Analítica Avanzada de Datos

Caso #3

Autor/a

Afiliación

Katherine M. Tajan Niebles

Universidad del Norte, Barranquilla

Introducción

Los ARN no codificantes largos (lncRNAs en inglés) comprenden un vasto repertorio de ARN que desempeñan una amplia variedad de funciones cruciales en la fisiología de los tejidos de manera específica para cada célula. A pesar de estar implicados en miles de mecanismos reguladores, todavía no se ha asignado ninguna función a muchos lncRNAs.

Clasificación de las funciones de los lncRNAs. Imagen tomada de aquí.

Cada vez hay más pruebas de que los lncRNAs se expresan de forma aberrante en la progresión de la Enfermedad de Alzheimer (EA). En nuestro grupo, hemos analizado la expresión de cientos de lncRNAs en 15 individuos diagnosticados con EA, y 15 voluntarios sin la enfermedad en el marco del proyecto de investigación “Nuevos ARN no codificantes exo- somales y su papel en la patogénesis de la Enfermedad de Alzheimer”, código 121584468097 (844/2019), contract 416-2020, otorgado al Grupo en Genética y Medicina Molecular (Departamento de Medicina), y al Grupo de Productividad y Competitividad (Departamento de Ingeniería Industrial) de la Universidad del Norte, Barranquilla, Colombia.

Una muestra de los datos puede obtenerse haciendo:

Las variables relevantes son

• id: identificador del participante.
• dx: diagnóstico del participante (0: no EA; 1: EA)
• x1, x2,…, x29809: niveles de expresión del lncRNA \(j\), con \(j=1,2,\ldots,29809\).

El objeto d tiene 30 filas y 29811 columnas.

Importante

A partir de los datos, el objetivo es identificar los lncRNAs cuya expresión incrementa o reduce la probabilidad de diagnóstico de EA.

Fecha de entrega: Junio 11, 2024.

Como la variable respuesta es binaria, podemos utilizar las funciones que ya hemos empleado con anterioridad en Regresión Logística (RL) para medir el desempeño del modelo.

This is a collection of functions created by 
Professor Jorge Vélez, PhD <https://jorgeivanvelez.netlify.app/> 
for the fitting and validation of Binary Classification models 
 
In case of problems, please an email to 
<jvelezv@uninorte.edu.co> with enough details 
 
Last modification:  April 23, 2024 
 

Preguntas

P1

Use una prueba \(t\) de Student de 2 colas para identificar los lncRNAs diferencialmente expresados en personas con EA. Extraiga el valor \(p\) y compárelo con el nivel de significancia \(\alpha = 0.05\). Cuántos de estos son estadísticamente significativos? Use la función p.adjust() para ajustar los valores \(p\) utilizando el método FDR. Compare.

A continuación se define función cal_t_test para implementar una prueba \(t\) de Student de 2 colas para identificar los lncRNAs diferencialmente expresados en personas con EA, a tráves del valor de \(p\).

#Definición de función para realizar prueba t-test
cal_t_test <- function(j){
    t.test(d[,2+j]~dx,d)$p.value
}

# Aplicación de función cal_t_test a cada columna relevante
t_test_resultado <- sapply(1:(ncol(d) - 2), cal_t_test)

t_test_resultado <- as.data.frame(t_test_resultado)
names(t_test_resultado) <- c('P_valor')

# Cuenta los p-valores menores a 0.05 usando sapply
p_value_sig <- sum(sapply(t_test_resultado$'P_valor', function(p_val) {
  if (p_val < 0.05) {
    return(1)
  } else {
    return(0)
  }
}))

Luego de ejecutar la función cal_t_test, extraer el valor \(p\) y compararlo con el nivel de significancia \(\alpha = 0.05\), se observa que existen 1718 valores \(p\) estadísticamente significativos.

Ahora, usaremos la función p.adjust() para ajustar los valores \(p\) utilizando el método FDR.

P_valor.adj <- p.adjust(t_test_resultado$P_valor, method = 'fdr', n = length(t_test_resultado$'P_valor'))

t_test_resultado$P_valor.adj <- P_valor.adj

A continuación se visualizan los primeras filas que contienen el valor \(p\) extraido despues de aplicar una \(t\) de student de 2 colas para identificar los IncRNAs diferencialmente expresados en personas con EA y el valor \(p\) ajustado usando la función p.adjust(), al compararlos se observa que ningún valor de \(p\) ajustado es significativo.

head(t_test_resultado)

       P_valor P_valor.adj
1 0.7273178923   0.9731852
2 0.6163451816   0.9690406
3 0.0008774605   0.4359370
4 0.8337503626   0.9850402
5 0.4156953303   0.9547774
6 0.0133455919   0.6887817

P2

Construya una función que, a partir del número del lncRNA, estime el modelo de RL, extraiga el coeficiente estimado, el estadístico \(t\) y el valor \(p\). Por ejemplo, para el lncRNA \(j\), aplicar my_magical_function(j) debe producir \(\hat{\beta}_j\), \(t_j\) y \(p_j\).

A continuación se define función my_magical_function para aplicarla sobre lncRNA \(j\) , estimar el modelo de RL y extraer el coeficiente estimado \(\hat{\beta}_j\), el estadístico \(t_j\) y el valor \(p_j\). Lo anterior, se puede visualizar a continuación el resultadao de los 6 primeros IncRNa:

my_magical_function <- function(j){
  model <- summary(glm(dx ~ d[, 2+j], family = 'binomial', data=d))
  model$coefficients[2,][c(1, 3, 4)]
}

coeff_model <- sapply(1:(ncol(d) - 2), my_magical_function)

coeff_model_t <- t(coeff_model)
coeff_model_df <- as.data.frame(coeff_model_t)
coeff_model_df$ID<- seq(1, 29809)
names(coeff_model_df)<- c('Coeficiente','valor z','P_valor','ID')

head(coeff_model_df)

  Coeficiente    valor z    P_valor ID
1  -0.2899068 -0.3636479 0.71612099  1
2  -0.7003055 -0.5201932 0.60292894  2
3   2.4160254  2.6872190 0.00720497  3
4   0.1373603  0.2190667 0.82659807  4
5   0.5914169  0.8277951 0.40778655  5
6   2.9116294  2.1904604 0.02849086  6

P3

Realice el mismo análisis que en P1, pero ahora estandarice los niveles de expresión. Apóyese en la función scale(). Considera usted que la estandarización es necesaria? Por qué si? Por qué no? Compare los resultados cuando usa estandarización y cuando no.

A continuación se define función cal_t_test_scale para implementar una prueba \(t\) de Student de 2 colas sobre las variables predictoras (niveles de expresión) estandarizadas y lograr identificar los lncRNAs diferencialmente expresados en personas con EA, a tráves del valor de \(p\).

cal_t_test_scale <- function(j){
    t.test(scale(d[,2+j])~dx,d)$p.value
}

# Aplicación de función cal_t_test a cada columna relevante
t_test_scale <- sapply(1:(ncol(d) - 2), cal_t_test_scale)

t_test_scale <- as.data.frame(t_test_scale)
names(t_test_scale) <- c('P_valor')

# Cuenta los p-valores menores a 0.05 usando sapply
p_value_sig_scale <- sum(sapply(t_test_scale$'P_valor', function(p_val) {
  if (p_val < 0.05) {
    return(1)
  } else {
    return(0)
  }
}))

Luego de ejecutar la función cal_t_test_scale, extraer el valor \(p\) y compararlo con el nivel de significancia \(\alpha = 0.05\), se observa que existen 1718 valores \(p\) estadísticamente significativos.

Sí comparamos los dos resultados se observa que se tiene igual número de valores \(p\) estadísticamente significativos, siendo innecesario el escalamiento de datos.

Lo anterior, se puede visualizar a tráves del siguiente gráfico:

data_box <- data.frame(P_valor_t_test = t_test_resultado$P_valor,
                       P_valor_t_test_scale = t_test_scale$P_valor,
                       sig_t_test = ifelse(t_test_resultado$P_valor <= 0.05, 'SI', 'NO'),
                       sig_t_test_scale = ifelse(t_test_scale$P_valor <= 0.05, 'SI', 'NO'))


fig_v_p3 <- subplot(
  data_box %>% 
    plot_ly() %>% 
    add_trace(
      x = ~sig_t_test,
      y = ~P_valor_t_test,
      text = ~P_valor_t_test,
      type = 'box',
      color = ~sig_t_test,
      name = "P- Valor Data Original",
      colors = c("#8C4D3F","#1B7F7A")
    ) %>%
    layout(yaxis = list(title = "P- Valor", range = c(0, 1))),  # Establecer el rango del eje Y
  data_box %>%
    plot_ly() %>%
    add_trace(
      x = ~sig_t_test_scale,
      y = ~P_valor_t_test_scale,
      text = ~P_valor_t_test_scale,
      type = 'box',
      color = ~sig_t_test_scale,
      name = "P- Valor Data Estandarizada",
      colors = c("#4682B4", "#DEB887")
    ) %>%
    layout(yaxis = list(title = "P- Valor data estandarizada", range = c(0, 1))),  # Establecer el rango del eje Y
  nrows = 1
)

fig_v_p3 <- fig_v_p3 %>% layout(title = "Boxplot P-valores t-student")

fig_v_p3

Tener en cuenta que:

El SI sugiere que el valor \(p\) es significativo, mientras que el NO sugiere que el valor \(p\) no es significativo.

P4

En P1 se usó una prueba \(t\) de Student y en P2 un modelo de RL. Explique por qué hay diferencias en el número de lncRNAs que resultan ser significativos al emplear estos métodos.

La prueba \(t\) de student se emplea para evaluar si hay diferencias significativas entre las medias de dos grupos, asumiendo homogeneidad en los datos. Por otro lado, un modelo de regresión logística puede enfocarse específicamente en la variable predictora de interés, sin tener en cuenta otras variables predictoras, mientras controla el efecto de posibles variables confundentes. Estas variables, que generalmente se consideran en la prueba t de Student, pueden influir en los resultados al introducir sesgos en el análisis.

P5

Ordene los resultados obtenidos de acuerdo con el valor \(p\) y extraiga los 10 lncRNAs más significativos de acuerdo con el valor \(p\). Cuántos de ellos confieren riesgo y cuántos protección contra EA? Concluya.

A continuación, se extraen los 10 IncRNAs más significativos de acuerdo al valor de \(p\), y posterior a ésto, calculamos la razón de probabilidades \(OR\), el cúal puede obtenerse como:

\[ \hat{OR}_j = e^{\hat{\beta}_j} \]

Lo anterior, para poder establecer cuántos y cuáles IncRNAs confieren riesgo y cuáles confieren protección.

coeff_model_df_sorte<-coeff_model_df%>% 
  arrange(P_valor)
  mas_sig_10 <- head(coeff_model_df_sorte, 10)
  mas_sig_10$OR <- sapply(mas_sig_10$Coeficiente, exp)

kbl(round(mas_sig_10,3), 
    caption = "", 
    align = "c", 
    position = "top") %>%
    kable_styling(bootstrap_options = c("striped", "hover"))%>% 
    scroll_box(width = "920px", height = "300px")

Coeficiente	valor z	P_valor	ID	OR
4.595	3.009	0.003	4634	99.015
-4.804	-2.965	0.003	14508	0.008
3.479	2.868	0.004	6948	32.425
2.135	2.864	0.004	23214	8.457
-7.161	-2.864	0.004	26435	0.001
2.405	2.851	0.004	28007	11.078
2.776	2.851	0.004	11576	16.048
4.361	2.850	0.004	2114	78.340
2.343	2.847	0.004	27199	10.409
1.687	2.829	0.005	2237	5.401

Con base en los resultados obtenidos, se observa que 8 de los 10 lncRNAs más significativos confieren riesgo, mientras que los 2 restantes confieren protección.

