Using Bootstrap to Conduct Inference about Two Means: Dependent Samples

We are going to work with the data in Table 1 from Section 11.2A.

First, install the Mosaic package, if necessary.

install.packages("mosaic")

Read the data from Table 1 into R.

Table1 <- read.csv("https://sullystats.github.io/Statistics6e/Data/Chapter11/Table1.csv")
head(Table1,n=3)

##   Dominant Nondominant
## 1    0.177       0.179
## 2    0.210       0.202
## 3    0.186       0.208

First use the transform command to obtain the differences and store the result in a new data frame. Call the new variable differences.

Table1_diff <- transform(Table1,difference=Dominant-Nondominant)
head(Table1_diff)

##   Dominant Nondominant difference
## 1    0.177       0.179     -0.002
## 2    0.210       0.202      0.008
## 3    0.186       0.208     -0.022
## 4    0.189       0.184      0.005
## 5    0.198       0.215     -0.017
## 6    0.194       0.193      0.001

Following along with Example 1, we want to test

\(H_0:\mu_d = 0\)
\(H_1:\mu_d < 0\)

where difference = Dominant - Nondominant

library(mosaic)
mean(~difference,data=Table1_diff)

## [1] -0.01316667

We must add 0.0132 to each observation so that the mean of the differenced data is 0.

Table1_adj <- transform(Table1_diff,diff_adj = difference + 0.0132)
mean(~diff_adj,data=Table1_adj)

## [1] 3.333333e-05

The mean of the adjusted data is very close to 0 (which is the goal). Now, let’s find 2000 resamples of this data, find the mean of each resample, and display the first four bootstrap means.

set.seed(43)      #Use a seed so everyone gets the same results. 
bootstrap <- do(2000)*mean(~diff_adj,data=resample(Table1_adj))  #Find 2000 bootstrap means
head(bootstrap,n=4)

##           mean
## 1 -0.002216667
## 2  0.002450000
## 3 -0.003633333
## 4  0.006616667

Now, find the proportion of bootstrap means that are -0.0132 or less.

prop(~(mean <= -0.0132),data=bootstrap)

## prop_TRUE 
##    0.0025

The estimated P-value is 0.0025. The P-value using Students t-distribution is 0.009.