T-Test Essentials: Definition, Formula and Calculation

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.

Start by loading the following required packages:

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

Research questions

Typical research questions are:

1. whether the mean difference (\(m\)) is equal to 0?
2. whether the mean difference (\(m\)) is less than 0?
3. whether the mean difference (\(m\)) is greather than 0?

statistical hypotheses

In statistics, we can define the corresponding null hypothesis (\(H_0\)) as follow:

1. \(H_0: m = 0\)
2. \(H_0: m \leq 0\)
3. \(H_0: m \geq 0\)

The corresponding alternative hypotheses (\(H_a\)) are as follow:

1. \(H_a: m \ne 0\) (different)
2. \(H_a: m > 0\) (greater)
3. \(H_a: m < 0\) (less)

Note that:

• Hypotheses 1) are called two-tailed tests
• Hypotheses 2) and 3) are called one-tailed tests

Formula

The paired t-test statistics value can be calculated using the following formula:

\[
t = \frac{m}{s/\sqrt{n}}
\]

where,

• m is the mean differences
• n is the sample size (i.e., size of d).
• s is the standard deviation of d

We can compute the p-value corresponding to the absolute value of the t-test statistics (|t|) for the degrees of freedom (df): \(df = n - 1\).

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

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

Visualization

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

Assumptions and preleminary tests

The paired samples t-test assume the following characteristics about the data:

• the two groups are paired. In our example, this is the case since the data have been collected from measuring twice the weight of the same mice.
• 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.

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

mice2 %>% identify_outliers(differences)

## [1] id          before      after       differences is.outlier  is.extreme 
## <0 rows> (or 0-length row.names)

mice2 %>% shapiro_test(differences) 

## # A tibble: 1 x 3
##   variable    statistic     p
##   <chr>           <dbl> <dbl>
## 1 differences     0.968 0.867

ggqqplot(mice2, "differences")

Computation

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

We’ll use the pipe-friendly t_test() function [rstatix package], a wrapper around the R base function t.test().

stat.test <- mice2.long  %>% 
  t_test(weight ~ group, paired = TRUE) %>%
  add_significance()
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.

To compute one tailed paired t-test, you can specify the option alternative as follow.

• if you want to test whether the average weight before treatment is less than the average weight after treatment, type this:

mice2.long  %>%
  t_test(weight ~ group, paired = TRUE, alternative = "less")

• Or, if you want to test whether the average weight before treatment is greater than the average weight after treatment, type this

mice2.long  %>%
  t_test(weight ~ group, paired = TRUE, alternative = "greater")

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

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.

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 describes the formula and the basics of the paired t-test or dependent t-test. Examples of R codes are provided to check the assumptions, computing the test and the effect size, interpreting and reporting the results.