Sesión 4. Pruebas T y barras de error para dos variables
Curso: POL278 - Estadística para el análisis político 1
Jefes de práctica: Reyna Osorio, Gabriel Ramos, Almendra Rodriguez y Alexander Benites
Ciclo 2021-2
Recordando lo avanzado
En la sesión anterior, nos adentramos al análisis inferencial bivariado de una BD, teniendo como base del tema el cálculo del Intervalo de Confianza (IC) para una media (variables numéricas) y para una proporción (variables categóricas). Recordemos que gracias al IC podemos determinar si la estimación es representativa de la población. En efecto, la comparación de los intervalos de confianza de 2 grupos es un método “informal” para saber si las diferencias en las medias o las proporciones entre dos grupos se pueden extrapolar a la población. La idea era calcular los intervalos de confianza para cada grupo y ver si los intervalos se interceptan o no. La regla era que si los intervalos de ambos grupos no se interceptaban, podíamos extrapolar que la diferencia muestral existe en la población al 95% de confianza.
Para profundizar sobre estadística inferencial, en esta sesión se verá cómo pasar de la evaluación “informal” a la “formal” mediante la introducción a la prueba t de diferencia de medias y desarrollaremos los cincos pasos de la inferencia estadística.Estos pasos normalmente se seguirán en toda prueba inferencial, donde se quiera extrapolar un resultado de la muestra a la población.
¿Qué es la prueba T de diferencia de medias?
Generalmente, cuando queremos comparar dos grupos centramos nuestra atención en el promedio de cada uno. Sin embargo, el hecho de que los promedios sean distintos no supone, necesariamente, que existe una diferencia estadísticamente significativa.
Para saber si la diferencia observada entre las medias de dos grupos es o no significativa se emplean métodos paramétricos como el de Z-scores o la distribución T-student. Estas técnicas calculan los intervalos de confianza de cada grupo y concluyen si existe una diferencia real entre las medias.
La prueba T compara la media de una variable de numérica para dos grupos o categorías de una variable nominal u ordinal. Los grupos que forman la variable nominal/ordinal tienen que ser independientes. Es decir, cada observación debe pertenecer a un grupo o al otro, pero no a ambos.
Pruebas T para muestras independientes
Condiciones
Independencia: Las muestras deben ser independientes. El muestreo debe ser aleatorio.
Igualdad de varianza: La varianza de ambas poblaciones comparadas debe ser igual.
IMPORTANTE
La condición de normalidad también es considerada si es que la muestra fuera pequeña (Agresti y Finlay proponen que se aplica con un n pequeño menor a 30 observaciones). A medida que el n se hace más grande, el supuesto de normalidad es menos importante pues con grandes n confiamos en el teorema del límite central que nos indica que la distribución muestral será siempre normal.
Pasos para realizar la Prueba T
- Establecer hipótesis
- Calcular el estadístico (parámetro estimado) que se va a emplear
- Determinar el nivel de significancia α (alpha)
- Calcular el p-value y realizar la prueba prop.test
- Interpretar
Recuerda El H0 de la prueba T es la siguiente:
Ho: No existe diferencia estadísticamente significativa entre las medias de los dos grupos comparados. H1: Sí existe diferencia estadísticamente significativa entre las medias de los dos grupos comparados.
La H0 es generalmente la hipótesis de no efecto, de no diferencias.
Carguemos la BD
Descripción del Proyecto: COVID Isolation on Sleep and Health in Healthcare Workers
Estudios anteriores realizados en todo el mundo han demostrado un mayor estrés, empeoramiento de la salud mental y empeoramiento de los patrones de sueño en los trabajadores de la salud de primera línea durante la pandemia de COVID-19. El fundamento de este estudio fue examinar si los comportamientos relacionados con el sueño, el estado de ánimo y la salud podrían diferir entre los trabajadores de la salud que pasaron a brindar atención desde el hogar y los que continuaron reportándose en persona a sus respectivos hospitales o instalaciones de atención médica.
Impacto del estudio: este estudio muestra un empeoramiento del estado de ánimo entre los trabajadores de la salud, independientemente de si trabajaban desde casa o continuaban presentándose al trabajo. Los hallazgos destacan la necesidad de programas de apoyo para monitorear el estado de ánimo y la salud durante la pandemia de COVID-19.
Más sobre el proyecto accediendo al siguiente enlace: https://www.openicpsr.org/openicpsr/project/127081/version/V1/view
#Antes que todo, debemos instalar el paquete escribiendo en la consola: **install.packages("rio")**
library(rio) #Convocamos el paquete
data=import("COVID Isolation.xlsx") #Utilizamos la función import que creará un objeto en R que contenga la base de datos del archivo "COVID Isolation.xlsx" que hemos descargado. Podemos verificar la creación en el environment.Exploramos las variables que tiene la base de datos:
str(data) #Visualice la estructura de la bbdd## 'data.frame': 915 obs. of 66 variables:
## $ StartDate : chr "Start Date" "43918.558425925927" "43920.353900462964" "43920.360729166663" ...
## $ EndDate : chr "End Date" "43918.560115740744" "43920.355613425927" "43920.362581018519" ...
## $ Status : chr "Response Type" "Survey Preview" "Survey Preview" "Survey Preview" ...
## $ IPAddress : chr "IP Address" NA NA NA ...
## $ Progress : chr "Progress" "100" "100" "100" ...
## $ Duration (in seconds): chr "Duration (in seconds)" "145" "147" "160" ...
## $ Finished : chr "Finished" "True" "True" "True" ...
## $ RecordedDate : chr "Recorded Date" "43918.560125717595" "43920.355617592591" "43920.362593877318" ...
## $ ResponseId : chr "Response ID" "R_D2zZmZzPFOTHGhj" "R_ZesvyNm3EjoBl2F" "R_2zT0jHR2F9OOQtk" ...
## $ RecipientLastName : chr "Recipient Last Name" NA NA NA ...
## $ RecipientFirstName : chr "Recipient First Name" NA NA NA ...
## $ RecipientEmail : chr "Recipient Email" NA NA NA ...
## $ ExternalReference : chr "External Data Reference" NA NA NA ...
## $ LocationLatitude : chr "Location Latitude" "42.270904541015625" "42.270904541015625" "42.270904541015625" ...
## $ LocationLongitude : chr "Location Longitude" "-83.726799011230469" "-83.726799011230469" "-83.726799011230469" ...
## $ DistributionChannel : chr "Distribution Channel" "preview" "preview" "preview" ...
## $ UserLanguage : chr "User Language" "EN" "EN" "EN" ...
## $ Q1 : chr "What is your age?" NA NA NA ...
## $ Q2 : chr "What is your gender?" NA NA NA ...
## $ Q3 : chr "50 States, D.C. and Puerto Rico" NA NA NA ...
## $ Q4 : chr "What ethnicity do you consider yourself?" NA NA NA ...
## $ Q5 : chr "Choose one or more races that you consider yourself to be: - Selected Choice" NA NA NA ...
## $ Q5_6_TEXT : chr "Choose one or more races that you consider yourself to be: - Other - Text" NA NA NA ...
## $ Q6 : chr "What is your role in the healthcare field? (e.g. psychologist, physician, nurse)" NA NA NA ...
## $ Q7 : chr "Are you a frontline worker providing care for acutely ill patients that may be infected with COVID-1" NA NA NA ...
## $ Q8 : chr "Are you currently conducting your job mostly from home now?" NA "No" "Yes" ...
## $ Q8a : chr "Which week in March 2020 did you start working\r\nfrom home?" NA NA NA ...
## $ Q9 : chr "Are you conducting clinical video visits with your patients from your home?" NA "2" "1" ...
## $ Q9a : chr "Please rate YOUR satisfaction with conducting\r\nvideo visits" NA NA NA ...
## $ Q9b : chr "Please rate your estimation of your PATIENT's satisfaction with video visits" NA NA NA ...
## $ Q10 : chr "Does\r\nyour work require you to follow a fixed work schedule while working from\r\nhome?" NA "No" "No" ...
## $ Q11 : chr "Are children home from school in the house?" NA "1" "1" ...
## $ Q11a : chr "How many children are in the home?" NA NA NA ...
## $ Q11b : chr "How old are the children in the home?" NA NA NA ...
## $ Q12 : chr "Approximately how many hours did you sleep on an\r\naverage work night in January 2020?" NA NA NA ...
## $ Q13 : chr "Approximately how many hours did you sleep on an\r\naverage work night in the last week?" NA NA NA ...
## $ Q14 : chr "Has the number of your work hours per week changed?" NA "Yes" "Yes" ...
## $ Q14a : chr "Please tell us how your work hours per week have changed" NA "I am working fewer hours" "I am working fewer hours" ...
## $ Q15 : chr "Have you varied your work schedule?" NA NA NA ...
## $ Q15a : chr "Please tell us how you have varied your work schedule." NA NA NA ...
## $ Q16 : chr "Have your sleep patterns changed?" "Yes" "No" "Yes" ...
## $ Q16a : chr "Please tell us how your bedtime has changed" NA NA NA ...
## $ Q16b : chr "Please tell us how your waketime has changed" NA NA NA ...
## $ Q17 : chr "Has the number of naps you are taking changed?" NA "No" NA ...
## $ Q17a : chr "Please tell us how the number of the naps have changed" NA NA NA ...
## $ Q18 : chr "Has the amount of alcohol you are consuming\r\nchanged?" "Yes" "No" "Yes" ...
## $ Q18a : chr "Please tell us how the amount of alcohol you are consuming has changed" NA NA "I am drinking less alcohol" ...
## $ Q19 : chr "In January 2020, approximately how often did you have a drink containing alcohol?" NA NA NA ...
## $ Q20 : chr "In the last month, approximately how often did you have a drink containing alcohol?" NA NA NA ...
## $ Q21 : chr "In January 2020, approximately how often did you use marijuana/cannabis (recreational or medical)?" NA NA NA ...
## $ Q22 : chr "In the last month, approximately how often did you use marijuana/cannabis (recreational or medical)?" NA NA NA ...
## $ Q23 : chr "Has\r\nthe amount of news you are consuming increased since the end of Feb, 2020?" NA "No" "No" ...
## $ Q24 : chr "How many hours of COVID-19 related news or\r\nsocial media are you consuming on average per day?" NA "0-.5 hour" "0-.5 hour" ...
## $ Q25 : chr "Have\r\nyou had more “screen time” (e.g. use of smartphone, tablet, etc.) around\r\nbedtime?" NA "No" "Yes" ...
## $ Q25a : chr "Please tell us how much more screen time you estimate you have been exposed to before bed" NA NA NA ...
## $ Q26 : chr "Have you changed your movement/exercise?" "Yes" "No" "Yes" ...
## $ Q26a : chr "Please tell us how your movement/exercise habits have changed" NA NA "I am exercising/moving less" ...
## $ Q27 : chr "Has the quality of your diet changed?" "Yes" "No" "Yes" ...
## $ Q27a : chr "Please tell us how your diet has changed?" NA NA "I am eating more healthy foods" ...
## $ Q28 : chr "Has the amount of food you have been eating per day changed?" "Yes" "No" "Yes" ...
## $ Q28a : chr "Please tell us how the amount of food you have been eating has changed" NA NA "I am eating more food each day" ...
## $ Q29 : chr "Has your mood changed?" "Yes" "No" "Yes" ...
## $ Q29a : chr "Please tell us how your mood has changed. My mood has been:" "Slightly worse" NA "Moderately worse" ...
## $ Q29b : chr "Please\r\ntell us how your mood has worsened" NA NA NA ...
## $ Q29c : chr "Please tell us how your mood has worsened" NA NA "I am moderately more depressed" ...
## $ Q29d : chr "Please tell us how your mood has worsened" NA NA NA ...names(data) #Visualice los nombres de las variables de la bbdd## [1] "StartDate" "EndDate" "Status"
## [4] "IPAddress" "Progress" "Duration (in seconds)"
## [7] "Finished" "RecordedDate" "ResponseId"
## [10] "RecipientLastName" "RecipientFirstName" "RecipientEmail"
## [13] "ExternalReference" "LocationLatitude" "LocationLongitude"
## [16] "DistributionChannel" "UserLanguage" "Q1"
## [19] "Q2" "Q3" "Q4"
## [22] "Q5" "Q5_6_TEXT" "Q6"
## [25] "Q7" "Q8" "Q8a"
## [28] "Q9" "Q9a" "Q9b"
## [31] "Q10" "Q11" "Q11a"
## [34] "Q11b" "Q12" "Q13"
## [37] "Q14" "Q14a" "Q15"
## [40] "Q15a" "Q16" "Q16a"
## [43] "Q16b" "Q17" "Q17a"
## [46] "Q18" "Q18a" "Q19"
## [49] "Q20" "Q21" "Q22"
## [52] "Q23" "Q24" "Q25"
## [55] "Q25a" "Q26" "Q26a"
## [58] "Q27" "Q27a" "Q28"
## [61] "Q28a" "Q29" "Q29a"
## [64] "Q29b" "Q29c" "Q29d"Limpieza de las variables previo al análisis:
- La variable Q8 (imputando valores perdidos)
Q8: Are you currently conducting your job mostly from home now?