Se concluye que:

• Al aumentar la cantidad de IncRNA(4634) en una unidad, es \(\thickapprox 99\%\) más probable que un paciente padezca EA, por lo tanto, se considera un factor de riesgo.

• Al aumentar la cantidad de IncRNA(14508) en una unidad, es \(\thickapprox 0%\) veces más probable que un paciente padezca EA, por lo tanto, se considera un factor de protección.

• Al aumentar la cantidad de IncRNA(6948) en una unidad, es \(\thickapprox 32\%\) más probable que un paciente padezca EA, por lo tanto, se considera un factor de riesgo.

• Al aumentar la cantidad de IncRNA(23214) en una unidad, es \(\thickapprox 8\%\) más probable que un paciente padezca EA, por lo tanto, se considera un factor de riesgo.

• Al aumentar la cantidad de IncRNA(26435) en una unidad, es \(\thickapprox 0\) veces más probable que un paciente padezca EA, por lo tanto, se considera un factor de protección.

• Al aumentar la cantidad de IncRNA(28007) en una unidad, es \(\thickapprox 11\%\) más probable que un paciente padezca EA, por lo tanto, se considera un factor de riesgo.

• Al aumentar la cantidad de IncRNA(11576) en una unidad, es \(\thickapprox 16\) veces más probable que un paciente padezca EA, por lo tanto, se considera un factor de riesgo.

• Al aumentar la cantidad de IncRNA(2114) en una unidad, es \(\thickapprox 78\) veces más probable que un paciente padezca EA, por lo tanto, se considera un factor de riesgo.

• Al aumentar la cantidad de IncRNA(27199) en una unidad, es \(\thickapprox 10\) veces más probable que un paciente padezca EA, por lo tanto, se considera un factor de riesgo.

• Al aumentar la cantidad de IncRNA(2237) en una unidad, es \(\thickapprox 5\) veces más probable que un paciente padezca EA, por lo tanto, se considera un factor de riesgo.

P6

Construya un modelo de RL que incluya los lncRNAs seleccionados en P5. Evalúe la idoneidad. Concluya.

modelp6 <- glm(dx ~ x4634 + x14508 + x6948 + x23214 + x26435 + x28007 + x11576 + x2114 + x27199 + x2237, family = 'binomial', data = d)

summary(modelp6)

Call:
glm(formula = dx ~ x4634 + x14508 + x6948 + x23214 + x26435 + 
    x28007 + x11576 + x2114 + x27199 + x2237, family = "binomial", 
    data = d)

Deviance Residuals: 
       Min          1Q      Median          3Q         Max  
-1.294e-05  -2.110e-08   0.000e+00   6.245e-07   9.404e-06  

Coefficients:
              Estimate Std. Error z value Pr(>|z|)
(Intercept) -6.234e+01  3.853e+06       0        1
x4634        1.604e+00  6.266e+05       0        1
x14508      -5.857e+01  2.001e+05       0        1
x6948        2.065e+01  1.444e+05       0        1
x23214       2.485e+01  1.879e+05       0        1
x26435       9.805e+00  5.700e+05       0        1
x28007      -3.074e+01  3.988e+05       0        1
x11576       2.158e+01  2.133e+05       0        1
x2114        2.786e+01  2.074e+05       0        1
x27199       1.370e+01  1.496e+05       0        1
x2237       -5.430e+00  2.308e+05       0        1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 4.1589e+01  on 29  degrees of freedom
Residual deviance: 5.2465e-10  on 19  degrees of freedom
AIC: 22

Number of Fisher Scoring iterations: 25

De acuerdo con el resultado anterior, el modelo estimado de Regresión Logística es:

\[ logit(\hat{\theta}) = -6.234 + 1.604x_{4634} - 58.57x_{14508} + 20.65x_{6948} + 24.85x_{23214}+9.805x_{26435} - 30.74x_{28007} + 21.58x_{11576} + 27.86x_{2114} + 13.70x_{27199} - 5.430x_{2237}\]

Para efectos de análisis y determinar la significancia estadística del coeficiente \(\beta_{4634}\) a nivel poblacional sobre la probabilidad de que un paciente padezca la enfermedad de Alzheimer (EA) , realizamos la siguiente prueba de hipótesis:

\[ H_0: \beta_{4634} = 0 \]

\[ H_1: \beta_{4634}\neq 0 \]

Dado que el valor \(p\) es 1, rechazamos \(H_0\) y concluimos que, a nivel poblacional, no hay suficiente evidencia para afirmar que al aumentrar la expresión del IncRNA con ID 4634 tiene un efecto significativo sobre la probabilidad de que el paciente padezca la Enfermedad de Alzheimer (EA), manteniendo constantes los otros IncRNA.

Ahora, a partir del modelo ajustado, estimamos las probabilidades estimadas \(\hat{\theta}\):

## probabilidades estimadas 
thetahat <- predict(modelp6, type = 'response')  
thetahat_df <- as.data.frame(thetahat)

## ahora incluyamos thetahat en los datos originales 
d$thetahat <- thetahat

kbl(round(thetahat_df,3), 
    caption = "", 
    align = "c", 
    position = "top") %>%
    kable_styling(bootstrap_options = c("striped", "hover"))%>% 
    scroll_box(width = "920px", height = "300px")

thetahat
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1

A continuación se realizan las pruebas correspondientes para evaluar la idoneidad del modelo:

Bondad de ajuste

Para verificar que el modelo de Regresión Logística tiene buen ajuste, utilizamos:

• La prueba de bondad de ajuste de Hosmer & Lemeshow.

En esta prueba, debe decidirse el parámetro \(g\). Teóricamente, \(g > p+1\), donde \(p\) es el número de predictores. En nuestro caso, \(p=10\) y, por lo tanto, \(g>11\). Seleccionaremos \(g=11\). La prueba se encuentra implementada en la función hoslem.test del paquete ResourceSelection.

## hosmer & lemeshow GoF test
if(!require(ResourceSelection)) install.packages('ResourceSelection')
require(ResourceSelection)

# Realizar el test de Hosmer-Lemeshow
hl <- hoslem.test(modelp6$y, fitted(modelp6), g = length(coef(modelp6)) + 1)

# Imprimir el p-valor
hl$p.value 

[1] 1

# Imprimir los valores observados
hl$observed 

                                       
cutyhat                                 y0 y1
  [2.22044604925e-16,3.19208348324e-12] 10  0
  (3.19208348324e-12,1.70221147337e-11]  3  0
  (1.70221147337e-11,0.50000000002]      2  0
  (0.50000000002,0.999999999979]         0  2
  (0.999999999979,0.999999999991]        0  3
  (0.999999999991,0.999999999998]        0  2
  (0.999999999998,1]                     0  8

Basados en la prueba de bondad de ajuste de Hosmer & Lemeshow, podemos decir que el modelo de Regresión Logística ajustado es correcto puesto que el valor \(p\) de la prueba es \(>0.05\), sin embargo debemos tener en cuenta el Warning que la prueba genera, posiblemente por el tamaño pequeño de la muestra.

Si se evalua la probabilidad de los datos observados en el rango entre 2.22e-16 y 0.50000000002 se observa que existen 15 personas que tienen la enfermedad EA (\(\text{y}_0\)) y si se evalua el rango entre 0.0000000002 y 1 se observa que existen 15 pacientes que no tienen la enfermedad EA (\(\text{y}_1\)).

Conclusión de la prueba: el modelo ajusta bien.

• Cálculo del pseudo \(R^2\)

Para estimar el \(R^2_\text{pseudo}\) en R utilizamos la función pR2() del paquete pscl:

## verificamos si el paquete pscl está disponible
if(!require(pscl)) install.packages('pscl')
require(pscl)

## pseudo R^2 para Regresión Logística
pR2(modelp6)['McFadden'] # Aplico la función y de la función extraigo el cuadrado de McFadden.

fitting null model for pseudo-r2

McFadden 
       1 

Según McFadden, un coeficiente de determinación \(R^2\) entre 0.2 y 0.4 sugiere un excelente ajuste del modelo, mientras que un valor inferior a 0.2 indica que el modelo tiene limitaciones para predecir., ver aquí en stackexchange. Con base a lo anterior, se puede afirmar que el modelo ajusta bien (\(R^2 = 1\)), lo que sugiere un buen ajuste, dado que está fuera del rango deseado de 0.2 a 0.4, se podría interpretar que el modelo tiene margen para mejorar su ajuste.

Conclusión de la prueba: el modelo ajusta bien, pero se puede mejorar.

P7

Utilizando el criterio \(S_e \approx S_p\), se determinó que el cutoff óptimo como \(\theta_0 = 0.3\) con base en el modelo modelo de RL ajustastado en P6. Está usted de acuerdo? Concluya.

Inicialmente cargamos algunas funciones útiles para realizar los análisis con autoria del Phd, Jorge I. Vélez:

## funciones para calcular las medidas de desempeño para diferentes puntos de corte
source('https://www.dropbox.com/s/xclvdugfbrf5ryn/logistic-functions.R?dl=1')

This is a collection of functions created by 
Professor Jorge Vélez, PhD <https://jorgeivanvelez.netlify.app/> 
for the fitting and validation of Binary Classification models 
 
In case of problems, please an email to 
<jvelezv@uninorte.edu.co> with enough details 
 
Last modification:  April 23, 2024 
 

Para construir la matriz de confusión, varias medidas de desempeño para 500 cutoffs posibles y graficar los resultados, procedemos de la siguiente manera:

##  definimos 500 posibles valores para el cutoff
cutoff <- seq(.1, .9, length = 500) 

##  cálculos
real <- d$dx
res <- data.frame(cutoff = cutoff, bycutoff2(modelp6, real = real, cutoff))

##  gráfico
with(res, matplot(cutoff, cbind(sensitivity, specificity, class_rate), 
                  type = 'l', las = 1, ylim = c(0, 1), lty = 1, ylab = 'Value (%)'))
legend('bottomright', c('Se', 'Sp', 'Acc'), lty = 1, 
       col = 1:3, bty = 'n', ncol = 3)

Todos los valores se encuentra en 1, lo que indica que el punto de corte será siempre le mismo. De igual forma se determinarán los cutoffs:

• Usando \(S_e \approx S_p\)

res[which.min(abs(res$sensitivity - res$specificity)), ] 

  cutoff  a b c  d sensitivity specificity ppv npv fdr fpr class_rate lift
1    0.1 15 0 0 15           1           1   1   1   0   0          1    2
  delta
1     0

• Maximizando la tasa de clasificación correcta:

## maximizando Acc 

res[which.max(res$class_rate), ] 

  cutoff  a b c  d sensitivity specificity ppv npv fdr fpr class_rate lift
1    0.1 15 0 0 15           1           1   1   1   0   0          1    2
  delta
1     0

• Maximizando el lift:

# maximizando lift 
res[which.max(res$lift), ] 

  cutoff  a b c  d sensitivity specificity ppv npv fdr fpr class_rate lift
1    0.1 15 0 0 15           1           1   1   1   0   0          1    2
  delta
1     0

• Usemos \(\theta = 0.3\)

La matriz de confusión puede obtenerse haciendo

## matriz de confusión para theta_0 = 0.3
y_pred <- 1*(thetahat >= 0.3)  
y_real <- d$dx 
confusion <- table(modelo = y_pred, real = y_real)[2:1, 2:1]
confusion

      real
modelo  1  0
     1 15  0
     0  0 15

Finalmente las medidas de desempeño, incluyendo el AUC, pueden calcularse como

## medidas de desempeño  
measureslog(confusion)

sensitivity specificity         ppv         npv         fdr         fpr 
          1           1           1           1           0           0 
 class_rate         auc 
          1           1 

Con base al Lift el modelo del P6 es 1 vez mejor que lanzar monedas para determinar quien padece EA o quien no EA. De igual forma todas las medidas de desempeño indican que el modelo es perfecto, es decir que tiene la capacidad de predecir de forma precisa. Se e obtuvo un cutoff de 0.1.

