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6.4: La prueba t de muestras pareadas contraataca

Última actualización

31 oct 2022
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- 6.3: Prueba t de muestras emparejadas
- 6.5: Prueba t de muestras independientes — ¿El retorno de la prueba t?

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Debes estar preguntándote si alguna vez terminaremos de hablar de muestras pareadas pruebas t... ¿por qué estamos haciendo la ronda 2, oh no! No se preocupe, solo vamos a 1) recordarle lo que estábamos haciendo con el estudio infantil, y 2) hacer una prueba t de muestras pareadas en todo el conjunto de datos y discutir.

Recuerde, nos preguntábamos si los infantes mirarían más tiempo hacia el cantante que cantó la canción familiar durante la fase de prueba en comparación con la fase basal. Te mostramos datos de 5 infantes, y recorrimos los cómputos para la $t$ prueba. Como recordatorio, se veía así:

infantil	Línea de base	Test	diferencias	diff_from_mean	Diferencias Cuadradas
1	0.44	0.6	0.16	0.106	0.011236
2	0.41	0.68	0.27	0.216	0.046656
3	0.75	0.72	-0.03	-0.084	0.00705600000000001
4	0.44	0.28	-0.16	-0.214	0.045796
5	0.47	0.5	0.03	-0.024	0.0005759999999999
Sumas	2.51	2.78	0.27	0	0.11132
Medios	0.502	0.556	0.054	0	0.022264
				sd	0.167
				SEM	0.075
				t	0.72

library(data.table)
suppressPackageStartupMessages(library(dplyr))
all_data <- fread(
  "https://stats.libretexts.org/@api/deki/files/10603/MehrSongSpelke2016.csv")
experiment_one <- all_data %>% filter(exp1==1)
paired_sample_df <-  data.frame(infant=1:5, 
	Baseline = round(experiment_one$Baseline_Proportion_Gaze_to_Singer[1:5],
				digits=2), 
	Test = round(experiment_one$Test_Proportion_Gaze_to_Singer[1:5],
				digits=2))
paired_sample_df <- cbind(paired_sample_df,
	differences = (paired_sample_df$Test-
	paired_sample_df$Baseline))
paired_sample_df <- paired_sample_df %>%
   rbind(c("Sums",colSums(paired_sample_df[1:5,2:4]))) %>%
   rbind(c("Means",colMeans(paired_sample_df[1:5,2:4])))

paired_sample_df <-  data.frame(infant=1:5, 
	Baseline = round(experiment_one$Baseline_Proportion_Gaze_to_Singer[1:5],
                     digits=2), 
	Test = round(experiment_one$Test_Proportion_Gaze_to_Singer[1:5],
                 digits=2))
differences <-  paired_sample_df$Test-paired_sample_df$Baseline
diff_from_mean <- differences-mean(differences)
Squared_differences <- diff_from_mean^2
paired_sample_df <- cbind(paired_sample_df, 
	differences, diff_from_mean, Squared_differences)
paired_sample_df <- paired_sample_df %>%
	rbind(c("Sums",colSums(paired_sample_df[1:5,2:6]))) %>%
	rbind(c("Means",colMeans(paired_sample_df[1:5,2:6]))) %>%
	rbind(c(" "," "," "," ","sd ",round(sd(paired_sample_df[1:5,4]),
                                        digits=3))) %>%
	rbind(c(" "," "," "," ","SEM ",round(sd(paired_sample_df[1:5,4])/sqrt(5),
                                         digits=3))) %>%
	rbind(c(" "," "," "," ","t",mean(differences)/round(
      sd(paired_sample_df[1:5,4])/sqrt(5), digits=3))
    )
paired_sample_df[6,5]<-0
paired_sample_df[7,5]<-0
t.test(round(differences, digits=2), mu=0)

	One Sample t-test

data:  round(differences, digits = 2)
t = 0.72381, df = 4, p-value = 0.5092
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
 -0.1531384  0.2611384
sample estimates:
mean of x 
    0.054

Escribamos el hallazgo una vez más: La diferencia de medias fue de 0.054, $t$ (4) = .72, $p$ =.509. Ahora también podemos confirmar, que el $p$ -valor era de una prueba de dos colas. Entonces, ¿qué significa todo esto realmente?

Podemos decir que un $t$ valor con un absoluto de .72 o mayor ocurre 50.9% de las veces. Más precisamente, la distribución de no diferencias (el nulo), producirá un $t$ valor tan grande o mayor 50.9% de las veces. En otras palabras, el azar solo bueno ha producido fácilmente el $t$ valor de nuestra muestra, y la diferencia de medias que observamos o .054, fácilmente podría haber sido resultado del azar.

Pongamos rápidamente todos los datos en la $t$ prueba y volvamos a ejecutar la prueba usando todos los sujetos infantiles.