Categoria de respuesta: (1=Yes,remoto/ 2=No,presencial)
class(data$Q8)## [1] "character"any(is.na(data$Q8)) #Confirma que sí hay NA's## [1] TRUEsum(is.na(data$Q8)) #Cantidad de valores perdidos en el vector## [1] 4BD<- data[!is.na(data$Q8), ] #primero, imputamos los NA's
sum(is.na(BD$Q8))## [1] 0class(BD$Q8)## [1] "character"BD$Q8=as.factor(BD$Q8)
class(BD$Q8)## [1] "factor"- Formateando variable Q13 (character to numeric)
Q13: Approximately how many hours did you sleep on an average work night in the last week?
class(BD$Q13)## [1] "character"BD$Q13## [1] "Approximately how many hours did you sleep on an\r\naverage work night in the last week?"
## [2] NA
## [3] NA
## [4] "9"
## [5] "8"
## [6] "8.5"
## [7] "8"
## [8] "8"
## [9] "9"
## [10] "5.5"
## [11] "8"
## [12] "9"
## [13] "7"
## [14] "4.5"
## [15] "7"
## [16] "6.5"
## [17] "7"
## [18] "6"
## [19] "7.5"
## [20] "6"
## [21] "7"
## [22] "8"
## [23] "6"
## [24] "8"
## [25] "7"
## [26] "8.5"
## [27] "7"
## [28] "7.5"
## [29] "7"
## [30] "5.5"
## [31] "7"
## [32] "7"
## [33] "8"
## [34] "8"
## [35] "8"
## [36] "5"
## [37] "5.5"
## [38] "8"
## [39] "8.5"
## [40] "8"
## [41] "8"
## [42] "6"
## [43] "7"
## [44] "8"
## [45] "7"
## [46] "8"
## [47] "7"
## [48] "5"
## [49] "7"
## [50] "6"
## [51] "8"
## [52] "8"
## [53] "6"
## [54] "8.5"
## [55] "7"
## [56] "9"
## [57] "5"
## [58] "4"
## [59] "7"
## [60] "6"
## [61] "6"
## [62] "6"
## [63] "8"
## [64] "7.5"
## [65] "8"
## [66] "7"
## [67] "6"
## [68] "7"
## [69] "4"
## [70] "7"
## [71] "9"
## [72] "12"
## [73] "7"
## [74] "7"
## [75] "7.5"
## [76] "7.5"
## [77] "7"
## [78] "7"
## [79] "5"
## [80] "8.5"
## [81] "10"
## [82] "9"
## [83] "7.5"
## [84] "8"
## [85] "7"
## [86] "6"
## [87] "5.5"
## [88] "9"
## [89] "7"
## [90] "8"
## [91] "7"
## [92] "9"
## [93] "8"
## [94] "5"
## [95] "8.5"
## [96] "8"
## [97] "7"
## [98] "8.5"
## [99] "7"
## [100] "8"
## [101] "6"
## [102] "9"
## [103] "6"
## [104] "10"
## [105] "8"
## [106] "7"
## [107] "8"
## [108] "7"
## [109] "8.25"
## [110] "5.25"
## [111] "4.5"
## [112] "4"
## [113] "7"
## [114] "6"
## [115] "6"
## [116] "8"
## [117] "8"
## [118] "7"
## [119] "7"
## [120] "5"
## [121] "6.5"
## [122] "6"
## [123] "8.5"
## [124] "7"
## [125] "5"
## [126] "8"
## [127] "6"
## [128] "7"
## [129] "6"
## [130] "5"
## [131] "7"
## [132] "4.5"
## [133] "7"
## [134] "8"
## [135] "7.5"
## [136] "7.5"
## [137] "7"
## [138] "6.5"
## [139] "7"
## [140] "8"
## [141] "7.5"
## [142] "6"
## [143] "7"
## [144] "7"
## [145] "9"
## [146] "8"
## [147] "6.5"
## [148] "5"
## [149] "7.5"
## [150] "7.5"
## [151] "8"
## [152] "5"
## [153] "6"
## [154] "7"
## [155] "8"
## [156] "6"
## [157] "6.75"
## [158] "6"
## [159] "9"
## [160] "6"
## [161] "9"
## [162] "7.25"
## [163] "9"
## [164] "9"
## [165] "6"
## [166] "9"
## [167] "5"
## [168] "8"
## [169] "7"
## [170] "7"
## [171] "8"
## [172] "5"
## [173] "7.5"
## [174] "8"
## [175] "6"
## [176] "6"
## [177] "4.5"
## [178] "7.5"
## [179] "9"
## [180] "7"
## [181] "7"
## [182] "5.5"
## [183] "7"
## [184] "8"
## [185] "6.5"
## [186] "8"
## [187] "9"
## [188] "6"
## [189] "6"
## [190] "7"
## [191] "7"
## [192] "7"
## [193] "7"
## [194] "6"
## [195] "8"
## [196] "7"
## [197] "8"
## [198] "10"
## [199] "8"
## [200] "10"
## [201] "5"
## [202] "8"
## [203] "8"
## [204] "8"
## [205] "6"
## [206] "6"
## [207] "7"
## [208] "8"
## [209] "6"
## [210] "6.5"
## [211] "8"
## [212] "5"
## [213] "6"
## [214] "6"
## [215] "7"
## [216] "9"
## [217] "8"
## [218] "7"
## [219] "7"
## [220] "8"
## [221] "4.5"
## [222] "8"
## [223] "8"
## [224] "6"
## [225] "2"
## [226] "8"
## [227] "5"
## [228] "6"
## [229] "8"
## [230] "4"
## [231] "5"
## [232] "8"
## [233] "8"
## [234] "8"
## [235] "7"
## [236] "8"
## [237] "8"
## [238] "8"
## [239] "8"
## [240] "7.5"
## [241] "7"
## [242] "9"
## [243] "6"
## [244] "7"
## [245] "5.5"
## [246] "7"
## [247] "5"
## [248] "7"
## [249] "7"
## [250] "7"
## [251] "8"
## [252] "6"
## [253] "8"
## [254] "7.5"
## [255] "8"
## [256] "7"
## [257] "7.5"
## [258] "7"