P8

Construya una función que, a partir de los niveles de expresión de lncRNAs en una persona, determine el diagnóstico de EA. Tenga en cuenta en valor de \(\theta_0\) derivado en P7. Si \(\mathbf{x}_0\) representa el vector de dichos niveles de expresión, entonces:

A continuación se define función alzheimer para que a partir de los niveles de expresión de lncRNAs en una persona, determine el diagnóstico, 1 sí la persona padece EA, o 0 sí no padece la enfermedad.

alzheimer <- function(x0, model, cutoff){

  thetahat <- predict(model, newdata = x0, type = 'response')
  y_predict <- ifelse(thetahat >= cutoff, 1, 0)
  diag <- ifelse(y_predict == 0, 'El paciente no padece la Enfermedad de Alzheimer', 
                 'El paciente padece la Enfermedad de Alzheimer')

  return(diag)
}

# Seleccionar la fila 1 del dataframe d
individuo_1 <- d[1, ]

# Crear el dataframe x0 con la fila 1 
x0 <- data.frame(x4634 = individuo_1$'x4634',
                 x14508 = individuo_1$'x14508',
                 x6948 = individuo_1$'x6948',
                 x23214 = individuo_1$'x23214',
                 x28007 = individuo_1$'x28007',
                 x11576 = individuo_1$'x11576',
                 x2114 = individuo_1$'x21114',
                 x27199 = individuo_1$'x27199',
                 x26435 = individuo_1$'x26435',
                 x2237 = individuo_1$'x2237')

predict_x0 <- alzheimer(x0, modelp6, 0.1)
predict_x0

                                                 1 
"El paciente no padece la Enfermedad de Alzheimer" 

Luego de ejecutar la función Alzheimer, bajo las condiciones de \(\text{x}_0\) usando el modelo del P6 se obtiene como resultado que: El paciente no padece la Enfermedad de Alzheimer.

P9

Repita P2 pero use un modelo de Regresión Lineal Simple(RLS) en lugar de un modelo de RL. Compare los resultados y concluya. Con base en los resultados obtenidos, cree usted que tiene sentido (o hace alguna diferencia) usar RLS en lugar de RL?. Justifique.

A continuación se define función my_magical_function_RLS para aplicarla sobre lncRNA \(j\) , estimar el modelo de RLS y extraer el coeficiente estimado \(\hat{\beta}_j\), el estadístico \(t_j\) y el valor \(p_j\). Lo anterior, se puede visualizar en la tabla a continuación:

my_magical_function_RLS <- function(j){
  model <- summary(lm(dx ~ d[, 2+j], data=d))
  model$coefficients[2,][c(1, 3, 4)]
}

coeff_model_RLS <- sapply(1:(ncol(d) - 2), my_magical_function_RLS)

coeff_model_RLS_t <- t(coeff_model_RLS)
coeff_model_RLS_df <- as.data.frame(coeff_model_RLS_t)
coeff_model_RLS_df$ID<- seq(1, 29810)
names(coeff_model_RLS_df)<- c('Coeficiente','valor z','P_valor','ID')

kbl(coeff_model_RLS_df, 
    caption = "", 
    align = "c", 
    position = "top") %>%
    kable_styling(bootstrap_options = c("striped", "hover"))%>% 
    scroll_box(width = "920px", height = "300px")

