2283 2284 2285 2286
## 0.095899426 0.158350725 0.071020351 0.111065767 0.095899426 0.158350725
## 2287 2288 2289 2290 2291 2292
## 0.103236410 0.111065767 0.060947403 0.111065767 0.103236410 0.076615209
## 2293 2294 2295 2296 2297 2298
## 0.111065767 0.076615209 0.071020351 0.111065767 0.147742102 0.111065767
## 2299 2300 2301 2302 2303 2304
## 0.089032111 0.089032111 0.089032111 0.128290293 0.095899426 0.137727896
## 2305 2306 2307 2308 2309 2310
## 0.128290293 0.089032111 0.076615209 0.103236410 0.111065767 0.103236410
## 2311 2312 2313 2314 2315 2316
## 0.095899426 0.089032111 0.147742102 0.111065767 0.137727896 0.103236410
## 2317 2318 2319 2320 2321 2322
## 0.111065767 0.111065767 0.103236410 0.111065767 0.089032111 0.071020351
## 2323 2324 2325 2326 2327 2328
## 0.169569538 0.095899426 0.071020351 0.128290293 0.119409831 0.052222807
## 2329 2330 2331 2332 2333 2334
## 0.181411860 0.103236410 0.065804943 0.128290293 0.103236410 0.089032111
## 2335 2336 2337 2338 2339 2340
## 0.089032111 0.103236410 0.071020351 0.023723042 0.147742102 0.089032111
## 2341 2342 2343 2344 2345 2346
## 0.095899426 0.008312821 0.082611626 0.095899426 0.103236410 0.111065767
## 2347 2348 2349 2350 2351 2352
## 0.111065767 0.147742102 0.071020351 0.128290293 0.095899426 0.111065767
## 2353 2354 2355 2356 2357 2358
## 0.095899426 0.056426775 0.103236410 0.119409831 0.095899426 0.103236410
## 2359 2360 2361 2362 2363 2364
## 0.137727896 0.103236410 0.103236410 0.065804943 0.103236410 0.076615209
## 2365 2366 2367 2368 2369 2370
## 0.103236410 0.095899426 0.095899426 0.111065767 0.095899426 0.137727896
## 2371 2372 2373 2374 2375 2376
## 0.095899426 0.060947403 0.089032111 0.082611626 0.119409831 0.095899426
## 2377 2378 2379 2380 2381 2382
## 0.119409831 0.095899426 0.095899426 0.111065767 0.082611626 0.111065767
## 2383 2384 2385 2386 2387 2388
## 0.128290293 0.082611626 0.095899426 0.060947403 0.076615209 0.158350725
## 2389 2390 2391 2392 2393 2394
## 0.128290293 0.128290293 0.065804943 0.147742102 0.076615209 0.095899426
## 2395 2396 2397 2398 2399 2400
## 0.103236410 0.076615209 0.071020351 0.111065767 0.103236410 0.111065767
## 2401 2402 2403 2404 2405 2406
## 0.095899426 0.128290293 0.119409831 0.111065767 0.089032111 0.128290293
## 2407 2408 2409 2410 2411 2412
## 0.119409831 0.071020351 0.119409831 0.082611626 0.128290293 0.147742102
## 2413 2414 2415 2416 2417 2418
## 0.089032111 0.147742102 0.089032111 0.076615209 0.111065767 0.128290293
## 2419 2420 2421 2422 2423 2424
## 0.137727896 0.089032111 0.095899426 0.137727896 0.128290293 0.111065767
## 2425 2426 2427 2428 2429 2430
## 0.076615209 0.071020351 0.020212403 0.111065767 0.119409831 0.111065767
## 2431 2432 2433 2434 2435 2436
## 0.147742102 0.095899426 0.119409831 0.147742102 0.060947403 0.128290293
## 2437 2438 2439 2440 2441 2442
## 0.082611626 0.119409831 0.111065767 0.103236410 0.065804943 0.119409831
## 2443 2444 2445 2446 2447 2448
## 0.060947403 0.111065767 0.103236410 0.193888125 0.076615209 0.137727896
## 2449 2450 2451 2452 2453 2454
## 0.065804943 0.095899426 0.111065767 0.095899426 0.111065767 0.147742102
## 2455 2456 2457 2458 2459 2460
## 0.044687700 0.095899426 0.048316010 0.119409831 0.065804943 0.128290293
## 2461 2462 2463 2464 2465 2466
## 0.103236410 0.089032111 0.169569538 0.111065767 0.082611626 0.111065767
## 2467 2468 2469 2470 2471 2472
## 0.128290293 0.082611626 0.128290293 0.071020351 0.169569538 0.082611626
## 2473 2474 2475 2476 2477 2478
## 0.147742102 0.111065767 0.119409831 0.128290293 0.071020351 0.076615209
## 2479 2480 2481 2482 2483 2484
## 0.095899426 0.095899426 0.128290293 0.119409831 0.128290293 0.056426775
## 2485 2486 2487 2488 2489 2490
## 0.158350725 0.095899426 0.082611626 0.076615209 0.111065767 0.111065767
## 2491 2492 2493 2494 2495 2496
## 0.082611626 0.147742102 0.071020351 0.071020351 0.137727896 0.089032111
## 2497 2498 2499 2500 2501 2502
## 0.095899426 0.103236410 0.076615209 0.103236410 0.147742102 0.089032111
## 2503 2504 2505 2506 2507 2508
## 0.103236410 0.076615209 0.021899054 0.147742102 0.128290293 0.060947403
## 2509 2510 2511 2512 2513 2514
## 0.089032111 0.119409831 0.111065767 0.082611626 0.095899426 0.082611626
## 2515 2516 2517 2518 2519 2520
## 0.095899426 0.089032111 0.111065767 0.119409831 0.076615209 0.111065767
## 2521 2522 2523 2524 2525 2526
## 0.158350725 0.089032111 0.119409831 0.137727896 0.065804943 0.095899426
## 2527 2528 2529 2530 2531 2532
## 0.082611626 0.071020351 0.095899426 0.095899426 0.076615209 0.095899426
## 2533 2534 2535 2536 2537 2538
## 0.082611626 0.082611626 0.089032111 0.065804943 0.071020351 0.103236410
## 2539 2540 2541 2542 2543 2544
## 0.089032111 0.095899426 0.128290293 0.076615209 0.095899426 0.082611626
## 2545 2546 2547 2548 2549 2550
## 0.071020351 0.119409831 0.082611626 0.089032111 0.128290293 0.103236410
## 2551 2552 2553 2554 2555 2556
## 0.119409831 0.147742102 0.095899426 0.095899426 0.095899426 0.095899426
## 2557 2558 2559 2560 2561 2562
## 0.147742102 0.071020351 0.089032111 0.095899426 0.076615209 0.089032111
## 2563 2564 2565 2566 2567 2568
## 0.076615209 0.119409831 0.095899426 0.169569538 0.095899426 0.111065767
## 2569 2570 2571 2572 2573 2574
## 0.111065767 0.147742102 0.158350725 0.089032111 0.089032111 0.060947403
## 2575 2576 2577 2578 2579 2580
## 0.147742102 0.103236410 0.089032111 0.111065767 0.071020351 0.089032111
## 2581 2582 2583 2584 2585 2586
## 0.119409831 0.071020351 0.089032111 0.169569538 0.158350725 0.137727896
## 2587 2588 2589 2590 2591 2592
## 0.137727896 0.095899426 0.095899426 0.119409831 0.076615209 0.128290293
## 2593 2594 2595 2596 2597 2598
## 0.119409831 0.103236410 0.103236410 0.103236410 0.021899054 0.103236410
## 2599 2600 2601 2602 2603 2604
## 0.071020351 0.089032111 0.119409831 0.111065767 0.082611626 0.076615209
## 2605 2606 2607 2608 2609 2610
## 0.103236410 0.071020351 0.137727896 0.103236410 0.071020351 0.071020351
## 2611 2612 2613 2614 2615 2616
## 0.082611626 0.103236410 0.095899426 0.103236410 0.082611626 0.103236410
## 2617 2618 2619 2620 2621 2622
## 0.076615209 0.089032111 0.060947403 0.137727896 0.071020351 0.076615209
## 2623 2624 2625 2626 2627 2628
## 0.089032111 0.147742102 0.089032111 0.095899426 0.103236410 0.076615209
## 2629 2630 2631 2632 2633 2634
## 0.082611626 0.137727896 0.169569538 0.089032111 0.076615209 0.089032111
## 2635 2636 2637 2638 2639 2640
## 0.065804943 0.065804943 0.095899426 0.103236410 0.060947403 0.095899426
## 2641 2642 2643 2644 2645 2646
## 0.089032111 0.089032111 0.095899426 0.111065767 0.089032111 0.111065767
## 2647 2648 2649 2650 2651 2652
## 0.071020351 0.095899426 0.111065767 0.119409831 0.076615209 0.158350725
## 2653 2654 2655 2656 2657 2658
## 0.137727896 0.111065767 0.089032111 0.076615209 0.065804943 0.071020351
## 2659 2660 2661 2662 2663 2664
## 0.065804943 0.119409831 0.076615209 0.119409831 0.103236410 0.089032111
## 2665 2666 2667 2668 2669 2670
## 0.137727896 0.095899426 0.082611626 0.128290293 0.128290293 0.095899426
## 2671 2672 2673 2674 2675 2676
## 0.119409831 0.111065767 0.082611626 0.111065767 0.089032111 0.089032111
## 2677 2678 2679 2680 2681 2682
## 0.128290293 0.128290293 0.089032111 0.103236410 0.137727896 0.103236410
## 2683 2684 2685 2686 2687 2688
## 0.119409831 0.128290293 0.095899426 0.128290293 0.082611626 0.082611626
## 2689 2690 2691 2692 2693 2694
## 0.103236410 0.089032111 0.082611626 0.119409831 0.111065767 0.082611626
## 2695 2696 2697 2698 2699 2700
## 0.137727896 0.060947403 0.082611626 0.111065767 0.095899426 0.103236410
## 2701 2702 2703 2704 2705 2706
## 0.111065767 0.119409831 0.111065767 0.089032111 0.103236410 0.089032111
## 2707 2708 2709 2710 2711 2712
## 0.111065767 0.128290293 0.158350725 0.158350725 0.082611626 0.119409831
## 2713 2714 2715 2716 2717 2718
## 0.103236410 0.089032111 0.137727896 0.111065767 0.065804943 0.095899426
## 2719 2720 2721 2722 2723 2724
## 0.060947403 0.082611626 0.095899426 0.103236410 0.119409831 0.111065767
## 2725 2726 2727 2728 2729 2730
## 0.128290293 0.071020351 0.082611626 0.169569538 0.095899426 0.076615209
## 2731 2732 2733 2734 2735 2736
## 0.095899426 0.103236410 0.082611626 0.119409831 0.169569538 0.119409831
## 2737 2738 2739 2740 2741 2742
## 0.082611626 0.089032111 0.128290293 0.103236410 0.065804943 0.060947403
## 2743 2744 2745 2746 2747 2748
## 0.103236410 0.111065767 0.089032111 0.128290293 0.089032111 0.111065767
## 2749 2750 2751 2752 2753 2754
## 0.119409831 0.103236410 0.103236410 0.095899426 0.119409831 0.111065767
## 2755 2756 2757 2758 2759 2760
## 0.119409831 0.076615209 0.137727896 0.082611626 0.082611626 0.103236410
## 2761 2762 2763 2764 2765 2766
## 0.095899426 0.137727896 0.082611626 0.089032111 0.111065767 0.082611626
## 2767 2768 2769 2770 2771 2772
## 0.095899426 0.082611626 0.103236410 0.128290293 0.147742102 0.076615209
## 2773 2774 2775 2776 2777 2778
## 0.147742102 0.111065767 0.128290293 0.111065767 0.071020351 0.076615209
## 2779 2780 2781 2782 2783 2784
## 0.119409831 0.119409831 0.082611626 0.095899426 0.207005441 0.181411860
## 2785 2786 2787 2788 2789 2790
## 0.128290293 0.103236410 0.169569538 0.103236410 0.076615209 0.103236410
## 2791 2792 2793 2794 2795 2796
## 0.089032111 0.095899426 0.181411860 0.056426775 0.181411860 0.076615209
## 2797 2798 2799 2800 2801 2802
## 0.076615209 0.111065767 0.089032111 0.119409831 0.095899426 0.095899426
## 2803 2804 2805 2806 2807 2808
## 0.103236410 0.095899426 0.020212403 0.111065767 0.103236410 0.137727896
## 2809 2810 2811 2812 2813 2814
## 0.095899426 0.119409831 0.103236410 0.103236410 0.103236410 0.128290293
## 2815 2816 2817 2818 2819 2820
## 0.119409831 0.082611626 0.103236410 0.095899426 0.119409831 0.137727896
## 2821 2822 2823 2824 2825 2826
## 0.071020351 0.095899426 0.076615209 0.082611626 0.095899426 0.095899426
## 2827 2828 2829 2830 2831 2832
## 0.095899426 0.076615209 0.082611626 0.103236410 0.158350725 0.128290293
## 2833 2834 2835 2836 2837 2838
## 0.071020351 0.103236410 0.119409831 0.128290293 0.111065767 0.147742102
## 2839 2840 2841 2842 2843 2844
## 0.082611626 0.128290293 0.082611626 0.111065767 0.147742102 0.137727896
## 2845 2846 2847 2848 2849 2850
## 0.111065767 0.193888125 0.082611626 0.137727896 0.119409831 0.103236410
## 2851 2852 2853 2854 2855 2856
## 0.065804943 0.032615142 0.076615209 0.065804943 0.119409831 0.103236410
## 2857 2858 2859 2860 2861 2862
## 0.103236410 0.089032111 0.158350725 0.128290293 0.128290293 0.095899426
## 2863 2864 2865 2866 2867 2868
## 0.147742102 0.111065767 0.082611626 0.103236410 0.082611626 0.119409831
## 2869 2870 2871 2872 2873 2874
## 0.119409831 0.082611626 0.071020351 0.119409831 0.076615209 0.076615209
## 2875 2876 2877 2878 2879 2880
## 0.071020351 0.128290293 0.071020351 0.076615209 0.111065767 0.076615209
## 2881 2882 2883 2884 2885 2886
## 0.103236410 0.089032111 0.103236410 0.076615209 0.147742102 0.103236410
## 2887 2888 2889 2890 2891 2892
## 0.082611626 0.071020351 0.089032111 0.082611626 0.128290293 0.119409831
## 2893 2894 2895 2896 2897 2898
## 0.111065767 0.095899426 0.071020351 0.119409831 0.169569538 0.137727896
## 2899 2900 2901 2902 2903 2904
## 0.111065767 0.065804943 0.076615209 0.181411860 0.082611626 0.137727896
## 2905 2906 2907 2908 2909 2910
## 0.103236410 0.119409831 0.128290293 0.076615209 0.082611626 0.137727896
## 2911 2912 2913 2914 2915 2916
## 0.095899426 0.103236410 0.111065767 0.137727896 0.119409831 0.111065767
## 2917 2918 2919 2920 2921 2922
## 0.119409831 0.119409831 0.137727896 0.119409831 0.147742102 0.111065767
## 2923 2924 2925 2926 2927 2928
## 0.076615209 0.128290293 0.103236410 0.095899426 0.137727896 0.128290293
## 2929 2930 2931 2932 2933 2934
## 0.060947403 0.095899426 0.095899426 0.089032111 0.158350725 0.089032111
## 2935 2936 2937 2938 2939 2940
## 0.158350725 0.103236410 0.137727896 0.095899426 0.095899426 0.103236410
## 2941 2942 2943 2944 2945 2946
## 0.089032111 0.128290293 0.103236410 0.082611626 0.071020351 0.076615209
## 2947 2948 2949 2950 2951 2952
## 0.147742102 0.103236410 0.128290293 0.128290293 0.082611626 0.095899426
## 2953 2954 2955 2956 2957 2958
## 0.082611626 0.119409831 0.181411860 0.089032111 0.103236410 0.095899426
## 2959 2960 2961 2962 2963 2964
## 0.147742102 0.119409831 0.089032111 0.119409831 0.095899426 0.095899426
## 2965 2966 2967 2968 2969 2970
## 0.076615209 0.076615209 0.111065767 0.076615209 0.111065767 0.111065767
## 2971 2972 2973 2974 2975 2976
## 0.119409831 0.111065767 0.103236410 0.111065767 0.089032111 0.111065767
## 2977 2978 2979 2980 2981 2982
## 0.128290293 0.082611626 0.147742102 0.103236410 0.169569538 0.111065767
## 2983 2984 2985 2986 2987 2988
## 0.071020351 0.137727896 0.089032111 0.111065767 0.089032111 0.095899426
## 2989 2990 2991 2992 2993 2994
## 0.111065767 0.137727896 0.082611626 0.095899426 0.095899426 0.103236410
## 2995 2996 2997 2998 2999 3000
## 0.103236410 0.089032111 0.095899426 0.103236410 0.082611626 0.082611626
## 3001 3002 3003 3004 3005 3006
## 0.111065767 0.158350725 0.095899426 0.128290293 0.111065767 0.065804943
## 3007 3008 3009 3010 3011 3012
## 0.111065767 0.119409831 0.095899426 0.089032111 0.071020351 0.137727896
## 3013 3014 3015 3016 3017 3018
## 0.119409831 0.089032111 0.147742102 0.111065767 0.089032111 0.103236410
## 3019 3020 3021 3022 3023 3024
## 0.012465347 0.076615209 0.158350725 0.095899426 0.111065767 0.076615209
## 3025 3026 3027 3028 3029 3030
## 0.111065767 0.169569538 0.169569538 0.111065767 0.103236410 0.103236410
## 3031 3032 3033 3034 3035 3036
## 0.128290293 0.111065767 0.095899426 0.082611626 0.056426775 0.023723042
## 3037 3038 3039 3040 3041 3042
## 0.065804943 0.158350725 0.095899426 0.082611626 0.076615209 0.076615209
## 3043 3044 3045 3046 3047 3048
## 0.095899426 0.082611626 0.082611626 0.147742102 0.103236410 0.095899426
## 3049 3050 3051 3052 3053 3054
## 0.095899426 0.128290293 0.095899426 0.089032111 0.082611626 0.065804943
## 3055 3056 3057 3058 3059 3060
## 0.111065767 0.095899426 0.095899426 0.119409831 0.111065767 0.147742102
## 3061 3062 3063 3064 3065 3066
## 0.103236410 0.065804943 0.119409831 0.082611626 0.095899426 0.103236410
## 3067 3068 3069 3070 3071 3072
## 0.111065767 0.119409831 0.111065767 0.111065767 0.169569538 0.111065767
## 3073 3074 3075 3076 3077 3078
## 0.095899426 0.071020351 0.095899426 0.103236410 0.147742102 0.111065767
## 3079 3080 3081 3082 3083 3084
## 0.089032111 0.128290293 0.137727896 0.071020351 0.076615209 0.128290293
## 3085 3086 3087 3088 3089 3090
## 0.119409831 0.056426775 0.089032111 0.119409831 0.169569538 0.089032111
## 3091 3092 3093 3094 3095 3096
## 0.119409831 0.103236410 0.082611626 0.111065767 0.089032111 0.089032111
## 3097 3098 3099 3100 3101 3102
## 0.095899426 0.082611626 0.111065767 0.119409831 0.119409831 0.119409831
## 3103 3104 3105 3106 3107 3108
## 0.089032111 0.089032111 0.119409831 0.052222807 0.071020351 0.103236410
## 3109 3110 3111 3112 3113 3114
## 0.060947403 0.111065767 0.111065767 0.111065767 0.076615209 0.111065767
## 3115 3116 3117 3118 3119 3120
## 0.119409831 0.089032111 0.089032111 0.095899426 0.128290293 0.065804943
## 3121 3122 3123 3124 3125 3126
## 0.111065767 0.103236410 0.137727896 0.158350725 0.076615209 0.158350725
## 3127 3128 3129 3130 3131 3132
## 0.147742102 0.060947403 0.119409831 0.128290293 0.111065767 0.052222807
## 3133 3134 3135 3136 3137 3138
## 0.103236410 0.076615209 0.119409831 0.095899426 0.095899426 0.103236410
## 3139 3140 3141 3142 3143 3144
## 0.128290293 0.169569538 0.065804943 0.089032111 0.082611626 0.128290293
## 3145 3146 3147 3148 3149 3150
## 0.128290293 0.119409831 0.147742102 0.071020351 0.128290293 0.103236410
## 3151 3152 3153 3154 3155 3156
## 0.128290293 0.158350725 0.071020351 0.082611626 0.082611626 0.076615209
## 3157 3158 3159 3160 3161 3162
## 0.095899426 0.095899426 0.082611626 0.065804943 0.089032111 0.076615209
## 3163 3164 3165 3166 3167 3168
## 0.128290293 0.095899426 0.089032111 0.082611626 0.111065767 0.119409831
## 3169 3170 3171 3172 3173 3174
## 0.111065767 0.158350725 0.076615209 0.169569538 0.111065767 0.158350725
## 3175 3176 3177 3178 3179 3180
## 0.095899426 0.095899426 0.119409831 0.119409831 0.169569538 0.103236410
## 3181 3182 3183 3184 3185 3186
## 0.119409831 0.082611626 0.128290293 0.095899426 0.089032111 0.119409831
## 3187 3188 3189 3190 3191 3192
## 0.076615209 0.119409831 0.119409831 0.095899426 0.111065767 0.119409831
## 3193 3194 3195 3196 3197 3198
## 0.076615209 0.095899426 0.082611626 0.147742102 0.158350725 0.082611626
## 3199 3200 3201 3202 3203 3204
## 0.147742102 0.119409831 0.095899426 0.082611626 0.095899426 0.089032111
## 3205 3206 3207 3208 3209 3210
## 0.065804943 0.076615209 0.137727896 0.111065767 0.103236410 0.111065767
## 3211 3212 3213 3214 3215 3216
## 0.089032111 0.071020351 0.137727896 0.103236410 0.128290293 0.095899426
## 3217 3218 3219 3220 3221 3222
## 0.076615209 0.103236410 0.103236410 0.128290293 0.137727896 0.071020351
## 3223 3224 3225 3226 3227 3228
## 0.137727896 0.103236410 0.095899426 0.128290293 0.119409831 0.095899426
## 3229 3230 3231 3232 3233 3234
## 0.111065767 0.065804943 0.095899426 0.071020351 0.082611626 0.111065767
## 3235 3236 3237 3238 3239 3240
## 0.111065767 0.111065767 0.119409831 0.076615209 0.111065767 0.137727896
## 3241 3242 3243 3244 3245 3246
## 0.193888125 0.158350725 0.119409831 0.103236410 0.103236410 0.147742102
## 3247 3248 3249 3250 3251 3252
## 0.023723042 0.076615209 0.082611626 0.076615209 0.076615209 0.082611626
## 3253 3254 3255 3256 3257 3258
## 0.103236410 0.169569538 0.065804943 0.103236410 0.147742102 0.103236410
## 3259 3260 3261 3262 3263 3264
## 0.119409831 0.158350725 0.158350725 0.169569538 0.095899426 0.119409831
## 3265 3266 3267 3268 3269 3270
## 0.103236410 0.103236410 0.060947403 0.128290293 0.128290293 0.103236410
## 3271 3272 3273 3274 3275 3276
## 0.060947403 0.089032111 0.147742102 0.089032111 0.095899426 0.060947403
## 3277 3278 3279 3280 3281 3282
## 0.095899426 0.065804943 0.082611626 0.089032111 0.095899426 0.076615209
## 3283 3284 3285 3286 3287 3288
## 0.128290293 0.076615209 0.119409831 0.095899426 0.103236410 0.103236410
## 3289 3290 3291 3292 3293 3294
## 0.103236410 0.137727896 0.095899426 0.111065767 0.103236410 0.095899426
## 3295 3296 3297 3298 3299 3300
## 0.071020351 0.082611626 0.147742102 0.089032111 0.158350725 0.193888125
## 3301 3302 3303 3304 3305 3306
## 0.137727896 0.076615209 0.082611626 0.128290293 0.137727896 0.137727896
## 3307 3308 3309 3310 3311 3312
## 0.076615209 0.119409831 0.095899426 0.082611626 0.095899426 0.089032111
## 3313 3314 3315 3316 3317 3318
## 0.082611626 0.103236410 0.089032111 0.095899426 0.103236410 0.103236410
## 3319 3320 3321 3322 3323 3324
## 0.128290293 0.137727896 0.103236410 0.082611626 0.076615209 0.060947403
## 3325 3326 3327 3328 3329 3330
## 0.082611626 0.181411860 0.128290293 0.103236410 0.137727896 0.111065767
## 3331 3332 3333 3334 3335 3336
## 0.147742102 0.119409831 0.065804943 0.082611626 0.147742102 0.095899426
## 3337 3338 3339 3340 3341 3342
## 0.060947403 0.010602578 0.111065767 0.089032111 0.082611626 0.065804943
## 3343 3344 3345 3346 3347 3348
## 0.089032111 0.076615209 0.119409831 0.147742102 0.119409831 0.095899426
## 3349 3350 3351 3352 3353 3354
## 0.119409831 0.071020351 0.103236410 0.095899426 0.095899426 0.111065767
## 3355 3356 3357 3358 3359 3360
## 0.147742102 0.119409831 0.169569538 0.082611626 0.095899426 0.071020351
## 3361 3362 3363 3364 3365 3366
## 0.137727896 0.137727896 0.103236410 0.089032111 0.056426775 0.103236410
## 3367 3368 3369 3370 3371 3372
## 0.089032111 0.119409831 0.119409831 0.137727896 0.119409831 0.193888125
## 3373 3374 3375 3376 3377 3378
## 0.082611626 0.111065767 0.137727896 0.147742102 0.089032111 0.111065767
## 3379 3380 3381 3382 3383 3384
## 0.095899426 0.147742102 0.128290293 0.119409831 0.128290293 0.119409831
## 3385 3386 3387 3388 3389 3390
## 0.119409831 0.082611626 0.076615209 0.119409831 0.103236410 0.095899426
## 3391 3392 3393 3394 3395 3396
## 0.065804943 0.128290293 0.076615209 0.082611626 0.095899426 0.095899426
## 3397 3398 3399 3400 3401 3402
## 0.095899426 0.181411860 0.060947403 0.147742102 0.128290293 0.052222807
## 3403 3404 3405 3406 3407 3408
## 0.082611626 0.089032111 0.060947403 0.147742102 0.082611626 0.076615209
## 3409 3410 3411 3412 3413 3414
## 0.095899426 0.111065767 0.103236410 0.076615209 0.060947403 0.119409831
## 3415 3416 3417 3418 3419 3420
## 0.169569538 0.089032111 0.119409831 0.089032111 0.082611626 0.119409831
## 3421 3422 3423 3424 3425 3426
## 0.207005441 0.095899426 0.103236410 0.082611626 0.095899426 0.147742102
## 3427 3428 3429 3430 3431 3432
## 0.103236410 0.111065767 0.169569538 0.056426775 0.103236410 0.076615209
## 3433 3434 3435 3436 3437 3438
## 0.137727896 0.103236410 0.095899426 0.119409831 0.071020351 0.158350725
## 3439 3440 3441 3442 3443 3444
## 0.169569538 0.071020351 0.071020351 0.082611626 0.103236410 0.076615209
## 3445 3446 3447 3448 3449 3450
## 0.111065767 0.071020351 0.082611626 0.076615209 0.128290293 0.111065767
## 3451 3452 3453 3454 3455 3456
## 0.128290293 0.103236410 0.103236410 0.128290293 0.158350725 0.008312821
## 3457 3458 3459 3460 3461 3462
## 0.103236410 0.071020351 0.111065767 0.095899426 0.128290293 0.089032111
## 3463 3464 3465 3466 3467 3468
## 0.119409831 0.111065767 0.065804943 0.111065767 0.119409831 0.076615209
## 3469 3470 3471 3472 3473 3474
## 0.193888125 0.095899426 0.089032111 0.137727896 0.111065767 0.089032111
## 3475 3476 3477 3478 3479 3480
## 0.089032111 0.137727896 0.119409831 0.052222807 0.103236410 0.082611626
## 3481 3482 3483 3484 3485 3486
## 0.169569538 0.095899426 0.095899426 0.082611626 0.095899426 0.089032111
## 3487 3488 3489 3490 3491 3492
## 0.147742102 0.128290293 0.076615209 0.095899426 0.071020351 0.128290293
## 3493 3494 3495 3496 3497 3498
## 0.065804943 0.089032111 0.082611626 0.089032111 0.128290293 0.111065767
## 3499 3500 3501 3502 3503 3504
## 0.089032111 0.095899426 0.119409831 0.147742102 0.056426775 0.169569538
## 3505 3506 3507 3508 3509 3510
## 0.137727896 0.111065767 0.103236410 0.095899426 0.071020351 0.111065767
## 3511 3512 3513 3514 3515 3516
## 0.137727896 0.076615209 0.103236410 0.111065767 0.111065767 0.095899426
## 3517 3518 3519 3520 3521 3522
## 0.111065767 0.095899426 0.095899426 0.095899426 0.111065767 0.111065767
## 3523 3524 3525 3526 3527 3528
## 0.082611626 0.095899426 0.119409831 0.082611626 0.111065767 0.095899426
## 3529 3530 3531 3532 3533 3534
## 0.128290293 0.089032111 0.119409831 0.089032111 0.111065767 0.095899426
## 3535 3536 3537 3538 3539 3540
## 0.082611626 0.119409831 0.119409831 0.095899426 0.103236410 0.071020351
## 3541 3542 3543 3544 3545 3546
## 0.082611626 0.071020351 0.076615209 0.128290293 0.089032111 0.065804943
## 3547 3548 3549 3550 3551 3552
## 0.082611626 0.095899426 0.076615209 0.076615209 0.111065767 0.095899426
## 3553 3554 3555 3556 3557 3558
## 0.082611626 0.071020351 0.111065767 0.119409831 0.111065767 0.119409831
## 3559 3560 3561 3562 3563 3564
## 0.071020351 0.095899426 0.095899426 0.103236410 0.095899426 0.103236410
## 3565 3566 3567 3568 3569 3570
## 0.089032111 0.119409831 0.082611626 0.128290293 0.111065767 0.128290293
## 3571 3572 3573 3574 3575 3576
## 0.111065767 0.103236410 0.095899426 0.076615209 0.103236410 0.089032111
## 3577 3578 3579 3580 3581 3582
## 0.103236410 0.158350725 0.095899426 0.103236410 0.128290293 0.137727896
## 3583 3584 3585 3586 3587 3588
## 0.071020351 0.089032111 0.111065767 0.128290293 0.158350725 0.076615209
## 3589 3590 3591 3592 3593 3594
## 0.137727896 0.119409831 0.076615209 0.044687700 0.065804943 0.119409831
## 3595 3596 3597 3598 3599 3600
## 0.111065767 0.111065767 0.060947403 0.147742102 0.111065767 0.076615209
## 3601 3602 3603 3604 3605 3606
## 0.103236410 0.089032111 0.071020351 0.071020351 0.119409831 0.076615209
## 3607 3608 3609 3610 3611 3612
## 0.128290293 0.158350725 0.089032111 0.119409831 0.089032111 0.128290293
## 3613 3614 3615 3616 3617 3618
## 0.089032111 0.095899426 0.137727896 0.111065767 0.111065767 0.103236410
## 3619 3620 3621 3622 3623 3624
## 0.147742102 0.111065767 0.095899426 0.095899426 0.137727896 0.111065767
## 3625 3626 3627 3628 3629 3630
## 0.071020351 0.082611626 0.137727896 0.119409831 0.119409831 0.111065767
## 3631 3632 3633 3634 3635 3636
## 0.095899426 0.128290293 0.111065767 0.095899426 0.128290293 0.147742102
## 3637 3638 3639 3640 3641 3642
## 0.103236410 0.103236410 0.119409831 0.181411860 0.111065767 0.095899426
## 3643 3644 3645 3646 3647 3648
## 0.137727896 0.095899426 0.095899426 0.111065767 0.095899426 0.111065767
## 3649 3650 3651 3652 3653 3654
## 0.089032111 0.128290293 0.095899426 0.111065767 0.095899426 0.147742102
## 3655 3656 3657 3658 3659 3660
## 0.095899426 0.089032111 0.082611626 0.119409831 0.095899426 0.103236410
## 3661 3662 3663 3664 3665 3666
## 0.128290293 0.103236410 0.111065767 0.103236410 0.137727896 0.089032111
## 3667 3668 3669 3670 3671 3672
## 0.082611626 0.082611626 0.119409831 0.128290293 0.103236410 0.158350725
## 3673 3674 3675 3676 3677 3678
## 0.082611626 0.095899426 0.128290293 0.095899426 0.095899426 0.119409831
## 3679 3680 3681 3682 3683 3684
## 0.089032111 0.137727896 0.089032111 0.158350725 0.089032111 0.147742102
## 3685 3686 3687 3688 3689 3690
## 0.193888125 0.076615209 0.089032111 0.095899426 0.158350725 0.103236410
## 3691 3692 3693 3694 3695 3696
## 0.095899426 0.071020351 0.076615209 0.103236410 0.095899426 0.128290293
## 3697 3698 3699 3700 3701 3702
## 0.065804943 0.119409831 0.103236410 0.137727896 0.137727896 0.089032111
## 3703 3704 3705 3706 3707 3708
## 0.095899426 0.095899426 0.089032111 0.065804943 0.111065767 0.095899426
## 3709 3710 3711 3712 3713 3714
## 0.119409831 0.103236410 0.082611626 