## [259] "9"
## [260] "8"
## [261] "9"
## [262] "7.5"
## [263] "7"
## [264] "9"
## [265] "6"
## [266] "7"
## [267] "5"
## [268] "9"
## [269] "7"
## [270] "12"
## [271] "5"
## [272] "5"
## [273] "8"
## [274] "7"
## [275] "8"
## [276] "7"
## [277] "7"
## [278] "6"
## [279] "6.5"
## [280] "9"
## [281] "8"
## [282] "7"
## [283] "8"
## [284] "6"
## [285] "2.6"
## [286] "6"
## [287] "4"
## [288] "7.5"
## [289] "6"
## [290] "6.5"
## [291] "8"
## [292] "6.5"
## [293] "6.5"
## [294] "6"
## [295] "6"
## [296] "9.5"
## [297] "6"
## [298] "9"
## [299] "5"
## [300] "6.8"
## [301] "7"
## [302] "7"
## [303] "8"
## [304] "6"
## [305] "6.5"
## [306] "6.5"
## [307] "8"
## [308] "9"
## [309] "7"
## [310] "7"
## [311] "7"
## [312] "9"
## [313] "5"
## [314] "7"
## [315] "5"
## [316] "6"
## [317] "5"
## [318] "7.5"
## [319] "8"
## [320] "6"
## [321] "7"
## [322] "8"
## [323] "8.5"
## [324] "7"
## [325] "5"
## [326] "9"
## [327] "7"
## [328] "10"
## [329] "7.5"
## [330] "8"
## [331] "6.30"
## [332] "8"
## [333] "7.5"
## [334] "7"
## [335] "8"
## [336] "8"
## [337] "8"
## [338] "7"
## [339] "8"
## [340] "7"
## [341] "NA"
## [342] "8"
## [343] "8"
## [344] "7.5"
## [345] "6.5"
## [346] "7"
## [347] "8"
## [348] "7.75"
## [349] "7"
## [350] "6.5"
## [351] "6"
## [352] "7.5"
## [353] "6"
## [354] "7"
## [355] "8"
## [356] "8"
## [357] "6"
## [358] "6.5"
## [359] "7"
## [360] "7.5"
## [361] "6"
## [362] "7"
## [363] "6"
## [364] "6"
## [365] "4"
## [366] "8"
## [367] "6"
## [368] "8"
## [369] "9"
## [370] "8"
## [371] "6"
## [372] "8.5"
## [373] "7.5"
## [374] "4.5"
## [375] "5"
## [376] "7"
## [377] "7"
## [378] "5"
## [379] "7"
## [380] "8.5"
## [381] "8"
## [382] "5"
## [383] "7"
## [384] "8"
## [385] "8"
## [386] "8"
## [387] "8"
## [388] "3.5"
## [389] "8"
## [390] "9"
## [391] "8"
## [392] "9"
## [393] "9"
## [394] "6"
## [395] "6.5"
## [396] "8"
## [397] "8"
## [398] "8"
## [399] "7"
## [400] "8"
## [401] "8"
## [402] "8"
## [403] "7"
## [404] "7"
## [405] "8"
## [406] "7"
## [407] "7"
## [408] "9"
## [409] "9"
## [410] "7"
## [411] "6"
## [412] "7"
## [413] "8"
## [414] "6"
## [415] "6"
## [416] "4"
## [417] "5"
## [418] "10"
## [419] "6.5"
## [420] "8"
## [421] "8"
## [422] "8"
## [423] "7"
## [424] "8"
## [425] "7"
## [426] "5"
## [427] "7.5"
## [428] "5"
## [429] "6.5"
## [430] "7"
## [431] "7"
## [432] "8"
## [433] "7"
## [434] "7"
## [435] "6"
## [436] "7"
## [437] "7"
## [438] "8"
## [439] "8.5"
## [440] "7"
## [441] "7"
## [442] "5"
## [443] "7"
## [444] "9.5"
## [445] "7"
## [446] "9.5"
## [447] "7"
## [448] "6"
## [449] "6"
## [450] "7"
## [451] "7"
## [452] "7.5"
## [453] "7"
## [454] "9"
## [455] "7.5"
## [456] "8"
## [457] "4.5"
## [458] "9"
## [459] "7"
## [460] "7"
## [461] "6"
## [462] "8.5"
## [463] "8"
## [464] "8"
## [465] "8"
## [466] "5"
## [467] "8"
## [468] "4"
## [469] "7"
## [470] "7"
## [471] "6.5"
## [472] "7"
## [473] "9"
## [474] "7.5"
## [475] "7"
## [476] "7"
## [477] "8"
## [478] "8"
## [479] "4"
## [480] "6"
## [481] "7.5"
## [482] "8"
## [483] "9"
## [484] "NA"
## [485] "7.5"
## [486] "6.5"
## [487] "8"
## [488] "6"
## [489] "5"
## [490] "9"
## [491] NA
## [492] NA
## [493] "8"
## [494] "7"
## [495] NA
## [496] NA
## [497] "7.5"
## [498] NA
## [499] NA
## [500] NA
## [501] "7"
## [502] NA
## [503] NA
## [504] NA
## [505] "8"
## [506] "6"
## [507] NA
## [508] NA
## [509] NA
## [510] NA
## [511] "5"
## [512] "8"
## [513] "5.5"
## [514] NA
## [515] "8"
## [516] "6.5"
## [517] NA
## [518] NA
## [519] "7"
## [520] "7.5"
## [521] "7"
## [522] "5"
## [523] "8"
## [524] "8"
## [525] "9"
## [526] "7"
## [527] "8"
## [528] "7.5"
## [529] "7.5"
## [530] "6.5"
## [531] "9"
## [532] "7"
## [533] "7"
## [534] "6"
## [535] "7"
## [536] "7"
## [537] "8.5"
## [538] "9"
## [539] "6.75"
## [540] "7.5"
## [541] "7"
## [542] "8.5"
## [543] "7.5"
## [544] NA
## [545] "7"
## [546] "6.5"
## [547] "7.5"
## [548] "8"
## [549] NA
## [550] "5"
## [551] "8"
## [552] "6.5"
## [553] "8"
## [554] "7.5"
## [555] "9"