Coeficiente	valor z	P_valor	ID
-0.0721663	-3.528503e-01	0.7268432	1
-0.1736234	-5.069467e-01	0.6161616	2
0.4375241	3.723634e+00	0.0008771	3
0.0342991	2.119346e-01	0.8336932	4
0.1415228	8.271466e-01	0.4151479	5
0.5815715	2.645271e+00	0.0132342	6
-0.1647325	-1.519918e+00	0.1397444	7
0.0454857	9.460070e-02	0.9253057	8
0.0143907	8.204730e-02	0.9351928	9
-0.0782690	-4.918417e-01	0.6266652	10
-0.1171518	-9.081998e-01	0.3715236	11
-0.1973622	-1.320035e+00	0.1975138	12
-0.0574629	-5.477732e-01	0.5881882	13
-0.0914435	-6.936828e-01	0.4935962	14
0.1792804	8.913463e-01	0.3803393	15
-0.0723195	-3.470097e-01	0.7311787	16
-0.1718743	-7.240377e-01	0.4750491	17
-0.0758493	-6.419213e-01	0.5261452	18
-0.1427583	-1.200046e+00	0.2401772	19
-0.4333373	-2.303411e+00	0.0288934	20
0.5959094	1.754507e+00	0.0902831	21
0.0138287	8.203440e-02	0.9352030	22
0.0292594	1.950477e-01	0.8467640	23
0.3274496	1.084327e+00	0.2874687	24
-0.0468946	-4.410260e-01	0.6625809	25
0.0644562	2.916136e-01	0.7727317	26
-0.1252844	-9.080643e-01	0.3715940	27
0.0888440	5.146874e-01	0.6108107	28
-0.2735593	-1.201225e+00	0.2397267	29
-0.0657919	-4.164999e-01	0.6802198	30
-0.0268529	-2.154425e-01	0.8309840	31
-0.0035714	-4.332410e-02	0.9657507	32
0.0694460	5.274780e-01	0.6020168	33
0.0723759	6.207781e-01	0.5397664	34
0.1812610	1.149287e+00	0.2601605	35
0.1076068	5.692839e-01	0.5737024	36
-0.3874304	-1.319534e+00	0.1976790	37
-0.0873317	-5.631906e-01	0.5777877	38
-0.0405473	-3.443463e-01	0.7331587	39
-0.0908012	-4.973338e-01	0.6228368	40
-0.0867008	-2.907166e-01	0.7734106	41
0.4923338	2.333915e+00	0.0270035	42
-0.0038150	-3.988900e-02	0.9684647	43
0.2353259	1.282121e+00	0.2103144	44
0.1265219	5.048991e-01	0.6175807	45
0.0167822	5.691090e-02	0.9550204	46
-0.2074353	-1.566555e+00	0.1284520	47
0.0908166	6.541362e-01	0.5183611	48
-0.0602501	-5.393157e-01	0.5939321	49
0.0194035	1.181697e-01	0.9067768	50
-0.1342543	-7.137794e-01	0.4812715	51
0.0679570	4.598981e-01	0.6491406	52
-0.0187053	-1.737215e-01	0.8633342	53
0.3186373	2.435363e+00	0.0214997	54
0.2070645	1.513880e+00	0.1412638	55
-0.1407378	-1.977557e+00	0.0578980	56
-0.2269302	-1.253573e+00	0.2203635	57
0.0678638	8.603882e-01	0.3968826	58
-0.1781949	-8.514453e-01	0.4017456	59
-0.8172090	-1.481793e+00	0.1495630	60
-0.0592167	-3.446007e-01	0.7329696	61
-0.0134709	-9.664830e-02	0.9236942	62
-0.2402823	-1.182821e+00	0.2468266	63
0.1628657	6.728688e-01	0.5065466	64
0.0837526	4.885978e-01	0.6289315	65
0.0351607	1.230680e-01	0.9029324	66
-0.0484422	-2.319356e-01	0.8182748	67
0.2241470	1.171126e+00	0.2514185	68
-0.1241782	-8.736626e-01	0.3897337	69
0.0085645	4.680730e-02	0.9629991	70
0.0327339	1.764304e-01	0.8612258	71
-0.0087083	-7.366670e-02	0.9417993	72
-0.0566468	-6.615687e-01	0.5136556	73
-0.3126006	-1.191607e+00	0.2434179	74
0.0559797	2.637063e-01	0.7939373	75
-0.1481929	-1.936465e+00	0.0629630	76
-0.2836048	-1.117413e+00	0.2733145	77
0.0611053	1.763631e-01	0.8612781	78
0.1646362	1.017654e+00	0.3175555	79
-0.2267546	-9.041520e-01	0.3736287	80
0.0884490	9.900335e-01	0.3306366	81
-0.4122580	-1.390887e+00	0.1752152	82
-0.0413109	-1.017657e-01	0.9196679	83
0.1889951	1.064436e+00	0.2962250	84
0.2320910	1.122177e+00	0.2713186	85
0.0497800	4.313990e-01	0.6694817	86
0.3710306	2.648824e+00	0.0131240	87
0.1702144	1.078586e+00	0.2899768	88
0.1691284	2.443090e+00	0.0211258	89
-0.1677643	-6.414024e-01	0.5264773	90
-0.1438308	-1.145747e+00	0.2615982	91
0.0757284	4.786109e-01	0.6359318	92
-0.0408345	-2.607929e-01	0.7961608	93
-0.1229715	-1.068402e+00	0.2944644	94
0.2745646	1.330660e+00	0.1940368	95
-0.1036104	-5.920665e-01	0.5585571	96
-0.0031443	-2.207420e-02	0.9825453	97
-0.0861072	-1.261897e+00	0.2173968	98
-0.0074911	-5.270650e-02	0.9583400	99
-0.0667862	-4.778388e-01	0.6364743	100
0.0440720	2.428789e-01	0.8098696	101
-0.1158655	-1.042432e+00	0.3061288	102
0.1096029	7.232389e-01	0.4755319	103
0.2228286	2.370593e+00	0.0248804	104
-0.0625168	-3.372951e-01	0.7384099	105
-0.0710886	-7.007612e-01	0.4892350	106
0.0397579	2.854982e-01	0.7773639	107
-0.0428563	-5.892468e-01	0.5604204	108
-0.2759476	-1.254970e+00	0.2198636	109
-0.4175143	-1.785698e+00	0.0849845	110
-0.0147187	-1.486269e-01	0.8829133	111
-0.0781098	-5.122288e-01	0.6125079	112
-0.1111455	-9.254380e-01	0.3626461	113
0.0697001	6.535286e-01	0.5187468	114
0.1026614	1.112091e+00	0.2755566	115
0.0968953	3.905347e-01	0.6990967	116
-0.1630394	-7.032710e-01	0.4876939	117
0.1619974	8.645268e-01	0.3946449	118
0.0543828	3.179530e-01	0.7528799	119
0.1888405	1.071792e+00	0.2929649	120
0.0932469	8.080241e-01	0.4258887	121
-0.1246403	-1.618554e+00	0.1167542	122
0.1908463	1.287524e+00	0.2084525	123
-0.1150655	-4.304947e-01	0.6701315	124
-0.0043199	-3.364530e-02	0.9733987	125
-0.1564133	-7.148624e-01	0.4806124	126
-0.2470274	-1.531319e+00	0.1369119	127
-0.0471926	-2.875549e-01	0.7758051	128
-0.0464357	-2.440111e-01	0.8090014	129
-0.1728907	-1.149809e+00	0.2599490	130
-0.1505420	-5.941763e-01	0.5571649	131
-0.3062994	-2.603361e+00	0.0146009	132
0.0994779	5.519347e-01	0.5853719	133
-0.1978913	-6.334194e-01	0.5316001	134
-0.0880623	-2.339961e-01	0.8166905	135
0.0600644	2.757688e-01	0.7847507	136
0.0030884	2.819350e-02	0.9777078	137
0.1882770	2.462500e+00	0.0202128	138
-0.1579984	-4.695371e-01	0.6423218	139
-0.0841679	-9.787623e-01	0.3360789	140
-0.0736846	-7.366810e-01	0.4674444	141
0.0500721	2.999723e-01	0.7664140	142
0.0356441	2.582836e-01	0.7980771	143
-0.2052200	-1.265611e+00	0.2160828	144
-0.1313807	-1.281260e+00	0.2106125	145
0.0322815	1.437006e-01	0.8867662	146
-0.0386705	-1.991584e-01	0.8435780	147
0.1305694	9.827271e-01	0.3341577	148
0.0014226	7.357100e-03	0.9941821	149
-0.1381506	-8.482058e-01	0.4035164	150
0.1341333	9.177370e-01	0.3665946	151
-0.1766162	-1.331429e+00	0.1937868	152
-0.3054398	-1.128232e+00	0.2687966	153
-0.0040373	-3.767720e-02	0.9702124	154
-0.1864742	-1.620237e+00	0.1163910	155
-0.2429848	-1.139750e+00	0.2640472	156
0.0327650	1.793939e-01	0.8589203	157
-0.2969398	-1.228981e+00	0.2293080	158
0.1327492	5.637483e-01	0.5774131	159
-0.1402155	-4.982742e-01	0.6221823	160
0.0495114	6.142927e-01	0.5439816	161
-0.0883940	-7.162557e-01	0.4797652	162
-0.1561729	-8.809140e-01	0.3858636	163
-0.1734784	-1.236311e+00	0.2266138	164
-0.1480594	-1.342058e+00	0.1903594	165
0.0885905	1.003477e+00	0.3242248	166
-0.0014241	-1.038570e-02	0.9917872	167
-0.2904576	-1.053840e+00	0.3009658	168
0.2156059	9.268677e-01	0.3619161	169
0.0769949	6.339126e-01	0.5312828	170
-0.3180988	-6.576035e-01	0.5161630	171
0.1935302	9.497909e-01	0.3503452	172
0.1238991	6.657904e-01	0.5109933	173
-0.0132879	-9.508250e-02	0.9249265	174
0.0810866	5.903737e-01	0.5596754	175
0.1141021	6.500763e-01	0.5209414	176
-0.0533835	-3.478519e-01	0.7305529	177
0.1897835	9.752901e-01	0.3377677	178
-0.2480621	-6.902200e-01	0.4957377	179
0.1961377	1.023471e+00	0.3148468	180
0.0754064	4.570430e-01	0.6511664	181
0.0204458	1.475766e-01	0.8837345	182
-0.1178974	-1.241465e+00	0.2247340	183
-0.0651861	-3.720651e-01	0.7126455	184
-0.0445711	-2.618223e-01	0.7953750	185
0.0236594	2.270764e-01	0.8220141	186
-0.1783398	-1.244088e+00	0.2237817	187
-0.4740527	-2.085729e+00	0.0462351	188
0.3027784	1.755713e+00	0.0900731	189
-0.1155907	-8.175419e-01	0.4205214	190
-0.0946024	-6.937375e-01	0.4935624	191
0.0679675	4.724987e-01	0.6402330	192
0.2087026	1.481594e+00	0.1496155	193
-0.1254699	-7.250891e-01	0.4744139	194
-0.0318387	-1.734783e-01	0.8635236	195
0.4169763	2.296651e+00	0.0293282	196
0.1318305	5.851698e-01	0.5631202	197
-0.1369359	-7.044609e-01	0.4869642	198
-0.0745819	-4.569780e-01	0.6512125	199
0.0365462	2.421136e-01	0.8104567	200
-0.1742911	-8.681680e-01	0.3926828	201
0.2770506	1.569249e+00	0.1278231	202
0.2869239	1.238266e+00	0.2258993	203
0.0794836	5.262310e-01	0.6028715	204
-0.2824515	-1.860172e+00	0.0733958	205
-0.1560195	-8.358875e-01	0.4102949	206
-0.0882782	-1.470263e+00	0.1526393	207
-0.2398417	-1.668290e+00	0.1064048	208
0.1993958	1.707863e+00	0.0987279	209
0.1941108	1.795491e+00	0.0833766	210
-0.1903063	-1.908433e+00	0.0666360	211
-0.1222890	-8.179443e-01	0.4202954	212
0.0254680	1.651908e-01	0.8699807	213
-0.0075889	-6.124220e-02	0.9516014	214
0.5758536	3.641462e+00	0.0010891	215
0.0326030	2.545629e-01	0.8009211	216
-0.0070723	-4.437760e-02	0.9649184	217
0.0108547	1.855619e-01	0.8541260	218
0.3594414	9.521405e-01	0.3491733	219
-0.0983375	-6.233136e-01	0.5381232	220
0.0502356	1.605758e-01	0.8735804	221
0.2318997	9.626578e-01	0.3439599	222
0.0712562	4.696516e-01	0.6422410	223
-0.1454190	-9.205918e-01	0.3651276	224
-0.0955551	-5.402022e-01	0.5933288	225
0.0218855	9.508940e-02	0.9249211	226
0.0771522	3.488648e-01	0.7298007	227
-0.5440927	-2.320116e+00	0.0278441	228
0.0356422	2.084188e-01	0.8364107	229
-0.0386476	-2.637795e-01	0.7938815	230
-0.0181657	-2.185593e-01	0.8285786	231
0.0651323	5.573669e-01	0.5817056	232
0.0766461	2.642206e-01	0.7935451	233
-0.0102443	-7.229960e-02	0.9428775	234
0.1784724	9.889011e-01	0.3311807	235
-0.0040310	-1.694500e-02	0.9866007	236
-0.0311629	-2.365420e-01	0.8147340	237
-0.0592832	-5.691064e-01	0.5738212	238
0.0498334	2.718438e-01	0.7877366	239
0.4297272	1.788721e+00	0.0844853	240
-0.0363277	-4.046752e-01	0.6887912	241
-0.2525537	-9.238334e-01	0.3634665	242
-0.0253947	-3.292375e-01	0.7444263	243
-0.0536439	-9.816243e-01	0.3346913	244
0.1809314	2.257802e+00	0.0319434	245
0.1299855	1.514290e+00	0.1411601	246
0.3158642	1.603081e+00	0.1201394	247
0.0142248	1.190567e-01	0.9060805	248
0.1670490	2.574464e+00	0.0156184	249
-0.0642148	-5.268039e-01	0.6024788	250
0.1312221	6.445989e-01	0.5244335	251
-0.0530833	-3.446233e-01	0.7329528	252
0.1872197	1.160673e+00	0.2555754	253
0.0191413	1.602998e-01	0.8737958	254
-0.1437472	-1.189150e+00	0.2443677	255
-0.0726641	-5.537046e-01	0.5841761	256
0.0274674	2.385849e-01	0.8131651	257
0.1951344	1.041582e+00	0.3065161	258
-0.0162204	-1.245468e-01	0.9017722	259
-0.3829474	-1.485915e+00	0.1484755	260
-0.0816906	-5.355182e-01	0.5965199	261
-0.1534851	-7.636310e-01	0.4514741	262
0.0936201	5.138721e-01	0.6113732	263
0.0515386	5.626295e-01	0.5781646	264
0.5249186	3.285397e+00	0.0027408	265
-0.2383442	-1.256769e+00	0.2192210	266
0.1837097	1.522025e+00	0.1392175	267
0.0635434	5.993285e-01	0.5537729	268
0.0771040	1.303708e+00	0.2029506	269
-0.1389648	-8.241102e-01	0.4168420	270
-0.2139444	-1.350507e+00	0.1876685	271
0.0941577	5.629518e-01	0.5779481	272
0.0346953	2.050583e-01	0.8390100	273
-0.3219719	-8.836106e-01	0.3844307	274
0.3926115	1.606980e+00	0.1192788	275
0.3274986	1.440824e+00	0.1607245	276
0.3788710	2.731131e+00	0.0107984	277
-0.0100576	-7.642910e-02	0.9396212	278
0.0630194	1.011131e+00	0.3206123	279
-0.1001584	-4.613910e-01	0.6480825	280
0.0513185	3.850571e-01	0.7031047	281
-0.1215175	-4.675245e-01	0.6437430	282
-0.0714180	-3.844602e-01	0.7035419	283
-0.0856140	-7.885679e-01	0.4369907	284
0.0030839	1.710830e-02	0.9864715	285
-0.3081631	-1.159674e+00	0.2559754	286
0.0677951	3.488189e-01	0.7298347	287
0.0087680	5.903600e-02	0.9533428	288
0.0158599	1.375932e-01	0.8915466	289
-0.1496305	-1.809533e+00	0.0811162	290
-0.0144675	-8.946260e-02	0.9293512	291
0.0734994	1.087000e+00	0.2863060	292
-0.3554349	-1.317745e+00	0.1982697	293
0.1376805	1.246873e+00	0.2227739	294
0.1391720	1.182755e+00	0.2468526	295
0.0261179	1.439336e-01	0.8865839	296
0.0693113	4.054344e-01	0.6882396	297
0.0434773	2.029895e-01	0.8406112	298
-0.1851704	-1.468311e+00	0.1531650	299
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0.0427420	3.527039e-01	0.7269518	1907
0.3805018	1.112536e+00	0.2753687	1908
-0.0078993	-3.752060e-02	0.9703361	1909
-0.2301482	-9.746062e-01	0.3381010	1910
-0.0871111	-4.765891e-01	0.6373531	1911
0.2251790	8.428669e-01	0.4064456	1912
-0.4884948	-2.472289e+00	0.0197663	1913
0.1812708	8.442081e-01	0.4057085	1914
0.0417811	1.747682e-01	0.8625194	1915
0.0367335	1.320444e-01	0.8958934	1916
0.0998105	1.007773e+00	0.3221937	1917
-0.3125598	-1.554191e+00	0.1313703	1918
-0.1446313	-9.875501e-01	0.3318306	1919
0.0089924	5.139170e-02	0.9593783	1920
0.3524425	1.151235e+00	0.2593717	1921
0.0271135	1.480975e-01	0.8833272	1922
0.5153767	4.329710e+00	0.0001726	1923
0.1903996	6.542153e-01	0.5183110	1924
0.1147136	2.211407e+00	0.0353416	1925
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0.0014666	1.040000e-02	0.9917759	1930
-0.3686957	-1.369378e+00	0.1817654	1931
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-0.0774169	-5.928703e-01	0.5580265	1933
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0.0320456	2.812638e-01	0.7805762	1936
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0.2897163	1.939925e+00	0.0625220	1939
0.0781916	1.897990e-01	0.8508359	1940
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0.0972544	3.620084e-01	0.7200636	1942
0.0769539	3.750054e-01	0.7104821	1943
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0.0081096	6.004700e-02	0.9525447	1946
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0.9169778	1.660283e+00	0.1080173	1951
0.3419876	1.731365e+00	0.0943933	1952
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0.2893740	1.170364e+00	0.2517198	1956
-0.0060850	-3.334150e-02	0.9736388	1957
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0.0961418	6.110912e-01	0.5460687	1962
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0.0730948	5.714004e-01	0.5722868	1965
0.1110909	9.135529e-01	0.3687517	1966
0.2414078	1.354320e+00	0.1864636	1967
-0.0535614	-5.652989e-01	0.5763725	1968
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0.0010652	9.159300e-03	0.9927570	1970
0.1098441	7.497271e-01	0.4596725	1971
0.0507736	7.182153e-01	0.4785751	1972
0.0717913	9.256266e-01	0.3625497	1973
0.4370069	1.440735e+00	0.1607495	1974
-0.4078696	-1.677590e+00	0.1045571	1975
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0.0420754	2.515346e-01	0.8032378	1977
0.1203457	6.704745e-01	0.5080483	1978
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0.2819879	1.860351e+00	0.0733696	1980
-0.2442059	-1.273439e+00	0.2133332	1981
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-0.1353428	-8.528860e-01	0.4009596	1983
-0.1338817	-1.014839e+00	0.3188720	1984
-0.0263658	-1.497394e-01	0.8820436	1985
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0.0775496	5.973122e-01	0.5550991	1987
-0.1165272	-1.379422e+00	0.1786831	1988
-0.3809930	-1.026297e+00	0.3135368	1989
0.0986550	7.421686e-01	0.4641660	1990
-0.0392504	-4.489574e-01	0.6569181	1991
0.0588714	5.075391e-01	0.6157513	1992
-0.3279772	-2.274897e+00	0.0307677	1993
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0.0009432	8.058300e-03	0.9936276	1996
-0.1813782	-1.473334e+00	0.1518149	1997
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0.0580325	4.033897e-01	0.6897256	1999