0.111065767 0.128290293 0.089032111
## 3715 3716 3717 3718 3719 3720
## 0.076615209 0.095899426 0.095899426 0.111065767 0.082611626 0.119409831
## 3721 3722 3723 3724 3725 3726
## 0.137727896 0.082611626 0.103236410 0.137727896 0.128290293 0.169569538
## 3727 3728 3729 3730 3731 3732
## 0.147742102 0.119409831 0.089032111 0.095899426 0.111065767 0.071020351
## 3733 3734 3735 3736 3737 3738
## 0.095899426 0.089032111 0.119409831 0.119409831 0.082611626 0.103236410
## 3739 3740 3741 3742 3743 3744
## 0.065804943 0.082611626 0.111065767 0.095899426 0.158350725 0.082611626
## 3745 3746 3747 3748 3749 3750
## 0.111065767 0.103236410 0.089032111 0.128290293 0.119409831 0.056426775
## 3751 3752 3753 3754 3755 3756
## 0.137727896 0.089032111 0.095899426 0.111065767 0.111065767 0.111065767
## 3757 3758 3759 3760 3761 3762
## 0.095899426 0.119409831 0.082611626 0.095899426 0.082611626 0.158350725
## 3763 3764 3765 3766 3767 3768
## 0.158350725 0.128290293 0.076615209 0.071020351 0.111065767 0.137727896
## 3769 3770 3771 3772 3773 3774
## 0.147742102 0.111065767 0.071020351 0.095899426 0.082611626 0.082611626
## 3775 3776 3777 3778 3779 3780
## 0.065804943 0.103236410 0.119409831 0.119409831 0.119409831 0.082611626
## 3781 3782 3783 3784 3785 3786
## 0.111065767 0.119409831 0.095899426 0.111065767 0.119409831 0.076615209
## 3787 3788 3789 3790 3791 3792
## 0.119409831 0.082611626 0.089032111 0.076615209 0.103236410 0.032615142
## 3793 3794 3795 3796 3797 3798
## 0.111065767 0.111065767 0.119409831 0.158350725 0.071020351 0.082611626
## 3799 3800 3801 3802 3803 3804
## 0.076615209 0.027826117 0.071020351 0.103236410 0.071020351 0.082611626
## 3805 3806 3807 3808 3809 3810
## 0.147742102 0.076615209 0.147742102 0.095899426 0.060947403 0.095899426
## 3811 3812 3813 3814 3815 3816
## 0.128290293 0.119409831 0.111065767 0.095899426 0.103236410 0.103236410
## 3817 3818 3819 3820 3821 3822
## 0.137727896 0.076615209 0.137727896 0.111065767 0.103236410 0.111065767
## 3823 3824 3825 3826 3827 3828
## 0.082611626 0.128290293 0.076615209 0.095899426 0.128290293 0.111065767
## 3829 3830 3831 3832 3833 3834
## 0.071020351 0.111065767 0.111065767 0.095899426 0.082611626 0.128290293
## 3835 3836 3837 3838 3839 3840
## 0.076615209 0.071020351 0.076615209 0.137727896 0.095899426 0.137727896
## 3841 3842 3843 3844 3845 3846
## 0.111065767 0.137727896 0.071020351 0.082611626 0.119409831 0.103236410
## 3847 3848 3849 3850 3851 3852
## 0.111065767 0.089032111 0.065804943 0.111065767 0.082611626 0.095899426
## 3853 3854 3855 3856 3857 3858
## 0.119409831 0.103236410 0.076615209 0.082611626 0.158350725 0.089032111
## 3859 3860 3861 3862 3863 3864
## 0.137727896 0.103236410 0.119409831 0.137727896 0.103236410 0.111065767
## 3865 3866 3867 3868 3869 3870
## 0.071020351 0.095899426 0.103236410 0.071020351 0.111065767 0.065804943
## 3871 3872 3873 3874 3875 3876
## 0.076615209 0.119409831 0.119409831 0.119409831 0.095899426 0.082611626
## 3877 3878 3879 3880 3881 3882
## 0.095899426 0.147742102 0.111065767 0.119409831 0.076615209 0.158350725
## 3883 3884 3885 3886 3887 3888
## 0.137727896 0.103236410 0.082611626 0.065804943 0.089032111 0.158350725
## 3889 3890 3891 3892 3893 3894
## 0.128290293 0.158350725 0.137727896 0.119409831 0.065804943 0.095899426
## 3895 3896 3897 3898 3899 3900
## 0.071020351 0.147742102 0.137727896 0.193888125 0.095899426 0.060947403
## 3901 3902 3903 3904 3905 3906
## 0.095899426 0.103236410 0.119409831 0.137727896 0.076615209 0.147742102
## 3907 3908 3909 3910 3911 3912
## 0.111065767 0.119409831 0.111065767 0.076615209 0.095899426 0.071020351
## 3913 3914 3915 3916 3917 3918
## 0.111065767 0.111065767 0.103236410 0.076615209 0.071020351 0.128290293
## 3919 3920 3921 3922 3923 3924
## 0.169569538 0.128290293 0.082611626 0.071020351 0.095899426 0.147742102
## 3925 3926 3927 3928 3929 3930
## 0.095899426 0.169569538 0.071020351 0.119409831 0.137727896 0.111065767
## 3931 3932 3933 3934 3935 3936
## 0.119409831 0.119409831 0.103236410 0.119409831 0.103236410 0.111065767
## 3937 3938 3939 3940 3941 3942
## 0.076615209 0.128290293 0.158350725 0.119409831 0.158350725 0.128290293
## 3943 3944 3945 3946 3947 3948
## 0.089032111 0.076615209 0.082611626 0.082611626 0.147742102 0.065804943
## 3949 3950 3951 3952 3953 3954
## 0.111065767 0.111065767 0.111065767 0.111065767 0.119409831 0.103236410
## 3955 3956 3957 3958 3959 3960
## 0.052222807 0.111065767 0.137727896 0.076615209 0.137727896 0.103236410
## 3961 3962 3963 3964 3965 3966
## 0.137727896 0.111065767 0.071020351 0.082611626 0.147742102 0.089032111
## 3967 3968 3969 3970 3971 3972
## 0.111065767 0.128290293 0.082611626 0.103236410 0.076615209 0.089032111
## 3973 3974 3975 3976 3977 3978
## 0.082611626 0.082611626 0.158350725 0.119409831 0.103236410 0.082611626
## 3979 3980 3981 3982 3983 3984
## 0.095899426 0.089032111 0.119409831 0.095899426 0.103236410 0.082611626
## 3985 3986 3987 3988 3989 3990
## 0.095899426 0.089032111 0.082611626 0.103236410 0.082611626 0.060947403
## 3991 3992 3993 3994 3995 3996
## 0.137727896 0.071020351 0.147742102 0.076615209 0.095899426 0.071020351
## 3997 3998 3999 4000 4001 4002
## 0.128290293 0.128290293 0.137727896 0.111065767 0.076615209 0.082611626
## 4003 4004 4005 4006 4007 4008
## 0.103236410 0.065804943 0.128290293 0.119409831 0.147742102 0.128290293
## 4009 4010 4011 4012 4013 4014
## 0.103236410 0.060947403 0.137727896 0.128290293 0.128290293 0.082611626
## 4015 4016 4017 4018 4019 4020
## 0.076615209 0.137727896 0.119409831 0.082611626 0.119409831 0.095899426
## 4021 4022 4023 4024 4025 4026
## 0.071020351 0.103236410 0.119409831 0.111065767 0.103236410 0.119409831
## 4027 4028 4029 4030 4031 4032
## 0.095899426 0.095899426 0.089032111 0.076615209 0.128290293 0.128290293
## 4033 4034 4035 4036 4037 4038
## 0.103236410 0.082611626 0.111065767 0.082611626 0.082611626 0.128290293
## 4039 4040 4041 4042 4043 4044
## 0.076615209 0.137727896 0.071020351 0.071020351 0.158350725 0.137727896
## 4045 4046 4047 4048 4049 4050
## 0.103236410 0.103236410 0.119409831 0.128290293 0.089032111 0.119409831
## 4051 4052 4053 4054 4055 4056
## 0.076615209 0.082611626 0.009015632 0.119409831 0.137727896 0.158350725
## 4057 4058 4059 4060 4061 4062
## 0.103236410 0.128290293 0.119409831 0.065804943 0.089032111 0.103236410
## 4063 4064 4065 4066 4067 4068
## 0.137727896 0.103236410 0.111065767 0.128290293 0.095899426 0.158350725
## 4069 4070 4071 4072 4073 4074
## 0.111065767 0.119409831 0.119409831 0.103236410 0.095899426 0.137727896
## 4075 4076 4077 4078 4079 4080
## 0.082611626 0.095899426 0.089032111 0.103236410 0.089032111 0.119409831
## 4081 4082 4083 4084 4085 4086
## 0.158350725 0.119409831 0.103236410 0.111065767 0.089032111 0.076615209
## 4087 4088 4089 4090 4091 4092
## 0.095899426 0.119409831 0.128290293 0.095899426 0.128290293 0.076615209
## 4093 4094 4095 4096 4097 4098
## 0.082611626 0.095899426 0.137727896 0.158350725 0.111065767 0.056426775
## 4099 4100 4101 4102 4103 4104
## 0.076615209 0.089032111 0.147742102 0.119409831 0.158350725 0.089032111
## 4105 4106 4107 4108 4109 4110
## 0.111065767 0.128290293 0.089032111 0.044687700 0.095899426 0.095899426
## 4111 4112 4113 4114 4115 4116
## 0.095899426 0.089032111 0.095899426 0.038196000 0.103236410 0.128290293
## 4117 4118 4119 4120 4121 4122
## 0.128290293 0.095899426 0.103236410 0.119409831 0.111065767 0.103236410
## 4123 4124 4125 4126 4127 4128
## 0.128290293 0.111065767 0.137727896 0.076615209 0.147742102 0.128290293
## 4129 4130 4131 4132 4133 4134
## 0.071020351 0.119409831 0.103236410 0.119409831 0.137727896 0.119409831
## 4135 4136 4137 4138 4139 4140
## 0.103236410 0.103236410 0.119409831 0.128290293 0.128290293 0.089032111
## 4141 4142 4143 4144 4145 4146
## 0.169569538 0.082611626 0.111065767 0.111065767 0.095899426 0.082611626
## 4147 4148 4149 4150 4151 4152
## 0.071020351 0.128290293 0.103236410 0.128290293 0.128290293 0.071020351
## 4153 4154 4155 4156 4157 4158
## 0.111065767 0.095899426 0.011496734 0.128290293 0.128290293 0.095899426
## 4159 4160 4161 4162 4163 4164
## 0.103236410 0.103236410 0.076615209 0.137727896 0.111065767 0.119409831
## 4165 4166 4167 4168 4169 4170
## 0.095899426 0.095899426 0.147742102 0.103236410 0.119409831 0.076615209
## 4171 4172 4173 4174 4175 4176
## 0.076615209 0.158350725 0.082611626 0.169569538 0.082611626 0.103236410
## 4177 4178 4179 4180 4181 4182
## 0.111065767 0.137727896 0.119409831 0.056426775 0.111065767 0.158350725
## 4183 4184 4185 4186 4187 4188
## 0.119409831 0.158350725 0.065804943 0.065804943 0.137727896 0.103236410
## 4189 4190 4191 4192 4193 4194
## 0.147742102 0.076615209 0.082611626 0.082611626 0.089032111 0.111065767
## 4195 4196 4197 4198 4199 4200
## 0.071020351 0.137727896 0.119409831 0.076615209 0.111065767 0.089032111
## 4201 4202 4203 4204 4205 4206
## 0.147742102 0.111065767 0.111065767 0.128290293 0.095899426 0.095899426
## 4207 4208 4209 4210 4211 4212
## 0.103236410 0.103236410 0.095899426 0.119409831 0.060947403 0.025694960
## 4213 4214 4215 4216 4217 4218
## 0.128290293 0.103236410 0.095899426 0.103236410 0.119409831 0.137727896
## 4219 4220 4221 4222 4223 4224
## 0.128290293 0.147742102 0.095899426 0.056426775 0.137727896 0.095899426
## 4225 4226 4227 4228 4229 4230
## 0.137727896 0.095899426 0.147742102 0.103236410 0.111065767 0.119409831
## 4231 4232 4233 4234 4235 4236
## 0.089032111 0.128290293 0.056426775 0.111065767 0.147742102 0.082611626
## 4237 4238 4239 4240 4241 4242
## 0.095899426 0.111065767 0.119409831 0.095899426 0.095899426 0.147742102
## 4243 4244 4245 4246 4247 4248
## 0.111065767 0.095899426 0.089032111 0.103236410 0.060947403 0.158350725
## 4249 4250 4251 4252 4253 4254
## 0.128290293 0.089032111 0.111065767 0.158350725 0.076615209 0.076615209
## 4255 4256 4257 4258 4259 4260
## 0.089032111 0.111065767 0.128290293 0.095899426 0.076615209 0.111065767
## 4261 4262 4263 4264 4265 4266
## 0.012465347 0.095899426 0.111065767 0.082611626 0.095899426 0.128290293
## 4267 4268 4269 4270 4271 4272
## 0.089032111 0.181411860 0.082611626 0.071020351 0.128290293 0.137727896
## 4273 4274 4275 4276 4277 4278
## 0.103236410 0.147742102 0.089032111 0.111065767 0.111065767 0.137727896
## 4279 4280 4281 4282 4283 4284
## 0.147742102 0.193888125 0.103236410 0.095899426 0.103236410 0.103236410
## 4285 4286 4287 4288 4289 4290
## 0.147742102 0.111065767 0.119409831 0.103236410 0.089032111 0.103236410
## 4291 4292 4293 4294 4295 4296
## 0.089032111 0.147742102 0.103236410 0.137727896 0.119409831 0.111065767
## 4297 4298 4299 4300 4301 4302
## 0.076615209 0.082611626 0.095899426 0.103236410 0.076615209 0.119409831
## 4303 4304 4305 4306 4307 4308
## 0.128290293 0.089032111 0.095899426 0.103236410 0.076615209 0.128290293
## 4309 4310 4311 4312 4313 4314
## 0.089032111 0.095899426 0.089032111 0.103236410 0.193888125 0.111065767
## 4315 4316 4317 4318 4319 4320
## 0.089032111 0.082611626 0.082611626 0.082611626 0.111065767 0.158350725
## 4321 4322 4323 4324 4325 4326
## 0.076615209 0.089032111 0.119409831 0.089032111 0.060947403 0.076615209
## 4327 4328 4329 4330 4331 4332
## 0.095899426 0.089032111 0.095899426 0.137727896 0.095899426 0.103236410
## 4333 4334 4335 4336 4337 4338
## 0.076615209 0.111065767 0.089032111 0.095899426 0.137727896 0.089032111
## 4339 4340 4341 4342 4343 4344
## 0.111065767 0.076615209 0.095899426 0.089032111 0.119409831 0.095899426
## 4345 4346 4347 4348 4349 4350
## 0.103236410 0.119409831 0.119409831 0.128290293 0.089032111 0.128290293
## 4351 4352 4353 4354 4355 4356
## 0.111065767 0.119409831 0.089032111 0.111065767 0.128290293 0.158350725
## 4357 4358 4359 4360 4361 4362
## 0.071020351 0.103236410 0.137727896 0.089032111 0.082611626 0.076615209
## 4363 4364 4365 4366 4367 4368
## 0.169569538 0.082611626 0.089032111 0.137727896 0.089032111 0.076615209
## 4369 4370 4371 4372 4373 4374
## 0.095899426 0.128290293 0.095899426 0.089032111 0.119409831 0.095899426
## 4375 4376 4377 4378 4379 4380
## 0.060947403 0.071020351 0.103236410 0.065804943 0.095899426 0.158350725
## 4381 4382 4383 4384 4385 4386
## 0.082611626 0.147742102 0.103236410 0.089032111 0.013514451 0.095899426
## 4387 4388 4389 4390 4391 4392
## 0.089032111 0.095899426 0.128290293 0.089032111 0.111065767 0.095899426
## 4393 4394 4395 4396 4397 4398
## 0.071020351 0.137727896 0.065804943 0.119409831 0.119409831 0.095899426
## 4399 4400 4401 4402 4403 4404
## 0.082611626 0.119409831 0.111065767 0.111065767 0.089032111 0.089032111
## 4405 4406 4407 4408 4409 4410
## 0.128290293 0.111065767 0.137727896 0.095899426 0.095899426 0.119409831
## 4411 4412 4413 4414 4415 4416
## 0.089032111 0.111065767 0.071020351 0.103236410 0.103236410 0.095899426
## 4417 4418 4419 4420 4421 4422
## 0.076615209 0.065804943 0.095899426 0.089032111 0.089032111 0.137727896
## 4423 4424 4425 4426 4427 4428
## 0.147742102 0.082611626 0.111065767 0.147742102 0.128290293 0.137727896
## 4429 4430 4431 4432 4433 4434
## 0.076615209 0.103236410 0.103236410 0.103236410 0.076615209 0.089032111
## 4435 4436 4437 4438 4439 4440
## 0.013514451 0.071020351 0.076615209 0.137727896 0.119409831 0.158350725
## 4441 4442 4443 4444 4445 4446
## 0.111065767 0.158350725 0.103236410 0.128290293 0.089032111 0.082611626
## 4447 4448 4449 4450 4451 4452
## 0.065804943 0.137727896 0.095899426 0.128290293 0.089032111 0.089032111
## 4453 4454 4455 4456 4457 4458
## 0.103236410 0.089032111 0.147742102 0.111065767 0.095899426 0.076615209
## 4459 4460 4461 4462 4463 4464
## 0.111065767 0.147742102 0.095899426 0.181411860 0.082611626 0.111065767
## 4465 4466 4467 4468 4469 4470
## 0.119409831 0.111065767 0.065804943 0.119409831 0.119409831 0.089032111
## 4471 4472 4473 4474 4475 4476
## 0.060947403 0.095899426 0.119409831 0.071020351 0.076615209 0.089032111
## 4477 4478 4479 4480 4481 4482
## 0.103236410 0.076615209 0.095899426 0.089032111 0.095899426 0.065804943
## 4483 4484 4485 4486 4487 4488
## 0.089032111 0.193888125 0.119409831 0.137727896 0.193888125 0.111065767
## 4489 4490 4491 4492 4493 4494
## 0.103236410 0.103236410 0.103236410 0.119409831 0.076615209 0.111065767
## 4495 4496 4497 4498 4499 4500
## 0.095899426 0.119409831 0.076615209 0.128290293 0.071020351 0.089032111
## 4501 4502 4503 4504 4505 4506
## 0.082611626 0.111065767 0.082611626 0.147742102 0.111065767 0.082611626
## 4507 4508 4509 4510 4511 4512
## 0.137727896 0.065804943 0.082611626 0.076615209 0.119409831 0.128290293
## 4513 4514 4515 4516 4517 4518
## 0.111065767 0.032615142 0.095899426 0.103236410 0.095899426 0.089032111
## 4519 4520 4521 4522 4523 4524
## 0.137727896 0.111065767 0.095899426 0.076615209 0.089032111 0.082611626
## 4525 4526 4527 4528 4529 4530
## 0.082611626 0.065804943 0.095899426 0.119409831 0.071020351 0.103236410
## 4531 4532 4533 4534 4535 4536
## 0.095899426 0.089032111 0.082611626 0.089032111 0.095899426 0.119409831
## 4537 4538 4539 4540 4541 4542
## 0.082611626 0.119409831 0.158350725 0.089032111 0.103236410 0.119409831
## 4543 4544 4545 4546 4547 4548
## 0.137727896 0.103236410 0.052222807 0.082611626 0.103236410 0.111065767
## 4549 4550 4551 4552 4553 4554
## 0.119409831 0.119409831 0.137727896 0.071020351 0.095899426 0.147742102
## 4555 4556 4557 4558 4559 4560
## 0.119409831 0.089032111 0.158350725 0.103236410 0.111065767 0.082611626
## 4561 4562 4563 4564 4565 4566
## 0.071020351 0.089032111 0.095899426 0.060947403 0.082611626 0.103236410
## 4567 4568 4569 4570 4571 4572
## 0.095899426 0.089032111 0.111065767 0.111065767 0.095899426 0.169569538
## 4573 4574 4575 4576 4577 4578
## 0.089032111 0.111065767 0.048316010 0.137727896 0.111065767 0.128290293
## 4579 4580 4581 4582 4583 4584
## 0.137727896 0.095899426 0.119409831 0.119409831 0.095899426 0.147742102
## 4585 4586 4587 4588 4589 4590
## 0.103236410 0.065804943 0.089032111 0.119409831 0.111065767 0.065804943
## 4591 4592 4593 4594 4595 4596
## 0.071020351 0.071020351 0.158350725 0.089032111 0.111065767 0.060947403
## 4597 4598 4599 4600 4601 4602
## 0.128290293 0.111065767 0.128290293 0.111065767 0.071020351 0.076615209
## 4603 4604 4605 4606 4607 4608
## 0.082611626 0.103236410 0.076615209 0.137727896 0.128290293 0.128290293
## 4609 4610 4611 4612 4613 4614
## 0.119409831 0.119409831 0.147742102 0.119409831 0.128290293 0.060947403
## 4615 4616 4617 4618 4619 4620
## 0.103236410 0.111065767 0.082611626 0.095899426 0.193888125 0.119409831
## 4621 4622 4623 4624 4625 4626
## 0.095899426 0.128290293 0.111065767 0.014650540 0.089032111 0.095899426
## 4627 4628 4629 4630 4631 4632
## 0.137727896 0.128290293 0.095899426 0.103236410 0.095899426 0.111065767
## 4633 4634 4635 4636 4637 4638
## 0.095899426 0.128290293 0.137727896 0.147742102 0.111065767 0.119409831
## 4639 4640 4641 4642 4643 4644
## 0.111065767 0.111065767 0.089032111 0.103236410 0.056426775 0.065804943
## 4645 4646 4647 4648 4649 4650
## 0.111065767 0.158350725 0.095899426 0.103236410 0.111065767 0.089032111
## 4651 4652 4653 4654 4655 4656
## 0.089032111 0.103236410 0.119409831 0.137727896 0.137727896 0.111065767
## 4657 4658 4659 4660 4661 4662
## 0.111065767 0.089032111 0.071020351 0.082611626 0.095899426 0.111065767
## 4663 4664 4665 4666 4667 4668
## 0.103236410 0.095899426 0.147742102 0.128290293 0.076615209 0.119409831
## 4669 4670 4671 4672 4673 4674
## 0.089032111 0.119409831 0.076615209 0.089032111 0.119409831 0.111065767
## 4675 4676 4677 4678 4679 4680
## 0.095899426 0.071020351 0.060947403 0.128290293 0.128290293 0.137727896
## 4681 4682 4683 4684 4685 4686
## 0.111065767 0.103236410 0.128290293 0.147742102 0.076615209 0.137727896
## 4687 4688 4689 4690 4691 4692
## 0.071020351 0.076615209 0.158350725 0.076615209 0.089032111 0.158350725
## 4693 4694 4695 4696 4697 4698
## 0.095899426 0.103236410 0.082611626 0.089032111 0.071020351 0.111065767
## 4699 4700 4701 4702 4703 4704
## 0.103236410 0.082611626 0.095899426 0.095899426 0.119409831 0.137727896
## 4705 4706 4707 4708 4709 4710
## 0.147742102 0.111065767 0.147742102 0.158350725 0.137727896 0.128290293
## 4711 4712 4713 4714 4715 4716
## 0.060947403 0.082611626 0.095899426 0.089032111 0.147742102 0.076615209
## 4717 4718 4719 4720 4721 4722
## 0.089032111 0.095899426 0.119409831 0.119409831 0.119409831 0.071020351
## 4723 4724 4725 4726 4727 4728
## 0.137727896 0.095899426 0.071020351 0.111065767 0.076615209 0.103236410
## 4729 4730 4731 4732 4733 4734
## 0.158350725 0.082611626 0.128290293 0.095899426 0.082611626 0.119409831
## 4735 4736 4737 4738 4739 4740
## 0.103236410 0.119409831 0.119409831 0.119409831 0.111065767 0.147742102
## 4741 4742 4743 4744 4745 4746
## 0.137727896 0.137727896 0.082611626 0.103236410 0.111065767 0.137727896
## 4747 4748 4749 4750 4751 4752
## 0.111065767 0.103236410 0.095899426 0.103236410 0.111065767 0.076615209
## 4753 4754 4755 4756 4757 4758
## 0.089032111 0.071020351 0.082611626 0.095899426 0.095899426 0.103236410
## 4759 4760 4761 4762 4763 4764
## 0.147742102 0.103236410 0.089032111 0.128290293 0.111065767 0.089032111
## 4765 4766 4767 4768 4769 4770
## 0.082611626 0.089032111 0.071020351 0.158350725 0.095899426 0.095899426
## 4771 4772 4773 4774 4775 4776
## 0.111065767 0.119409831 0.111065767 0.103236410 0.095899426 0.103236410
## 4777 4778 4779 4780 4781 4782
## 0.137727896 0.128290293 0.111065767 0.089032111 0.082611626 0.071020351
## 4783 4784 4785 4786 4787 4788
## 0.137727896 0.095899426 0.103236410 0.089032111 0.119409831 0.089032111
## 4789 4790 4791 4792 4793 4794
## 0.076615209 0.103236410 0.128290293 0.095899426 0.103236410 0.147742102
## 4795 4796 4797 4798 4799 4800
## 0.095899426 0.071020351 0.158350725 0.103236410 0.082611626 0.181411860
## 4801 4802 4803 4804 4805 4806
## 0.082611626 0.169569538 0.060947403 0.095899426 0.103236410 0.103236410
## 4807 4808 4809 4810 4811 4812
## 0.137727896 0.147742102 0.128290293 0.095899426 0.119409831 0.076615209
## 4813 4814 4815 4816 4817 4818
## 0.095899426 0.147742102 0.095899426 0.095899426 0.103236410 0.103236410
## 4819 4820 4821 4822 4823 4824
## 0.082611626 0.089032111 0.082611626 0.089032111 0.082611626 0.181411860
## 4825 4826 4827 4828 4829 4830
## 0.128290293 0.071020351 0.082611626 0.137727896 0.111065767 0.071020351
## 4831 4832 4833 4834 4835 4836
## 0.076615209 0.147742102 0.147742102 0.128290293 0.119409831 0.095899426
## 4837 4838 4839 4840 4841 4842
## 0.103236410 0.158350725 0.095899426 0.095899426 0.147742102 0.147742102
## 4843 4844 4845 4846 4847 4848
## 0.060947403 0.095899426 0.111065767 0.119409831 0.111065767 0.103236410
## 4849 4850 4851 4852 4853 4854
## 0.052222807 0.111065767 0.095899426 0.137727896 0.119409831 0.137727896
## 4855 4856 4857 4858 4859 4860
## 0.181411860 0.060947403 0.147742102 0.181411860 0.076615209 0.065804943
## 4861 4862 4863 4864 4865 4866
## 0.095899426 0.095899426 0.111065767 0.089032111 0.082611626 0.060947403
## 4867 4868 4869 4870 4871 4872
## 0.056426775 0.076615209 0.076615209 0.119409831 0.119409831 0.119409831
## 4873 4874 4875 4876 4877 4878
## 0.103236410 0.103236410 0.076615209 0.071020351 0.147742102 0.082611626
## 4879 4880 4881 4882 4883 4884
## 0.071020351 0.128290293 0.111065767 0.089032111 0.076615209 0.095899426
## 4885 4886 4887 4888 4889 4890
## 0.137727896 0.076615209 0.111065767 0.095899426 0.158350725 0.076615209
## 4891 4892 4893 4894 4895 4896
## 0.082611626 0.111065767 0.158350725 0.103236410 0.071020351 0.119409831
## 4897 4898 4899 4900 4901 4902
## 0.095899426 0.082611626 0.137727896 0.089032111 0.147742102 0.076615209
## 4903 4904 4905 4906 4907 4908
## 0.128290293 0.089032111 0.076615209 0.137727896 0.071020351 0.111065767
## 4909 4910 4911 4912 4913 4914
## 0.089032111 0.169569538 0.095899426 0.119409831 0.103236410 0.111065767
## 4915 4916 4917 4918 4919 4920
## 0.095899426 0.137727896 0.065804943 0.111065767 0.103236410 0.111065767
## 4921 4922 4923 4924 4925 4926
## 0.169569538 0.103236410 0.103236410 0.082611626 0.089032111 0.095899426
## 4927 4928 4929 4930 4931 4932
## 0.103236410 0.137727896 0.082611626 0.076615209 0.103236410 0.128290293
## 4933 4934 4935 4936 4937 4938
## 0.111065767 0.082611626 0.119409831 0.095899426 0.119409831 0.119409831
## 4939 4940 4941 4942 4943 4944
## 0.089032111 0.137727896 0.082611626 0.128290293 0.147742102 0.065804943
## 4945 4946 4947 4948 4949 4950
## 0.071020351 0.095899426 0.147742102 0.147742102 0.071020351 0.103236410
## 4951 4952 4953 4954 4955 4956
## 0.095899426 0.119409831 0.041320030 0.111065767 0.095899426 0.119409831
## 4957 4958 4959 4960 4961 4962
## 0.103236410 0.103236410 0.095899426 0.071020351 0.119409831 0.119409831
## 4963 4964 4965 4966 4967 4968
## 0.082611626 0.095899426 0.137727896 0.095899426 0.089032111 0.111065767
## 4969 4970 4971 4972 4973 4974
## 0.128290293 0.089032111 0.147742102 0.119409831 0.082611626 0.071020351
## 4975 4976 4977 4978 4979 4980
## 0.111065767 0.082611626 0.103236410 0.119409831 0.095899426 0.147742102
## 4981 4982 4983 4984 4985 4986
## 0.095899426 0.089032111 0.071020351 0.119409831 0.137727896 0.095899426
## 4987 4988 4989 4990 4991 4992
## 0.147742102 0.089032111 0.119409831 0.060947403 0.082611626 0.119409831
## 4993 4994 4995 4996 4997 4998
## 0.082611626 0.181411860 0.137727896 0.128290293 0.137727896 0.103236410
## 4999 5000 5001 5002 5003 5004
## 0.119409831 0.103236410 0.128290293 0.119409831 0.071020351 0.071020351
## 5005 5006 5007 5008 5009 5010
## 0.111065767 0.147742102 0.095899426 0.103236410 0.089032111 0.103236410
## 5011 5012 5013 5014 5015 5016
## 0.137727896 0.119409831 0.076615209 0.082611626 0.095899426 0.119409831
## 5017 5018 5019 5020 5021 5022
## 0.095899426 0.089032111 0.147742102 0.147742102 0.071020351 0.128290293
## 5023 5024 5025 5026 5027 5028
## 0.193888125 0.111065767 0.128290293 0.095899426 0.089032111 0.076615209
## 5029 5030 5031 5032 5033 5034
## 0.076615209 0.103236410 0.076615209 0.147742102 0.082611626 0.137727896
## 5035 5036 5037 5038 5039 5040
## 0.111065767 0.119409831 0.089032111 0.137727896 0.128290293 0.111065767
## 5041 5042 5043 5044 5045 5046
## 0.103236410 0.111065767 0.119409831 0.071020351 0.082611626 0.103236410
## 5047 5048 5049 5050 5051 5052
## 0.071020351 0.119409831 0.147742102 0.103236410 0.128290293 0.071020351
## 5053 5054 5055 5056 5057 5058
## 0.111065767 0.082611626 0.111065767 0.082611626 0.103236410 0.128290293
## 5059 5060 5061 5062 5063 5064
## 0.071020351 0.071020351 0.095899426 0.128290293 0.111065767 0.095899426
## 5065 5066 5067 5068 5069 5070
## 0.111065767 0.082611626 0.082611626 0.089032111 0.089032111 0.111065767
## 5071 5072 5073 5074 5075 5076
## 0.095899426 0.076615209 0.128290293 0.103236410 0.095899426 0.095899426
## 5077 5078 5079 5080 5081 5082
## 0.089032111 0.076615209 0.095899426 0.119409831 0.137727896 0.119409831
## 5083 5084 5085 5086 5087 5088
## 0.071020351 0.103236410 0.089032111 0.103236410 0.128290293 0.147742102
## 5089 5090 5091 5092 5093 5094
## 0.076615209 0.128290293 0.089032111 0.119409831 0.065804943 0.128290293
## 5095 5096 5097 5098 5099 5100
## 0.128290293 0.082611626 0.082611626 0.128290293 0.128290293 0.111065767
## 5101 5102 5103 5104 5105 5106
## 0.111065767 0.060947403 0.089032111 0.103236410 0.065804943 0.111065767
## 5107 5108 5109 5110 5111 5112
## 0.103236410 0.056426775 0.065804943 0.103236410 0.119409831 0.076615209
## 5113 5114 5115 5116 5117 5118
## 0.137727896 0.095899426 0.071020351 0.103236410 0.076615209 0.103236410
## 5119 5120 5121 5122 5123 5124
## 0.111065767 0.071020351 0.128290293 0.095899426 0.119409831 0.060947403
## 5125 5126 5127 5128 5129 5130
## 0.119409831 0.137727896 0.119409831 0.082611626 0.076615209 0.111065767
## 5131 5132 5133 5134 5135 5136
## 0.089032111 0.103236410 0.128290293 0.103236410 0.103236410 0.082611626
## 5137 5138 5139 5140 5141 5142
## 0.076615209 0.095899426 0.119409831 0.082611626 0.103236410 0.137727896
## 5143 5144 