## [556] "8.5"
## [557] "6.5"
## [558] "7"
## [559] NA
## [560] "9"
## [561] NA
## [562] "6"
## [563] "7"
## [564] "8"
## [565] "7"
## [566] "6.5"
## [567] "8"
## [568] "7"
## [569] "8"
## [570] "8"
## [571] "7.5"
## [572] "7.5"
## [573] "4"
## [574] "6"
## [575] "8"
## [576] "7"
## [577] "7"
## [578] "7"
## [579] "11"
## [580] "8"
## [581] "5.5"
## [582] NA
## [583] "8"
## [584] "9"
## [585] "7"
## [586] "6"
## [587] "7.5"
## [588] NA
## [589] "7.5"
## [590] "7"
## [591] "4.5"
## [592] "6.5"
## [593] "7"
## [594] "7.5"
## [595] "18"
## [596] "6"
## [597] "8"
## [598] "7.5"
## [599] "8"
## [600] "7"
## [601] "7"
## [602] "7.5"
## [603] "6.75"
## [604] NA
## [605] NA
## [606] "6"
## [607] "8"
## [608] "7"
## [609] "8"
## [610] "5"
## [611] "7"
## [612] "8"
## [613] "8"
## [614] "6"
## [615] "7"
## [616] "7"
## [617] "7.5"
## [618] "7"
## [619] "5"
## [620] "8"
## [621] "9"
## [622] "6"
## [623] "7.5"
## [624] "8.5"
## [625] "5.5"
## [626] "8"
## [627] "7"
## [628] "9"
## [629] "7"
## [630] "7.5"
## [631] "8"
## [632] "6"
## [633] "8"
## [634] "5"
## [635] "10"
## [636] "6"
## [637] "5"
## [638] "8"
## [639] "8"
## [640] "5"
## [641] "10"
## [642] "6.5"
## [643] "6"
## [644] "6"
## [645] "5"
## [646] "8"
## [647] "7.5"
## [648] "9"
## [649] "8.5"
## [650] "5"
## [651] "7"
## [652] "8"
## [653] "7"
## [654] "7.5"
## [655] "8"
## [656] "8"
## [657] "4.5"
## [658] "4"
## [659] "8"
## [660] "9"
## [661] "7.5"
## [662] "5"
## [663] "9.5"
## [664] "13"
## [665] "7"
## [666] "8.75"
## [667] "8"
## [668] "6.5"
## [669] "6.5"
## [670] "4"
## [671] "7"
## [672] "8"
## [673] "6"
## [674] "4"
## [675] "5"
## [676] "5"
## [677] "7"
## [678] "9"
## [679] "6"
## [680] "8"
## [681] "7"
## [682] "10"
## [683] "6"
## [684] "10"
## [685] "6"
## [686] "6"
## [687] "6.5"
## [688] "6.5"
## [689] "7"
## [690] "8"
## [691] "7"
## [692] "8"
## [693] "7"
## [694] "4.5"
## [695] "7"
## [696] "6.5"
## [697] "8"
## [698] "7.5"
## [699] "4.5"
## [700] "9.5"
## [701] "3.5"
## [702] "8"
## [703] "6"
## [704] "6"
## [705] "7"
## [706] "7.5"
## [707] "7"
## [708] "5"
## [709] "7"
## [710] "5"
## [711] "7.5"
## [712] NA
## [713] "7"
## [714] "7"
## [715] "8"
## [716] "8"
## [717] "5.5"
## [718] "4.3"
## [719] "7"
## [720] "7.5"
## [721] "8"
## [722] "7"
## [723] "6"
## [724] "6.5"
## [725] "8"
## [726] "5"
## [727] "8"
## [728] "8"
## [729] "8"
## [730] "6"
## [731] "8"
## [732] "5.5"
## [733] "8"
## [734] "7"
## [735] "7.5"
## [736] "7"
## [737] "8"
## [738] "7"
## [739] "6"
## [740] "4.5"
## [741] "6.5"
## [742] "4"
## [743] "3"
## [744] "9"
## [745] "5"
## [746] "9"
## [747] "8"
## [748] "6"
## [749] "5"
## [750] "4"
## [751] "4"
## [752] "6"
## [753] "6"
## [754] "4.5"
## [755] "5"
## [756] "4.5"
## [757] "5"
## [758] "8"
## [759] "10"
## [760] "6.5"
## [761] "6"
## [762] "7.5"
## [763] "5"
## [764] "8"
## [765] "5"
## [766] "5"
## [767] "7.45"
## [768] "8"
## [769] "7"
## [770] "5.5"
## [771] "6"
## [772] "6"
## [773] NA
## [774] "8"
## [775] "7.5"
## [776] "8.5"
## [777] "7"
## [778] "3.5"
## [779] NA
## [780] "6.5"
## [781] "7"
## [782] "6"
## [783] "10"
## [784] "8"
## [785] "5.5"
## [786] "6"
## [787] "5"
## [788] "6"
## [789] "7"
## [790] "6"
## [791] "6"
## [792] "5"
## [793] "5"
## [794] "5"
## [795] "7"
## [796] "8"
## [797] "5"
## [798] "7.5"
## [799] "5"
## [800] "8"
## [801] "5"
## [802] "4"
## [803] "4"
## [804] "8"
## [805] "8"
## [806] "3"
## [807] "7"
## [808] "4"
## [809] "4.5"
## [810] "7"
## [811] "5"
## [812] "4.5"
## [813] "7"
## [814] "4"
## [815] "6.5"
## [816] "6.5"
## [817] "7"
## [818] "6"
## [819] "6"
## [820] "7.5"
## [821] "4"
## [822] "6"
## [823] "4.5"
## [824] "6"
## [825] "7.25"
## [826] "7.5"
## [827] "6.5"
## [828] "5"
## [829] "5"
## [830] "5"
## [831] "8"
## [832] "8"
## [833] "5"
## [834] "6"
## [835] NA
## [836] "7"
## [837] NA
## [838] "7"
## [839] "9"
## [840] "6"
## [841] NA
## [842] "7"
## [843] NA
## [844] "8"
## [845] NA
## [846] "7.5"
## [847] "7"
## [848] "4"
## [849] NA
## [850] "5"
## [851] "5"