-0.0794838	-4.923605e-01	0.6263031	2000
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0.0142772	1.066711e-01	0.9158105	2002
-0.0716747	-4.463659e-01	0.6587660	2003
-0.4702245	-1.456645e+00	0.1563379	2004
0.1601117	6.849243e-01	0.4990228	2005
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0.0126923	1.539451e-01	0.8787572	2009
-0.0710194	-7.893810e-01	0.4365232	2010
-0.0255972	-1.543466e-01	0.8784437	2011
0.1396182	9.831435e-01	0.3339563	2012
-0.0072080	-1.247186e-01	0.9016374	2013
-0.0973379	-7.508591e-01	0.4590018	2014
-0.0000099	-6.180000e-05	0.9999511	2015
0.3614694	2.120301e+00	0.0429741	2016
0.3148487	2.039141e+00	0.0509757	2017
-0.0235057	-1.876520e-01	0.8525027	2018
0.0191155	1.995662e-01	0.8432621	2019
-0.0180106	-9.228270e-02	0.9271305	2020
-0.3840842	-1.088971e+00	0.2854508	2021
0.1750928	1.109675e+00	0.2765790	2022
0.2177404	1.437227e+00	0.1617353	2023
0.2148291	6.700633e-01	0.5083064	2024
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0.0064201	4.712790e-02	0.9627458	2026
0.1156886	3.850918e-01	0.7030792	2027
0.0968465	5.651713e-01	0.5764581	2028
0.0560393	8.529221e-01	0.4009400	2029
0.0947983	4.722142e-01	0.6404336	2030
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0.1570422	8.731794e-01	0.3899925	2033
0.2605877	1.812619e+00	0.0806267	2034
0.3271717	1.142668e+00	0.2628533	2035
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0.0010154	4.447800e-03	0.9964827	2037
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0.3993712	1.525842e+00	0.1382669	2041
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0.2665150	1.756423e+00	0.0899497	2045
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0.1433319	5.780031e-01	0.5678819	2047
0.2509805	1.803317e+00	0.0821103	2048
0.4090605	2.366815e+00	0.0250918	2049
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0.0523135	4.502666e-01	0.6559853	2052
0.0851985	4.226247e-01	0.6757971	2053
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0.0261831	1.436619e-01	0.8867965	2056
0.0521794	3.368887e-01	0.7387130	2057
0.2351212	1.229592e+00	0.2290823	2058
0.1576000	8.504022e-01	0.4023153	2059
0.1165083	1.299434e+00	0.2043927	2060
0.0115643	1.049485e-01	0.9171649	2061
0.0699251	7.956967e-01	0.4329026	2062
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0.3775900	2.221442e+00	0.0345802	2064
0.0290208	2.935833e-01	0.7712415	2065
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0.0259874	1.187847e-01	0.9062940	2067
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0.1362475	1.836967e+00	0.0768514	2069
0.0330461	5.101045e-01	0.6139761	2070
0.0211143	1.369429e-01	0.8920559	2071
0.0697635	2.577943e-01	0.7984510	2072
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0.0026518	2.725060e-02	0.9784531	2075
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0.1457935	9.480896e-01	0.3511954	2078
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0.0440194	5.556487e-01	0.5828640	2081
0.1690188	8.385537e-01	0.4088217	2082
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0.2336000	7.618157e-01	0.4525395	2084
0.3308867	9.698404e-01	0.3404297	2085
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0.0394993	1.356368e-01	0.8930789	2090
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0.0017648	1.799210e-02	0.9857727	2094
0.1611246	9.044108e-01	0.3734938	2095
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0.1315606	9.304286e-01	0.3601023	2097
0.0921955	4.087901e-01	0.6858036	2098
0.2208935	1.226210e+00	0.2303324	2099
0.0797419	3.477720e-01	0.7306123	2100
0.1571335	5.481180e-01	0.5879546	2101
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0.2859324	1.621031e+00	0.1162197	2103
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0.2898176	2.182599e+00	0.0376120	2105
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0.3051596	2.539646e+00	0.0169316	2110
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0.0252930	4.021947e-01	0.6905946	2112
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0.6877228	4.666934e+00	0.0000689	2114
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0.0743186	3.364937e-01	0.7390076	2160
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0.2373470	1.723249e+00	0.0958716	2165
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0.3994878	1.578365e+00	0.1257140	2179
0.0705537	1.245717e+00	0.2231919	2180
0.0041153	3.685210e-02	0.9708644	2181
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0.0653396	2.787540e-01	0.7824820	2183
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0.1758570	9.753909e-01	0.3377186	2186
0.0871507	5.454059e-01	0.5897932	2187
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0.1827923	1.539472e+00	0.1349152	2193
0.0802942	4.307253e-01	0.6699658	2194
0.1144447	5.500303e-01	0.5866599	2195
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0.0531484	4.234101e-01	0.6752309	2197
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0.5226949	3.353769e+00	0.0023006	2200
0.0931991	1.546822e+00	0.1331354	2201
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0.0466459	4.880139e-01	0.6293398	2203
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0.4677919	2.213913e+00	0.0351501	2205
0.1401936	1.225102e+00	0.2307433	2206
0.1560023	5.783500e-01	0.5676510	2207
0.0651274	4.101488e-01	0.6848182	2208
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0.3501566	1.755332e+00	0.0901393	2210
0.0018275	1.141470e-02	0.9909735	2211
0.1738856	7.792177e-01	0.4423880	2212
0.0721696	5.317648e-01	0.5990831	2213
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0.2267017	1.153994e+00	0.2582579	2217
0.0106678	8.381240e-02	0.9338020	2218
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0.1017529	8.485690e-01	0.4033176	2220
0.0361634	2.859915e-01	0.7769899	2221
0.3886896	1.087022e+00	0.2862966	2222
0.0331114	2.734855e-01	0.7864873	2223
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0.2668515	9.504204e-01	0.3500309	2225
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0.0113961	9.911720e-02	0.9217515	2227
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0.0373052	2.139658e-01	0.8321242	10146
0.0355730	2.385941e-01	0.8131580	10147
0.1308284	1.125270e+00	0.2700281	10148
0.0689048	1.252596e+00	0.2207140	10149
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0.0728135	2.314904e-01	0.8186173	10151
0.1135819	6.807262e-01	0.5016357	10152
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0.1974130	1.076321e+00	0.2909707	10161
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0.0777358	1.084365e+00	0.2874522	10164
0.0541011	4.653548e-01	0.6452766	10165
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0.0201816	1.193500e-01	0.9058503	10170
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0.1173445	6.767567e-01	0.5041133	10172
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0.5042578	1.556717e+00	0.1307698	10178
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0.0031818	1.390570e-02	0.9890038	10181
0.0684397	7.734309e-01	0.4457484	10182
0.0600320	3.717428e-01	0.7128828	10183
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0.0449725	3.741309e-01	0.7111252	10185
0.1268587	1.847698e+00	0.0752363	10186
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0.2427035	1.625030e+00	0.1153611	10189
0.0437281	3.413631e-01	0.7353788	10190
0.5198588	2.419489e+00	0.0222869	10191
0.0963041	1.053076e+00	0.3013097	10192
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0.3386691	1.264559e+00	0.2164543	10195
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0.0553154	3.708274e-01	0.7135570	10199
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0.2580320	2.224484e+00	0.0343523	10201
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0.2675890	1.071606e+00	0.2930471	10203
0.2815857	1.031835e+00	0.3109802	10204
0.3706802	2.434996e+00	0.0215176	10205
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0.1569003	1.608321e+00	0.1189840	10209
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0.1189788	1.048082e+00	0.3035640	10211
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0.2545845	1.583704e+00	0.1244921	10214
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0.0484630	5.556160e-01	0.5828861	10216
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0.2119276	1.084222e+00	0.2875144	10218
0.0544951	1.793876e-01	0.8589252	10219
0.3735383	1.471848e+00	0.1522135	10220
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0.0682138	5.831215e-01	0.5644790	10222
0.3139895	2.065041e+00	0.0482897	10223
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0.5093422	3.702066e+00	0.0009285	10225
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0.1601859	1.445630e+00	0.1593816	10227
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0.5096711	2.664708e+00	0.0126416	10231
0.3242731	1.508889e+00	0.1425296	10232
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0.1777599	6.755980e-01	0.5048378	10238
0.0804198	4.414098e-01	0.6623064	10239
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0.0141656	1.143894e-01	0.9097455	10244
0.1870447	8.927943e-01	0.3795766	10245
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0.1421634	1.163643e+00	0.2543893	10248
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0.1542859	1.044083e+00	0.3053779	10251
0.1603883	2.188279e+00	0.0371543	10252
0.0162627	1.479266e-01	0.8834609	10253
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0.1428722	9.727131e-01	0.3390247	10255
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0.0651747	1.055867e+00	0.3000545	10259
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0.0430607	4.000992e-01	0.6921196	10261
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0.1852662	9.790500e-01	0.3359393	10264
0.2191817	1.289050e+00	0.2079289	10265
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0.0859177	1.400699e+00	0.1722896	10268
0.3598084	2.021668e+00	0.0528615	10269
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0.2671401	1.579828e+00	0.1253783	10274
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0.0187251	1.572027e-01	0.8762132	10277
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0.0758804	5.817489e-01	0.5653905	10282
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0.1608236	1.250897e+00	0.2213239	10289
0.0859687	3.012705e-01	0.7654343	10290
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0.0980519	7.284874e-01	0.4723646	10292
0.0107881	1.610805e-01	0.8731866	10293
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0.2209143	1.110401e+00	0.2762714	10297
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0.1610839	2.095394e+00	0.0453020	10299
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0.1523918	1.557813e+00	0.1305098	10302
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0.0390882	3.362651e-01	0.7391781	10309
0.0086193	7.158450e-02	0.9434415	10310
0.0304795	2.857801e-01	0.7771501	10311
0.3749634	3.167746e+00	0.0036942	10312
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0.1445446	7.721890e-01	0.4464716	10316
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0.7917825	2.603456e+00	0.0145976	10319
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0.1491762	1.211794e+00	0.2357186	10321
0.0459495	1.294100e-01	0.8979584	10322
0.2040790	6.958070e-01	0.4922851	10323
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0.0930500	4.453872e-01	0.6594645	10329
0.1837765	1.025343e+00	0.3139785	10330
0.1000803	1.618442e+00	0.1167785	10331
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0.3557404	1.856781e+00	0.0738923	10334
0.0673727	1.094050e+00	0.2832558	10335
0.0732339	8.532190e-01	0.4007781	10336
0.0481692	4.849650e-01	0.6314739	10337
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0.0719269	1.648956e-01	0.8702108	10340
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0.4660680	1.786188e+00	0.0849034	10343
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0.4577061	3.929891e+00	0.0005070	10346
0.0511346	3.152605e-01	0.7549015	10347
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0.3605483	3.010636e+00	0.0054714	10349
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0.2086412	1.973509e+00	0.0583806	10351
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0.1297923	1.005154e+00	0.3234307	10353
0.1504170	9.041252e-01	0.3736426	10354
0.2000315	1.289708e+00	0.2077034	10355
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0.1991790	1.644192e+00	0.1113195	10357
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0.0405366	2.779784e-01	0.7830713	10360
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0.0194004	1.233010e-01	0.9027496	10362
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0.1314691	9.860631e-01	0.3325469	10366
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0.0981920	2.130762e+00	0.0420286	10368
0.3835398	2.757330e+00	0.0101431	10369
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0.1114794	8.512667e-01	0.4018431	10371
0.0472891	3.374081e-01	0.7383257	10372
0.2805059	1.369287e+00	0.1817935	10373
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0.0531381	3.928114e-01	0.6974334	10377
0.0946652	8.092315e-01	0.4252055	10378
0.1008825	8.452199e-01	0.4051529	10379
0.1933087	1.167951e+00	0.2526757	10380
0.1724383	6.392692e-01	0.5278436	10381
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0.1945175	9.548747e-01	0.3478129	10387
0.1454636	5.738231e-01	0.5706686	10388
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0.0946833	1.091721e+00	0.2842611	10390
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0.1259934	6.607315e-01	0.5141844	10393
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0.5402990	2.380637e+00	0.0243261	10395
0.4290450	2.733135e+00	0.0107469	10396
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0.0432748	2.865719e-01	0.7765500	10400
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0.0048870	4.910990e-02	0.9611803	10408
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0.6685828	2.938609e+00	0.0065350	10410
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0.0729958	5.246484e-01	0.6039570	10412
0.0704687	4.228705e-01	0.6756198	10413
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0.2944305	1.818935e+00	0.0796324	10418
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0.0100723	8.661080e-02	0.9315973	10420
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0.2134509	1.161759e+00	0.2551411	10423
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0.0187375	7.384900e-02	0.9416556	10426
0.1439429	6.956189e-01	0.4924011	10427
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0.1590645	6.896051e-01	0.4961185	10429