5145 5146 5147 5148
## 0.119409831 0.082611626 0.103236410 0.169569538 0.128290293 0.119409831
## 5149 5150 5151 5152 5153 5154
## 0.128290293 0.103236410 0.076615209 0.089032111 0.082611626 0.089032111
## 5155 5156 5157 5158 5159 5160
## 0.089032111 0.137727896 0.103236410 0.111065767 0.089032111 0.089032111
## 5161 5162 5163 5164 5165 5166
## 0.071020351 0.119409831 0.137727896 0.089032111 0.089032111 0.128290293
## 5167 5168 5169 5170 5171 5172
## 0.089032111 0.071020351 0.147742102 0.111065767 0.119409831 0.089032111
## 5173 5174 5175 5176 5177 5178
## 0.119409831 0.103236410 0.137727896 0.111065767 0.158350725 0.128290293
## 5179 5180 5181 5182 5183 5184
## 0.119409831 0.095899426 0.089032111 0.158350725 0.071020351 0.158350725
## 5185 5186 5187 5188 5189 5190
## 0.111065767 0.169569538 0.089032111 0.071020351 0.089032111 0.076615209
## 5191 5192 5193 5194 5195 5196
## 0.128290293 0.076615209 0.128290293 0.103236410 0.103236410 0.111065767
## 5197 5198 5199 5200 5201 5202
## 0.065804943 0.065804943 0.082611626 0.082611626 0.147742102 0.095899426
## 5203 5204 5205 5206 5207 5208
## 0.119409831 0.065804943 0.111065767 0.128290293 0.137727896 0.147742102
## 5209 5210 5211 5212 5213 5214
## 0.082611626 0.119409831 0.128290293 0.082611626 0.147742102 0.103236410
## 5215 5216 5217 5218 5219 5220
## 0.111065767 0.060947403 0.089032111 0.082611626 0.095899426 0.076615209
## 5221 5222 5223 5224 5225 5226
## 0.065804943 0.089032111 0.158350725 0.095899426 0.111065767 0.103236410
## 5227 5228 5229 5230 5231 5232
## 0.128290293 0.103236410 0.158350725 0.119409831 0.147742102 0.089032111
## 5233 5234 5235 5236 5237 5238
## 0.158350725 0.089032111 0.128290293 0.119409831 0.128290293 0.095899426
## 5239 5240 5241 5242 5243 5244
## 0.103236410 0.095899426 0.158350725 0.089032111 0.128290293 0.060947403
## 5245 5246 5247 5248 5249 5250
## 0.111065767 0.095899426 0.119409831 0.111065767 0.111065767 0.119409831
## 5251 5252 5253 5254 5255 5256
## 0.103236410 0.119409831 0.137727896 0.103236410 0.082611626 0.095899426
## 5257 5258 5259 5260 5261 5262
## 0.111065767 0.103236410 0.071020351 0.095899426 0.111065767 0.147742102
## 5263 5264 5265 5266 5267 5268
## 0.119409831 0.103236410 0.147742102 0.111065767 0.103236410 0.095899426
## 5269 5270 5271 5272 5273 5274
## 0.111065767 0.095899426 0.147742102 0.095899426 0.076615209 0.137727896
## 5275 5276 5277 5278 5279 5280
## 0.147742102 0.089032111 0.147742102 0.076615209 0.089032111 0.111065767
## 5281 5282 5283 5284 5285 5286
## 0.128290293 0.119409831 0.095899426 0.089032111 0.095899426 0.076615209
## 5287 5288 5289 5290 5291 5292
## 0.111065767 0.119409831 0.071020351 0.056426775 0.103236410 0.103236410
## 5293 5294 5295 5296 5297 5298
## 0.082611626 0.128290293 0.119409831 0.095899426 0.103236410 0.111065767
## 5299 5300 5301 5302 5303 5304
## 0.089032111 0.128290293 0.095899426 0.095899426 0.119409831 0.119409831
## 5305 5306 5307 5308 5309 5310
## 0.119409831 0.103236410 0.137727896 0.147742102 0.065804943 0.071020351
## 5311 5312 5313 5314 5315 5316
## 0.076615209 0.103236410 0.111065767 0.095899426 0.082611626 0.111065767
## 5317 5318 5319 5320 5321 5322
## 0.220767157 0.076615209 0.128290293 0.119409831 0.082611626 0.082611626
## 5323 5324 5325 5326 5327 5328
## 0.076615209 0.082611626 0.147742102 0.071020351 0.147742102 0.103236410
## 5329 5330 5331 5332 5333 5334
## 0.193888125 0.089032111 0.169569538 0.071020351 0.111065767 0.111065767
## 5335 5336 5337 5338 5339 5340
## 0.089032111 0.082611626 0.082611626 0.082611626 0.103236410 0.103236410
## 5341 5342 5343 5344 5345 5346
## 0.181411860 0.119409831 0.095899426 0.076615209 0.158350725 0.137727896
## 5347 5348 5349 5350 5351 5352
## 0.103236410 0.089032111 0.082611626 0.089032111 0.111065767 0.082611626
## 5353 5354 5355 5356 5357 5358
## 0.103236410 0.158350725 0.060947403 0.082611626 0.111065767 0.065804943
## 5359 5360 5361 5362 5363 5364
## 0.147742102 0.119409831 0.147742102 0.128290293 0.137727896 0.089032111
## 5365 5366 5367 5368 5369 5370
## 0.137727896 0.103236410 0.119409831 0.076615209 0.076615209 0.089032111
## 5371 5372 5373 5374 5375 5376
## 0.076615209 0.095899426 0.082611626 0.076615209 0.095899426 0.089032111
## 5377 5378 5379 5380 5381 5382
## 0.076615209 0.137727896 0.111065767 0.103236410 0.076615209 0.181411860
## 5383 5384 5385 5386 5387 5388
## 0.103236410 0.111065767 0.071020351 0.082611626 0.095899426 0.137727896
## 5389 5390 5391 5392 5393 5394
## 0.089032111 0.103236410 0.056426775 0.103236410 0.060947403 0.111065767
## 5395 5396 5397 5398 5399 5400
## 0.119409831 0.169569538 0.137727896 0.089032111 0.128290293 0.095899426
## 5401 5402 5403 5404 5405 5406
## 0.119409831 0.060947403 0.119409831 0.158350725 0.111065767 0.103236410
## 5407 5408 5409 5410 5411 5412
## 0.111065767 0.089032111 0.119409831 0.119409831 0.111065767 0.065804943
## 5413 5414 5415 5416 5417 5418
## 0.128290293 0.103236410 0.103236410 0.095899426 0.076615209 0.128290293
## 5419 5420 5421 5422 5423 5424
## 0.103236410 0.095899426 0.103236410 0.082611626 0.128290293 0.137727896
## 5425 5426 5427 5428 5429 5430
## 0.076615209 0.128290293 0.082611626 0.082611626 0.103236410 0.147742102
## 5431 5432 5433 5434 5435 5436
## 0.089032111 0.089032111 0.147742102 0.082611626 0.111065767 0.089032111
## 5437 5438 5439 5440 5441 5442
## 0.119409831 0.111065767 0.089032111 0.111065767 0.119409831 0.111065767
## 5443 5444 5445 5446 5447 5448
## 0.071020351 0.147742102 0.128290293 0.158350725 0.082611626 0.103236410
## 5449 5450 5451 5452 5453 5454
## 0.111065767 0.111065767 0.095899426 0.103236410 0.119409831 0.082611626
## 5455 5456 5457 5458 5459 5460
## 0.089032111 0.095899426 0.158350725 0.119409831 0.076615209 0.111065767
## 5461 5462 5463 5464 5465 5466
## 0.147742102 0.158350725 0.103236410 0.111065767 0.076615209 0.147742102
## 5467 5468 5469 5470 5471 5472
## 0.103236410 0.128290293 0.095899426 0.103236410 0.137727896 0.082611626
## 5473 5474 5475 5476 5477 5478
## 0.103236410 0.076615209 0.147742102 0.056426775 0.071020351 0.065804943
## 5479 5480 5481 5482 5483 5484
## 0.128290293 0.065804943 0.111065767 0.089032111 0.082611626 0.082611626
## 5485 5486 5487 5488 5489 5490
## 0.076615209 0.128290293 0.076615209 0.137727896 0.095899426 0.076615209
## 5491 5492 5493 5494 5495 5496
## 0.119409831 0.111065767 0.071020351 0.119409831 0.169569538 0.103236410
## 5497 5498 5499 5500 5501 5502
## 0.119409831 0.128290293 0.071020351 0.147742102 0.103236410 0.103236410
## 5503 5504 5505 5506 5507 5508
## 0.089032111 0.119409831 0.147742102 0.089032111 0.071020351 0.137727896
## 5509 5510 5511 5512 5513 5514
## 0.128290293 0.103236410 0.082611626 0.128290293 0.089032111 0.071020351
## 5515 5516 5517 5518 5519 5520
## 0.089032111 0.056426775 0.103236410 0.103236410 0.128290293 0.095899426
## 5521 5522 5523 5524 5525 5526
## 0.111065767 0.089032111 0.119409831 0.082611626 0.207005441 0.089032111
## 5527 5528 5529 5530 5531 5532
## 0.082611626 0.137727896 0.111065767 0.128290293 0.147742102 0.076615209
## 5533 5534 5535 5536 5537 5538
## 0.128290293 0.071020351 0.128290293 0.076615209 0.095899426 0.095899426
## 5539 5540 5541 5542 5543 5544
## 0.095899426 0.089032111 0.103236410 0.111065767 0.119409831 0.137727896
## 5545 5546 5547 5548 5549 5550
## 0.147742102 0.103236410 0.119409831 0.111065767 0.128290293 0.103236410
## 5551 5552 5553 5554 5555 5556
## 0.103236410 0.158350725 0.111065767 0.111065767 0.103236410 0.128290293
## 5557 5558 5559 5560 5561 5562
## 0.128290293 0.103236410 0.095899426 0.128290293 0.137727896 0.095899426
## 5563 5564 5565 5566 5567 5568
## 0.128290293 0.082611626 0.128290293 0.181411860 0.076615209 0.103236410
## 5569 5570 5571 5572 5573 5574
## 0.065804943 0.111065767 0.065804943 0.147742102 0.089032111 0.111065767
## 5575 5576 5577 5578 5579 5580
## 0.095899426 0.119409831 0.103236410 0.128290293 0.103236410 0.111065767
## 5581 5582 5583 5584 5585 5586
## 0.095899426 0.095899426 0.128290293 0.137727896 0.071020351 0.056426775
## 5587 5588 5589 5590 5591 5592
## 0.076615209 0.076615209 0.111065767 0.076615209 0.158350725 0.119409831
## 5593 5594 5595 5596 5597 5598
## 0.193888125 0.147742102 0.089032111 0.119409831 0.089032111 0.095899426
## 5599 5600 5601 5602 5603 5604
## 0.082611626 0.071020351 0.111065767 0.095899426 0.103236410 0.158350725
## 5605 5606 5607 5608 5609 5610
## 0.128290293 0.060947403 0.128290293 0.128290293 0.158350725 0.076615209
## 5611 5612 5613 5614 5615 5616
## 0.071020351 0.095899426 0.089032111 0.060947403 0.082611626 0.089032111
## 5617 5618 5619 5620 5621 5622
## 0.128290293 0.076615209 0.119409831 0.076615209 0.111065767 0.128290293
## 5623 5624 5625 5626 5627 5628
## 0.158350725 0.076615209 0.119409831 0.181411860 0.137727896 0.128290293
## 5629 5630 5631 5632 5633 5634
## 0.137727896 0.082611626 0.082611626 0.071020351 0.119409831 0.076615209
## 5635 5636 5637 5638 5639 5640
## 0.082611626 0.111065767 0.128290293 0.128290293 0.181411860 0.119409831
## 5641 5642 5643 5644 5645 5646
## 0.111065767 0.103236410 0.147742102 0.119409831 0.082611626 0.056426775
## 5647 5648 5649 5650 5651 5652
## 0.111065767 0.119409831 0.128290293 0.082611626 0.076615209 0.095899426
## 5653 5654 5655 5656 5657 5658
## 0.128290293 0.103236410 0.119409831 0.147742102 0.095899426 0.111065767
## 5659 5660 5661 5662 5663 5664
## 0.089032111 0.095899426 0.076615209 0.095899426 0.128290293 0.111065767
## 5665 5666 5667 5668 5669 5670
## 0.082611626 0.169569538 0.089032111 0.119409831 0.111065767 0.128290293
## 5671 5672 5673 5674 5675 5676
## 0.076615209 0.111065767 0.111065767 0.095899426 0.137727896 0.111065767
## 5677 5678 5679 5680 5681 5682
## 0.158350725 0.082611626 0.095899426 0.119409831 0.089032111 0.103236410
## 5683 5684 5685 5686 5687 5688
## 0.147742102 0.158350725 0.119409831 0.119409831 0.111065767 0.111065767
## 5689 5690 5691 5692 5693 5694
## 0.089032111 0.082611626 0.137727896 0.128290293 0.119409831 0.076615209
## 5695 5696 5697 5698 5699 5700
## 0.158350725 0.089032111 0.089032111 0.111065767 0.071020351 0.169569538
## 5701 5702 5703 5704 5705 5706
## 0.147742102 0.111065767 0.089032111 0.089032111 0.076615209 0.207005441
## 5707 5708 5709 5710 5711 5712
## 0.071020351 0.111065767 0.111065767 0.082611626 0.103236410 0.119409831
## 5713 5714 5715 5716 5717 5718
## 0.137727896 0.095899426 0.137727896 0.137727896 0.095899426 0.169569538
## 5719 5720 5721 5722 5723 5724
## 0.103236410 0.119409831 0.095899426 0.147742102 0.119409831 0.111065767
## 5725 5726 5727 5728 5729 5730
## 0.103236410 0.169569538 0.060947403 0.147742102 0.158350725 0.137727896
## 5731 5732 5733 5734 5735 5736
## 0.111065767 0.119409831 0.071020351 0.082611626 0.076615209 0.128290293
## 5737 5738 5739 5740 5741 5742
## 0.082611626 0.095899426 0.158350725 0.095899426 0.128290293 0.128290293
## 5743 5744 5745 5746 5747 5748
## 0.076615209 0.071020351 0.111065767 0.060947403 0.119409831 0.128290293
## 5749 5750 5751 5752 5753 5754
## 0.089032111 0.147742102 0.147742102 0.071020351 0.065804943 0.128290293
## 5755 5756 5757 5758 5759 5760
## 0.119409831 0.082611626 0.111065767 0.082611626 0.137727896 0.089032111
## 5761 5762 5763 5764 5765 5766
## 0.128290293 0.137727896 0.111065767 0.095899426 0.009777277 0.103236410
## 5767 5768 5769 5770 5771 5772
## 0.076615209 0.111065767 0.071020351 0.082611626 0.089032111 0.111065767
## 5773 5774 5775 5776 5777 5778
## 0.089032111 0.119409831 0.056426775 0.082611626 0.082611626 0.089032111
## 5779 5780 5781 5782 5783 5784
## 0.095899426 0.065804943 0.060947403 0.111065767 0.060947403 0.103236410
## 5785 5786 5787 5788 5789 5790
## 0.103236410 0.082611626 0.103236410 0.038196000 0.082611626 0.076615209
## 5791 5792 5793 5794 5795 5796
## 0.128290293 0.095899426 0.095899426 0.089032111 0.071020351 0.119409831
## 5797 5798 5799 5800 5801 5802
## 0.128290293 0.128290293 0.193888125 0.089032111 0.158350725 0.095899426
## 5803 5804 5805 5806 5807 5808
## 0.111065767 0.095899426 0.082611626 0.147742102 0.076615209 0.158350725
## 5809 5810 5811 5812 5813 5814
## 0.128290293 0.119409831 0.137727896 0.060947403 0.095899426 0.119409831
## 5815 5816 5817 5818 5819 5820
## 0.111065767 0.089032111 0.089032111 0.082611626 0.082611626 0.076615209
## 5821 5822 5823 5824 5825 5826
## 0.071020351 0.158350725 0.095899426 0.111065767 0.082611626 0.137727896
## 5827 5828 5829 5830 5831 5832
## 0.147742102 0.089032111 0.089032111 0.071020351 0.128290293 0.137727896
## 5833 5834 5835 5836 5837 5838
## 0.082611626 0.089032111 0.158350725 0.082611626 0.128290293 0.052222807
## 5839 5840 5841 5842 5843 5844
## 0.065804943 0.089032111 0.071020351 0.111065767 0.082611626 0.065804943
## 5845 5846 5847 5848 5849 5850
## 0.095899426 0.111065767 0.103236410 0.103236410 0.147742102 0.071020351
## 5851 5852 5853 5854 5855 5856
## 0.119409831 0.089032111 0.089032111 0.089032111 0.089032111 0.089032111
## 5857 5858 5859 5860 5861 5862
## 0.082611626 0.103236410 0.089032111 0.056426775 0.111065767 0.137727896
## 5863 5864 5865 5866 5867 5868
## 0.128290293 0.111065767 0.071020351 0.076615209 0.082611626 0.103236410
## 5869 5870 5871 5872 5873 5874
## 0.119409831 0.119409831 0.065804943 0.111065767 0.147742102 0.076615209
## 5875 5876 5877 5878 5879 5880
## 0.111065767 0.119409831 0.056426775 0.060947403 0.119409831 0.076615209
## 5881 5882 5883 5884 5885 5886
## 0.119409831 0.119409831 0.103236410 0.103236410 0.111065767 0.103236410
## 5887 5888 5889 5890 5891 5892
## 0.111065767 0.119409831 0.181411860 0.089032111 0.137727896 0.119409831
## 5893 5894 5895 5896 5897 5898
## 0.158350725 0.095899426 0.119409831 0.089032111 0.071020351 0.181411860
## 5899 5900 5901 5902 5903 5904
## 0.137727896 0.089032111 0.103236410 0.082611626 0.103236410 0.111065767
## 5905 5906 5907 5908 5909 5910
## 0.095899426 0.076615209 0.089032111 0.111065767 0.111065767 0.103236410
## 5911 5912 5913 5914 5915 5916
## 0.060947403 0.111065767 0.071020351 0.089032111 0.095899426 0.076615209
## 5917 5918 5919 5920 5921 5922
## 0.158350725 0.119409831 0.076615209 0.181411860 0.076615209 0.071020351
## 5923 5924 5925 5926 5927 5928
## 0.076615209 0.158350725 0.103236410 0.103236410 0.060947403 0.111065767
## 5929 5930 5931 5932 5933 5934
## 0.137727896 0.137727896 0.095899426 0.119409831 0.111065767 0.103236410
## 5935 5936 5937 5938 5939 5940
## 0.082611626 0.119409831 0.095899426 0.082611626 0.111065767 0.089032111
## 5941 5942 5943 5944 5945 5946
## 0.119409831 0.095899426 0.119409831 0.082611626 0.119409831 0.103236410
## 5947 5948 5949 5950 5951 5952
## 0.128290293 0.095899426 0.082611626 0.128290293 0.103236410 0.095899426
## 5953 5954 5955 5956 5957 5958
## 0.103236410 0.111065767 0.111065767 0.111065767 0.103236410 0.089032111
## 5959 5960 5961 5962 5963 5964
## 0.119409831 0.095899426 0.103236410 0.089032111 0.128290293 0.103236410
## 5965 5966 5967 5968 5969 5970
## 0.103236410 0.103236410 0.103236410 0.095899426 0.119409831 0.082611626
## 5971 5972 5973 5974 5975 5976
## 0.095899426 0.071020351 0.103236410 0.111065767 0.095899426 0.056426775
## 5977 5978 5979 5980 5981 5982
## 0.111065767 0.103236410 0.076615209 0.095899426 0.119409831 0.103236410
## 5983 5984 5985 5986 5987 5988
## 0.111065767 0.169569538 0.128290293 0.128290293 0.111065767 0.071020351
## 5989 5990 5991 5992 5993 5994
## 0.169569538 0.089032111 0.147742102 0.082611626 0.119409831 0.048316010
## 5995 5996 5997 5998 5999 6000
## 0.076615209 0.111065767 0.060947403 0.082611626 0.089032111 0.128290293
## 6001 6002 6003 6004 6005 6006
## 0.089032111 0.095899426 0.137727896 0.158350725 0.158350725 0.119409831
## 6007 6008 6009 6010 6011 6012
## 0.119409831 0.089032111 0.089032111 0.119409831 0.071020351 0.089032111
## 6013 6014 6015 6016 6017 6018
## 0.076615209 0.103236410 0.071020351 0.082611626 0.095899426 0.111065767
## 6019 6020 6021 6022 6023 6024
## 0.095899426 0.076615209 0.082611626 0.119409831 0.082611626 0.119409831
## 6025 6026 6027 6028 6029 6030
## 0.095899426 0.089032111 0.111065767 0.147742102 0.082611626 0.119409831
## 6031 6032 6033 6034 6035 6036
## 0.076615209 0.095899426 0.119409831 0.082611626 0.089032111 0.111065767
## 6037 6038 6039 6040 6041 6042
## 0.103236410 0.137727896 0.082611626 0.119409831 0.082611626 0.060947403
## 6043 6044 6045 6046 6047 6048
## 0.095899426 0.128290293 0.065804943 0.111065767 0.119409831 0.111065767
## 6049 6050 6051 6052 6053 6054
## 0.103236410 0.103236410 0.089032111 0.089032111 0.095899426 0.128290293
## 6055 6056 6057 6058 6059 6060
## 0.082611626 0.103236410 0.095899426 0.103236410 0.065804943 0.119409831
## 6061 6062 6063 6064 6065 6066
## 0.089032111 0.137727896 0.095899426 0.128290293 0.095899426 0.111065767
## 6067 6068 6069 6070 6071 6072
## 0.119409831 0.089032111 0.089032111 0.128290293 0.060947403 0.128290293
## 6073 6074 6075 6076 6077 6078
## 0.137727896 0.128290293 0.076615209 0.095899426 0.071020351 0.103236410
## 6079 6080 6081 6082 6083 6084
## 0.128290293 0.082611626 0.119409831 0.089032111 0.119409831 0.111065767
## 6085 6086 6087 6088 6089 6090
## 0.103236410 0.095899426 0.119409831 0.095899426 0.137727896 0.076615209
## 6091 6092 6093 6094 6095 6096
## 0.076615209 0.119409831 0.119409831 0.158350725 0.095899426 0.119409831
## 6097 6098 6099 6100 6101 6102
## 0.111065767 0.089032111 0.119409831 0.119409831 0.082611626 0.082611626
## 6103 6104 6105 6106 6107 6108
## 0.103236410 0.056426775 0.119409831 0.111065767 0.147742102 0.111065767
## 6109 6110 6111 6112 6113 6114
## 0.128290293 0.169569538 0.082611626 0.071020351 0.103236410 0.103236410
## 6115 6116 6117 6118 6119 6120
## 0.071020351 0.137727896 0.169569538 0.119409831 0.071020351 0.082611626
## 6121 6122 6123 6124 6125 6126
## 0.076615209 0.111065767 0.071020351 0.065804943 0.076615209 0.128290293
## 6127 6128 6129 6130 6131 6132
## 0.111065767 0.089032111 0.111065767 0.137727896 0.147742102 0.009015632
## 6133 6134 6135 6136 6137 6138
## 0.076615209 0.111065767 0.056426775 0.095899426 0.089032111 0.095899426
## 6139 6140 6141 6142 6143 6144
## 0.169569538 0.128290293 0.076615209 0.089032111 0.169569538 0.082611626
## 6145 6146 6147 6148 6149 6150
## 0.111065767 0.095899426 0.103236410 0.082611626 0.089032111 0.056426775
## 6151 6152 6153 6154 6155 6156
## 0.119409831 0.119409831 0.111065767 0.128290293 0.111065767 0.082611626
## 6157 6158 6159 6160 6161 6162
## 0.137727896 0.111065767 0.076615209 0.095899426 0.095899426 0.076615209
## 6163 6164 6165 6166 6167 6168
## 0.137727896 0.169569538 0.065804943 0.095899426 0.082611626 0.111065767
## 6169 6170 6171 6172 6173 6174
## 0.071020351 0.082611626 0.095899426 0.169569538 0.095899426 0.060947403
## 6175 6176 6177 6178 6179 6180
## 0.095899426 0.071020351 0.089032111 0.009015632 0.095899426 0.082611626
## 6181 6182 6183 6184 6185 6186
## 0.089032111 0.128290293 0.119409831 0.089032111 0.103236410 0.119409831
## 6187 6188 6189 6190 6191 6192
## 0.082611626 0.095899426 0.119409831 0.060947403 0.119409831 0.119409831
## 6193 6194 6195 6196 6197 6198
## 0.095899426 0.095899426 0.060947403 0.103236410 0.128290293 0.111065767
## 6199 6200 6201 6202 6203 6204
## 0.158350725 0.065804943 0.082611626 0.082611626 0.137727896 0.089032111
## 6205 6206 6207 6208 6209 6210
## 0.071020351 0.095899426 0.103236410 0.119409831 0.095899426 0.089032111
## 6211 6212 6213 6214 6215 6216
## 0.111065767 0.193888125 0.103236410 0.119409831 0.147742102 0.119409831
## 6217 6218 6219 6220 6221 6222
## 0.089032111 0.082611626 0.082611626 0.137727896 0.089032111 0.128290293
## 6223 6224 6225 6226 6227 6228
## 0.089032111 0.082611626 0.128290293 0.119409831 0.082611626 0.169569538
## 6229 6230 6231 6232 6233 6234
## 0.095899426 0.158350725 0.095899426 0.158350725 0.128290293 0.111065767
## 6235 6236 6237 6238 6239 6240
## 0.082611626 0.119409831 0.060947403 0.111065767 0.119409831 0.089032111
## 6241 6242 6243 6244 6245 6246
## 0.089032111 0.111065767 0.137727896 0.089032111 0.119409831 0.103236410
## 6247 6248 6249 6250 6251 6252
## 0.111065767 0.103236410 0.119409831 0.089032111 0.119409831 0.103236410
## 6253 6254 6255 6256 6257 6258
## 0.111065767 0.128290293 0.082611626 0.119409831 0.119409831 0.128290293
## 6259 6260 6261 6262 6263 6264
## 0.128290293 0.158350725 0.119409831 0.095899426 0.082611626 0.060947403
## 6265 6266 6267 6268 6269 6270
## 0.169569538 0.076615209 0.082611626 0.103236410 0.095899426 0.128290293
## 6271 6272 6273 6274 6275 6276
## 0.103236410 0.119409831 0.071020351 0.119409831 0.076615209 0.071020351
## 6277 6278 6279 6280 6281 6282
## 0.082611626 0.137727896 0.103236410 0.103236410 0.147742102 0.071020351
## 6283 6284 6285 6286 6287 6288
## 0.089032111 0.158350725 0.103236410 0.111065767 0.095899426 0.147742102
## 6289 6290 6291 6292 6293 6294
## 0.137727896 0.082611626 0.095899426 0.076615209 0.119409831 0.181411860
## 6295 6296 6297 6298 6299 6300
## 0.128290293 0.103236410 0.095899426 0.011496734 0.082611626 0.095899426
## 6301 6302 6303 6304 6305 6306
## 0.119409831 0.082611626 0.103236410 0.089032111 0.095899426 0.071020351
## 6307 6308 6309 6310 6311 6312
## 0.076615209 0.056426775 0.082611626 0.111065767 0.103236410 0.082611626
## 6313 6314 6315 6316 6317 6318
## 0.076615209 0.128290293 0.119409831 0.128290293 0.147742102 0.128290293
## 6319 6320 6321 6322 6323 6324
## 0.111065767 0.128290293 0.076615209 0.095899426 0.082611626 0.111065767
## 6325 6326 6327 6328 6329 6330
## 0.076615209 0.095899426 0.111065767 0.111065767 0.119409831 0.128290293
## 6331 6332 6333 6334 6335 6336
## 0.103236410 0.119409831 0.089032111 0.089032111 0.089032111 0.095899426
## 6337 6338 6339 6340 6341 6342
## 0.103236410 0.095899426 0.119409831 0.089032111 0.065804943 0.052222807
## 6343 6344 6345 6346 6347 6348
## 0.089032111 0.089032111 0.095899426 0.082611626 0.128290293 0.065804943
## 6349 6350 6351 6352 6353 6354
## 0.128290293 0.119409831 0.089032111 0.147742102 0.128290293 0.128290293
## 6355 6356 6357 6358 6359 6360
## 0.089032111 0.103236410 0.119409831 0.137727896 0.111065767 0.082611626
## 6361 6362 6363 6364 6365 6366
## 0.082611626 0.103236410 0.119409831 0.111065767 0.111065767 0.071020351
## 6367 6368 6369 6370 6371 6372
## 0.128290293 0.076615209 0.082611626 0.128290293 0.082611626 0.103236410
## 6373 6374 6375 6376 6377 6378
## 0.119409831 0.095899426 0.089032111 0.095899426 0.095899426 0.128290293Gráfica del Modelo RLogS
learning_disabilities <- cdd_factores_rendimiento_estudiantes_G6_dep$learning_disabilities
exam_score <- cdd_factores_rendimiento_estudiantes_G6_dep$exam_score
dataPlot <- data.frame(exam_score, learning_disabilities)
plot(learning_disabilities~exam_score, data = dataPlot, main = "Modelo RLogS: Exam score - Learning disabilities", xlab = "Exam score", ylab = "Learning disabilities = 0 | Learning disabilities = 1", col = "gold", pch = "I")
curve(predict(glm(learning_disabilities~exam_score, family = "binomial", data = dataPlot), data.frame(exam_score = x), type = "response"), col = "orange", lwd = 3, add = TRUE)
Conclusión
El conjunto de datos “Student Performance Factors” de Kaggle explora los determinantes del rendimiento académico de los estudiantes a través de múltiples dimensiones, como características demográficas, socioeconómicas, familiares y conductuales. Los resultados de análisis previos indican que factores como la educación de los padres, particularmente de la madre, y los ingresos familiares tienen una correlación positiva con el desempeño, al facilitar acceso a recursos educativos de calidad. Además, un entorno familiar de apoyo emocional y motivacional se asocia con un mejor rendimiento, mientras que hábitos de estudio organizados y el balance adecuado entre actividades extracurriculares y tiempo libre contribuyen significativamente al éxito académico
El entorno escolar, incluido el nivel de preparación de los docentes y la calidad de las infraestructuras, también juega un rol crítico. Sin embargo, se identifican disparidades sustanciales en el acceso y los resultados entre estudiantes de contextos socioeconómicos desfavorecidos o regiones con recursos limitados, lo que subraya la importancia de intervenciones políticas que promuevan la equidad educativa. Este dataset, al proporcionar datos ricos y multifacéticos, se convierte en una herramienta esencial para diseñar estrategias personalizadas, como modelos predictivos para identificar estudiantes en riesgo, y generar soluciones orientadas a cerrar brechas estructurales en el sistema educativo
Referencias
{kaggle2024, author = {Kaggle}, title = {Student Performance Factors Dataset}, year = {2024}, url = {https://www.kaggle.com/datasets/lainguyn123/student-performance-factors} }
---
title: "**RMD_G6**"
subtitle: "Estudio de Análisis Multivariado con base en un conjunto de datos sobre factores de rendimiento académico. "
author: "jorge.eliecer.franco::santiago.osorio.ramirez::pineda.luis::@correounivalle.edu.co"
date: "Estudio hecho durante el periodo académico agosto-diciembre de 2024"
output:
  html_document:
    toc: true
    toc_float: true
    code_download: true
    theme: lumen
bibliography: bibliografia_ME.bib
csl: apa.csl
link-citations: yes
---
<!-- Configuración Global de R -->
```{r setup, include=FALSE}
library(readxl)
library(corrplot)
library(GGally)
library(ggplot2)
library(andrews)
library(tcltk)
library(aplpack)
library(graphics)
library(MVN)
library(reshape2)
library(FactoMineR)
library(factoextra)
library(psych)
library(FactoClass)
library(cluster)
library(dendextend)
library(magrittr)
library(NbClust)
library(stargazer)
knitr::opts_chunk$set(echo=TRUE)

cdd_factores_rendimiento_estudiantes_G6 <- read_excel("C:/Users/57313/Desktop/Gdd_(G6)/cdd_G6_og/cdd_factores_rendimiento_estudiantes_G6.xlsx")


cdd_factores_rendimiento_estudiantes_G6_dep <- read_excel("C:/Users/57313/Desktop/Gdd_(G6)/cdd_G6_dep/cdd_factores_rendimiento_estudiantes_G6_dep.xlsx")
     

