# T-Test Essentials: Definition, Formula and Calculation

## How to Do Paired T-test in R

This article describes how to do a paired t-test in R (or in Rstudio). Note that the paired t-test is also referred as dependent t-test, related samples t-test, matched pairs t test or paired sample t test.

You will learn how to:

• Perform the paired t-test in R using the following functions :
• t_test() [rstatix package]: the result is a data frame for easy plotting using the ggpubr package.
• t.test() [stats package]: R base function.
• Interpret and report the paired t-test
• Add p-values and significance levels to a plot
• Calculate and report the paired t-test effect size using Cohen’s d. The d statistic redefines the difference in means as the number of standard deviations that separates those means. T-test conventional effect sizes, proposed by Cohen, are: 0.2 (small effect), 0.5 (moderate effect) and 0.8 (large effect) (Cohen 1998).

Contents:

#### Related Book

Practical Statistics in R II - Comparing Groups: Numerical Variables

## Prerequisites

Make sure you have installed the following R packages:

• tidyverse for data manipulation and visualization
• ggpubr for creating easily publication ready plots
• rstatix provides pipe-friendly R functions for easy statistical analyses.
• datarium: contains required data sets for this chapter.

library(tidyverse)
library(ggpubr)
library(rstatix)

## Demo data

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

We want to know, if there is any significant difference in the mean weights after treatment?

## Summary statistics

Compute some summary statistics (mean and sd) by groups:

mice2.long %>%
group_by(group) %>%
get_summary_stats(weight, type = "mean_sd")
## # A tibble: 2 x 5
##   group  variable     n  mean    sd
##   <chr>  <chr>    <dbl> <dbl> <dbl>
## 1 after  weight      10  400.  30.1
## 2 before weight      10  201.  20.0

## Calculation

### Using the R base function

There are two options for computing the independent t-test depending whether the two groups data are saved either in two different vectors or in a data frame.

Option 1. The data are saved in two different numeric vectors:

# Save the data in two different vector
before <- mice2$before after <- mice2$after
# Compute t-test
res <- t.test(before, after, paired = TRUE)
res
##
##  Paired t-test
##
## data:  before and after
## t = -30, df = 9, p-value = 1e-09
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -217 -182
## sample estimates:
## mean of the differences
##                    -199

Option 2. The data are saved in a data frame.

# Compute t-test
res <- t.test(weight ~ group, data = mice2.long, paired = TRUE)
res

As you can see, the two methods give the same results.

In the result above :

• t is the t-test statistic value (t = -25.55),
• df is the degrees of freedom (df= 9),
• p-value is the significance level of the t-test (p-value = 1.03910^{-9}).
• conf.int is the confidence interval of the mean of the differences at 95% (conf.int = [-217.1442, -181.8158]);
• sample estimates is the mean of the differences (mean = -199.48).

### Using the rstatix package

We’ll use the pipe-friendly t_test() function [rstatix package], a wrapper around the R base function t.test(). The results can be easily added to a plot using the ggpubr R package.

stat.test <- mice2.long  %>%
t_test(weight ~ group, paired = TRUE) %>%
stat.test
## # A tibble: 1 x 9
##   .y.    group1 group2    n1    n2 statistic    df             p p.signif
##   <chr>  <chr>  <chr>  <int> <int>     <dbl> <dbl>         <dbl> <chr>
## 1 weight after  before    10    10      25.5     9 0.00000000104 ****

The results above show the following components:

• .y.: the y variable used in the test.
• group1,group2: the compared groups in the pairwise tests.
• statistic: Test statistic used to compute the p-value.
• df: degrees of freedom.
• p: p-value.

Note that, you can obtain a detailed result by specifying the option detailed = TRUE.

mice2.long %>%
t_test(weight ~ group, paired = TRUE, detailed = TRUE) %>%
add_significance()
## # A tibble: 1 x 14
##   estimate .y.    group1 group2    n1    n2 statistic             p    df conf.low conf.high method alternative p.signif
##      <dbl> <chr>  <chr>  <chr>  <int> <int>     <dbl>         <dbl> <dbl>    <dbl>     <dbl> <chr>  <chr>       <chr>
## 1     199. weight after  before    10    10      25.5 0.00000000104     9     182.      217. T-test two.sided   ****

## Interpretation

The p-value of the test is 1.0410^{-9}, which is less than the significance level alpha = 0.05. We can then reject null hypothesis and conclude that the average weight of the mice before treatment is significantly different from the average weight after treatment with a p-value = 1.0410^{-9}.

## Effect size

The effect size for a paired-samples t-test can be calculated by dividing the mean difference by the standard deviation of the difference, as shown below.

Cohen’s d formula:

$d = \frac{mean_D}{SD_D}$

Where D is the differences of the paired samples values.

Calculation:

mice2.long  %>% cohens_d(weight ~ group, paired = TRUE)
## # A tibble: 1 x 7
##   .y.    group1 group2 effsize    n1    n2 magnitude
## * <chr>  <chr>  <chr>    <dbl> <int> <int> <ord>
## 1 weight after  before    8.08    10    10 large

There is a large effect size, Cohen’s d = 8.07.

## Report

We could report the result as follow: The average weight of mice was significantly increased after treatment, t(9) = 25.5, p < 0.0001, d = 8.07.

Visualize the results:

# Create a box plot
bxp <- ggpaired(mice2.long, x = "group", y = "weight",
order = c("before", "after"),
ylab = "Weight", xlab = "Groups")

# Add p-value and significance levels
stat.test <- stat.test %>% add_xy_position(x = "group")
bxp +
stat_pvalue_manual(stat.test, tip.length = 0) +
labs(subtitle = get_test_label(stat.test, detailed= TRUE)) ## Summary

This article shows how to perform the paired t-test in R/Rstudio using two different ways: the R base function t.test() and the t_test() function in the rstatix package. We also describe how to interpret and report the t-test results.

• Ezequiel

This package has made my life easier several times. I was wondering how would you run a paired t-test if you had before and after but for 2 datasets

• Ezequiel

And is it possible to do it when you have 2 subgroups?

#### Course Curriculum

##### Teacher 