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0.1487917	1.317208e+00	0.1984471	10432
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0.3166125	1.462893e+00	0.1546322	10434
0.2692542	1.508827e+00	0.1425454	10435
0.2128091	1.798068e+00	0.0829577	10436
0.0101834	8.199270e-02	0.9352358	10437
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0.1008776	7.905818e-01	0.4358334	10439
0.1282465	7.409435e-01	0.4648967	10440
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0.1218867	6.293335e-01	0.5342324	10442
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0.1737644	2.304335e+00	0.0288344	10444
0.0625066	4.493733e-01	0.6566217	10445
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0.0570047	3.636577e-01	0.7188451	10447
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0.1104349	5.123497e-01	0.6124244	10449
0.3011138	1.491694e+00	0.1469611	10450
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0.1454522	7.597670e-01	0.4537437	10452
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0.0627799	3.137527e-01	0.7560345	10454
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0.0596904	2.781023e-01	0.7829772	10456
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0.0432272	4.007547e-01	0.6916424	10462
0.2288934	1.500446e+00	0.1446918	10463
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0.2125866	1.278057e+00	0.2117235	10465
0.0851493	1.012334e+00	0.3200470	10466
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0.0636494	4.266081e-01	0.6729270	10472
0.0326685	1.146035e-01	0.9095773	10473
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0.1382275	1.130402e+00	0.2678973	10477
0.1740786	6.141218e-01	0.5440928	10478
0.1116665	8.556309e-01	0.3994649	10479
0.0769335	1.054629e+00	0.3006107	10480
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0.3629027	1.523016e+00	0.1389701	10482
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0.1635411	8.452091e-01	0.4051589	10516
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0.1890492	1.261262e+00	0.2176218	10636
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0.0430447	2.870205e-01	0.7762100	10686
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0.0676279	6.920116e-01	0.4946291	10689
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0.0091844	6.096350e-02	0.9518214	10694
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0.0608867	3.499640e-01	0.7289846	10705
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0.0607755	6.855457e-01	0.4986367	10707
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0.1390553	9.937946e-01	0.3288341	10710
0.0912234	8.532404e-01	0.4007664	10711
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0.3633351	1.138975e+00	0.2643646	10716
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0.0131021	6.489400e-02	0.9487195	10718
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0.0413084	2.346091e-01	0.8162194	10720
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1.1055268	2.953340e+00	0.0063027	10722
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0.0675639	6.924467e-01	0.4943601	10724
0.0107693	8.953350e-02	0.9292953	10725
0.3053331	1.456500e+00	0.1563778	10726
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0.4419319	2.849362e+00	0.0081253	10729
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0.1589614	2.100419e+00	0.0448235	10734
0.0970959	4.605553e-01	0.6486748	10735
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0.1953269	5.431689e-01	0.5913118	10737
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0.3710054	3.857150e+00	0.0006155	10739
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0.2184232	1.450793e+00	0.1579492	10742
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0.3934763	2.453441e+00	0.0206343	10746
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0.0108505	7.373590e-02	0.9417448	10749
0.3613974	2.804547e+00	0.0090554	10750
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0.1158054	5.053412e-01	0.6172741	10752
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0.1237410	1.476302e+00	0.1510218	10754
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0.1861954	1.409784e+00	0.1696151	10756
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0.2945803	9.912210e-01	0.3300668	10761
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0.0637097	9.179287e-01	0.3664960	10764
0.1110965	8.375323e-01	0.4093857	10765
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0.2514888	9.240604e-01	0.3633503	10767
0.0398073	4.184395e-01	0.6788179	10768
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0.4752327	2.342632e+00	0.0264845	10770
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0.0622195	4.952240e-01	0.6243062	10773
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0.0430343	2.300603e-01	0.8197174	10778
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0.1904212	6.129943e-01	0.5448275	10780
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0.0186100	2.410230e-01	0.8112935	10782
0.2303106	2.724399e+00	0.0109731	10783
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0.0233583	1.712835e-01	0.8652327	10786
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0.0245740	1.800940e-01	0.8583759	10791
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0.0702704	7.159154e-01	0.4799720	10795
0.0141957	4.414550e-02	0.9651018	10796
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0.2705500	1.357189e+00	0.1855614	10799
0.0294466	1.211222e-01	0.9044593	10800
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0.3367954	2.254245e+00	0.0321932	10803
0.3848675	2.952527e+00	0.0063153	10804
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0.1420610	6.555604e-01	0.5174576	10807
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0.1679310	2.697118e+00	0.0117083	10809
0.0902521	6.383302e-01	0.5284456	10810
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0.0863606	8.177367e-01	0.4204120	10812
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0.0021148	1.613410e-02	0.9872418	10814
0.6741510	1.746231e+00	0.0917352	10815
0.1370640	1.084711e+00	0.2873012	10816
0.2064611	6.687423e-01	0.5091363	10817
0.1069648	4.040847e-01	0.6892204	10818
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0.1851386	1.467032e+00	0.1535104	10820
0.0171271	1.319512e-01	0.8959665	10821
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0.0879596	6.348669e-01	0.5306693	10824
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0.3292152	2.532004e+00	0.0172332	10826
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0.1092151	1.505457e+00	0.1434053	10828
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0.0148200	1.343188e-01	0.8941113	10830
0.0190242	2.146074e-01	0.8316288	10831
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0.0471551	5.687380e-01	0.5740678	10833
0.1537969	8.953947e-01	0.3782094	10834
0.0455415	3.205388e-01	0.7509399	10835
0.0618721	4.761879e-01	0.6376353	10836
0.0068842	4.443540e-02	0.9648727	10837
0.0330507	4.284606e-01	0.6715940	10838
0.6431682	2.056334e+00	0.0491784	10839
0.1148609	1.692236e+00	0.1017020	10840
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0.0186969	1.722543e-01	0.8644766	10845
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0.1083065	6.060864e-01	0.5493398	10848
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0.0404099	3.250247e-01	0.7475786	10851
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0.4285722	1.446962e+00	0.1590111	10853
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0.1219284	9.587294e-01	0.3459010	10859
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0.0673671	1.600240e+00	0.1207695	10875
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0.3336196	2.260524e+00	0.0317535	10877
0.3206764	1.914380e+00	0.0658416	10878
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0.1461208	1.753726e+00	0.0904192	10883
0.0088262	8.273440e-02	0.9346514	10884
0.1764413	1.175728e+00	0.2496041	10885
0.2775588	1.623167e+00	0.1157606	10886
0.2911277	1.340503e+00	0.1908578	10887
0.0549621	4.017750e-01	0.6908999	10888
0.0760701	4.397078e-01	0.6635240	10889
0.0262465	1.822635e-01	0.8566892	10890
0.0713220	1.129087e+00	0.2684421	10891
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0.0455607	3.874286e-01	0.7013684	10893
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0.2252309	7.375314e-01	0.4669355	10896
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0.4049650	2.184584e+00	0.0374515	10898
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0.3427774	1.407949e+00	0.1701525	10902
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0.2542001	1.712116e+00	0.0979313	10928
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0.0494294	6.254137e-01	0.5367641	10934
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0.2336619	1.724759e+00	0.0955950	10936
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0.3013600	8.744229e-01	0.3893268	10974
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0.0711208	4.021470e-01	0.6906294	10977
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0.2719146	1.875261e+00	0.0712214	10987
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0.1497695	6.779727e-01	0.5033536	10989
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0.0653008	4.787397e-01	0.6358412	11000
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0.2886315	1.217892e+00	0.2334289	11020
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0.4723596	3.288683e+00	0.0027179	11022
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0.0411819	4.266754e-01	0.6728786	11028
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0.1144442	8.491340e-01	0.4030085	11030
0.0890732	7.617634e-01	0.4525702	11031
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0.0781626	1.441017e+00	0.1606704	11033
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0.2035822	1.223883e+00	0.2311957	11035
0.1477244	7.500593e-01	0.4594756	11036
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0.0730646	3.476558e-01	0.7306986	11039
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0.0646782	3.382045e-01	0.7377319	11041
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0.2528414	1.920436e+00	0.0650410	11044
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0.1426825	7.195840e-01	0.4777449	11047
0.0358825	1.033213e-01	0.9184445	11048
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0.4202746	1.252514e+00	0.2207434	11052
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0.0208282	1.692629e-01	0.8668068	11057
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0.3461265	1.596151e+00	0.1216815	11059
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0.1626768	5.662891e-01	0.5757085	11061
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0.4374536	2.344187e+00	0.0263929	11064
0.2096423	1.125864e+00	0.2697808	11065
0.2700630	1.041654e+00	0.3064830	11066
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0.0683007	3.308668e-01	0.7432085	11068
0.0663949	3.708828e-01	0.7135161	11069
0.2925751	1.525196e+00	0.1384273	11070
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0.1407247	6.166303e-01	0.5424602	11073
0.0277845	2.461351e-01	0.8073731	11074
0.1529584	1.034487e+00	0.3097613	11075
0.5049078	2.048214e+00	0.0500202	11076
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0.4038325	1.297989e+00	0.2048822	11084
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0.0998098	6.527210e-01	0.5192597	11087
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0.1428542	7.914635e-01	0.4353273	11089
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0.0223115	1.189570e-01	0.9061588	11093
0.1138458	8.719078e-01	0.3906740	11094
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0.2167346	1.240420e+00	0.2251140	11096
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0.0183614	1.196679e-01	0.9056008	11098
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0.4381136	2.860023e+00	0.0079177	11100
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0.2193828	1.055773e+00	0.3000967	11109
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0.0835188	7.478894e-01	0.4607626	11122
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0.1841947	1.448788e+00	0.1585044	11320
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0.1031997	7.325922e-01	0.4698959	11322
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0.2842633	1.700900e+00	0.1000439	11324
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0.0325785	1.716823e-01	0.8649221	11326
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0.0862003	6.804478e-01	0.5018092	11329
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0.2250838	1.709480e+00	0.0984244	11333
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0.0368434	2.641916e-01	0.7935671	11340
0.0967118	6.082030e-01	0.5479551	11341
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0.0349011	1.709116e-01	0.8655223	11346
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0.5209851	1.966880e+00	0.0591785	11348
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0.2823978	1.717133e+00	0.0969986	11351
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0.1207042	1.251664e+00	0.2210483	11354
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0.0088966	5.760750e-02	0.9544704	11358
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0.0316044	3.539376e-01	0.7260370	11360
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0.1044296	5.515817e-01	0.5856105	11363
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0.1359551	6.743122e-01	0.5056425	11365
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0.5821240	2.201543e+00	0.0361048	11367
0.3554607	2.060568e+00	0.0487444	11368
0.0670379	5.947720e-01	0.5567722	11369
0.0062538	4.159550e-02	0.9671164	11370
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0.1423878	1.022615e+00	0.3152444	11374
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0.0965582	3.568809e-01	0.7238566	11377
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0.0058167	2.621140e-02	0.9792747	11379
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0.0248661	1.676078e-01	0.8680966	11387
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0.0579732	3.260904e-01	0.7467807	11389
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0.1343893	9.019046e-01	0.3748008	11391
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0.1440680	9.724794e-01	0.3391389	11393
0.3417906	2.120099e+00	0.0429926	11394
0.0759741	8.010438e-01	0.4298516	11395
0.5628670	1.911578e+00	0.0662148	11396
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0.0296147	1.345034e-01	0.8939667	11398
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0.3214216	2.437621e+00	0.0213898	11401
