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 (\(m\)) of the sample is equal to the theoretical mean (\(\mu\))?
2. whether the mean (\(m\)) of the sample is less than the theoretical mean (\(\mu\))?
3. whether the mean (\(m\)) of the sample is greater than the theoretical mean (\(\mu\))?

Statistical hypotheses

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

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

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

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

Formula

The the one-sample t-test formula can be written as follow:

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

where,

• \(m\) is the sample mean
• \(n\) is the sample size
• \(s\) is the sample standard deviation with \(n-1\) degrees of freedom
• \(\mu\) is the theoretical mean

The p-value, corresponding to the absolute value of the t-test statistics (|t|), is computed for the degrees of freedom (df): df = n - 1.

How to interpret the one-sample t-test results?

Demo data

Demo dataset: mice [in datarium package]. Contains the weight of 10 mice:

# Load and inspect the data
data(mice, package = "datarium")
head(mice, 3)

## # A tibble: 3 x 2
##   name  weight
##   <chr>  <dbl>
## 1 M_1     18.9
## 2 M_2     19.5
## 3 M_3     23.1

Summary statistics

Compute some summary statistics: count (number of subjects), mean and sd (standard deviation)

mice %>% get_summary_stats(weight, type = "mean_sd")

## # A tibble: 1 x 4
##   variable     n  mean    sd
##   <chr>    <dbl> <dbl> <dbl>
## 1 weight      10  20.1  1.90

Visualization

Create a boxplot to visualize the distribution of mice weights. Add also jittered points to show individual observations. The big dot represents the mean point.

bxp <- ggboxplot(
  mice$weight, width = 0.5, add = c("mean", "jitter"), 
  ylab = "Weight (g)", xlab = FALSE
  )
bxp

Assumptions and preleminary tests

The one-sample t-test assumes the following characteristics about the data:

• No significant outliers in the data
• Normality. the data should be approximately normally distributed

In this section, we’ll perform some preliminary tests to check whether these assumptions are met.

mice %>% identify_outliers(weight)

## [1] name       weight     is.outlier is.extreme
## <0 rows> (or 0-length row.names)

mice %>% shapiro_test(weight)

## # A tibble: 1 x 3
##   variable statistic     p
##   <chr>        <dbl> <dbl>
## 1 weight       0.923 0.382

ggqqplot(mice, x = "weight")

Calculate one-sample t-test in R

We want to know, whether the average weight of the mice differs from 25g (two-tailed test)?

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

stat.test <- mice %>% t_test(weight ~ 1, mu = 25)
stat.test

## # A tibble: 1 x 7
##   .y.    group1 group2         n statistic    df       p
## * <chr>  <chr>  <chr>      <int>     <dbl> <dbl>   <dbl>
## 1 weight 1      null model    10     -8.10     9 0.00002

The results above show the following components:

• .y.: the outcome variable used in the test.
• group1,group2: generally, the compared groups in the pairwise tests. Here, we have null model (one-sample test).
• statistic: test statistic (t-value) used to compute the p-value.
• df: degrees of freedom.
• p: p-value.

Note that:

1. if you want to test whether the mean weight of mice is less than 25g (one-tailed test), type this:

mice %>% t_test(weight ~ 1, mu = 25, alternative = "less")

2. Or, if you want to test whether the mean weight of mice is greater than 25g (one-tailed test), type this:

mice %>% t_test(weight ~ 1, mu = 25, alternative = "greater")

To calculate t-test using the R base function, type this:

t.test(mice$weight, mu = 25)

Effect size

To calculate an effect size, called Cohen's d, for the one-sample t-test you need to divide the mean difference by the standard deviation of the difference, as shown below. Note that, here: sd(x-mu) = sd(x).

Cohen’s d formula:

\[
d = \frac{m-\mu}{s}
\]

• \(m\) is the sample mean
• \(s\) is the sample standard deviation with \(n-1\) degrees of freedom
• \(\mu\) is the theoretical mean against which the mean of our sample is compared (default value is mu = 0).

Calculation:

mice %>% cohens_d(weight ~ 1, mu = 25)

## # A tibble: 1 x 6
##   .y.    group1 group2     effsize     n magnitude
## * <chr>  <chr>  <chr>        <dbl> <int> <ord>    
## 1 weight 1      null model    10.6    10 large

Report

We could report the result as follow:

A one-sample t-test was computed to determine whether the recruited mice average weight was different to the population normal mean weight (25g).

The mice weight value were normally distributed, as assessed by Shapiro-Wilk’s test (p > 0.05) and there were no extreme outliers in the data, as assessed by boxplot method.

The measured mice mean weight (20.14 +/- 1.94) was statistically significantly lower than the population normal mean weight 25 (t(9) = -8.1, p < 0.0001, d = 2.56); where t(9) is shorthand notation for a t-statistic that has 9 degrees of freedom.

Create a box plot with p-value:

bxp + labs(
  subtitle = get_test_label(stat.test, detailed = TRUE)
  )

Create a density plot with p-value:

• Red line corresponds to the observed mean
• Blue line corresponds to the theoretical mean

ggdensity(mice, x = "weight", rug = TRUE, fill = "lightgray") +
  scale_x_continuous(limits = c(15, 27)) +
  stat_central_tendency(type = "mean", color = "red", linetype = "dashed") +
  geom_vline(xintercept = 25, color = "blue", linetype = "dashed") + 
  labs(subtitle = get_test_label(stat.test,  detailed = TRUE))

Summary

This article describes the basics and the formula of the on-sample t-test. Additionally, it provides an example for:

• checking the on-sample t-test assumptions,
• calculating the one-sample t-test in R using the t_test() function [rstatix package],
• computing Cohen’s d for one-sample t-test
• Interpreting and reporting the results