This article describes the paired t-test assumptions and provides examples of R code to check whether the assumptions are met before calculating the t-test. This also referred as:
- paired sample t test assumptions,
- assumptions for matched pairs t test and
- assumptions of dependent t test
The procedure of the paired t-test analysis is as follow:
- Calculate the difference (\(d\)) between each pair of value
- Compute the mean (\(m\)) and the standard deviation (\(s\)) of \(d\)
- Compare the average difference to 0. If there is any significant difference between the two pairs of samples, then the mean of d (\(m\)) is expected to be far from 0.
The paired samples t-test assume the following characteristics about the data:
- the two groups are paired.
- No significant outliers in the difference between the two related groups
- Normality. the difference of pairs follow a normal distribution.
In this section, we’ll perform some preliminary tests to check whether these assumptions are met.
Check paired t-test assumptions in R
Make sure you have installed the following R packages:
tidyversefor data manipulation and visualization
ggpubrfor creating easily publication ready plots
rstatixprovides pipe-friendly R functions for easy statistical analyses.
datarium: contains required data sets for this chapter.
Start by loading the following required packages:
library(tidyverse) library(ggpubr) library(rstatix)
Here, we’ll use a demo dataset
mice2 [datarium package], which contains the weight of 10 mice before and after the treatment.
# Wide format data("mice2", package = "datarium") head(mice2, 3)
## id before after ## 1 1 187 430 ## 2 2 194 404 ## 3 3 232 406
# Transform into long data: # gather the before and after values in the same column mice2.long <- mice2 %>% gather(key = "group", value = "weight", before, after) head(mice2.long, 3)
## id group weight ## 1 1 before 187 ## 2 2 before 194 ## 3 3 before 232
First, start by computing the difference between groups:
mice2 <- mice2 %>% mutate(differences = before - after) head(mice2, 3)
## id before after differences ## 1 1 187 430 -242 ## 2 2 194 404 -210 ## 3 3 232 406 -174
Outliers can be easily identified using boxplot methods, implemented in the R function
identify_outliers() [rstatix package].
mice2 %>% identify_outliers(differences)
##  id before after differences is.outlier is.extreme ## <0 rows> (or 0-length row.names)
There were no extreme outliers.
Note that, in the situation where you have extreme outliers, this can be due to: 1) data entry errors, measurement errors or unusual values.
You can include the outlier in the analysis anyway if you do not believe the result will be substantially affected. This can be evaluated by comparing the result of the t-test with and without the outlier.
It’s also possible to keep the outliers in the data and perform Wilcoxon test or robust t-test using the WRS2 package.
Check normality by groups
The normality assumption can be checked by computing the Shapiro-Wilk test for each group. If the data is normally distributed, the p-value should be greater than 0.05.
mice2 %>% shapiro_test(differences)
## # A tibble: 1 x 3 ## variable statistic p ## <chr> <dbl> <dbl> ## 1 differences 0.968 0.867
From the output, the two p-values are greater than the significance level 0.05 indicating that the distribution of the data are not significantly different from the normal distribution. In other words, we can assume the normality.
You can also create QQ plots for each group. QQ plot draws the correlation between a given data and the normal distribution.
All the points fall approximately along the (45-degree) reference line, for each group. So we can assume normality of the data.
Note that, if your sample size is greater than 50, the normal QQ plot is preferred because at larger sample sizes the Shapiro-Wilk test becomes very sensitive even to a minor deviation from normality.
In the situation where the data are not normally distributed, it’s recommended to use the non parametric Wilcoxon test.