0.0794048	5.805573e-01	0.5661825	11402
0.2164674	1.229806e+00	0.2290035	11403
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0.0369503	1.457982e-01	0.8851253	11405
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0.0435593	4.815080e-01	0.6338975	11408
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0.0394789	5.199174e-01	0.6072077	11660
0.0553384	2.417410e-01	0.8107426	11661
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1.0284917	3.091446e+00	0.0044745	11682
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0.1283287	2.141690e+00	0.0410608	11728
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0.1191141	8.561244e-01	0.3991965	11730
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0.0640200	7.230696e-01	0.4756343	11732
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0.0769888	1.411214e+00	0.1691972	11734
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0.1462009	1.377068e+00	0.1794018	11749
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0.0959159	8.590410e-01	0.3976128	11756
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0.0359869	1.969403e-01	0.8452968	12975
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0.0641344	6.660180e-01	0.5108500	13000
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0.1127053	6.248616e-01	0.5371212	13039
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0.7641158	3.865736e+00	0.0006016	13079
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0.0896728	5.340381e-01	0.5975300	13289
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0.0272374	1.879978e-01	0.8522342	13309
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0.0588943	3.062481e-01	0.7616814	13315
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0.5480533	3.146400e+00	0.0038983	13320
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0.1022461	8.290226e-01	0.4141033	13322
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0.1306319	1.785284e+00	0.0850531	13598
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0.1384489	3.294178e-01	0.7442916	13625
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0.0555837	3.108145e-01	0.7582438	13633
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0.0964059	1.117223e+00	0.2733943	13635
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0.3024022	1.120439e+00	0.2720454	13640
0.3611312	1.842021e+00	0.0760871	13641
0.7617072	2.361134e+00	0.0254128	13642
0.2098634	7.814152e-01	0.4411159	13643
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0.0810529	1.146535e+00	0.2612779	13653
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0.4027752	1.433384e+00	0.1628209	13655
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0.0740773	1.006664e+00	0.3227169	13685
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0.0636574	6.267291e-01	0.5359138	13688
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0.1355311	6.314743e-01	0.5328523	13851
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0.1368760	1.192559e+00	0.2430510	13859
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0.3746828	1.661703e+00	0.1077298	13891
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0.1252071	8.396494e-01	0.4082173	13893
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0.0148101	6.898000e-02	0.9454958	13910
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0.0963453	5.309255e-01	0.5996569	13922
0.3063483	3.120103e+00	0.0041646	13923
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0.2286771	9.231485e-01	0.3638170	13925
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0.0152620	1.231766e-01	0.9028472	13928
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0.0295986	1.563157e-01	0.8769058	13930
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0.2553564	2.678901e+00	0.0122247	13941
0.0330414	3.388866e-01	0.7372236	13942
0.1747165	1.036760e+00	0.3087189	13943
0.2111557	2.157579e+00	0.0396891	13944
0.0661838	5.733415e-01	0.5709900	13945
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0.3337578	1.889248e+00	0.0692556	14576
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0.0286280	4.533378e-01	0.6537994	14580
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0.1441906	7.141715e-01	0.4810328	16803
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0.0940922	1.598445e+00	0.1211691	16805
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0.3322250	1.497887e+00	0.1453524	16810
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0.0945941	1.170476e+00	0.2516757	16813
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0.2050724	1.223258e+00	0.2314279	16821
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0.2441683	1.282788e+00	0.2100841	16827
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0.1894921	1.124848e+00	0.2702041	16832
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0.0673878	5.413160e-01	0.5925711	16839
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0.0206113	5.514430e-02	0.9564151	16859
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0.2377963	1.330493e+00	0.1940908	16867
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0.1562840	9.131580e-01	0.3689557	16878
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0.0210118	1.549643e-01	0.8779611	16881
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0.1457834	5.533743e-01	0.5843991	17023
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0.1405688	1.012795e+00	0.3198306	17026
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0.0140563	1.804034e-01	0.8581353	17080
0.3472074	1.629900e+00	0.1143226	17081
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0.0194055	7.691260e-02	0.9392400	17087
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0.0782959	7.120603e-01	0.4823189	17098
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0.0233734	2.067938e-01	0.8376674	17100
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0.1160762	8.623636e-01	0.3958135	17113
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0.6094589	1.825206e+00	0.0786556	17116
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0.3257999	1.044563e+00	0.3051599	18090
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0.1185147	5.429390e-01	0.5914680	18370
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0.0703730	2.513328e-01	0.8033922	18386
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0.1637524	7.276357e-01	0.4728777	18389
0.1508444	1.159503e+00	0.2560440	18390
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0.1306739	1.022460e+00	0.3153166	18686
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0.0857537	6.786917e-01	0.5029047	18935
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1.0356437	2.659439e+00	0.0127998	18982
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0.2245389	1.665397e+00	0.1069850	19658
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0.2085953	1.286707e+00	0.2087333	19660
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0.3738506	1.503569e+00	0.1438890	19664
0.1953564	9.454748e-01	0.3525047	19665
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0.0144181	7.520320e-02	0.9405878	19668
0.0553218	4.093817e-01	0.6853745	19669
0.1301766	1.734067e+00	0.0939053	19670
0.1516860	1.033163e+00	0.3103693	19671
0.1475291	6.134312e-01	0.5445428	19672
0.3374665	2.878641e+00	0.0075671	19673
0.0640194	5.861435e-01	0.5624748	19674
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0.3186074	2.309337e+00	0.0285171	19677
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0.1217733	4.676038e-01	0.6436869	19687
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0.1616072	7.288339e-01	0.4721558	19689
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0.2559576	7.911913e-01	0.4354836	19710
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0.0130797	5.351660e-02	0.9577003	19725
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0.1693569	1.027356e+00	0.3130466	19739
0.0225779	2.296385e-01	0.8200420	19740
0.1781486	1.407412e+00	0.1703103	19741
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0.0681387	4.197409e-01	0.6778781	19744
0.1346903	7.399443e-01	0.4654932	19745
0.1717131	7.829201e-01	0.4402460	19746
0.0038791	3.231120e-02	0.9744531	19747
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0.4269200	2.091962e+00	0.0456314	19749
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0.2875877	1.119985e+00	0.2722353	19763
0.0308508	2.205694e-01	0.8270282	19764
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0.0821494	5.831181e-01	0.5644813	19771
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0.4632251	2.052184e+00	0.0496070	19776
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0.2195935	1.292580e+00	0.2067217	19783
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0.0460979	5.027365e-01	0.6190811	19786
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0.0642611	4.260868e-01	0.6733023	19790
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0.0448267	2.723849e-01	0.7873248	19794
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0.0790137	4.342866e-01	0.6674087	19801
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0.1529684	1.278449e+00	0.2115873	19803
0.0662162	4.598007e-01	0.6492097	19804
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0.0644409	4.130273e-01	0.6827325	19810
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0.0882269	4.793444e-01	0.6354164	19860
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0.0032355	2.673100e-02	0.9788639	19885
0.2692399	1.717277e+00	0.0969718	19886
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0.2547941	1.483742e+00	0.1490478	19888
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0.1586176	8.458659e-01	0.4047986	19894
0.2812685	1.467340e+00	0.1534271	19895
0.0557419	3.118795e-01	0.7574427	19896
0.0327520	1.847968e-01	0.8547205	19897
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0.0029904	3.226420e-02	0.9744903	19899
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0.0273229	1.545990e-01	0.8782464	19902
0.0040221	1.343280e-02	0.9893777	19903
0.5809681	3.629752e+00	0.0011231	19904
0.1088421	1.224564e+00	0.2309428	19905
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0.0722662	6.079489e-01	0.5481213	19909
0.1130932	1.087459e+00	0.2861067	19910
0.0443152	5.491505e-01	0.5872554	19911
0.4544456	2.050322e+00	0.0498005	19912
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0.1259605	6.691557e-01	0.5088765	19915
0.1329971	1.232118e+00	0.2281519	19916
0.0253594	1.477162e-01	0.8836253	19917
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0.0082364	4.733180e-02	0.9625848	19919
0.1106366	1.574729e+00	0.1265518	19920
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0.0057580	3.778730e-02	0.9701254	19926
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0.1977896	1.233060e+00	0.2278055	19934
0.1475924	5.902713e-01	0.5597430	19935
0.0118732	5.482030e-02	0.9566710	19936
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0.0209106	1.054630e-01	0.9167604	19938
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0.0232739	1.519261e-01	0.8803346	19941
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0.1227106	5.976942e-01	0.5548477	19943
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0.0538474	1.844388e-01	0.8549986	19948
0.6206183	3.004946e+00	0.0055491	19949
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0.9079318	3.642625e+00	0.0010858	19951
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0.0995178	4.821235e-01	0.6334656	19953
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0.0549838	3.534890e-01	0.7263696	19956
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0.1536498	1.117547e+00	0.2732579	19958
0.0370662	2.706838e-01	0.7886196	19959
0.2088021	1.766549e+00	0.0882051	19960
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0.0525651	7.533067e-01	0.4575534	26623
0.0430187	5.479999e-01	0.5880346	26624
0.1506799	1.160499e+00	0.2556452	26625
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0.0866066	5.538440e-01	0.5840820	26628
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0.0241347	1.933533e-01	0.8480780	26640
0.3247122	2.278517e+00	0.0305238	26641
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1.0053120	3.286122e+00	0.0027357	26695
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0.2852065	1.855194e+00	0.0741256	27256
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0.1520333	7.472495e-01	0.4611426	27258
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0.0048685	1.683100e-02	0.9866908	27260
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0.0318346	2.832039e-01	0.7791039	27265
0.1144873	4.112109e-01	0.6840484	27266
0.1154908	6.963291e-01	0.4919632	27267
0.3441583	1.931214e+00	0.0636373	27268
0.1722377	1.185140e+00	0.2459236	27269
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0.2563646	1.711159e+00	0.0981101	27573
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0.1545382	6.481980e-01	0.5221374	28836
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0.0866590	5.332539e-01	0.5980656	29423
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0.0008185	6.015200e-03	0.9952433	29451
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0.1105456	7.632549e-01	0.4516947	29456
0.7528085	3.454683e+00	0.0017733	29457
0.3830737	1.526312e+00	0.1381501	29458
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0.3227361	2.570964e+00	0.0157460	29460
0.1813484	1.041795e+00	0.3064191	29461
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0.3646177	1.203534e+00	0.2388466	29467
0.2434415	2.516791e+00	0.0178484	29468
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0.1243158	1.589117e+00	0.1232633	29475
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0.0898290	9.581305e-01	0.3461976	29477
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-0.0699524	-2.476143e-01	0.8062397	29791
0.0165405	1.218514e-01	0.9038870	29792
0.0403045	3.449785e-01	0.7326886	29793
0.0324905	2.331059e-01	0.8173749	29794
-0.2145088	-1.772232e+00	0.0872387	29795
-0.0495253	-5.130103e-01	0.6119682	29796
-0.0788439	-8.454477e-01	0.4050280	29797
-0.0258196	-2.359233e-01	0.8152094	29798
-0.3522567	-1.923107e+00	0.0646907	29799
-0.1942241	-9.167127e-01	0.3671219	29800
-0.2365660	-9.280432e-01	0.3613167	29801
0.0640218	7.984228e-01	0.4313455	29802
0.1966533	1.551711e+00	0.1319624	29803
0.1190734	1.382181e+00	0.1778438	29804
-0.1046358	-7.820862e-01	0.4407279	29805
0.0047464	3.112740e-02	0.9753888	29806
-0.0059404	-3.800450e-02	0.9699537	29807
-0.0519649	-8.122084e-01	0.4235239	29808
0.0862626	5.673938e-01	0.5749681	29809
1.0000000	1.536273e+11	0.0000000	29810

