# T-Test Essentials: Definition, Formula and Calculation

## How to Do a T-test in R: Calculation and Reporting

This article describes how to do a t-test in R (or in Rstudio). You will learn how to:

• Perform a t-test in R using the following functions :
• t_test() [rstatix package]: a wrapper around the R base function t.test(). The result is a data frame, which can be easily added to a plot using the ggpubr R package.
• t.test() [stats package]: R base function to conduct a t-test.
• Interpret and report the t-test
• Add p-values and significance levels to a plot
• Calculate and report the 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).

We will provide examples of R code to run the different types of t-test in R, including the:

• one-sample t-test
• two-sample t-test (also known as independent t-test or unpaired t-test)
• paired t-test (also known as dependent t-test or matched pairs t test)

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.

Start by loading the following required packages:

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

## One-sample t-test

### 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

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

### 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

### Calculation

#### Using the R base function

res <- t.test(mice$weight, mu = 25) res ## ## One Sample t-test ## ## data: mice$weight
## t = -8, df = 9, p-value = 2e-05
## alternative hypothesis: true mean is not equal to 25
## 95 percent confidence interval:
##  18.8 21.5
## sample estimates:
## mean of x
##      20.1

In the result above :

• t is the t-test statistic value (t = -8.105),
• df is the degrees of freedom (df= 9),
• p-value is the significance level of the t-test (p-value = 1.99510^{-5}).
• conf.int is the confidence interval of the mean at 95% (conf.int = [18.7835, 21.4965]);
• sample estimates is the mean value of the sample (mean = 20.14).

#### 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 <- 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.

You can obtain a detailed result by specifying the option detailed = TRUE in the function t_test().

mice %>% t_test(weight ~ 1, mu = 25, detailed = TRUE)
## # A tibble: 1 x 12
##   estimate .y.    group1 group2         n statistic       p    df conf.low conf.high method alternative
## *    <dbl> <chr>  <chr>  <chr>      <int>     <dbl>   <dbl> <dbl>    <dbl>     <dbl> <chr>  <chr>
## 1     20.1 weight 1      null model    10     -8.10 0.00002     9     18.8      21.5 T-test two.sided

### Interpretation

The p-value of the test is 210^{-5}, which is less than the significance level alpha = 0.05. We can conclude that the mean weight of the mice is significantly different from 25g with a p-value = 210^{-5}.

### 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

Recall that, t-test conventional effect sizes, proposed by Cohen J. (1998), are: 0.2 (small effect), 0.5 (moderate effect) and 0.8 (large effect) (Cohen 1998). As the effect size, d, is 2.56 you can conclude that there is a large effect.

### Reporting

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 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.

The results can be visualized using either a box plot or a density plot.

#### Box Plot

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.

# Create the box-plot
bxp <- ggboxplot(
mice\$weight, width = 0.5, add = c("mean", "jitter"),
ylab = "Weight (g)", xlab = FALSE
)
# Add significance levels
bxp + labs(subtitle = get_test_label(stat.test, detailed = TRUE))

#### Density plot

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))

## Two-sample t-test

The two-sample t-test is also known as the independent t-test. The independent samples t-test comes in two different forms:

• the standard Student’s t-test, which assumes that the variance of the two groups are equal.
• the Welch’s t-test, which is less restrictive compared to the original Student’s test. This is the test where you do not assume that the variance is the same in the two groups, which results in the fractional degrees of freedom.

The two methods give very similar results unless both the group sizes and the standard deviations are very different.

### Demo data

Demo dataset: genderweight [in datarium package] containing the weight of 40 individuals (20 women and 20 men).

Load the data and show some random rows by groups:

# Load the data
data("genderweight", package = "datarium")
# Show a sample of the data by group
set.seed(123)
genderweight %>% sample_n_by(group, size = 2)
## # A tibble: 4 x 3
##   id    group weight
##   <fct> <fct>  <dbl>
## 1 6     F       65.0
## 2 15    F       65.9
## 3 29    M       88.9
## 4 37    M       77.0

We want to know, whether the average weights are different between groups.

### Summary statistics

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

genderweight %>%
group_by(group) %>%
get_summary_stats(weight, type = "mean_sd")
## # A tibble: 2 x 5
##   group variable     n  mean    sd
##   <fct> <chr>    <dbl> <dbl> <dbl>
## 1 F     weight      20  63.5  2.03
## 2 M     weight      20  85.8  4.35

### Calculation

Recall that, by default, R computes the Welch t-test, which is the safer one. This is the test where you do not assume that the variance is the same in the two groups, which results in the fractional degrees of freedom. If you want to assume the equality of variances (Student t-test), specify the option var.equal = TRUE.