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library(data.table)
suppressPackageStartupMessages(library(dplyr))
all_data <- fread(
  "https://stats.libretexts.org/@api/deki/files/10603/MehrSongSpelke2016.csv")
experiment_one <- all_data %>% filter(exp1==1)
paired_sample_df <-  data.frame(infant=1:5, 
	Baseline = round(experiment_one$Baseline_Proportion_Gaze_to_Singer[1:5],
				digits=2), 
	Test = round(experiment_one$Test_Proportion_Gaze_to_Singer[1:5],
				digits=2))
paired_sample_df <- cbind(paired_sample_df,
	differences = (paired_sample_df$Test-
	paired_sample_df$Baseline))
paired_sample_df <- paired_sample_df %>%
   rbind(c("Sums",colSums(paired_sample_df[1:5,2:4]))) %>%
   rbind(c("Means",colMeans(paired_sample_df[1:5,2:4])))
paired_sample_df <-  data.frame(infant=1:5, 
	Baseline = round(experiment_one$Baseline_Proportion_Gaze_to_Singer[1:5],
                     digits=2), 
	Test = round(experiment_one$Test_Proportion_Gaze_to_Singer[1:5],
                 digits=2))
differences <-  paired_sample_df$Test-paired_sample_df$Baseline
diff_from_mean <- differences-mean(differences)
Squared_differences <- diff_from_mean^2
paired_sample_df <- cbind(paired_sample_df, 
	differences, diff_from_mean, Squared_differences)
paired_sample_df <- paired_sample_df %>%
	rbind(c("Sums",colSums(paired_sample_df[1:5,2:6]))) %>%
	rbind(c("Means",colMeans(paired_sample_df[1:5,2:6]))) %>%
	rbind(c(" "," "," "," ","sd ",round(sd(paired_sample_df[1:5,4]),
                                        digits=3))) %>%
	rbind(c(" "," "," "," ","SEM ",round(sd(paired_sample_df[1:5,4])/sqrt(5),
                                         digits=3))) %>%
	rbind(c(" "," "," "," ","t",mean(differences)/round(
      sd(paired_sample_df[1:5,4])/sqrt(5), digits=3))
    )
paired_sample_df[6,5]<-0
paired_sample_df[7,5]<-0

paired_sample_df <-  data.frame(infant=1:32, 
	Baseline = round(experiment_one$Baseline_Proportion_Gaze_to_Singer[1:32],
                     digits=2), 
	Test = round(experiment_one$Test_Proportion_Gaze_to_Singer[1:32],
                 digits=2))
differences <-  paired_sample_df$Test-paired_sample_df$Baseline
t.test(differences,mu=0)

library(data.table)
suppressPackageStartupMessages(library(dplyr))
all_data <- fread(
  "https://stats.libretexts.org/@api/deki/files/10603/MehrSongSpelke2016.csv")
experiment_one <- all_data %>% filter(exp1==1)
paired_sample_df <-  data.frame(infant=1:5, 
	Baseline = round(experiment_one$Baseline_Proportion_Gaze_to_Singer[1:5],
				digits=2), 
	Test = round(experiment_one$Test_Proportion_Gaze_to_Singer[1:5],
				digits=2))
paired_sample_df <- cbind(paired_sample_df,
	differences = (paired_sample_df$Test-
	paired_sample_df$Baseline))
paired_sample_df <- paired_sample_df %>%
   rbind(c("Sums",colSums(paired_sample_df[1:5,2:4]))) %>%
   rbind(c("Means",colMeans(paired_sample_df[1:5,2:4])))
paired_sample_df <-  data.frame(infant=1:5, 
	Baseline = round(experiment_one$Baseline_Proportion_Gaze_to_Singer[1:5],
                     digits=2), 
	Test = round(experiment_one$Test_Proportion_Gaze_to_Singer[1:5],
                 digits=2))
differences <-  paired_sample_df$Test-paired_sample_df$Baseline
diff_from_mean <- differences-mean(differences)
Squared_differences <- diff_from_mean^2
paired_sample_df <- cbind(paired_sample_df, 
	differences, diff_from_mean, Squared_differences)
paired_sample_df <- paired_sample_df %>%
	rbind(c("Sums",colSums(paired_sample_df[1:5,2:6]))) %>%
	rbind(c("Means",colMeans(paired_sample_df[1:5,2:6]))) %>%
	rbind(c(" "," "," "," ","sd ",round(sd(paired_sample_df[1:5,4]),
                                        digits=3))) %>%
	rbind(c(" "," "," "," ","SEM ",round(sd(paired_sample_df[1:5,4])/sqrt(5),
                                         digits=3))) %>%
	rbind(c(" "," "," "," ","t",mean(differences)/round(
      sd(paired_sample_df[1:5,4])/sqrt(5), digits=3))
    )
paired_sample_df[6,5]<-0
paired_sample_df[7,5]<-0

paired_sample_df <-  data.frame(infant=1:32, 
	Baseline = round(experiment_one$Baseline_Proportion_Gaze_to_Singer[1:32],
                     digits=2), 
	Test = round(experiment_one$Test_Proportion_Gaze_to_Singer[1:32],
                 digits=2))
differences <-  paired_sample_df$Test-paired_sample_df$Baseline
t.test(differences,mu=0)

	One Sample t-test

data:  differences
t = 2.4388, df = 31, p-value = 0.02066
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
 0.01192088 0.13370412
sample estimates:
mean of x 
0.0728125

Ahora obtenemos una respuesta muy diferente. Resumiríamos los resultados diciendo que la diferencia de medias fue de .073, t (31) = 2.44, p = 0.020. ¿Cuántos infantes totales fueron sus? Bueno los grados de libertad fueron 31, por lo que debió haber 32 infantes en el estudio. Ahora vemos un $p$ valor mucho menor. Esta también fue una prueba de dos colas, por lo que nosotros que observar un $t$ valor de 2.4 o mayor (valor absoluto) solo ocurre 2% del tiempo. Es decir, la distribución de no diferencias producirá muy raramente el valor t observado. Entonces, es poco probable que la diferencia de medias observada de .073 se deba al azar (podría haber sido por casualidad, pero eso es muy poco probable). Como resultado, podemos tener cierta confianza en concluir que algo acerca de ver y escuchar a una persona desconocida cantar una canción familiar, hace que un infante llame su atención hacia el cantante, y esto potencialmente beneficia el aprendizaje social por parte del infante.

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