Al igual que en el P5, se extraen los 10 IncRNAs más significativos de acuerdo al valor de \(p\) :

coeff_model_RLS_sorte<-coeff_model_RLS_df%>% 
  arrange(P_valor)
  mas_sig_10_RLS <- head(coeff_model_RLS_sorte, 10)

kbl(mas_sig_10_RLS, 
    caption = "", 
    align = "c", 
    position = "top") %>%
    kable_styling(bootstrap_options = c("striped", "hover"))%>% 
    scroll_box(width = "920px", height = "300px")

Coeficiente	valor z	P_valor	ID
1.0000000	1.536273e+11	0.00e+00	29810
0.6220172	5.470868e+00	7.70e-06	4634
0.5999949	5.024129e+00	2.60e-05	24198
0.8053454	4.875338e+00	3.90e-05	3994
-1.3783024	-4.752761e+00	5.45e-05	12278
0.7057086	4.746792e+00	5.54e-05	22359
0.6877228	4.666934e+00	6.89e-05	2114
0.2524093	4.602604e+00	8.21e-05	6101
0.3652637	4.582242e+00	8.68e-05	1457
0.4283546	4.581874e+00	8.69e-05	6948

Al comparar los resultados con los IncRNA más significativos obtenidos en el P5, se evidencia que en el modelo de Regresión Logística (RL), los valores de \(p\) asociados a todos los IncRNA son notablemente bajos. Sin embargo, al examinar detenidamente los coeficientes, se observa que el impacto en la variable dependiente es mínimo comparado con el modelo RL. Esto se debe a que la implementación adecuada de un Modelo de Regresión Logística requiere que la variable sea numérica y continua. Dado que nuestra variable dependiente posee una naturaleza binaria, no es apropiado seleccionar esta técnica para inferencia sobre éstos.

P10

Suponga que se ajustan los modelos

\(M_1:\) glm(dx ~ x1, family = 'binomial', data = d)

\(M_2:\) glm(dx ~ x1 + x2, family = 'binomial', data = d)

tal que \(M_1 \subseteq M_2\).

Después del proceso de estimación, se obtuvieron los coeficientes \((\hat{\beta}_{0,M_1}, \hat{\beta}_{1,M_1})\) y \((\hat{\beta}_{0,M_2}, \hat{\beta}_{1,M_2}, \hat{\beta}_{2,M_2})\) con base en una muestra de tamaño \(n\) almacenada en el objeto d.

Interprete \(\hat{\beta}_{1,M_1}\) y \(\hat{\beta}_{1,M_2}\). Qué representa \(\hat{\beta}_{2,M_2}\)?.

• Modelo \(M_1\)

m1 <- glm(dx ~ x1, family = 'binomial', data = d)
summary(m1)

Call:
glm(formula = dx ~ x1, family = "binomial", data = d)

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-1.25986  -1.17146  -0.01248   1.16388   1.30854  

Coefficients:
            Estimate Std. Error z value Pr(>|z|)
(Intercept)   0.9932     2.7555   0.360    0.719
x1           -0.2899     0.7972  -0.364    0.716

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 41.589  on 29  degrees of freedom
Residual deviance: 41.456  on 28  degrees of freedom
AIC: 45.456

Number of Fisher Scoring iterations: 3

Dado lo anterior, calculamos el odds ratio (OR) para determinar la probabilidad de que un paciente padezca la enfermedad de Alzheimer (EA) a medida que aumente la expresión del IncRNA \(\text{x}_1\).

OR_m1 <- exp(m1$coefficients)

De lo anterior, se puede inferir que al aumentar la cantidad de IncRNA(1) en una unidad, es \(\thickapprox\) 0.2516667 veces menos probable que un paciente padezca EA, por lo tanto, se considera un factor de protección.

• Modelo \(M_2\)

m2 <- glm(dx ~ x1+x2, family = 'binomial', data = d)
summary(m2)

Call:
glm(formula = dx ~ x1 + x2, family = "binomial", data = d)

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-1.35406  -1.16660   0.00077   1.14138   1.39492  

Coefficients:
            Estimate Std. Error z value Pr(>|z|)
(Intercept)   4.8105     9.7775   0.492    0.623
x1           -0.1422     0.8754  -0.162    0.871
x2           -0.6017     1.4762  -0.408    0.684

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 41.589  on 29  degrees of freedom
Residual deviance: 41.288  on 27  degrees of freedom
AIC: 47.288

Number of Fisher Scoring iterations: 4

Dado lo anterior, calculamos el odds ratio (OR) para determinar la probabilidad de que un paciente padezca la enfermedad de Alzheimer (EA) a medida que aumente la expresión del IncRNA \(\text{x}_2\).

OR_m2 <- exp(m2$coefficients)
coef_x1 <- OR_m2["x1"]
coef_x2 <- OR_m2["x2"]

De lo anterior, se puede inferir que al aumentar la cantidad de expresión del IncRNA(1) en una unidad, es \(\thickapprox\) 0.8674099 veces menos probable que un paciente padezca EA manteniendo constante la cantidad de expresión del IncNRA(2), por lo tanto, se considera un factor de protección.

De igual forma, cada que aumente la cantidad de expresión del IncRNA(2) en una unidad, es \(\thickapprox\) 0.5478544 veces menos probable que un paciente padezca EA manteniendo constante la cantidad de expresión del IncNRA(1), por lo tanto, se considera un factor de protección.