## [852] "6"
## [853] "8"
## [854] "6"
## [855] "5"
## [856] "6"
## [857] "7.5"
## [858] "5"
## [859] "4.5"
## [860] "4"
## [861] NA
## [862] "7"
## [863] "6"
## [864] NA
## [865] NA
## [866] "6"
## [867] NA
## [868] "10"
## [869] NA
## [870] "6"
## [871] "5"
## [872] "7"
## [873] NA
## [874] NA
## [875] "8"
## [876] "7.5"
## [877] "6.5"
## [878] NA
## [879] "6"
## [880] "4.5"
## [881] NA
## [882] "6"
## [883] "8"
## [884] "7.5"
## [885] "7.25"
## [886] "11"
## [887] NA
## [888] "8"
## [889] "6"
## [890] NA
## [891] "7"
## [892] "7.5"
## [893] "7"
## [894] "9.5"
## [895] "6"
## [896] NA
## [897] NA
## [898] NA
## [899] NA
## [900] "6.5"
## [901] NA
## [902] "4"
## [903] NA
## [904] NA
## [905] "8"
## [906] NA
## [907] NA
## [908] "7.5"
## [909] NA
## [910] NA
## [911] "5"any(is.na(BD$Q13)) #Confirma que sí hay NA's## [1] TRUEsum(is.na(BD$Q13)) #Cantidad de valores perdidos en el vector## [1] 58BD1<- BD[!is.na(BD$Q13), ] #primero, imputamos los NA's
sum(is.na(BD1$Q13))## [1] 0BD1$Q13=as.numeric(BD1$Q13)## Warning: NAs introduced by coercionclass(BD1$Q13)## [1] "numeric"BD1$Q13## [1] NA 9.00 8.00 8.50 8.00 8.00 9.00 5.50 8.00 9.00 7.00 4.50
## [13] 7.00 6.50 7.00 6.00 7.50 6.00 7.00 8.00 6.00 8.00 7.00 8.50
## [25] 7.00 7.50 7.00 5.50 7.00 7.00 8.00 8.00 8.00 5.00 5.50 8.00
## [37] 8.50 8.00 8.00 6.00 7.00 8.00 7.00 8.00 7.00 5.00 7.00 6.00
## [49] 8.00 8.00 6.00 8.50 7.00 9.00 5.00 4.00 7.00 6.00 6.00 6.00
## [61] 8.00 7.50 8.00 7.00 6.00 7.00 4.00 7.00 9.00 12.00 7.00 7.00
## [73] 7.50 7.50 7.00 7.00 5.00 8.50 10.00 9.00 7.50 8.00 7.00 6.00
## [85] 5.50 9.00 7.00 8.00 7.00 9.00 8.00 5.00 8.50 8.00 7.00 8.50
## [97] 7.00 8.00 6.00 9.00 6.00 10.00 8.00 7.00 8.00 7.00 8.25 5.25
## [109] 4.50 4.00 7.00 6.00 6.00 8.00 8.00 7.00 7.00 5.00 6.50 6.00
## [121] 8.50 7.00 5.00 8.00 6.00 7.00 6.00 5.00 7.00 4.50 7.00 8.00
## [133] 7.50 7.50 7.00 6.50 7.00 8.00 7.50 6.00 7.00 7.00 9.00 8.00
## [145] 6.50 5.00 7.50 7.50 8.00 5.00 6.00 7.00 8.00 6.00 6.75 6.00
## [157] 9.00 6.00 9.00 7.25 9.00 9.00 6.00 9.00 5.00 8.00 7.00 7.00
## [169] 8.00 5.00 7.50 8.00 6.00 6.00 4.50 7.50 9.00 7.00 7.00 5.50
## [181] 7.00 8.00 6.50 8.00 9.00 6.00 6.00 7.00 7.00 7.00 7.00 6.00
## [193] 8.00 7.00 8.00 10.00 8.00 10.00 5.00 8.00 8.00 8.00 6.00 6.00
## [205] 7.00 8.00 6.00 6.50 8.00 5.00 6.00 6.00 7.00 9.00 8.00 7.00
## [217] 7.00 8.00 4.50 8.00 8.00 6.00 2.00 8.00 5.00 6.00 8.00 4.00
## [229] 5.00 8.00 8.00 8.00 7.00 8.00 8.00 8.00 8.00 7.50 7.00 9.00
## [241] 6.00 7.00 5.50 7.00 5.00 7.00 7.00 7.00 8.00 6.00 8.00 7.50
## [253] 8.00 7.00 7.50 7.00 9.00 8.00 9.00 7.50 7.00 9.00 6.00 7.00
## [265] 5.00 9.00 7.00 12.00 5.00 5.00 8.00 7.00 8.00 7.00 7.00 6.00
## [277] 6.50 9.00 8.00 7.00 8.00 6.00 2.60 6.00 4.00 7.50 6.00 6.50
## [289] 8.00 6.50 6.50 6.00 6.00 9.50 6.00 9.00 5.00 6.80 7.00 7.00
## [301] 8.00 6.00 6.50 6.50 8.00 9.00 7.00 7.00 7.00 9.00 5.00 7.00
## [313] 5.00 6.00 5.00 7.50 8.00 6.00 7.00 8.00 8.50 7.00 5.00 9.00
## [325] 7.00 10.00 7.50 8.00 6.30 8.00 7.50 7.00 8.00 8.00 8.00 7.00
## [337] 8.00 7.00 NA 8.00 8.00 7.50 6.50 7.00 8.00 7.75 7.00 6.50
## [349] 6.00 7.50 6.00 7.00 8.00 8.00 6.00 6.50 7.00 7.50 6.00 7.00
## [361] 6.00 6.00 4.00 8.00 6.00 8.00 9.00 8.00 6.00 8.50 7.50 4.50
## [373] 5.00 7.00 7.00 5.00 7.00 8.50 8.00 5.00 7.00 8.00 8.00 8.00
## [385] 8.00 3.50 8.00 9.00 8.00 9.00 9.00 6.00 6.50 8.00 8.00 8.00
## [397] 7.00 8.00 8.00 8.00 7.00 7.00 8.00 7.00 7.00 9.00 9.00 7.00
## [409] 6.00 7.00 8.00 6.00 6.00 4.00 5.00 10.00 6.50 8.00 8.00 8.00
## [421] 7.00 8.00 7.00 5.00 7.50 5.00 6.50 7.00 7.00 8.00 7.00 7.00
## [433] 6.00 7.00 7.00 8.00 8.50 7.00 7.00 5.00 7.00 9.50 7.00 9.50
## [445] 7.00 6.00 6.00 7.00 7.00 7.50 7.00 9.00 7.50 8.00 4.50 9.00
## [457] 7.00 7.00 6.00 8.50 8.00 8.00 8.00 5.00 8.00 4.00 7.00 7.00
## [469] 6.50 7.00 9.00 7.50 7.00 7.00 8.00 8.00 4.00 6.00 7.50 8.00
## [481] 9.00 NA 7.50 6.50 8.00 6.00 5.00 9.00 8.00 7.00 7.50 7.00
## [493] 8.00 6.00 5.00 8.00 5.50 8.00 6.50 7.00 7.50 7.00 5.00 8.00