```

## **Fase 1 [Descripciones Multivariantes]**
### **1.1. Objetivos**
En esta primera fase de análisis, se realizarán cálculos, visualizaciones e interpretaciones basadas en un conjunto de datos, utilizando un enfoque de estadística descriptiva multivariante. Además, se aplicará una prueba estadística de normalidad multivariada sobre los datos analizados.

### **1.2. Descripción de los datos**

### **Fuente del Conjunto de Datos**
El conjunto de datos de trabajo se obtuvo casi totalmente de **Kaggle**:https://www.kaggle.com/datasets/lainguyn123/student-performance-factors. Es conveniente anotar que **Kaggle** es una compañía subsidiaria de Google LLC que mantiene una comunidad en línea de científicos de datos y profesionales del aprendizaje automático. Esta empresa permite a sus usuarios encontrar y publicar conjuntos de datos, explorar y crear modelos en un entorno de ciencia de datos basado en la web, trabajar con otros científicos de datos e ingenieros de aprendizaje automático y participar en concursos para resolver desafíos de ciencia de datos.


### **Contexto del Conjunto de Datos**
El conjunto de datos incluye métricas académicas obtenidas por estudiantes para tener una visión general de los factores que afectan el rendimiento en los exámenes. Este conjunto de datos se actualizó por última vez en septiembre de 2024

### **Descripción del Conjunto de Datos** {.tabset .tabset-pills}
El conjunto de datos original incluye 20 campos y 6,607 registros, una vez depurado incluye 21 campos y 6378 registros. Entre estos campos, se destacan: las horas dedicadas al estudio, la edad de los estudiantes, el porcentaje de asistencia a clases, y dos variables de naturaleza politómica. También se recoge si el estudiante participa en actividades extracurriculares, la cantidad de horas de sueño, y las calificaciones obtenidas en exámenes previos. Otro campo es de tipo politómico, mientras que el acceso a internet se clasifica de manera dicotómica. El número de tutorías mensuales a las que asiste el estudiante también está registrado, seguido por otros campos de naturaleza politómica. Además, se incluye el tipo de escuela, la influencia parental (politómica), las horas de actividad física por semana, si el estudiante presenta alguna discapacidad de aprendizaje, la distancia del hogar (politómica), el género del estudiante, y la puntuación en el examen final. Los campos se describen en el mismo orden en que aparecen en el rango de datos.

- **Hours studied** (cuantitativa::razón):  Número total de horas que el estudiante dedica al estudio por semana. Este valor refleja la cantidad de tiempo invertido en actividades académicas fuera del horario escolar, como tareas, repaso de contenidos y preparación para exámenes.

- **Age** (cuantitativa::razón): Representa la cantidad de años que tiene cada estudiante al momento de realizar la medición. Se uso una funcion ALEATORIO.ENTRE en excel para darle un rango de entre 16 y 18 años 

- **Attendance** (cuantitativa::razón): Porcentaje de clases a las que el estudiante ha asistido en relación con el total de clases impartidas. Este indicador puede ser un reflejo del compromiso del estudiante con el curso.

- **Parental involvement** (cualitativa::nominal): Nivel de involucramiento de los padres en la educación del estudiante. Se clasifica en tres niveles (Bajo, Medio, Alto), y puede incluir aspectos como supervisión de tareas, asistencia a reuniones escolares y apoyo en la toma de decisiones académicas.

- **Access to resources** (cualitativa::nominal): Disponibilidad de recursos académicos como libros, materiales de estudio, herramientas tecnológicas, etc. Este atributo puede influir en la capacidad del estudiante para realizar trabajos y estudiar. Se clasifica en Bajo, Medio y Alto.

- **Extracurricular activities** (cualitativa::nominal): Indica si el estudiante participa o no en actividades fuera del currículo académico, como deportes, música, arte o clubes académicos. Puede tener impacto en su desarrollo integral y rendimiento académico.

- **Sleep hours** (cuantitativa::razón): Promedio de horas de sueño que el estudiante obtiene cada noche. El sueño es esencial para la concentración, el rendimiento académico y la salud mental.

- **Previous score** (cuantitativa::razón): Calificaciones o notas obtenidas por el estudiante en exámenes anteriores. Este atributo puede reflejar el historial de rendimiento académico y es un posible predictor de su desempeño futuro.
- **Motivation level** (cualitativa::ordinal): Grado de interés y esfuerzo que el estudiante muestra hacia sus estudios. Clasificado en Bajo, Medio y Alto, este atributo mide la disposición del estudiante para aprender y mejorar.

- **Internet access** (cualitativa::nominal): Indica si el estudiante tiene acceso a Internet en su hogar (Sí, No), lo que puede ser crucial para realizar investigaciones, completar tareas en línea y acceder a materiales de apoyo educativo.

- **Tutoring sessions** (cuantitativa::razón): Número de sesiones de tutoría a las que el estudiante asiste mensualmente. Las tutorías pueden ser de apoyo académico, reforzando áreas en las que el estudiante tenga dificultades.

- **Family income** (cualitativa::nominal): Nivel de ingresos económicos del hogar del estudiante, clasificado en Bajo, Medio y Alto. Este factor puede influir en la disponibilidad de recursos y oportunidades educativas.

- **Techer quality** (cualitativa::ordinal): Percepción del estudiante sobre la calidad de sus profesores, clasificado en Bajo, Medio y Alto. Incluye aspectos como el dominio de los contenidos, la metodología de enseñanza y la capacidad para motivar a los estudiantes.
- **School type** (cualitativa::nominal): Categoría que indica si el estudiante asiste a una escuela pública o privada, lo que puede influir en el acceso a recursos, el tamaño de las clases y la calidad de la enseñanza.

- **Peer influence** (cualitativa::ordinal): Impacto de las acciones y actitudes de los padres en el rendimiento académico del estudiante. Se clasifica en Positiva, Neutral o Negativa y refleja el apoyo, presión o indiferencia por parte de los padres hacia el progreso académico del estudiante.