## [505] 8.00 9.00 7.00 8.00 7.50 7.50 6.50 9.00 7.00 7.00 6.00 7.00
## [517] 7.00 8.50 9.00 6.75 7.50 7.00 8.50 7.50 7.00 6.50 7.50 8.00
## [529] 5.00 8.00 6.50 8.00 7.50 9.00 8.50 6.50 7.00 9.00 6.00 7.00
## [541] 8.00 7.00 6.50 8.00 7.00 8.00 8.00 7.50 7.50 4.00 6.00 8.00
## [553] 7.00 7.00 7.00 11.00 8.00 5.50 8.00 9.00 7.00 6.00 7.50 7.50
## [565] 7.00 4.50 6.50 7.00 7.50 18.00 6.00 8.00 7.50 8.00 7.00 7.00
## [577] 7.50 6.75 6.00 8.00 7.00 8.00 5.00 7.00 8.00 8.00 6.00 7.00
## [589] 7.00 7.50 7.00 5.00 8.00 9.00 6.00 7.50 8.50 5.50 8.00 7.00
## [601] 9.00 7.00 7.50 8.00 6.00 8.00 5.00 10.00 6.00 5.00 8.00 8.00
## [613] 5.00 10.00 6.50 6.00 6.00 5.00 8.00 7.50 9.00 8.50 5.00 7.00
## [625] 8.00 7.00 7.50 8.00 8.00 4.50 4.00 8.00 9.00 7.50 5.00 9.50
## [637] 13.00 7.00 8.75 8.00 6.50 6.50 4.00 7.00 8.00 6.00 4.00 5.00
## [649] 5.00 7.00 9.00 6.00 8.00 7.00 10.00 6.00 10.00 6.00 6.00 6.50
## [661] 6.50 7.00 8.00 7.00 8.00 7.00 4.50 7.00 6.50 8.00 7.50 4.50
## [673] 9.50 3.50 8.00 6.00 6.00 7.00 7.50 7.00 5.00 7.00 5.00 7.50
## [685] 7.00 7.00 8.00 8.00 5.50 4.30 7.00 7.50 8.00 7.00 6.00 6.50
## [697] 8.00 5.00 8.00 8.00 8.00 6.00 8.00 5.50 8.00 7.00 7.50 7.00
## [709] 8.00 7.00 6.00 4.50 6.50 4.00 3.00 9.00 5.00 9.00 8.00 6.00
## [721] 5.00 4.00 4.00 6.00 6.00 4.50 5.00 4.50 5.00 8.00 10.00 6.50
## [733] 6.00 7.50 5.00 8.00 5.00 5.00 7.45 8.00 7.00 5.50 6.00 6.00
## [745] 8.00 7.50 8.50 7.00 3.50 6.50 7.00 6.00 10.00 8.00 5.50 6.00
## [757] 5.00 6.00 7.00 6.00 6.00 5.00 5.00 5.00 7.00 8.00 5.00 7.50
## [769] 5.00 8.00 5.00 4.00 4.00 8.00 8.00 3.00 7.00 4.00 4.50 7.00
## [781] 5.00 4.50 7.00 4.00 6.50 6.50 7.00 6.00 6.00 7.50 4.00 6.00
## [793] 4.50 6.00 7.25 7.50 6.50 5.00 5.00 5.00 8.00 8.00 5.00 6.00
## [805] 7.00 7.00 9.00 6.00 7.00 8.00 7.50 7.00 4.00 5.00 5.00 6.00
## [817] 8.00 6.00 5.00 6.00 7.50 5.00 4.50 4.00 7.00 6.00 6.00 10.00
## [829] 6.00 5.00 7.00 8.00 7.50 6.50 6.00 4.50 6.00 8.00 7.50 7.25
## [841] 11.00 8.00 6.00 7.00 7.50 7.00 9.50 6.00 6.50 4.00 8.00 7.50
## [853] 5.00Primer paso: Establecer la hipótesis.
La hipotesis de la prueba T queda establecida de la siguiente forma:
Ho: No hay diferencia entre las medias de horas de sueño del personal de la salud que labora de manera remota vs presencial.
H1: Si existen diferencias entre las medias de horas de sueño del personal de la salud que labora de manera remota vs presencial.
Ambas hipótesis son acerca de los parámetros.
Segundo paso: Calcular el estadístico a emplear
En este caso, se calcula las diferencias entre las medias muestrales, ya que es lo quiero extrapolar y por tanto saber si existe o no una diferencia significativa entre las medias poblacionales de ambos grupos:
remoto= BD1[BD1$Q8=="Yes", "Q13"] #creo un objeto que contenga el promedio de horas de sueño del personal de la salud que labora de manera remota
presencial= BD1[BD1$Q8=="No", "Q13"] #creo un objeto que contenga el promedio de horas de sueño del personal de la salud que labora de manera presencial
mean(remoto, na.rm = TRUE) - mean(presencial, na.rm = TRUE) #calculo la diferencia de medias. Lo que quiero saber en adelante es si esta diferencia de media es estadísticamente significativa o no.## [1] 0.6013614#na.rm=TRUE para saber que tenemos NAs y se deben omitir al momento de generar el promedio (eliminar valores no asignados)Tercer paso: Determinar el nivel de significancia
Se trata de la probabilidad que define qué tan inusual debe ser la diferencia de medias muestrales para rechazar la H0 (que la diferencia de medias poblacionales sea 0). El valor más común es de α=0.05 a un nivel de confianza de 95%. De manera convencional establecemos la siguiente regla para nuestra prueba T:
- p-value<=0.05 Rechazo la H0 y acepto H1
- p-value>0.05 No rechazo la H0
Cuarto paso: Calcular el p-value y realizar la prueba prop.test
El p-value mide la probabilidad de observar en una muestra una diferencia de medias como la observada, si la diferencia de medias poblacional fuera cero.