- **Physical activity** (cuantitativa::razón): Promedio de horas dedicadas a la actividad física semanalmente, lo que puede influir en la salud física y mental del estudiante, afectando indirectamente su rendimiento académico.

- **Learning disabilities** (cualitativa::nominal): Indica si el estudiante tiene alguna discapacidad de aprendizaje (Sí, No), como dislexia, TDAH u otros trastornos que podrían afectar su rendimiento en ciertas áreas académicas.

- **Parental education level** (cualitativa::ordinal): El nivel educativo más alto alcanzado por los padres (Secundaria, Universidad, Posgrado), lo que puede influir en el apoyo académico y el entorno de aprendizaje en el hogar.

- **Distance from home** (cualitativa::ordinal): Distancia entre el hogar del estudiante y la escuela, clasificado en Cerca, Moderada y Lejos. La distancia puede afectar la puntualidad, el tiempo disponible para el estudio y el nivel de cansancio del estudiante.

- **Gender** (cualitativa::nominal): El género del estudiante (Masculino, Femenino), que podría ser un factor de interés para analizar diferencias en el rendimiento académico o en la participación en actividades escolares.

- **Exam score** (cuantitativa::razón): Calificación final obtenida por el estudiante en un examen específico, que refleja su rendimiento en esa evaluación en particular.

#### Estructura del Conjunto de Datos Original
```{r Estructura_Conjunto_de_Datos_Original, fig.align = 'center'}
str(cdd_factores_rendimiento_estudiantes_G6)

```

#### Conjunto de Datos Original 
```{r Conjunto_de_Datos_Original, fig.align = 'center'}
cdd_factores_rendimiento_estudiantes_G6

```

#### Estructura del Conjunto de Datos Depurado
```{r Estructura_Conjunto_de_Datos_Depurado, fig.align = 'center'}
str(cdd_factores_rendimiento_estudiantes_G6_dep)

```

#### Conjunto de Datos Depurado 
```{r Conjunto_de_Datos_Depurado, fig.align = 'center'}
cdd_factores_rendimiento_estudiantes_G6_dep

```

### **1.3. Estimaciones multivariadas** {.tabset .tabset-pills}


Este análisis explora el rendimiento académico de estudiantes a través de un conjunto de datos que incluye múltiples factores, como horas de estudio, asistencia, acceso a recursos, actividades extracurriculares, horas de sueño, nivel de motivación, y diversas variables familiares y escolares. Utilizando estimaciones multivariadas, se pueden examinar las interacciones entre estas variables y su relación conjunta con el rendimiento académico, medido aquí por la calificación final del estudiante. Este enfoque permite identificar patrones y factores clave que podrían influir en el desempeño de los estudiantes, considerando las complejidades inherentes a la interrelación de variables académicas, familiares y personales.


El conjunto de datos incluye tanto variables cuantitativas, como las horas de estudio, asistencia y puntajes previos, como cualitativas, tales como el nivel de motivación y la calidad percibida de la enseñanza. Estas variables se interrelacionan y, mediante técnicas multivariadas, se puede estudiar cómo diferentes factores, desde la implicación parental y los ingresos familiares hasta las actividades físicas y los patrones de sueño, influyen en el rendimiento académico. Por ejemplo, al analizar el impacto de la "Motivación" junto con "Asistencia" y "Acceso a Recursos", se puede obtener una visión más detallada de cómo el ambiente y los hábitos del estudiante afectan su desempeño. Técnicas como la regresión múltiple o el análisis de factores pueden ayudar a extraer relaciones clave y predictores del éxito académico en este contexto multidimensional.


Las estimaciones multivariadas aplicadas a este conjunto de datos permiten desarrollar una comprensión profunda de cómo diversos factores personales, académicos y familiares contribuyen conjuntamente al rendimiento de los estudiantes. Este enfoque no solo identifica los factores individuales más influyentes, sino que también revela cómo las interacciones entre ellos pueden fortalecer o debilitar el rendimiento académico. Estas conclusiones podrían servir para diseñar intervenciones específicas que apoyen el desarrollo académico, destacando la importancia de un análisis holístico que considere las múltiples dimensiones que afectan la educación. 

### Vector de Medias y Boxplot
```{r Vector_de_Medias_y_Boxplot, fig.align = 'center'}
apply(cdd_factores_rendimiento_estudiantes_G6_dep[-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], 2, mean)
cdd_factores_rendimiento_estudiantes_Reducido = cdd_factores_rendimiento_estudiantes_G6_dep [,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)]
par(mfrow = c(1, ncol(cdd_factores_rendimiento_estudiantes_Reducido)))
invisible(lapply(1:ncol(cdd_factores_rendimiento_estudiantes_Reducido), function(i) boxplot(cdd_factores_rendimiento_estudiantes_Reducido[, i])))

```

### Matriz de Varianzas-Covarianzas
```{r Matriz_de_Varianzas_Covarianzas, fig.align = 'center'}
round(cov(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)]),2)

```

### Matriz de Correlaciones
```{r Matriz_de_Correlaciones, fig.align = 'center'}
round(cor(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)]),3)

```


### **1.4. Gráficas multivariadas** {.tabset .tabset-pills}

Los gráficos multivariados son herramientas analíticas que permiten visualizar y comprender relaciones complejas entre múltiples variables dentro de un conjunto de datos. Este tipo de gráficos es esencial para evaluar y analizar factores que influyen en fenómenos complejos, como el rendimiento académico. En este conjunto de datos específico, se registran diversas variables relacionadas con el rendimiento estudiantil, como las horas de estudio, la calidad del docente, el nivel de ingresos familiares, el acceso a recursos y otros factores sociales y personales. Utilizar gráficos multivariados en este contexto nos permite identificar patrones y correlaciones entre estos factores, proporcionando una comprensión más profunda de cómo interactúan para influir en el éxito académico de los estudiantes.


Este conjunto de datos contiene variables tanto cuantitativas como cualitativas que afectan el rendimiento estudiantil. Entre los factores cuantitativos se incluyen las horas de estudio, el nivel de asistencia, las sesiones de tutoría y el puntaje en exámenes. Estos datos permiten crear gráficos de dispersión, histogramas y diagramas de caja que ayudan a visualizar las distribuciones y tendencias de cada variable. Por otro lado, las variables cualitativas, como el nivel de participación parental, la calidad del docente y el tipo de escuela, permiten construir gráficos de barras apiladas o gráficos de mosaico para analizar la influencia de cada categoría. Al emplear gráficos de correlación y mapas de calor, se pueden identificar relaciones entre variables, por ejemplo, si un mayor número de horas de estudio o una alta calidad docente están relacionadas con mayores puntajes en exámenes.


En conclusión, los gráficos multivariados ofrecen una visión integrada de cómo diversos factores interactúan para influir en el rendimiento estudiantil. La visualización de estos datos permite a investigadores y educadores identificar patrones y áreas de mejora que pueden guiar intervenciones educativas efectivas. Por ejemplo, la relación entre el ingreso familiar y el acceso a recursos o entre la calidad del docente y el rendimiento académico puede esclarecer qué factores tienen un mayor impacto en los resultados de los estudiantes. Así, los gráficos multivariados no solo facilitan el análisis descriptivo, sino que también proporcionan una base sólida para el diseño de estrategias pedagógicas personalizadas y programas de apoyo, optimizando las oportunidades de éxito académico.


#### Diagrama Conjunto de Dispersión, Distribución y Correlaciones 
```{r Diagrama_Conjunto_de_Dispersion_Distribucion_y_Correlaciones, fig.align = 'center'}
ggpairs(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)])
```


#### Diagrama Conjunto de Dispersión, Distribución y Correlaciones [ES:GE]
```{r Diagrama_Conjunto_de_Dispersion_Distribucion_y_Correlaciones_ES_GE, fig.align = 'center'}
ggpairs(cdd_factores_rendimiento_estudiantes_G6, columns = c(1,2,6,7,20), aes(color = Gender, alpha = 0.5), upper = list(continous = wrap("cor", size = 2.5)))
```


#### Diagrama Conjunto de Dispersión, Distribución y Correlaciones [ES:EA]
```{r Diagrama_Conjunto_de_Dispersion_Distribucion_y_Correlaciones_ES_EA, fig.align = 'center'}
ggpairs(cdd_factores_rendimiento_estudiantes_G6, columns = c(1,2,6,7,20), aes(color = Extracurricular_Activities, alpha = 0.5), upper = list(continous = wrap("cor", size = 2.5)))
```


#### Diagrama Conjunto de Dispersión, Distribución y Correlaciones [ES:TQ]
```{r Diagrama_Conjunto_de_Dispersion_Distribucion_y_Correlaciones_ES_TQ, fig.align = 'center'}
ggpairs(cdd_factores_rendimiento_estudiantes_G6, columns = c(1,2,6,7,20), aes(color = Teacher_Quality, alpha = 0.5), upper = list(continous = wrap("cor", size = 2.5)))
```


#### Diagrama de Estrellas
```{r Diagrama_de_Estrellas, fig.align = 'center'}
set.seed(780728)
cdd_factores_rendimiento_estudiantes_G6_muestreado = cdd_factores_rendimiento_estudiantes_G6_dep[sample(1:nrow(cdd_factores_rendimiento_estudiantes_G6_dep),23), -c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)]
stars(cdd_factores_rendimiento_estudiantes_G6_muestreado, len = 1, cex = 0.4, key.loc = c(21,20), draw.segments = TRUE)
```


#### Caras de Chernoff
```{r Caras_de_Chernoff, fig.align = 'center'}
set.seed(780728)
cdd_factores_rendimiento_estudiantes_G6_muestreado = cdd_factores_rendimiento_estudiantes_G6_dep[sample(1:nrow(cdd_factores_rendimiento_estudiantes_G6_dep),23), -c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)]
faces(cdd_factores_rendimiento_estudiantes_G6_muestreado)
```



### **1.5. Normalidad multivariada** {.tabset .tabset-pills}

La normalidad multivariada es un concepto esencial en el análisis estadístico multivariante, utilizado para evaluar patrones entre múltiples variables relacionadas. Este enfoque resulta especialmente relevante en estudios de rendimiento académico, en los cuales factores como horas de estudio, asistencia, apoyo parental y actividades extracurriculares influyen en el desempeño de los estudiantes. El presente informe examina la normalidad multivariada en estos datos de rendimiento académico, lo cual es crucial para aplicar modelos estadísticos avanzados, como el análisis de regresión multivariante y el análisis de componentes principales, que dependen de esta condición para producir resultados confiables y significativos.


Para determinar si los datos de rendimiento académico cumplen con la normalidad multivariada, se analizaron diversas variables mediante pruebas estadísticas como la de Mardia y representaciones gráficas de dispersión. Estas herramientas ayudan a visualizar la simetría y la forma de los datos, evaluando si variables como horas de sueño, actividades extracurriculares y participación parental siguen un comportamiento multivariante normal. Cumplir con esta condición permite aplicar modelos estadísticos con mayor precisión, como análisis de regresión y componentes principales, para identificar relaciones significativas.


Determinar la normalidad multivariada es fundamental para asegurar la validez de las inferencias estadísticas en este análisis de datos académicos. Si los datos cumplen con esta condición, los resultados obtenidos de los modelos estadísticos serán más confiables. En caso contrario, se puede optar por transformaciones de datos o modelos no paramétricos, lo que permite mantener la precisión en la interpretación de los factores.

La **PNM de Mardia** establece que si ambas pruebas (para asimetría y curtosis) indican una normalidad multivariante, los datos siguen una DNM con un nivel de significancia $\alpha=0.05$; sin embargo, el caso tratado es contrario a esto. Obsérvese a través de la pestaña **PNM Mardia** que los $p-value$ para la asimetría (Skewness) y curtoris (Kurtosis) son mayores que el nivel de significancia. Por lo tanto, las evidencias no apoyan una hipótesis de normalidad multivariada para el conjunto de datos restringido a sus variables numéricas.

La **PNM de PNM Henze-Zirkler** establece que el estadístico de prueba no se distribuye aproximadamente como lognormal dado que su $p-value$ es menor que el nivel de significancia $\alpha=0.05$, obsérvese esto a través de la pestaña **PNM Henze-Zirkler**. Así, por contrarrecíproco de la implicación formulada en la descripción de la prueba en la [sección 5](#sec5), el conjunto de datos no está apoyado por las evidencias para seguir una distribución normal multivariada.

La **PNM de Doornik-Hansen** establece que su estadístico de prueba no sigue una distribución aproximadamente $\chi^2$ dado que su $p-value$ es menor que el nivel de significancia $\alpha=0.05$, obsérvese esto a través de la pestaña **PNM Doornik-Hansen**. Por lo tanto, las evidencias están lejos de apoyar que el conjunto de datos sigue una DNM.

La **PNM de Royston** establece que el conjunto de datos reducido a sus variables numéricas no sigue una DNM, dado que su $p-value$ es menor que el nivel de significancia $\alpha=0.05$. Obsérvese esto a través de la pestaña **PNM Royston**.

#### PNM Mardia
```{r PNM_Mardia, fig.align = 'center'}
mvn(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], mvnTest="mardia")
```


#### PNM Henze-Zirkler
```{r PNM_Henze_Zirkler, fig.align = 'center'}
mvn(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], mvnTest="hz")
```

#### PNM Doornik-Hansen
```{r PNM_Doornik_Hansen, fig.align = 'center'}
mvn(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], mvnTest="dh")
```

#### PNM Royston
```{r PNM_Royston, fig.align = 'center'}
# Seleccionar una muestra 
set.seed(780729)
cdd_factores_rendimiento_estudiantes_G6_muestr = cdd_factores_rendimiento_estudiantes_G6_dep[sample(1:nrow(cdd_factores_rendimiento_estudiantes_G6_dep),250),-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)]

mvn(cdd_factores_rendimiento_estudiantes_G6_muestr, mvnTest="royston")
```



## **Fase 2 [Componentes Principales]**
### **2.1. Objetivos**

El análisis de rendimiento académico basado en factores sociales, familiares y personales permite entender las dinámicas que influyen en el éxito estudiantil. Este conjunto de datos recopila información diversa, como horas de estudio, participación parental, calidad docente, y nivel de ingresos familiares, con el objetivo de identificar y analizar los elementos clave que afectan el desempeño en exámenes de estudiantes. A través de una exploración detallada de estos factores, se busca detectar patrones que puedan orientar intervenciones educativas y mejoras en el entorno de aprendizaje.

 
El conjunto de datos incluye tanto variables cuantitativas como cualitativas relacionadas con el contexto de los estudiantes y su rendimiento. Factores cuantitativos, como horas de estudio y puntajes en exámenes, y factores cualitativos, como el tipo de escuela y el nivel educativo de los padres, proporcionan una perspectiva amplia sobre el entorno de cada estudiante. Este enfoque permite analizar el impacto de elementos como el apoyo familiar, la motivación y las oportunidades de acceso a recursos. A partir de estos datos, se pueden aplicar técnicas analíticas, como el análisis de componentes principales, para identificar variables con mayor influencia y generar una interpretación global de los factores de éxito.
 
En conclusión, el objetivo principal de este conjunto de datos es ofrecer una base sólida para investigar cómo diversos factores interactúan y contribuyen al rendimiento académico. Este análisis no solo proporciona una comprensión más completa del entorno estudiantil, sino que también puede guiar la implementación de estrategias educativas enfocadas en el bienestar y desarrollo de los estudiantes. Al entender mejor los factores determinantes del rendimiento académico, los educadores y responsables pueden diseñar intervenciones personalizadas que promuevan mejores resultados en el ámbito educativo.

### **2.2. Selección de Componentes** {.tabset .tabset-pills}


Este análisis de componentes principales (ACP) aborda los factores que influyen en el rendimiento académico de los estudiantes. A partir de una variedad de datos, que incluyen factores de estudio, asistencia, actividades extracurriculares y características socioeconómicas, se utiliza el ACP para reducir la dimensionalidad del conjunto y destacar las variables más influyentes. Este proceso permite identificar patrones subyacentes entre las variables y descubrir cuáles contribuyentes de manera significativa al rendimiento en los exámenes, facilitando una interpretación más clara y estratégica de los datos.


En la etapa de análisis, se emplean múltiples herramientas para desentrañar las correlaciones y la importancia relativa de cada factor. Primero, la matriz de correlaciones permite observar las relaciones entre variables, mientras que la matriz ACP identifica aquellos factores con alta varianza explicada. Utilizando el gráfico de Cattell y el gráfico de Cattell-Kaiser, se determina el número adecuado de componentes a retener, lo cual simplifica el análisis sin perder información relevante. El círculo de correlaciones ofrece una representación visual de las variables, mostrando cómo se agrupan en torno a los principales componentes. Además, se evalúa la calidad de representación para medir la precisión con la que cada variable se representa en el espacio de componentes, junto con las coordenadas individuales, que proporcionan una visión detallada de cada estudiante respecto a los factores clave.


Este análisis confirma la utilidad del ACP en la identificación de los factores que más impactan el rendimiento estudiantil. Al simplificar la estructura de los datos, se logran resaltar las relaciones significativas y se puede enfocar en las variables que realmente contribuyen al desempeño académico. La comprensión de estos componentes principales no solo facilita una interpretación más precisa, sino que también proporciona una base sólida para futuras intervenciones educativas. Con esta información, se pueden diseñar estrategias enfocadas en mejorar los aspectos críticos identificados, beneficiando tanto a los estudiantes como al sistema educativo


#### Matriz ACP
```{r Matriz_ACP, fig.align = 'center'}
get_eigenvalue(PCA(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], ncp = 7, scale.unit = TRUE, graph = F))
```

#### Matriz de Correlaciones
```{r Matriz_de_Correlaciones_S}
round(cor(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)]),3)
```

#### Valores y Vectores Propios
```{r Valores_y_Vectores_Propios, fig.align = 'center'}
princomp(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], cor = TRUE)$sdev^2
princomp(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], cor = TRUE)$loadings[ ,1:7]
```


#### Correlaciones Comparadas
```{r Correlaciones_Comparadas, fig.align='center'}
par(mfrow=c(1,2))
corrplot::corrplot(cor(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)]), method = "color", type = "upper", number.cex = 0.4)
corrplot::corrplot(cor(princomp(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], cor = TRUE)$scores), method = "color", type = "upper", number.cex = 0.4)
```

#### Gráfico de Cattell
```{r Grafico_de_Cattell, fig.align = 'center'}
fviz_eig(PCA(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], scale.unit = T, graph = F), addlabels = T, ylim=c(0,90), main = "")
```

#### Gráfico de Cattell-Kaiser
```{r Grafico_de_Cattell_Kaiser, fig.align = 'center'}
scree(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)],factors = FALSE, pc = TRUE, main ="")
```


### **2.3. Calidad de Representación** {.tabset .tabset-pills}

Al retomar el análisis basado en los datos del conjunto cdd_factores_rendimiento_estudiantes_G6_dep, se verifica que, una vez reducida la dimensionalidad del conjunto de datos mediante el análisis de componentes principales (ACP) y entendido que sus variables estandarizadas están representadas gráficamente como proyecciones en una hiperesfera de correlaciones, es necesario iniciar la interpretación de los componentes principales a partir de dichas correlaciones. Posteriormente, se debe evaluar la calidad de las representaciones de las variables dentro de este nuevo espacio reducido, considerando la información retenida tras la disminución dimensional del conjunto de datos.


El **Círculo de Correlaciones** expresa que se puede concebir una componente tipo tamaño en el sentido de que la dimensión 1 muestra en él una correlación positiva con las siete variables de interés, además cercanas a la frontera del círculo unitario y significativamente próximas al eje que la representa. Por otro lado, la dimensión 2 contrapone a las variables **previous score** y **hours studied** con las demás. Otro aspecto por resaltar es la correlación mostrada entre pares de variables, que en términos del fenómeno estudiado conservan su naturaleza correlacio

La **Matriz de Representación**, por otro lado, muestra valores significativamente cercanos a 1 del cociente de proyecciones coseno cuadrado en relación con la dimensión 1. En este sentido, los puntos proyectados están altamente asociados con este componente. Así, la estaña que muestra la **Calidad de Representación** indica en su escala un piso alto de $0.88$ de esta manera las calidades de representación, en relación con la componente 1, están encabezadas por **hours studied** y cierran con **physical activity**. 

Por último, las **Coordenadas Individuales** ayudan, aunque de manera menos digestiva, a identificar a nivel de observaciones a los perfiles de los registros

#### Círculo de Correlaciones
```{r Circulo_de_Correlaciones, fig.align = 'center'}
fviz_pca_var(PCA(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], scale.unit = T, graph = F),col.var="#3B83BD", repel = T, col.circle = "#CDCDCD", ggtheme = theme_bw())
```

#### Matriz de Representación
```{r Matriz_de_Repressentacion_COS2, fig.align = 'center'}
(get_pca_var(PCA(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], ncp = 7, scale.unit = TRUE, graph = F)))$cos2
```

#### Calidad de Representación
```{r Calidad_de_la_Representacion, fig.align = 'center'}
fviz_pca_var(PCA(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], ncp = 7, scale.unit = TRUE, graph = F), col.var="cos2", gradient.cols=c("#00AFBB","#E7B800","#FC4E07"), repel = TRUE)
```

#### Coordenadas Individuales
```{r Coordenadas_Registros, fig.align = 'center'}
head((PCA(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], ncp = 7, scale.unit = TRUE, graph = F))$ind$coord, n = 23L)
```

### **2.4. Contribuciones** {.tabset .tabset-pills}

Este trabajo presenta un análisis estadístico multivariante detallado, aplicando metodologías clave para explorar y modelar relaciones complejas entre variables. El objetivo principal es demostrar cómo estas técnicas pueden proporcionar insights significativos en la interpretación de datos reales, promoviendo una comprensión profunda de los fenómenos analizados.

El análisis se llevó a cabo utilizando herramientas estadísticas como análisis de componentes principales (PCA), análisis discriminante, y clustering jerárquico. Cada técnica fue aplicada de manera rigurosa a un conjunto de datos específicos, seleccionados por su relevancia en contextos industriales y académicos. Los resultados obtenidos destacan patrones ocultos en los datos, identificando agrupaciones naturales y estableciendo relaciones lineales que permiten reducir la dimensionalidad de los mismos sin perder información clave. Además, las visualizaciones generadas proporcionan una interpretación clara y accesible, lo que facilita la comunicación de hallazgos incluso a audiencias no especializadas.

En conjunto, este trabajo contribuye significativamente al campo del análisis multivariante al ofrecer un caso práctico que combina técnicas robustas con una presentación clara de resultados. Los métodos y conclusiones expuestos no solo demuestran su aplicabilidad en situaciones concretas, sino que también establecen una base para futuras investigaciones. Esto refuerza la utilidad de estas herramientas para resolver problemas complejos en ingeniería y otras disciplinas, promoviendo su integración en prácticas cotidianas de análisis de datos.


#### Matriz de Contribuciones
```{r Matriz_de_Contribuciones, fig.align = 'center'}
(get_pca_var(PCA(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], ncp = 7, scale.unit = TRUE, graph = F)))$contrib
```

#### Contribuciones a D1
```{r Contribuciones_DIM_1, fig.align = 'center'}
fviz_contrib(PCA(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], ncp = 7, scale.unit = TRUE, graph = F), choice = "var", axes = 1, top = 10)
```

#### Contribuciones a D2
```{r Contribuciones_DIM_2, fig.align = 'center'}
fviz_contrib(PCA(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], ncp = 7, scale.unit = TRUE, graph = F), choice = "var", axes = 2, top = 10)
```

#### Contribuciones a D3
```{r Contribuciones_DIM_3, fig.align = 'center'}
fviz_contrib(PCA(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], ncp = 7, scale.unit = TRUE, graph = F), choice = "var", axes = 3, top = 10)
```

#### Contribuciones a D4
```{r Contribuciones_DIM_4, fig.align = 'center'}
fviz_contrib(PCA(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], ncp = 7, scale.unit = TRUE, graph = F), choice = "var", axes = 4, top = 10)
```

#### Contribuciones a D5
```{r Contribuciones_DIM_5, fig.align = 'center'}
fviz_contrib(PCA(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], ncp = 7, scale.unit = TRUE, graph = F), choice = "var", axes = 5, top = 10)
```

#### Contribuciones a D6
```{r Contribuciones_DIM_6, fig.align = 'center'}
fviz_contrib(PCA(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], ncp = 7, scale.unit = TRUE, graph = F), choice = "var", axes = 6, top = 10)
```

#### Contribuciones a D7
```{r Contribuciones_DIM_7, fig.align = 'center'}
fviz_contrib(PCA(cdd_factores_rendimiento_estudiantes_G6_dep[,-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], ncp = 7, scale.unit = TRUE, graph = F), choice = "var", axes = 7, top = 10)
```


### **2.5 Interpretación**{.tabset .tabset-pills}

Según lo señalado, las coordenadas obtenidas de registros tras una reducción dimensional permiten ubicarlos en un plano de factores, facilitando el análisis e interpretación. En este plano, las variables reducidas corresponden a los componentes principales, que se representan como ejes, mientras que los valores asociados a estos componentes son los puntajes obtenidos. Este enfoque subraya que las distancias entre los puntos, definidas por los puntajes de los componentes, poseen un significado importante, ya que ayudan a identificar similitudes entre los perfiles observados. Sin embargo, dichas similitudes pueden no abarcar todas las variables, sino solo algunas. En consecuencia, se espera que las distancias originales en el espacio multidimensional sean adecuadamente reflejadas en el espacio reducido formado por los componentes principales.

Con base en las variables cuantitativas del conjunto de datos se requiere definir e interpretar sus componentes principales.


#### Biplot de Variables y Registros [Total GN]
```{r Biplot_de_Variables_y_Registros_Total_GN, fig.align = 'center'}
# Seleccionar una muestra 
set.seed(780729)
cdd_factores_rendimiento_estudiantes_G6_muestr = cdd_factores_rendimiento_estudiantes_G6_dep[sample(1:nrow(cdd_factores_rendimiento_estudiantes_G6_dep),100),-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19)]

# Convertir en campo categorico
cdd_factores_rendimiento_estudiantes_G6_muestr$gender <- as.factor(cdd_factores_rendimiento_estudiantes_G6_muestr$gender)

#Graficar el biplot PCA
fviz_pca_biplot(PCA(cdd_factores_rendimiento_estudiantes_G6_muestr[,], ncp = 7, scale.unit = TRUE, graph = F, quali.sup = "gender"), axes = c(1, 2), repel = TRUE, habillage = "gender")

```


#### Biplot de Variables y Registros [Total SH]
```{r Biplot_de_Variables_y_Registros_Total_SH, fig.align = 'center'}
# Seleccionar una muestra 
set.seed(780729)
cdd_factores_rendimiento_estudiantes_G6_muestr = cdd_factores_rendimiento_estudiantes_G6_dep[sample(1:nrow(cdd_factores_rendimiento_estudiantes_G6_dep),100),-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)]

# Convertir en campo categorico
cdd_factores_rendimiento_estudiantes_G6_muestr$sleep_hours <- as.factor(cdd_factores_rendimiento_estudiantes_G6_muestr$sleep_hours)

#Graficar el biplot PCA
fviz_pca_biplot(PCA(cdd_factores_rendimiento_estudiantes_G6_muestr[,], ncp = 7, scale.unit = TRUE, graph = F, quali.sup = "sleep_hours"), axes = c(1, 2), repel = TRUE, habillage = "sleep_hours")

```


#### Biplot de Variables y Registros [Total TQ]
```{r Biplot_de_Variables_y_Registros_Total_TQ, fig.align = 'center'}
# Seleccionar una muestra 
set.seed(780729)
cdd_factores_rendimiento_estudiantes_G6_muestr = cdd_factores_rendimiento_estudiantes_G6_dep[sample(1:nrow(cdd_factores_rendimiento_estudiantes_G6_dep),100),-c(2, 4, 5, 6, 9, 10, 12, 14, 15, 17, 18, 19, 20)]

# Convertir en campo categorico
cdd_factores_rendimiento_estudiantes_G6_muestr$teacher_quality <- as.factor(cdd_factores_rendimiento_estudiantes_G6_muestr$teacher_quality)

#Graficar el biplot PCA
fviz_pca_biplot(PCA(cdd_factores_rendimiento_estudiantes_G6_muestr[,], ncp = 7, scale.unit = TRUE, graph = F, quali.sup = "teacher_quality"), axes = c(1, 2), repel = TRUE, habillage = "teacher_quality")

```

#### Coordenadas individuales [Subconjunto SH] 
```{r coordenadas_individuales_subconjunto, fig.align = 'center'}
set.seed(780729)
data_61_SH <- cdd_factores_rendimiento_estudiantes_G6_dep[sample(1:nrow(cdd_factores_rendimiento_estudiantes_G6_dep),61),-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)]
set.seed(780728)
data_61 <- cbind(cdd_factores_rendimiento_estudiantes_G6_dep[sample(1:nrow(cdd_factores_rendimiento_estudiantes_G6_dep),61),-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], data_61_SH$sleep_hours)
head(PCA(data_61, ncp = 7, scale.unit = T, graph = F, quali.sup = "sleep_hours")$ind$coord, n = 61L)
```



#### Biplot de variables y registros [Subconjunto SH]
```{r biplot_de_variables_y_registros_subconjunto, fig.align = 'center'}
set.seed(780729)
data_61_SH <- cdd_factores_rendimiento_estudiantes_G6_dep[sample(1:nrow(cdd_factores_rendimiento_estudiantes_G6_dep),61),-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)]
set.seed(780728)
data_61 <- cbind(cdd_factores_rendimiento_estudiantes_G6_dep[sample(1:nrow(cdd_factores_rendimiento_estudiantes_G6_dep),61),-c(2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20)], data_61_SH$sleep_hours)
fviz_pca_biplot(PCA(data_61, ncp = 7, scale.unit = T, graph = F, quali.sup = "sleep_hours"), axes = c(1, 2), repel = T, habillage = "sleep_hours")
```



## **Fase 3 [Correspondencias]**

### **3.1. Objetivos**

En términos generales, los objetivos del análisis de correspondencias simples y múltiples se centran en explorar y visualizar relaciones entre variables cualitativas de un conjunto de datos. Esta metodología permite identificar patrones, similitudes y diferencias significativas mediante herramientas como tablas de contingencia y disyuntivas completas. Además, se busca evaluar las calidades de representación y las contribuciones de cada variable o categoría, facilitando interpretaciones que aporten un entendimiento más profundo de las estructuras y asociaciones presentes en los datos analizados.

### **3.2. Correspondencias Simples ** {.tabset .tabset-pills}

El análisis de correspondencias simples (ACS) es una técnica estadística que permite visualizar relaciones entre categorías de variables categóricas en un espacio reducido y multidimensional. En este trabajo, se implementa el ACS para explorar las asociaciones entre pares de variables categóricas provenientes del conjunto de datos analizado. Este método no solo facilita la interpretación gráfica de tablas de contingencia, sino que también ofrece una comprensión intuitiva de las relaciones entre los niveles de las variables.

El ACS se aplicó a pares seleccionados de variables categóricas, destacando aquellas con relevancia en contextos multivariantes. A través del cálculo de perfiles fila y columna, y su representación gráfica en dos dimensiones, se identificaron proximidades entre categorías que sugieren patrones de asociación significativos. El número de dimensiones consideradas se determinó en función del menor número de categorías entre las variables analizadas, siguiendo la metodología estándar. Los resultados muestran cómo las categorías agrupadas tienden a compartir características comunes o a reflejar asociaciones que serían menos evidentes en el análisis directo de tablas de frecuencias. Además, se generaron gráficos bidimensionales que ilustran estas relaciones, aportando claridad visual a los hallazgos.

El análisis de correspondencias simples realizado en este trabajo aporta una perspectiva única sobre las relaciones entre las categorías de las variables del conjunto de datos. Al permitir la reducción de dimensiones y la visualización de patrones de proximidad, el ACS demuestra ser una herramienta poderosa para identificar asociaciones clave en datos categóricos. Este análisis sienta las bases para extender la metodología hacia análisis de correspondencias múltiples (ACM), ofreciendo la posibilidad de examinar conjuntos de datos más complejos en futuras investigaciones.

#### Análisis de Correspondencias {.tabset .tabset-pills}


##### Tablas de contingencia 
```{r ablas_de_contingencia, fig.align = 'center'}

addmargins(table(cdd_factores_rendimiento_estudiantes_G6_dep$gender,cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours))
addmargins(table(cdd_factores_rendimiento_estudiantes_G6_dep$gender,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality))
addmargins(table(cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality))

```

##### Probabilidades
```{r Probabilidades, fig.align = 'center'}

addmargins(prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$gender,cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours)) * 100)
addmargins(prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$gender,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality)) * 100)
addmargins(prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality)) * 100)

```

##### Frecuencias [CPF y CPC]
```{r Frecuencias, fig.align = 'center'}
round(addmargins(prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$gender,cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours),1) * 100,2), 2)
round(addmargins(prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$gender,cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours),2) * 100,1), 2)

round(addmargins(prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$gender,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality),1) * 100,2), 2)
round(addmargins(prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$gender,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality),2) * 100,1), 2)

round(addmargins(prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality),1) * 100,2), 2)
round(addmargins(prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality),2) * 100,1), 2)
```



##### Perfiles [CPF y CPC]
```{r Graficos_de_perfiles, fig.align = 'center'}
plotct(table(cdd_factores_rendimiento_estudiantes_G6_dep$gender,cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours),"row")
plotct(table(cdd_factores_rendimiento_estudiantes_G6_dep$gender,cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours),"col")

plotct(table(cdd_factores_rendimiento_estudiantes_G6_dep$gender,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality),"row")
plotct(table(cdd_factores_rendimiento_estudiantes_G6_dep$gender,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality),"col")

plotct(table(cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality),"row")
plotct(table(cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality),"col")

```

##### Pruebas de Hipótesis
```{r Pruebas_de_Correspondencia, fig.align = 'center'}
chisq.test(table(cdd_factores_rendimiento_estudiantes_G6_dep$gender,cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours))
chisq.test(table(cdd_factores_rendimiento_estudiantes_G6_dep$gender,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality))
chisq.test(table(cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality))
```

#### AC Pareja Única {.tabset .tabset-pills}

##### Contingencias y Residuales [SH-TQ]
```{r Contingencias_y_Residuales, fig.align = 'center'}
chisq.test(table(cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality))$observed
chisq.test(table(cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality))$expected 
chisq.test(table(cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality))$residuals
chisq.test(table(cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality))$stdres
```


##### Contribuciones [SH-TQ]
```{r Contribuciones_R-UR, fig.align = 'center'}
chisq.test(table(cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours, cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality))$residuals^2/chisq.test(table(cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality))$statistic*100
```




##### Correspondencia Simple Unidimensional [SH-TQ]
```{r Biplot_Correspondencia_Simple_R-UR, fig.align = 'center'}
CA(table(cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality), graph = FALSE)$eig
CA(table(cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality), graph = FALSE)$col
CA(table(cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours,cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality), graph = FALSE)$row
```



### **3.3. Correspondencias Múltiples** {.tabset .tabset-pills}

El análisis de correspondencias múltiples (ACM) es una técnica estadística multivariante diseñada para analizar relaciones entre varias variables cualitativas. Esta metodología amplía los principios del análisis de correspondencias simples, permitiendo estudiar conjuntos más complejos de datos categóricos. Su objetivo principal es descubrir patrones subyacentes, simplificar la representación de los datos y visualizar asociaciones en un espacio de dimensiones reducidas.

El ACM utiliza como base las tablas de contingencia y las convierte en representaciones gráficas que proyectan las categorías de las variables en un espacio factorial. Esto se logra calculando factores principales que explican la mayor parte de la varianza observada. Entre los elementos clave de esta técnica se encuentran las calidades de representación, que miden la precisión con la que cada categoría es representada en el espacio reducido, y las contribuciones, que indican la relevancia de cada variable o categoría en la formación de los ejes factoriales. Estas herramientas permiten interpretar de manera detallada las relaciones entre las categorías, ayudando a identificar asociaciones significativas y redundancias entre las variables cualitativas.


En conclusión, el análisis de correspondencias múltiples es una herramienta poderosa para la exploración y comprensión de datos cualitativos complejos. Su capacidad para simplificar y visualizar las relaciones entre múltiples variables lo convierte en una técnica esencial en diversos campos, desde el marketing y las ciencias sociales hasta la biomedicina y la industria. Al ofrecer representaciones claras y accesibles, el ACM facilita la toma de decisiones informadas basadas en los patrones identificados en los datos.

#### ACM
```{r ACM, fig.align = 'center'}
round(MCA(cdd_factores_rendimiento_estudiantes_G6[1:400, -c(1,2,6,7,10,15,20)], graph = FALSE)$eig,2)
```

#### Biplot ACM
```{r Biplot_ACM, fig.align = 'center'}
fviz_mca_biplot(MCA(cdd_factores_rendimiento_estudiantes_G6[1:400, -c(1,2,6,7,10,15,20)], graph = FALSE), repel = TRUE)
```

#### Calidad de Representación
```{r Calidad_de_Representacion_ACM, fig.align = 'center'}
fviz_mca_var(MCA(cdd_factores_rendimiento_estudiantes_G6[1:400, -c(1,2,6,7,10,15,20)], graph = FALSE), col.var ="cos2", gradient.cols = c("#00AFBB", "#E7B800", "#FC4E07"), repel = TRUE)
MCA(cdd_factores_rendimiento_estudiantes_G6[1:400, -c(1,2,6,7,10,15,20)], graph = FALSE)$var$cos2
```

#### Contribuciones
```{r Contribuciones_ACM, fig.align = 'center'}
fviz_contrib(MCA(cdd_factores_rendimiento_estudiantes_G6[1:400, -c(1,2,6,7,10,15,20)], graph = FALSE), choice = "var", axes = 1, top = 15)
fviz_contrib(MCA(cdd_factores_rendimiento_estudiantes_G6[1:400, -c(1,2,6,7,10,15,20)], graph = FALSE), choice = "var", axes = 2, top = 15)
fviz_contrib(MCA(cdd_factores_rendimiento_estudiantes_G6[1:400, -c(1,2,6,7,10,15,20)], graph = FALSE), choice = "var", axes = 3, top = 15)
fviz_contrib(MCA(cdd_factores_rendimiento_estudiantes_G6[1:400, -c(1,2,6,7,10,15,20)], graph = FALSE), choice = "var", axes = 4, top = 15)
fviz_contrib(MCA(cdd_factores_rendimiento_estudiantes_G6[1:400, -c(1,2,6,7,10,15,20)], graph = FALSE), choice = "var", axes = 5, top = 15)
```

#### Biplot con Contribuciones
```{r Biplot_con_Contribuciones_ACM, fig.align = 'center'}
fviz_mca_var(MCA(cdd_factores_rendimiento_estudiantes_G6[1:400, -c(1,2,6,7,10,15,20)], graph = FALSE), col.var ="contrib", gradient.cols = c("#00AFBB", "#E7B800", "#FC4E07"), repel = TRUE)
```



## **Fase 4 [Conglomerados]**

### **4.1. Objetivos**

En términos generales, los objetivos del análisis de conglomerados se centran en agrupar observaciones o elementos con características similares dentro de un conjunto de datos, con base en los resultados obtenidos en las fases previas. Esta etapa incluye tanto el enfoque jerárquico, representado mediante dendogramas, como el no jerárquico, utilizando el método de K-medias. A través de estos métodos, se busca identificar patrones subyacentes, clasificar elementos en grupos homogéneos y facilitar interpretaciones que contribuyan a un mejor entendimiento de la estructura y organización de los datos.


### **4.2. Agrupación Jerárquica** {.tabset .tabset-pills}

La agrupación jerárquica es una técnica de análisis multivariante utilizada para clasificar elementos o individuos en grupos con características similares. Este método organiza los datos en una estructura jerárquica en forma de árbol, conocida como dendograma, lo que permite explorar las relaciones entre las observaciones de manera visual y progresiva. Es una herramienta clave para identificar patrones y establecer relaciones entre los elementos de un conjunto de datos.

El proceso de agrupación jerárquica comienza con cada elemento considerado como un grupo independiente. Gradualmente, los grupos se van fusionando en función de su similitud, hasta formar un único conglomerado que incluye todos los elementos. Existen diferentes métodos para medir la similitud o distancia entre los elementos y los grupos, como la distancia euclidiana y los métodos de enlace (simple, completo o promedio). El resultado del análisis se representa en un dendograma, que muestra cómo se forman los conglomerados a diferentes niveles de similitud. Este enfoque permite no solo determinar cuántos grupos son más adecuados para el análisis, sino también interpretar la estructura interna de cada conglomerado.

En síntesis, la agrupación jerárquica es una técnica eficaz para analizar estructuras complejas y revelar relaciones inherentes en los datos. Su capacidad para representar las agrupaciones en un formato visualmente intuitivo la convierte en una herramienta valiosa en campos como la biología, la psicología, la mercadotecnia y la industria. Al permitir la identificación de patrones y similitudes, la agrupación jerárquica facilita la toma de decisiones basada en una comprensión profunda de las características de los datos.

#### Campo clasificador
```{r Campo_clasificador, fig.align='center'}
cdd_factores_rendimiento_estudiantes_G6_dep_CLAS_EST_Estados_costa_este_Average <- read_excel("C:/Users/57313/Desktop/Gdd_(G6)/cdd_G6_dep/cdd_factores_rendimiento_estudiantes_G6_dep_CLAS_EST_Estados_costa_este_Average.xlsx") 

head(as.data.frame(cdd_factores_rendimiento_estudiantes_G6_dep_CLAS_EST_Estados_costa_este_Average))

```

#### Disimilaridad
```{r Disimilaridad, fig.align='center'}
data_ = as.data.frame(cdd_factores_rendimiento_estudiantes_G6_dep_CLAS_EST_Estados_costa_este_Average)[, -c (1)]
rownames(data_) = unclass(cdd_factores_rendimiento_estudiantes_G6_dep_CLAS_EST_Estados_costa_este_Average$estados_costa_este)
fviz_dist(get_dist(data_, stand = T, method = "euclidean"), gradient = list(low = "#00AFBB", mid = "white", high = "#FC4E07"))
```


#### Optimización de Mojena  {.tabset .tabset-pills}

##### Unión simple
```{r Union_simple, fig.align = 'center'}
hc_single = hclust(get_dist(data_, stand = T, method = "euclidean"), method = "single")

mojena = function(hc){
  n_hd = length(hc$height)
  alp_g = 0 ; alpha = hc$height[n_hd:1]
  for(i in 1:(n_hd-1)){
    alp_g[i] = mean(alpha[(n_hd-i+1):1])+1.25*sd(alpha[(n_hd-i+1):1])
  }
  nog = sum(alp_g<= alpha[-n_hd]) + 1
  plot(alpha[-n_hd], pch=20, col=(alp_g>alpha[-n_hd])+1, main = paste("Optimal number of groups =",nog),
       ylab = expression(alpha[g]), xlab="Nodes")}

mojena(hc_single)
```



##### Unión completa
```{r Union_completa, fig.align = 'center'}
hc_complete = hclust(get_dist(data_, stand = T, method = "euclidean"), method = "complete")

mojena = function(hc){
  n_hd = length(hc$height)
  alp_g = 0 ; alpha = hc$height[n_hd:1]
  for(i in 1:(n_hd-1)){
    alp_g[i] = mean(alpha[(n_hd-i+1):1])+1.25*sd(alpha[(n_hd-i+1):1])
  }
  nog = sum(alp_g<= alpha[-n_hd]) + 1
  plot(alpha[-n_hd], pch=20, col=(alp_g>alpha[-n_hd])+1, main = paste("Optimal number of groups =",nog),
       ylab = expression(alpha[g]), xlab="Nodes")}

mojena(hc_complete)
```


##### Unión Promedio
```{r Optimizacion_Mojena_Promedio, fig.align = 'center'}
hc_average = hclust(get_dist(data_, stand = T, method = "euclidean"), method = "average")

mojena = function(hc){
  n_hd = length(hc$height)
  alp_g = 0 ; alpha = hc$height[n_hd:1]
  for(i in 1:(n_hd-1)){
    alp_g[i] = mean(alpha[(n_hd-i+1):1])+1.25*sd(alpha[(n_hd-i+1):1])
  }
  nog = sum(alp_g<= alpha[-n_hd]) + 1
  plot(alpha[-n_hd], pch=20, col=(alp_g>alpha[-n_hd])+1, main = paste("Optimal number of groups =",nog),
       ylab = expression(alpha[g]), xlab="Nodes")}

mojena(hc_average)
```

#### Dendograma optimizado {.tabset .tabset-pills}

##### Enlace simple
```{r Enlace_simple, fig.align = 'center'}

suppressWarnings(fviz_dend(hc_single, k = 3, cex = 0.5, k_colors = "npg", color_labels_by_k = T, rect = T))
```

##### Enlace Completo
```{r Enlace_completo, fig.align = 'center'}
fviz_dend(hc_complete, k = 3, cex = 0.5, k_colors = "npg", color_labels_by_k = T, rect = T)
```

##### Enlace promedio
```{r Enlace_promedio, fig.align = 'center'}
fviz_dend(hc_average, k = 4, cex = 0.5, k_colors = "npg", color_labels_by_k = T, rect = T)
```

### **4.3. Agrupación No-Jerárquica** {.tabset .tabset-pills}

La agrupación no jerárquica es un enfoque utilizado para clasificar datos en grupos sin necesidad de una estructura jerárquica preestablecida. A diferencia de la agrupación jerárquica, donde los datos se organizan en forma de un árbol, la agrupación no jerárquica divide los datos en un número fijo de grupos que no tienen una relación jerárquica entre sí. El método más utilizado en este tipo de análisis es el de K-medias, que se basa en asignar cada dato al grupo más cercano según ciertas características.

El proceso de agrupación no jerárquica comienza con una asignación inicial de los datos a un número predefinido de grupos. Luego, se calcula el centro de cada grupo, que sería el promedio de las características de los elementos dentro de ese grupo. A partir de ahí, se reasignan los datos a los grupos en función de qué tan cerca están de los centros. Este proceso se repite varias veces hasta que las asignaciones no cambian más. La principal ventaja de este método es su simplicidad y rapidez, lo que lo hace ideal para manejar grandes cantidades de datos. Sin embargo, uno de los retos es que debemos decidir de antemano cuántos grupos queremos, lo que puede influir en los resultados que obtengamos.

En resumen, la agrupación no jerárquica es una técnica muy útil cuando necesitamos segmentar un conjunto de datos en grupos específicos sin la necesidad de una jerarquía compleja. Es ampliamente utilizada en áreas como el análisis de clientes, la segmentación de mercados y el análisis de patrones. Aunque es necesario definir previamente el número de grupos, su simplicidad y eficiencia la convierten en una herramienta valiosa, especialmente cuando queremos obtener resultados rápidos y claros sin complicarnos con estructuras demasiado complejas.


#### K-óptimos {.tabset .tabset-pills}

##### Elbow
```{r WSS, fig.align = 'center'}
fviz_nbclust(data_, kmeans, method = "wss") + geom_vline(xintercept = 3, linetype = 2)
```

##### Silhouette
```{r Silhouette, fig.align = 'center'}
fviz_nbclust(data_, kmeans, method = "silhouette")
```

##### Gap Statistic
```{r GAP_STAT, fig.align = 'center'}
fviz_nbclust(data_, kmeans, method = "gap_stat")
```

##### Majority Rule
```{r Majority_Rule, fig.align = 'center'}
suppressWarnings(NbClust(data = data_, diss = NULL, distance = "euclidean", min.nc = 2, max.nc = 10, method = "kmeans")$Best.nc)
```

#### Resultados K-means {.tabset .tabset-pills}

##### K-óptimo [Sil_Ma-Rul 2]
```{r Sil_Ma-Rul, fig.align = 'center'}
set.seed(780728)
print(kmeans(data_, 2, nstart = 25))
```

##### K-óptimo [Elb 3]
```{r Elb, fig.align = 'center'}
set.seed(780728)
print(kmeans(data_, 3, nstart = 25))
```

#### Gráficos K-means {.tabset .tabset-pills}

##### K-óptimo [Sil_Ma-Rul 2]
```{r Graf_Sil_Ma-Rul, fig.align = 'center'}
fviz_cluster(kmeans(data_, 2, nstart = 25), data = data_, palette = c("#2E9FDF", "#00AFBB", "#E7B800", "#E7B801"), ellipse.type = "euclid", star.plot = TRUE, repel = TRUE, ggtheme = theme_minimal()
)
```

##### K-óptimo [Elb 3]
```{r Graf_Elb, fig.align = 'center'}
fviz_cluster(kmeans(data_, 3, nstart = 25), data = data_, palette = c("#2E9FDF", "#00AFBB", "#E7B800", "#E7B801"), ellipse.type = "euclid", star.plot = TRUE, repel = TRUE, ggtheme = theme_minimal()
)
```



## **Fase 5 [Análisis de Regresión]**

El análisis de regresión es un proceso estadístico utilizado para estimar las relaciones entre variables (una dependiente o de respuesta y otras independientes o predictoras) mediante técnicas de modelado y análisis que permiten comprender cómo varía el valor de la variable dependiente al modificar el valor de una o más variables independientes. Los modelos de regresión que se analizarán en este documento son: lineales (simples y Múltiples) y logísticos, los cuales se entienden como casos del modelo de regresión lineal generalizado.

### **5.1. Objetivos**

El análisis de regresión es una herramienta estadística fundamental cuyo principal objetivo es entender y modelar la relación entre una variable dependiente, también conocida como variable respuesta, y una o más variables independientes, llamadas predictoras. Este análisis permite explorar cómo las variables independientes afectan a la variable dependiente, prop}orcionando información sobre la dirección y la magnitud de estas relaciones. A través de esta herramienta, es posible identificar patrones, establecer tendencias y obtener conclusiones significativas basadas en los datos.

Uno de los objetivos centrales del análisis de regresión es la predicción, es decir, usar el modelo desarrollado para estimar o prever el valor de la variable dependiente en función de los valores conocidos de las variables independientes. Esto resulta especialmente útil en contextos donde se requiere anticipar resultados, como en la planeación industrial, el análisis financiero, la evaluación de riesgos o la toma de decisiones estratégicas.


### **5.2. Regresión Lineal Simple** {.tabset .tabset-pills}

En este estudio, se utilizará un modelo de Regresión Lineal Simple (RLS) compuesto por dos variables estadísticas $x$ y $Y$, donde se asume que $Y$ depende de $x$.La relación entre estas variables se expresa matemáticamente como: $$Y = \beta_0 + \beta_1 x + \varepsilon \hspace{10mm} \hspace{10mm}(1)$$ donde:

+ $Y$: representa una variable de respuesta, caracterizada por su naturaleza aleatoria.
+ $x$: es una variable predictora, no aleatoria, que influye sobre $Y$
+ $\varepsilon$: simboliza una variable aleatoria no observable, que recoge el error o la variabilidad no explicada por el modelo.
+ $\beta_0$ y $\beta_1$: son parámetros reales desconocidos que deben ser estimados a partir de los datos..

A diferencia del modelo lineal simple determinístico, representado por  $y = \beta_0 + \beta_1 x$, el probablístico supone que el valor esperado de $Y$ es una función lineal de $x$, Sin embargo, cuando x es fija, la variable Y puede diferir de su valor esperado debido a una cantidad aleatoria, representada como $\varepsilon$. Además, el término $\varepsilon$ en la ecuación del modelo $(1)$ se considera normalmente distribuido, con $E(\varepsilon)=0$ y $V(\varepsilon)=\sigma^2$. Este término aleatorio $\varepsilon$ también se conoce como el error aleatorio o desviación aleatoria dentro del modelo.

De manera complementaria, los valores de $\beta_0$, $\beta_1$ y $\sigma^2$, generalmente no son conocidos. En su lugar, se dispone de una muestra de datos compuesta por pares ordenados $( (x_1, y_1), (x_2, y_2), \dots, (x_n, y_n) )$, que permite estimar los parámetros del modelo y la línea de regresión verdadera, asumiendo que las observaciones son independientes. En este contexto, $y_i$ representa el valor observado de una variable aleatoria $Y_i$, definida como $Y_i = \beta_0 + \beta_1 x_i + \varepsilon_i$. Además, las $n$ desviaciones $\varepsilon_1, \varepsilon_2, \dots, \varepsilon_n$ se consideran variables aleatorias independientes. 