t.test(BD1$Q13 ~ BD1$Q8)##
## Welch Two Sample t-test
##
## data: BD1$Q13 by BD1$Q8
## t = -5.4814, df = 495.41, p-value = 6.732e-08
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -0.8169153 -0.3858074
## sample estimates:
## mean in group No mean in group Yes
## 6.596358 7.197719Otra forma de hacerlo:
t.test(
x = remoto,
y = presencial,
alternative = "two.sided", # intervalo de confianza entre la máximo y mínimo
mu = 0,
conf.level = 0.95 #nivel de confianza (95%)
)##
## Welch Two Sample t-test
##
## data: remoto and presencial
## t = 5.4814, df = 495.41, p-value = 6.732e-08
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## 0.3858074 0.8169153
## sample estimates:
## mean of x mean of y
## 7.197719 6.596358Sexto paso: Interpretar
¿Cómo interpreto?
Recordando nuestras hipotesis:
Ho: No hay diferencia entre las medias de horas de sueño del personal de la salud que labora de manera remota vs presencial.
H1: Si existen diferencias entre las medias de horas de sueño del personal de la salud que labora de manera remota vs presencial.
Asimismo, en el paso 4, determinamos el nivel de significancia de la siguiente manera:
Si el p-value del t test es <=0.05 Rechazo la H0 y se afirma H1.
Si el p-value del t test es >0.05 No rechazo la H0
Entonces, vemos que el p-value es 6.732e-08, y es menor al alpha (0.05), entonces acepto la H1. Por tanto, existe una diferencia estadísticamente significativa entre las medias de horas de sueño del personal de la salud que labora de manera remota vs presencial con un 95% de confianza.
Paso FINAL: Graficar
Otro método para evaluar la comparación entre grupos es realizar un gráfico de medias con intervalos de confianza de cada grupo.
#install.packages("Rmisc") #necesitamos instalar "Rmisc" para poder activar la función CI (intervalos de confianza)
library(Rmisc)## Loading required package: lattice## Loading required package: plyrci.indicador1 <- CI(remoto, ci=0.95)
ci.indicador2 <- CI(presencial, ci=0.95)library(gplots)##
## Attaching package: 'gplots'## The following object is masked from 'package:stats':
##
## lowessnames(BD1)## [1] "StartDate" "EndDate" "Status"
## [4] "IPAddress" "Progress" "Duration (in seconds)"
## [7] "Finished" "RecordedDate" "ResponseId"
## [10] "RecipientLastName" "RecipientFirstName" "RecipientEmail"
## [13] "ExternalReference" "LocationLatitude" "LocationLongitude"
## [16] "DistributionChannel" "UserLanguage" "Q1"
## [19] "Q2" "Q3" "Q4"
## [22] "Q5" "Q5_6_TEXT" "Q6"
## [25] "Q7" "Q8" "Q8a"
## [28] "Q9" "Q9a" "Q9b"
## [31] "Q10" "Q11" "Q11a"
## [34] "Q11b" "Q12" "Q13"
## [37] "Q14" "Q14a" "Q15"
## [40] "Q15a" "Q16" "Q16a"
## [43] "Q16b" "Q17" "Q17a"
## [46] "Q18" "Q18a" "Q19"
## [49] "Q20" "Q21" "Q22"
## [52] "Q23" "Q24" "Q25"
## [55] "Q25a" "Q26" "Q26a"
## [58] "Q27" "Q27a" "Q28"
## [61] "Q28a" "Q29" "Q29a"
## [64] "Q29b" "Q29c" "Q29d"plotmeans(BD1$Q13 ~ BD1$Q8, BD1 = BD1) #gráfico de medias## Warning in plot.window(...): "BD1" is not a graphical parameter## Warning in plot.xy(xy, type, ...): "BD1" is not a graphical parameter## Warning in axis(side = side, at = at, labels = labels, ...): "BD1" is not a
## graphical parameter
## Warning in axis(side = side, at = at, labels = labels, ...): "BD1" is not a
## graphical parameter## Warning in box(...): "BD1" is not a graphical parameter## Warning in title(...): "BD1" is not a graphical parameter## Warning in plot.xy(xy.coords(x, y), type = type, ...): "BD1" is not a graphical
## parameter## Warning in axis(1, at = 1:length(means), labels = legends, ...): "BD1" is not a
## graphical parameter## Warning in plot.xy(xy.coords(x, y), type = type, ...): "BD1" is not a graphical
## parameter
#Añadiendo labels y removiendo líneas
plotmeans(BD1$Q13 ~ BD1$Q8, BD1 = BD1,
mean.labels = TRUE, connect = FALSE)## Warning in plot.window(...): "BD1" is not a graphical parameter## Warning in plot.xy(xy, type, ...): "BD1" is not a graphical parameter## Warning in axis(side = side, at = at, labels = labels, ...): "BD1" is not a
## graphical parameter
## Warning in axis(side = side, at = at, labels = labels, ...): "BD1" is not a
## graphical parameter## Warning in box(...): "BD1" is not a graphical parameter## Warning in title(...): "BD1" is not a graphical parameter## Warning in text.default(x, y, label = labels, col = col, ...): "BD1" is not a
## graphical parameter## Warning in plot.xy(xy.coords(x, y), type = type, ...): "BD1" is not a graphical
## parameter## Warning in axis(1, at = 1:length(means), labels = legends, ...): "BD1" is not a
## graphical parameter
Interpretación: Tal como se observa ambos intervalos de confianza no se traslapan, por lo que se puede concluir gráficamente que existe una diferencia estadísticamente significativa entre las medias de horas de sueño del personal de la salud que labora de manera remota vs presencial con un 95% de confianza en la población.
Finalmente, no olvidemos exportar el Rmd en formato PDF o Html, usando Knit
Para Html, seleccionamos en el menú Knit > Knit to html
Para pdf, seleccionamos Knit > Knit to PDF, pero antes de exportar en pdf deberá instalar lo siguiente desde su consola:
#install.packages('tinytex')
tinytex::install_tinytex()## The directory /usr/local/bin is not writable. I recommend that you make it writable. See https://github.com/yihui/tinytex/issues/24 for more info.