Según el modelo, los puntos observados se distribuirán de manera aleatoria alrededor de la línea de regresión verdadera. En este contexto, la estimación de $y = \beta_0 + \beta_1 x$ debe representar una línea que se ajuste de la mejor manera posible a los puntos de la muestra. Esta línea debe cumplir con la característica de minimizar las desviaciones verticales (o distancias) entre los puntos observados y la línea, manteniéndolas lo más pequeñas posible. 

El principio de los mínimos cuadrados establece que la desviación vertical de cada punto $(x_i, y_i)$ con respecto a la recta de regresión $y= b_0+b_1x$ es $y_i-(b_0+b_1x)$. La suma de las desviaciones verticales al cuadrado de todos los puntos $(x_i, y_i)$ respecto a la línea de regresión se expresa como: $f(b_0,b_1)=\sum_{i=1}^n(y_i-(b_0+b_1x_i))^2$. De esta manera, las estimaciones de los parámetros $\beta_0$ y $\beta_1$, denotadas como $\hat{\beta}_0$ y $\hat{\beta}_1$, son los valores que minimizan la función $f(b_0,b_1)$. Es decir, la suma de los cuadrados de las desviaciones alcanza su valor más bajo cuando se utilizan estas estimaciones, de modo que $f(\hat{\beta}_0,\hat{\beta}_1)\leq f(b_0,b_1)$ para cualquier valor de $\beta_0$ y $\beta_1$. Por lo tanto, la línea de regresión estimada, o línea de mínimos cuadrados, es: $y=\hat{\beta}_0 + \hat{\beta}_1x$.

Después de calcular y resolver las ecuaciones en derivadas parciales de $f(b_0,b_1)$ con respecto a $b_0$ y $b_1$, y luego igualarlas a cero, se obtiene un sistema de ecuaciones conocido como ecuaciones normales. Este sistema es lineal en los parámetros $b_0$ y $b_1$, y siempre que al menos dos de los valores de $x_i$ sean diferentes, las estimaciones de los mínimos cuadrados serán la única solución para el sistema. Así, la estimación de $\beta_1$ de la línea de regresión verdadera se calcula como: $$\hat{\beta}_1=\dfrac{\sum_{i=1}^n(x_i-\bar{x})(y_i-\bar{y})}{\sum_{i=1}^n(x_i-\bar{x})^2}=\dfrac{S_{xy}}{S_{xx}}\hspace{10mm}(2)$$ Y la estimación de \beta_0 se calcula como:
$$\hat{\beta}_0=\dfrac{\sum_{i=1}^ny_i-\hat{\beta}_1\sum_{i=1}^nx_i}{n}=\bar{y}-\hat{\beta}_1\bar{x}\hspace{10mm}(3)$$ Para obtener estos cálculos con precisión, es crucial minimizar los efectos del redondeo. Además, antes de calcular $\hat{\beta}_1$ y $\hat{\beta}_0$, es importante revisar visualmente el conjunto de datos antes de aplicar el modelo para evaluar si es adecuado utilizar un modelo lineal probabilístico. En otras palabras, si los puntos no parecen agruparse alrededor de una línea recta con una dispersión similar en todas las $x_i$, se debe considerar la posibilidad de utilizar otros tipos de modelos.

El principio de los mínimos cuadrados establece que la desviación vertical de cada punto $(x_i, y_i)$ respecto a la recta de regresión $y=b_0+b_1x$ es $y_i-(b_0 +b_1x_i)$. La suma de las desviaciones verticales al cuadrado de todos los puntos $(x_i, y_i)$ con respecto a la línea de regresión se expresa como:$f(b_0,b_1)=\sum_{i=1}^n(y_i-(b_0+b_1x_i))^2$. Asi, Las estimaciones de los parámetros $\beta_0$ y $\beta_1$, denotadas como $\hat{\beta}_0$ y $\hat{\beta}_1$, son los valores que minimizan la función $f(b_0,b_1)$. Es decir, la suma de los cuadrados de las desviaciones alcanza su valor más bajo cuando se utilizan estas estimaciones, lo que implica que:$f(\hat{\beta}_0,\hat{\beta}_1)\leq f(b_0,b_1)$ Por lo tanto, la línea de regresión estimada, o línea de mínimos cuadrados, es:$y=\hat{\beta}_0 + \hat{\beta}_1x$.

Es importante tener en cuenta que la línea de mínimos cuadrados debe usarse con precaución para predecir valores de $x$ **_fuera_** del rango de los datos observados, ya que la relación lineal ajustada podría no ser válida en esos casos.

El parámetro $\sigma^2$ refleja la cantidad de variabilidad en el modelo de regresión, y su valor determina cuánta dispersión de los valores observados se produce en torno a la línea de regresión verdadera. Los residuos $y_i - \hat{y_i}$ son las desviaciones con respecto a la línea estimada. Si los residuos son pequeños, indica que la relación lineal entre $x$ y $y$ explica en gran medida la variabilidad en $y$. Si son grandes, sugiere que la variabilidad de $y$ no se debe únicamente a esta relación lineal. La estimación de $\sigma^2$ en un análisis de regresión se basa en la suma de los cuadrados residuales (SCE), que se calcula como:$$SCE=\sum_{i=1}^ny_i^2-\hat{\beta}_0\sum_{i=1}^ny_i-\hat{\beta}_1\sum_{i=1}^nx_iy_i\hspace{10mm}(4)$$ $$\hat\sigma^2=s^2=\dfrac{SCE}{n-2}\hspace{10mm}(5)$$La suma de cuadrados del error (SCE) mide la porción de la variación en $y$ que no es explicada por el modelo de regresión lineal, es decir, la variación que no está asociada con la relación lineal entre las variables. Además de la SCE, existe otra medida conocida como la suma total de cuadrados (STC), que representa la variación total observada en los valores de $y$. La STC se calcula como:$$STC=\sum_{i=1}^ny_i^2-\frac{(\sum_{i=1}^ny_i)^2}{n}\hspace{10mm}(6)$$ La proporción de la variación total que no es explicada por el modelo de regresión lineal simple se puede calcular mediante la razón $SCE/STC$. A partir de esto, se define el coeficiente de determinación $r^2$ como: $$r^2=1-\frac{SCE}{STC}\hspace{10mm}(7)$$ Este coeficiente se interpreta como la proporción de la variación observada en $y$ que puede ser explicada por el modelo de regresión lineal simple, es decir, la variación atribuible a una relación lineal aproximada entre $x$ y $y$. Cuanto más cercano sea $r^2$ a 1, mejor será el modelo en explicar la variación de $y$. Otra forma de calcular $r^2$ utiliza la suma de cuadrados debida a la regresión (SCR), que mide la porción de la variación total explicada por el modelo. En este caso, el coeficiente de determinación se expresa como: $$r^2=1-\frac{SCE}{STC}=\frac{STC-SCE}{STC}=\frac{SCR}{STC}\hspace{10mm}(8)$$ Es bien conocido que cualquier cantidad calculada a partir de datos muestrales puede variar entre muestras. Por ello, los procedimientos inferenciales estandarizan un estimador al restarle su valor medio y dividirlo entre su desviación estándar estimada. En el caso de un modelo de regresión lineal simple, se considera que las variables estandarizadas:  $t_{(n-2)}=\dfrac{\hat{\beta}_0-\beta_0}{\hat{\sigma} \sqrt{1/n+\bar{x}^2/S_{xx}}}$ y $t_{(n-2)}=\dfrac{\hat{\beta}_1-\beta_1}{ \hat{\sigma} \sqrt{1/S_{xx}}}$ siguen una distribución $t$ con $n-2$ grados de libertad. A partir de esto, los intervalos de confianza al $100*(1-\alpha)\%$ para los parámetros $\beta_0$ (intercepto) y $\beta_1$ (pendiente) de la línea de regresión verdadera son: $$\hat{\beta}_0 \pm t_{\alpha/2, n-2} \cdot  \hat{\sigma} \sqrt{1/n+\bar{x}^2/S_{xx}}\hspace{10mm}(9)$$ $$\hat{\beta}_1 \pm t_{\alpha/2, n-2} \cdot  \hat{\sigma} \sqrt{1/S_{xx}} \hspace{10mm}  (10)$$Estos intervalos están centrados en las estimaciones puntuales de cada parámetro, y su amplitud depende tanto del nivel de confianza seleccionado como de la variabilidad asociada a cada estimador.

De acuerdo con lo explicado, en los procedimientos de prueba de hipótesis para el modelo de regresión lineal simple, las hipótesis nulas sobre los parámetros $\beta_0$ y $\beta_1$ generalmente se formulan como igualdades. Los valores hipotéticos de estos parámetros se denotan como $\beta_{00}$ ("beta cero cero") para el intercepto y $\beta_{10}$ ("beta uno cero") para la pendiente.  Dado que los estadísticos de prueba tienen una distribución $t$ con $n-2$ grados de libertad bajo la suposición de que $H_0$ es verdadera, la probabilidad de cometer un error Tipo I se mantiene en el nivel de significancia $\alpha$ al emplear el valor crítico $t$ correspondiente. En este contexto, las hipótesis más comunes para el intercepto $\beta_0$ son:  $$H_0: \beta_0 = \beta_{00}\hspace{10mm}(11)$$ $$H_1: \beta_0 \neq \beta_{00}\hspace{10mm}(12)$$y el estadístico de prueba asociado se define como:  $$t_{(n-2)}=\dfrac{\hat{\beta}_0-\beta_{00}}{\hat{\sigma} \sqrt{1/n+\bar{x}^2/S_{xx}}}\hspace{10mm}(13)$$De manera similar, para la pendiente $\beta_1$, las hipótesis se formulan como:
$$H_0: \beta_1 = \beta_{10}\hspace{10mm}(14)$$ 
$$H_1: \beta_1 \neq \beta_{10}\hspace{10mm}(15)$$y el estadístico de prueba correspondiente es:  $$t_{(n-2)}=\dfrac{\hat{\beta}_1-\beta_{10}}{\hat{\sigma} \sqrt{1/S_{xx}}}\hspace{10mm}(16)$$Este conjunto de hipótesis y estadísticos se conoce como **_prueba de utilidad del modelo de regresión lineal simple_**. Las regiones de rechazo para la hipótesis nula $H_0$ en una prueba al nivel $\alpha$ son las siguientes:

- Para $H_1$: $\beta_1$>\beta_{10}\), se rechaza $H_0$ si $t\geq t_{\alpha,n-2}$.
- Para $H_1$: $\beta_1$ < \beta_{10}\), se rechaza $H_0$ si $t\leq -t_{\alpha,n-2}$.
- Para $H_1$: $\beta_1$\neq \beta_{10}\), se rechaza $H_0$ si $t\leq -t_{\alpha/2,n-2}$ o  $t\geq t_{\alpha/2,n-2}$.

Además, la utilidad del modelo de regresión lineal simple también puede evaluarse mediante una tabla ANOVA. En este caso, se rechaza $H_0$ si $f \geq F_{\alpha,1,n-2}$. Es importante destacar que la prueba $F$ produce el mismo resultado que la prueba $t$ de utilidad del modelo.

Por último, en un modelo de regresión lineal simple, un valor futuro de $Y$ no es un parámetro fijo, sino una variable aleatoria. Por lo tanto, es necesario referirse a un rango de valores posibles para este valor futuro, denominado **intervalo de predicción**. Cuando se predice utilizando el modelo de regresión lineal simple, el error de predicción se define como: $Y-( \hat{\beta}_0+ \hat{\beta}_1 x^*)$, Este error corresponde a la diferencia entre dos variables aleatorias, lo que implica que hay mayor incertidumbre en comparación con una estimación puntual. Por consiguiente, un intervalo de predicción es más amplio que un intervalo de confianza.A partir de la varianza del error de predicción, se establece que la variable estandarizada: $$T=\dfrac{Y-(\hat{\beta}_0+ \hat{\beta}_1 x^*)}{S \displaystyle\sqrt{1+\dfrac{1}{n}  + \dfrac{(x^*-\bar{x})^2}{S_{xx}}}}\hspace{10mm}(17)$$sigue una distribución $t$ con $n-2$ grados de libertad. Esto permite calcular un intervalo de predicción con un nivel de confianza del $100*(1-\alpha)\%$ para un valor futuro de $Y$ cuando $x=x^*$:$$\hat{\beta}_0+\hat{\beta}_1 x^*\pm t_{n-2,\alpha/2}\cdot s \displaystyle\sqrt{1+\dfrac{1}{n}+\dfrac{(x^*-\bar{x})^2}{S_{xx}}}\hspace{10mm}(18)$$La interpretación del nivel de predicción del $100 \cdot (1-\alpha)\%$ indica que, si se aplica repetidamente la fórmula correspondiente en la ecuación $(18)$, los intervalos calculados incluirán los valores observados de $Y$ en aproximadamente el $100 \cdot (1-\alpha)\%$ de los casos. Además, el término adicional $1$ en la raíz cuadrada hace que el intervalo de predicción sea más amplio en comparación con los intervalos de confianza para los parámetros, como los descritos en las ecuaciones $(9)$ y $(10)$. Por otro lado, a medida que el tamaño de la muestra $n\to\infty$, el ancho del intervalo de predicción no se reduce a cero, ya que siempre existirá una incertidumbre inherente en la predicción de valores futuros, incluso si los parámetros $\beta_0$ y $\beta_1$ se conocen con exactitud.


##### 5.2.1. Planteamiento del problema

Se planteará un modelo de regresión lineal simple basado en el conjunto de datos descrito anteriormente, con el propósito de examinar la relación lineal entre las variables **Hours studied** (variable independiente) y **Exam score** (variable dependiente).

##### 5.2.2. Desarrollo del análisis 

El estudio de regresión lineal simple ha sido procesado con `r R.version.string` mediado por RStudio 2024.04.2

##### 5.2.2.1. Resumen estadistico de las variables por estudiar. {.tabset .tabset-pills}

La navegación a través de las pestañas proporciona un análisis descriptivo de las variables de interés: **Exam score** (variable dependiente) y **Hours studied** (variable independiente). Este análisis incluye sus estadísticas resumen, diagramas de caja para identificar distribuciones y posibles valores atípicos, así como un diagrama de dispersión que ilustra la relación conjunta entre ambas variables.

Con base en la pestaña **Resumen de Exam score**, se observa que la variable **Exam score** muestra una distribución con baja dispersión, evidenciada por un rango intercuartílico estrecho (65-69). Aunque la mayoría de los valores se concentran en un rango alto dentro del intervalo de medición, se identifica un valor atípico por encima de 100. Esto indica que la variable tiende hacia valores elevados, con una variabilidad mínima en los datos principales.En comparación, según la pestaña **Resumen de Hours studied**, la variable **Hours studied** presenta una distribución más equilibrada y simétrica, sin presencia de valores atípicos. Su rango intercuartílico (16-24) y mediana (20) sugieren una ligera concentración de los datos hacia el extremo superior del intervalo, aunque la dispersión es más amplia que en Exam score.

Complementariamente, en el **Diagrama de Dispersión Hours studied vs. Exam score**, se observa que existe una correlación positiva de naturaleza ligeramente lineal entre las variables **hours studied** y **exam score**. Esto sugiere que a medida que los estudiantes dedican más horas al estudio, generalmente obtienen puntajes más altos en los exámenes.Sin embargo, si se analiza el gráfico de **Diagramas Totales de Dispersión**, que incluye diversas variables cuantitativas relacionadas con el rendimiento estudiantil, se puede constatar que otras variables, como **previous_scores** y **attendance**, podrían mostrar una relación más fuerte con el **exam score**. Esto indica que, aunque las horas de estudio son importantes, otros factores también influyen significativamente en el desempeño académico de los estudiantes.

###### Resumen de Exam score 
```{r resumen_exam_score, fig.align='center'}
summary(cdd_factores_rendimiento_estudiantes_G6_dep$exam_score)
boxplot(cdd_factores_rendimiento_estudiantes_G6_dep$exam_score, main = "Diagrama de caja de Exam score", col = c("orange"))
```


###### Diagrama de dispersión HS vs. ES 
```{r dispersion_hs_es, fig.align = 'center'}
plot(cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied, cdd_factores_rendimiento_estudiantes_G6_dep$exam_score, main = "Diagrama de Dispersión Hours studied vs. Exam score")
```


###### Diagramas totales de dispersión
```{r dispersiones_totales, fig.align = 'center'}
pairs(~hours_studied + age + attendance + sleep_hours + previous_scores + tutoring_sessions + physical_activity + exam_score , data = cdd_factores_rendimiento_estudiantes_G6_dep)
```

##### 5.2.2.2. Formulación del modelo de RLS entre las variables de estudio. {.tabset .tabset-pills}

Al explorar las diferentes pestañas, se presentan los coeficientes del modelo de regresión lineal simple, junto con su resumen estadístico y la tabla ANOVA. Cabe destacar que las variables principales analizadas son: **exam score** como la variable dependiente y **hours studied** como la variable independiente.

Con base en los resultados obtenidos del análisis del **Modelo RLS**, se establece que el modelo de regresión lineal simple que relaciona las variables de interés, **Exam Score** y **Hours Studied**, tiene la siguiente formulación:  $$\hat{\text{Exam Score}} = 61.437683 + 0.291047 \cdot \text{Hours Studied    (19)} $$ En este modelo, el intercepto $(61.437683)$ representa el puntaje esperado en el examen cuando el número de **hours studied** es igual a $0$. Sin embargo, su interpretación puede no ser significativa en este contexto, ya que no es común que un estudiante no estudie absolutamente nada antes de un examen. Por otro lado, el coeficiente asociado a la variable **Hours Studied** $(0.291047)$ indica que, por cada hora adicional de estudio, el puntaje del examen aumenta en promedio $(0.291)$ puntos. Esto sugiere una relación directa y positiva entre las horas de estudio y el puntaje obtenido, aunque el incremento es moderado. 

Complementariamente, en la pestaña **Resumen Estadístico del Modelo RLS**, se constata que, para cualquier nivel de significancia, las evidencias respaldan la existencia de una correlación positiva entre las variables **Hours Studied** y **Exam Score**. Además, el coeficiente de determinación $(R^2)$ establece que una porción considerable de la variabilidad en **Exam Score** es explicada por **Hours Studied**, confirmando la utilidad del modelo.  Esto también queda ratificado en la pestaña **Tabla ANOVA para el Modelo RLS**, donde el valor de $(F)$ obtenido $(1576.2)$ y el valor $(p < 2.2 \cdot 10^{-16})$ refuerzan la significancia estadística del modelo. Estos resultados demuestran que las horas de estudio son un predictor significativo y clave para explicar las variaciones en los puntajes del examen.  


###### Coeficientes del Modelo RLS
```{r coeficientes_regresion_rineal_simple, fig.align = 'center'}
modelo_RL_Simple = lm(cdd_factores_rendimiento_estudiantes_G6_dep$exam_score~cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied)
coef(modelo_RL_Simple)
```
###### Resumen Estadístico del Modelo RLS
```{r resumen_regresion_lineal_simple, fig.align = 'center'}
summary(modelo_RL_Simple)
```
###### Tabla ANOVA para el Modelo RLS
```{r ANOVA_regresion_lineal_simple, fig.align = 'center'}
anova(modelo_RL_Simple)
```

##### 5.2.2.3. Análisis del modelo RLS. {.tabset .tabset-pills}

La navegación a través de las pestañas muestra el intervalo de confianza para $\beta_1$ y para la predicción del modelo de regresión lineal simple, ambos al 95%. Las variables de interés en este análisis son: **Exam Score** (variable dependiente) y **Hours Studied** (variable independiente).

El análisis del modelo RLS muestra que es estadísticamente significativo y, en consecuencia, aporta información relevante para estimar **Exam Score** a partir de **Hours Studied**. Esto se debe a que el intervalo de confianza para el coeficiente de **Hours Studied** en el modelo RLS excluye al cero:  $$ 0.276676 < \beta_1 < 0.305418 \hspace{10mm}(20) $$  

Por último, la pestaña **Predicciones y sus Intervalos de Predicción** muestra los cálculos basados en el modelo, bajo intervalos de predicción al 95%, para las predicciones de **Exam Score** en todas las observaciones del conjunto de datos. Cabe destacar que estos intervalos son más amplios que aquellos calculados con base en intervalos de confianza al mismo nivel de significancia, reflejando la incertidumbre asociada a las predicciones individuales.


###### Intervalo de Confianza para B1
```{r intervalo_confianza_B1, fig.align = 'center'}
confint(modelo_RL_Simple, level = 0.95)
```
###### Predicciones y sus Intervalos de Predicción
```{r intervalos_prediccion_P, fig.align = 'center'}
predict(modelo_RL_Simple, data.frame(seq(1,400)), interval='prediction', level = 0.95)
```
###### Predicciones y sus Intervalos de Confianza
```{r intervalos_prediccion_C, fig.align = 'center'}
predict(modelo_RL_Simple, data.frame(seq(1,400)), interval='confidence', level = 0.95)
```


### **5.3. Regresión Lineal Múltiple** {.tabset .tabset-pills}

La regresión lineal múltiple (RLM) puede considerarse inicialmente como una extensión de la regresión lineal simple, lo que facilita su comprensión. En este estudio, se representará de forma general mediante la ecuación:
$${yi=β0+β1xi1+⋯+βkxik+εi,i=1,2,…,n(21)}$$
En este modelo, se supone que el término de error $(ϵ)$ tiene media igual a cero $E(ϵ)=0$ y varianza constante $V(ϵ)=σ2$. Además, para realizar pruebas de hipótesis y calcular intervalos de confianza y predicción, se asume que ϵ sigue una distribución normal. La estimación de los parámetros del modelo $(β_0+β_1⋯β_k)$ e lleva a cabo mediante el método de mínimos cuadrados ordinarios (MCO). Este enfoque consiste en minimizar una función de prueba que calcula la suma de los cuadrados de las desviaciones entre los valores observados y los valores predichos. La función de prueba se expresa como: 
$f(b0,b1,...,bk)=∑j[yi−(b0+b1x1j+b2x2j+...+bkxkj)]2$
Resolver esta función genera un sistema de ecuaciones lineales conocido como ecuaciones normales, cuya solución proporciona las estimaciones de mínimos cuadrados para los coeficientes $β0^,β1^,...,,βk^..$
El desempeño del modelo se evalúa mediante el coeficiente de determinación múltiple〖$(R)〗^2$, que mide la proporción de la variabilidad total explicada por las variables independientes. Este coeficiente se ajusta considerando el número de parámetros incluidos en el modelo.

Para verificar la utilidad del modelo, se realiza una prueba de hipótesis basada en un estadístico $F$. La hipotesis nula $〖(H〗_0)$ establece que todos los coeficientes son iguales a cero $$H0:β1=β2=⋯=βk=0	(22)$$, mientras que la hipótesis alternativa $H1 :al menos una βi≠0(i=1,...,k)		(23)$ afirma que al menos uno de los coeficientes es diferente de cero. El estadístico de prueba se calcula como: 

$$f=R2/k(1−R2)(n−(k+1))=SCR/kSCE/(n−(k+1))=RMCCME	(24)$$
Donde:

SCR es la suma de cuadrados de la regresión, obtenida como la diferencia entre la suma total de cuadrados (STC) y la suma de cuadrados de los errores (SCE).
El modelo se considera útil si el valor calculado de F es mayor o igual al valor crítico para un nivel de significancia α.
$$f ≥ Fα,k,n−(k+1)	(25)$$
Los intervalos de confianza para los coeficientes $(β_1)$ se construyen como:
$$β^i±tα/2,n−(k+1)⋅sβ^i 	(26)$$
Asimismo, los intervalos de predicción para valores futuros de la variable dependiente $(y)$ se determinan mediante:
$$y^±tα/2,n−(k+1)⋅s2+s2Y^−−−−−−√	(27)$$
Finalmente, en los análisis de regresión múltiple pueden surgir dificultades, como la no linealidad, la multicolinealidad (relaciones fuertes entre variables independientes), problemas en la selección de variables, y la identificación de observaciones influyentes, entre otros. Estas situaciones requieren técnicas específicas de solución, como transformaciones de datos o métodos de selección de variables.


##### 5.3.1. Planteamiento del Problema

A partir del conjunto de datos descrito en la sección 2, se desarrollará un modelo de regresión lineal múltiple con el objetivo de analizar la relación lineal que se supone entre las variables definidas. En este caso, la variable dependiente será **exam score** mientras que las variables independientes serán **hours_studied**, **age**, **attendance**, **parental_involvement**, **access_to_resources**, **extracurricular_activities**, **sleep_hours**, **previous_scores**, **motivation_level**, **internet_access**, **tutoring_sessions**, **family_income**, **teacher_quality**, **school_type**, **peer_influence**, **physical_activity**, **learning_disabilities**, **parental_education_level**, **distance_from_home**, **gender**

##### 5.3.2. Desarrollo del Análisis

El estudio de regresión lineal multiple ha sido procesado con `r R.version.string` mediado por RStudio 2024.04.2

##### 5.3.2.1. Resumen estadístico de las variables de estudio. {.tabset .tabset-pills}

La navegación entre las pestañas permite visualizar el resumen estadístico de todas las variables del conjunto de datos. Para las variables cuantitativas (de tipo razón), el resumen presentado será el tradicional, mientras que para las variables cualitativas (de tipo nominal y ordinal), el análisis incluirá conteos, proporciones y gráficos de barras. Cabe destacar que la variable dependiente en este caso es **Exam Score**


###### Resumen Variables Cuantitativas
```{r resumen_variables_cuantitativas, fig.align = 'center'}
summary(cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied)
summary(cdd_factores_rendimiento_estudiantes_G6_dep$age)
summary(cdd_factores_rendimiento_estudiantes_G6_dep$attendance)
summary(cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours)
summary(cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores)
summary(cdd_factores_rendimiento_estudiantes_G6_dep$physical_activity)
summary(cdd_factores_rendimiento_estudiantes_G6_dep$exam_score)
```


###### Resumen Variables Cualitativas
```{r resumen_Variables_Cualitativas, fig.align = 'center'}
table(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement)
prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement))
barplot(table(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement))

table(cdd_factores_rendimiento_estudiantes_G6_dep$access_to_resources)
prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$access_to_resources))
barplot(table(cdd_factores_rendimiento_estudiantes_G6_dep$access_to_resources))

table(cdd_factores_rendimiento_estudiantes_G6_dep$extracurricular_activities)
prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$extracurricular_activities))
barplot(table(cdd_factores_rendimiento_estudiantes_G6_dep$extracurricular_activities))

table(cdd_factores_rendimiento_estudiantes_G6_dep$motivation_level)
prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$motivation_level))
barplot(table(cdd_factores_rendimiento_estudiantes_G6_dep$motivation_level))

table(cdd_factores_rendimiento_estudiantes_G6_dep$internet_access)
prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$internet_access))
barplot(table(cdd_factores_rendimiento_estudiantes_G6_dep$internet_access))

table(cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions)
prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions))
barplot(table(cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions))

table(cdd_factores_rendimiento_estudiantes_G6_dep$family_income)
prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$family_income))
barplot(table(cdd_factores_rendimiento_estudiantes_G6_dep$family_income))

table(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality)
prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality))
barplot(table(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality))

table(cdd_factores_rendimiento_estudiantes_G6_dep$school_type)
prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$school_type))
barplot(table(cdd_factores_rendimiento_estudiantes_G6_dep$school_type))

table(cdd_factores_rendimiento_estudiantes_G6_dep$peer_influence)
prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$peer_influence))
barplot(table(cdd_factores_rendimiento_estudiantes_G6_dep$peer_influence))

table(cdd_factores_rendimiento_estudiantes_G6_dep$learning_disabilities)
prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$learning_disabilities))
barplot(table(cdd_factores_rendimiento_estudiantes_G6_dep$learning_disabilities))

table(cdd_factores_rendimiento_estudiantes_G6_dep$parental_education_level)
prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$parental_education_level))
barplot(table(cdd_factores_rendimiento_estudiantes_G6_dep$parental_education_level))

table(cdd_factores_rendimiento_estudiantes_G6_dep$distance_from_home)
prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$distance_from_home))
barplot(table(cdd_factores_rendimiento_estudiantes_G6_dep$distance_from_home))

table(cdd_factores_rendimiento_estudiantes_G6_dep$gender)
prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$gender))
barplot(table(cdd_factores_rendimiento_estudiantes_G6_dep$gender))

```


###### Diagramas de Dispersión Variables Cuantitativas
```{r dispersiones_variables_cuantitativas, fig.align = 'center'}
pairs(~hours_studied + age + attendance + sleep_hours + previous_scores + tutoring_sessions + physical_activity + exam_score , data = cdd_factores_rendimiento_estudiantes_G6_dep)
```


##### 5.3.2.2. Formulación del modelo de RLM entre las variables de estudio. {.tabset .tabset-pills}

De acuerdo con los resultados presentados en la pestaña Coeficientes del Modelo RLM Total, se establece que el modelo de regresión lineal múltiple completo relaciona las variables de interés consideradas en el análisis. Estas variables se resumen como:
$ES$ (Exam Score), $HS$ (hours_studied), $A$ (age), $AT$ (attendance), $PI$ (parental_involvement), $AR$ (access_to_resources), $EA$ (extracurricular_activities), $SH$ (sleep_hours), $PS$ (previous_scores), $ML$ (motivation_level), $IA$ (internet_access), $TS$ (tutoring_sessions), $FI$ (family_income), $TQ$ (teacher_quality), $ST$ (school_type), $PI_1$ (peer_influence::1), $PA$ (physical_activity), $LD$ (learning_disabilities), $PEL$ (parental_education_level), $DH$ (distance_from_home), y $G_1$ (gender::male).
El modelo ajustado toma la siguiente forma (con coeficientes redondeados a cuatro cifras decimales para facilitar su lectura):
ES^=−1.2425+0.0017⋅HS+0.0031⋅A−0.0026⋅AT+0.0227⋅PI+0.1188⋅AR−0.0009⋅EA+0.0244⋅SH−0.0146⋅PS−0.0093⋅ML−0.0072⋅IA+0.0105⋅TS(28)\hat{ES} = -1.2425 + 0.0017 \cdot HS + 0.0031 \cdot A - 0.0026 \cdot AT + 0.0227 \cdot PI + 0.1188 \cdot AR - 0.0009 \cdot EA + 0.0244 \cdot SH - 0.0146 \cdot PS - 0.0093 \cdot ML - 0.0072 \cdot IA + 0.0105 \cdot TS \hspace{10mm}(28)ES^=−1.2425+0.0017⋅HS+0.0031⋅A−0.0026⋅AT+0.0227⋅PI+0.1188⋅AR−0.0009⋅EA+0.0244⋅SH−0.0146⋅PS−0.0093⋅ML−0.0072⋅IA+0.0105⋅TS(28)
En este modelo, no se interpreta el valor del intercepto debido a su falta de relevancia práctica, ya que no es realista asumir que todas las variables predictoras sean simultáneamente iguales a cero. En este caso, Exam Score tendría un valor negativo, lo cual carece de sentido.
Este modelo será comparado con una versión reducida en la que se omiten algunas variables como gender, attendance, y teacher quality, con el propósito de analizar el impacto que estas tienen en la predicción de la variable dependiente.
$$ES=−3.4567+0.0152⋅HS+0.0028⋅A+0.0186⋅AT+0.0342⋅PI−0.0105⋅AR+0.0223⋅EA+0.0056⋅SH+0.0458⋅PS+0.0294⋅ML−0.0072⋅IA +0.0191⋅TS+0.0087⋅FI+0.0315⋅TQ−0.0039⋅ST+0.0048⋅PI1+0.0126⋅PA−0.0142⋅LD+0.0253⋅PEL−0.0097⋅DH−0.0011⋅G1+ 0.0191 \cdot TS + 0.0087 \cdot FI + 0.0315 \cdot TQ - 0.0039 \cdot ST + 0.0048 \cdot PI_1 + 0.0126 \cdot PA - 0.0142 \cdot LD + 0.0253 \cdot PEL - 0.0097 \cdot DH - 0.0011 \cdot G_1+0.0191⋅TS+0.0087⋅FI+0.0315⋅TQ−0.0039⋅ST+0.0048⋅PI1​+0.0126⋅PA−0.0142⋅LD+0.0253⋅PEL−0.0097⋅DH−0.0011⋅G1​ $$(28)
Las variables que serán excluidas podrían no tener una relación significativa con el resultado de interés, que en este caso es ES. Si los análisis previos muestran que estas variables no contribuyen de manera relevante al modelo o que su impacto en el valor de ESESES es muy bajo, puede ser conveniente excluirlas para simplificar el modelo y hacerlo más eficiente. 
$$ES=−3.4567+0.0152⋅HS+0.0186⋅AT+0.0342⋅PI−0.0105⋅AR+0.0223⋅EA+0.0056⋅SH+0.0458⋅PS+0.0294⋅ML−0.0072⋅IA+0.0191⋅TS+0.0315⋅TQ−0.0039⋅ST+0.0048⋅P$$I1(29)
Excluyendo estas variables, el modelo se enfoca en las variables que se consideran más relevantes o que tienen un impacto más directo sobre el resultado de interés. Esta simplificación puede mejorar la interpretación de los resultados y hacer que el modelo sea más eficiente, siempre que se justifique adecuadamente la exclusión de las variables en base a la evidencia empírica, la teoría subyacente y la disponibilidad de datos.




###### Resumen y ANOVA del Modelo RLM Total
```{r, resumen_y_ANOVA_RLM_Total, fig.align = 'center'}
summary(lm(cdd_factores_rendimiento_estudiantes_G6_dep$exam_score ~ cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied + cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores + cdd_factores_rendimiento_estudiantes_G6_dep$attendance + cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours + cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$gender) + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality) + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement)))
anova(lm(cdd_factores_rendimiento_estudiantes_G6_dep$exam_score ~ cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied + cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores + cdd_factores_rendimiento_estudiantes_G6_dep$attendance + cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours + cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$gender) + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality) + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement)))
```


###### Coeficientes del Modelo RLM Total
```{r, coeficientes_RLM_Total, fig.align = 'center'}
coefficients(lm(cdd_factores_rendimiento_estudiantes_G6_dep$exam_score ~ cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied + cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores + cdd_factores_rendimiento_estudiantes_G6_dep$attendance + cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours + cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$gender) + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality) + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement)))
```


###### Coeficientes del Modelo RLM Reducido
```{r coeficientes_RLM_Reducido, fig.align = 'center'}
coefficients(lm(cdd_factores_rendimiento_estudiantes_G6_dep$exam_score ~ cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied + cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores + cdd_factores_rendimiento_estudiantes_G6_dep$attendance  + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement)))
```

##### 5.3.2.3. Análisis del modelo RLM. {.tabset .tabset-pills}

Excluir las variables PA (Physical Activity), LD (Learning Disabilities), PEL (Parental Education Level), DH (Distance from Home) y G (Gender) podría ser acertado en ciertos contextos, dependiendo de varios factores, como el objetivo de tu estudio, la relación empírica de cada variable con ESESES, y el impacto que tiene la simplificación del modelo.
Excluyendo estas variables puede ser acertado si se basan en un análisis previo que demuestra que tienen poca influencia sobre ESESES o si simplificar el modelo mejora su interpretación sin perder precisión. Sin embargo, si el estudio requiere una comprensión más completa de los factores que afectan ESESES, como la actividad física o el nivel educativo de los padres, la exclusión podría no ser lo más adecuado.

###### Mejor Modelo Iterado según AIC
```{r, mejor_Modelo_AIC, fig.align = 'center'}
modelo_Iterado_STEP = step(lm(cdd_factores_rendimiento_estudiantes_G6_dep$exam_score ~ cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied + cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores + cdd_factores_rendimiento_estudiantes_G6_dep$attendance + cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours + cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$gender) + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality) + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement)))
coefficients(modelo_Iterado_STEP)
```

###### Bondades de Ajuste, Significancias y Criterios de Información Comparados
```{r}
modelo_RLM_TOTAL = lm(cdd_factores_rendimiento_estudiantes_G6_dep$exam_score ~ cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied + cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores + cdd_factores_rendimiento_estudiantes_G6_dep$attendance + cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours + cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$gender) + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality) + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement))
modelo_RLM_REDUCIDO = lm(cdd_factores_rendimiento_estudiantes_G6_dep$exam_score ~ cdd_factores_rendimiento_estudiantes_G6_dep$hours_studied + cdd_factores_rendimiento_estudiantes_G6_dep$previous_scores + cdd_factores_rendimiento_estudiantes_G6_dep$attendance + cdd_factores_rendimiento_estudiantes_G6_dep$sleep_hours + cdd_factores_rendimiento_estudiantes_G6_dep$tutoring_sessions + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$gender) + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$teacher_quality) + as.factor(cdd_factores_rendimiento_estudiantes_G6_dep$parental_involvement))

stargazer(modelo_RLM_TOTAL, modelo_RLM_REDUCIDO, modelo_Iterado_STEP, type = "text", df = TRUE)
AIC(modelo_RLM_TOTAL, modelo_RLM_REDUCIDO, modelo_Iterado_STEP)
BIC(modelo_RLM_TOTAL, modelo_RLM_REDUCIDO, modelo_Iterado_STEP)
```



### **5.4. Regresión Logística Simple** {.tabset .tabset-pills}

Este modelo que en este estudio será llamado **RlogS**, que difiere de un modelo de regresión lineal simple que relaciona una variable cuantitativa dependiente $y$ con una variable cuantitativa dependiente $x$, relaciona una variable categórica dicotomica independiente $y$ con valor de probabilidad $p(x)\in [0, 1]$ que depende de una variable cuantitativa $x$.

Los modelos de regresión utilizados en este estudio pueden considerarse como casos específicos del Modelo Lineal Generalizado (**GLM**, por sus siglas en inglés). Este modelo amplía el modelo lineal general al permitir que la variable dependiente se relacione linealmente con sus factores y covariables mediante una función de enlace, además de permitir que la variable dependiente tenga una distribución distinta a la normal. Aparte de los modelos aplicados en este estudio, el **GLM** también abarca: modelos loglineales para datos de conteo, modelos log-log complementarios para datos de supervivencia censurados por intervalos y otros modelos estadísticos, gracias a la formulación general que ofrece.

El **GLM** permite especificar distribuciones distintas a la normal y utilizar una función de enlace diferente a la identidad, lo que posibilita una amplia variedad de combinaciones de distribuciones y funciones de enlace. Varias de estas combinaciones pueden ser apropiadas para un conjunto de datos específico. Por lo tanto, la elección de la combinación más adecuada dependerá de consideraciones teóricas previas, de la naturaleza de las variables, de la experiencia del investigador y de los resultados obtenidos al comparar diferentes combinaciones.

En el caso que se aborda, se utilizará una distribución binomial (apropiada para variables que representan una respuesta dicotómica) con una función de enlace logit::$$\pi(x)=\dfrac{e^{\beta_0+\beta_1 x}}{1+ e^{\beta_0 +\beta_1 x}}=  \dfrac{1}{1+ e^{-(\beta_0+\beta_1 x)}}\hspace{10mm}(30)$$ 
Un nombre mas apropiado podria ser Regresión Lineal Binaria.

Para facilitar la interpretación, se considera que la función de enlace $\pi(x)$ proviene de una razón de probabilidades (denominada en inglés como ODDS ratio (OR)), que es el argumento de un logaritmo: $\log\left (\frac{\pi(x)}{1-\pi(x)}\right)$. De esta manera, se modela la probabilidad de que la variable de respuesta pertenezca al nivel de referencia $1$ en función de los valores de los predictores. Además, la transformación de probabilidades a razones de probabilidad mantiene la monotonicidad de los sentidos. También convierte el intervalo de probabilidades $[0,1]$ en el intervalo $(-\infty,\infty)$. Las propiedades que se dan entre las probabilidades complementarias de éxito y fracaso, sus razones y la función de enlace logit son:

||||
:-:|:-:|:-:|
$p(éxito)=p(fracaso)$|$OR=1$|$Logit\left(OR\right)=0$
$p(éxito)<p(fracaso)$|$OR<1$|$Logit\left(OR\right)<0$
$p(éxito)>p(fracaso)$|$OR>1$|$Logit\left(OR\right)>0$
||||


Se entiende que la transformación $Logit$ carece de sentido para la certeza del éxito o del fracaso.

##### 5.4.1. Planteamiento del Problema
Con base en el conjunto de dasto descrito anteriormente se formulará un modelo de regresión logística simple para supuest aentre las variables definidas por los campos: **Exam score** y **Extracurricular activities**, con base en una distribución binomial y la función de enlace $Logit$

##### 5.4.2. Desarrollo del Análisis
El estudio de regresión logistica simple ha sido procesado con `r R.version.string` mediado por RStudio 2024.04.2

##### 5.4.2.1. Resumen estadístico de las variables de estudio. {.tabset .tabset-pills}

La navegación a través de las pestañas muestra el resumen estadístico de la variable dependiente **Exam Score**, su boxplot e histograma. De la variable independiente **Learning disabilities** se mostrará su diagrama de barras, así como su media y mediana. Además, se exhibirá un Diagrama de Cajas conjunto entre ambas.

Con base en la pestaña **Resumen y Boxplot de Exam Score**, se puede comentar que la variable **Exam Score** presenta una asimetría de sesgo negativo, lo cual se puede observar en la curva de la logística gráfica. El rango intercuartílico muestra una dispersión moderada, aunque no se visualizan valores atípicos significativos. Así, puede decirse que la variable **Exam Score** tiende a registrar valores altos en relación con su intervalo de medición. Lo expuesto también se puede constatar a través de la pestaña Histograma de Exam Score.

Asimismo, según la pestaña **Resumen y Diagrama de Barras de Learning disabiities**, la variable cualitativa nominal **Learning disabiities** muestra una proporcionalidad mayor para los casos desfavorables (sin discapacidad) representados por el valor 0, en comparación con los casos favorables (con discapacidad) representados por el valor 1.

Complementariamente, la pestaña Resumen y Diagrama de Cajas Conjunto muestra que las observaciones son consistentes con el contexto del problema. Es decir, para los casos desfavorables de Learning Disabilities (sin discapacidad), los resultados de Exam Score tienden a ser mayores en comparación con los casos favorables (con discapacidad). Además, ambos casos muestran sesgo negativo, aunque es más notorio para los casos desfavorables. En el rango intercuartílico, se visualizan dispersiones opuestas: para los casos favorables, los valores tienden hacia la mitad inferior de los datos, mientras que para los desfavorables, tienden hacia la mitad superior. Asimismo, los valores atípicos, en ambos casos, se presentan en los extremos inferiores de las distribuciones, lo cual indica que los valores bajos resultaron más extraños.


###### Resumen y Boxplot de Exam score
```{r resumen__exam__score, fig.align = 'center'}
summary(cdd_factores_rendimiento_estudiantes_G6_dep$exam_score)
boxplot(cdd_factores_rendimiento_estudiantes_G6_dep$exam_score, main = "Diagrama de Caja de Exam score", col = c("orange"))
```

###### Histograma de Exam score
```{r histograma_exam_score, fig.align = 'center'}
summary(cdd_factores_rendimiento_estudiantes_G6_dep$exam_score)
hist(cdd_factores_rendimiento_estudiantes_G6_dep$exam_score, main = "Histograma de Exam score", col = c("gold"))
```

###### Resumen y Diagrama de Barras de Learning disabilities
```{r resumen_learning-disabilities, fig.align = 'center'}
table(cdd_factores_rendimiento_estudiantes_G6_dep$learning_disabilities)
prop.table(table(cdd_factores_rendimiento_estudiantes_G6_dep$learning_disabilities))
barplot(table(cdd_factores_rendimiento_estudiantes_G6_dep$learning_disabilities))
```

###### Resumen y Diagrama de Cajas Conjunto
```{r boxplot_Conjunto_exam-score_learning-disabilities, fig.align = 'center'}
tapply(cdd_factores_rendimiento_estudiantes_G6_dep$exam_score, cdd_factores_rendimiento_estudiantes_G6_dep$learning_disabilities, mean)
tapply(cdd_factores_rendimiento_estudiantes_G6_dep$exam_score, cdd_factores_rendimiento_estudiantes_G6_dep$learning_disabilities, median)
boxplot(cdd_factores_rendimiento_estudiantes_G6_dep$exam_score~cdd_factores_rendimiento_estudiantes_G6_dep$learning_disabilities, main = "Boxplot Conjunto: Exam score - Learning disabilities", col = c("orange", "gold"))
```

##### 5.4.2.2. Formulación del modelo de RLogS entre las variables de estudio. {.tabset .tabset-pills}

La navegación a través de las pestañas muestra los coeficientes del modelo RLogS y su resumen estadístico. Se menciona de nuevo que las variables de interés son: **Exam score** (variable dependiente) y **Learning disabiities** (variable independiente).

La pestaña **Coeficientes del Modelo RLogS** permite establecer que el modelo RLogS relaciona a $\pi(x)$ con $x$ a través de la función de enlace $Logit$ de la siguiente manera:$$\frac{\pi(x)}{1-\pi(x)}=e^{-7,658709+10,886959\cdot x}\hspace{10mm}(31)$$

Asimismo, la pestaña **Resumen Estadístico del Modelo RLogS** muestra los resúmenes de los modelos estudiados, uno basado en la variable **Learning Disabilities** y otro con la variable **Gender** , ambas en relación con el **Exam Score** . En términos del criterio de información de Akaike (AIC), que evalúa la bondad de ajuste del modelo considerando su precisión y complejidad, se observa que el modelo basado en Discapacidades de Aprendizaje es más ajustado que el de Género . Esto se debe a que el AIC para Learning Disabilities ($AIC_{LD} = 4231.8$) es más bajo que el de Gender ($AIC_{G} = 8688.8$), lo que sugiere que el modelo con Learning Disabilities tiene un mejor equilibrio entre sesgo y varianza.
Además, el análisis de las desviaciones muestra una diferencia notable en las desviaciones nula y residual. El modelo con Discapacidades de Aprendizaje presenta una disminución más pronunciada en la desviación residual (de $4277.9$ a $4227.8$), lo que implica un mejor ajuste del modelo en comparación con el modelo basado en Género , cuya reducción en la desviación es mínima (de $8685.0 $ a $ 8684,8 $).
De los coeficientes del modelo, se observa que el **Exam Score** tiene un efecto significativo en **Learning Disabilities**, con un valor p muy pequeño (prácticamente 0), lo que indica que el puntaje del examen es un predictor importante de la presencia o ausencia de discapacidades. de aprendizaje. En contraste, el modelo con **Gender** muestra un valor p más alto (0.266), lo que sugiere que el Exam Score no es un predictor significativo de género en este contexto.


###### Coeficientes del Modelo RLogS
```{r coeficientes_regresion_logistica_simple, fig.align = 'center'}
modelo_RLog_Simple = glm(cdd_factores_rendimiento_estudiantes_G6_dep$learning_disabilities~cdd_factores_rendimiento_estudiantes_G6_dep$exam_score, family = "binomial", data = data.frame(cdd_factores_rendimiento_estudiantes_G6_dep$learning_disabilities, cdd_factores_rendimiento_estudiantes_G6_dep$exam_score))
coef(modelo_RLog_Simple)
```
###### Resumen Estadístico del Modelo RLogS
```{r resumen_regresion_logistica_simple, fig.align = 'center'}
summary(modelo_RLog_Simple)
modelo_RLog_Simple_S = glm(cdd_factores_rendimiento_estudiantes_G6_dep$gender~cdd_factores_rendimiento_estudiantes_G6_dep$exam_score, family = "binomial", data = data.frame(cdd_factores_rendimiento_estudiantes_G6_dep$gender, cdd_factores_rendimiento_estudiantes_G6_dep$exam_score))
summary(modelo_RLog_Simple_S)
```

##### 5.4.2.3. Análisis del modelo RLogS. {.tabset .tabset-pills}

Se presentarán, mediante las pestañas, los resultados de algunas predicciones generadas por el modelo RLogS, con el fin de identificar en sus respuestas la relación de sentido en las razones de probabilidades (odds) a favor o en contra del evento en cuestión: $\frac{\pi}{1-\pi}$ y $\frac{1-\pi}{\pi}$, respectivamente. Se menciona de nuevo que las variables de interés son: **Exam score** (variable dependiente) y **Learning disabilities** (variable independiente).

La pestaña **Variable Predictora igual a Cero** plantea dos situaciones interpretativas. La primera permite comprender que el coeficiente del factor en el cual está presente la variable predictora, estima un valor base para la puntuación en el examen. En la primera, el intercepto tiene un valor de 3.32, lo que indica que, cuando la variable predictora no tiene influencia (es cero), la puntuación inicial estimada en el examen sería de aproximadamente 3.32 puntos. Sin embargo, el coeficiente asociado a la variable exam_score es negativo (-0.0819), lo que sugiere que un aumento en una unidad de esta variable se asocia con una leve disminución en el puntaje del examen.

Por otro lado, en la segunda, el intercepto tiene un valor de 27.76, lo que indica una puntuación inicial significativamente mayor en ausencia de influencia de la variable predictora. En este modelo, el coeficiente de exam_score es positivo (0.9214), lo que implica que cada incremento de una unidad en la variable predictora se traduce en un aumento promedio de 0.92 puntos en el resultado del examen.


Con base en los valores calculados en la pestaña de **Probabilidades Estimadas**, se puede observar entre los registros $79$ y $80$ un cambio absoluto de $0.050139$ en las probabilidades (lo que representa un incremento relativo aproximado del $41.45%$). Este incremento ocurre al aumentar la variable predictora en una unidad, en línea con las definiciones establecidas previamente en el análisis.

El gráfico muestra que a medida que aumenta el puntaje en el examen, la probabilidad de tener una discapacidad de aprendizaje disminuye, con una relación negativa típica de un modelo logístico. Las observaciones se concentran en puntajes bajos para quienes tienen discapacidad (1) y en puntajes altos para quienes no la tienen (0). Esto sugiere que las discapacidades de aprendizaje pueden estar asociadas a un menor rendimiento académico.

###### Variable Predictora igual a Cero
```{r variable_Predictora_Cero, fig.align = 'center'}
coef(modelo_RLog_Simple)
round(exp(coef(modelo_RLog_Simple)),6)
```
###### Probabilidades Estimadas
```{r probabilidades_Estimadas, fig.align = 'center'}
predict(modelo_RLog_Simple, data.frame(seq(1, 400)), type = "response")
```
###### Gráfica del Modelo RLogS
```{r grafica_Modelo_RLogS, fig.align = 'center'}
learning_disabilities <- cdd_factores_rendimiento_estudiantes_G6_dep$learning_disabilities
exam_score <- cdd_factores_rendimiento_estudiantes_G6_dep$exam_score
dataPlot <- data.frame(exam_score, learning_disabilities)
plot(learning_disabilities~exam_score, data = dataPlot, main = "Modelo RLogS: Exam score - Learning disabilities", xlab = "Exam score", ylab = "Learning disabilities = 0 | Learning disabilities = 1", col = "gold", pch = "I")
curve(predict(glm(learning_disabilities~exam_score, family = "binomial", data = dataPlot), data.frame(exam_score = x), type = "response"), col = "orange", lwd = 3, add = TRUE)
```


## **Conclusión**

El conjunto de datos "Student Performance Factors" de Kaggle explora los determinantes del rendimiento académico de los estudiantes a través de múltiples dimensiones, como características demográficas, socioeconómicas, familiares y conductuales. Los resultados de análisis previos indican que factores como la educación de los padres, particularmente de la madre, y los ingresos familiares tienen una correlación positiva con el desempeño, al facilitar acceso a recursos educativos de calidad. Además, un entorno familiar de apoyo emocional y motivacional se asocia con un mejor rendimiento, mientras que hábitos de estudio organizados y el balance adecuado entre actividades extracurriculares y tiempo libre contribuyen significativamente al éxito académico

El entorno escolar, incluido el nivel de preparación de los docentes y la calidad de las infraestructuras, también juega un rol crítico. Sin embargo, se identifican disparidades sustanciales en el acceso y los resultados entre estudiantes de contextos socioeconómicos desfavorecidos o regiones con recursos limitados, lo que subraya la importancia de intervenciones políticas que promuevan la equidad educativa. Este dataset, al proporcionar datos ricos y multifacéticos, se convierte en una herramienta esencial para diseñar estrategias personalizadas, como modelos predictivos para identificar estudiantes en riesgo, y generar soluciones orientadas a cerrar brechas estructurales en el sistema educativo

## **Referencias**

{kaggle2024,
  author = {Kaggle},
  title = {Student Performance Factors Dataset},
  year = {2024},
  url = {https://www.kaggle.com/datasets/lainguyn123/student-performance-factors}
}






  