#### Using the R base function

res <- t.test(weight ~ group, data = genderweight)
res
##
##  Welch Two Sample t-test
##
## data:  weight by group
## t = -20, df = 30, p-value <2e-16
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -24.5 -20.1
## sample estimates:
## mean in group F mean in group M
##            63.5            85.8

In the result above :

• t is the t-test statistic value (t = -20.79),
• df is the degrees of freedom (df= 26.872),
• p-value is the significance level of the t-test (p-value = 4.29810^{-18}).
• conf.int is the confidence interval of the means difference at 95% (conf.int = [-24.5314, -20.1235]);
• sample estimates is the mean value of the sample (mean = 63.499, 85.826).

#### Using the rstatix package

stat.test <- genderweight %>%
t_test(weight ~ group) %>%
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 F      M         20    20     -20.8  26.9 4.30e-18 ****

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.

genderweight %>%
t_test(weight ~ group, detailed = TRUE) %>%
add_significance()
## # A tibble: 1 x 16
##   estimate estimate1 estimate2 .y.    group1 group2    n1    n2 statistic        p    df conf.low conf.high method alternative p.signif
##      <dbl>     <dbl>     <dbl> <chr>  <chr>  <chr>  <int> <int>     <dbl>    <dbl> <dbl>    <dbl>     <dbl> <chr>  <chr>       <chr>
## 1    -22.3      63.5      85.8 weight F      M         20    20     -20.8 4.30e-18  26.9    -24.5     -20.1 T-test two.sided   ****

### Interpretation

The p-value of the test is 4.310^{-18}, which is less than the significance level alpha = 0.05. We can conclude that men’s average weight is significantly different from women’s average weight with a p-value = 4.310^{-18}.

### Cohen’s d for Student t-test

There are multiple version of Cohen’s d for Student t-test. The most commonly used version of the Student t-test effect size, comparing two groups ($A$ and $B$), is calculated by dividing the mean difference between the groups by the pooled standard deviation.

Cohen’s d formula:

$d = \frac{m_A - m_B}{SD_{pooled}}$

where,

• $m_A$ and $m_B$ represent the mean value of the group A and B, respectively.
• $n_A$ and $n_B$ represent the sizes of the group A and B, respectively.
• $SD_{pooled}$ is an estimator of the pooled standard deviation of the two groups. It can be calculated as follow :
$SD_{pooled} = \sqrt{\frac{\sum{(x-m_A)^2}+\sum{(x-m_B)^2}}{n_A+n_B-2}}$

Calculation. If the option var.equal = TRUE, then the pooled SD is used when computing the Cohen’s d.

genderweight %>%  cohens_d(weight ~ group, var.equal = TRUE)
## # A tibble: 1 x 7
##   .y.    group1 group2 effsize    n1    n2 magnitude
## * <chr>  <chr>  <chr>    <dbl> <int> <int> <ord>
## 1 weight F      M        -6.57    20    20 large

There is a large effect size, d = 6.57.

Note that, for small sample size (< 50), the Cohen’s d tends to over-inflate results. There exists a Hedge’s Corrected version of the Cohen’s d (Hedges and Olkin 1985), which reduces effect sizes for small samples by a few percentage points. The correction is introduced by multiplying the usual value of d by (N-3)/(N-2.25) (for unpaired t-test) and by (n1-2)/(n1-1.25) for paired t-test; where N is the total size of the two groups being compared (N = n1 + n2).

### Cohen’s d for Welch t-test

The Welch test is a variant of t-test used when the equality of variance can’t be assumed. The effect size can be computed by dividing the mean difference between the groups by the “averaged” standard deviation.

Cohen’s d formula:

$d = \frac{m_A - m_B}{\sqrt{(Var_1 + Var_2)/2}}$

where,

• $m_A$ and $m_B$ represent the mean value of the group A and B, respectively.
• $Var_1$ and $Var_2$ are the variance of the two groups.

Calculation:

genderweight %>% cohens_d(weight ~ group, var.equal = FALSE)
## # A tibble: 1 x 7
##   .y.    group1 group2 effsize    n1    n2 magnitude
## * <chr>  <chr>  <chr>    <dbl> <int> <int> <ord>
## 1 weight F      M        -6.57    20    20 large

Note that, when group sizes are equal and group variances are homogeneous, Cohen’s d for the standard Student and Welch t-tests are identical.

### Report

We could report the result as follow:

The mean weight in female group was 63.5 (SD = 2.03), whereas the mean in male group was 85.8 (SD = 4.3). A Welch two-samples t-test showed that the difference was statistically significant, t(26.9) = -20.8, p < 0.0001, d = 6.57; where, t(26.9) is shorthand notation for a Welch t-statistic that has 26.9 degrees of freedom.

Visualize the results:

# Create a box-plot
bxp <- ggboxplot(
genderweight, x = "group", y = "weight",
ylab = "Weight", xlab = "Groups", add = "jitter"
)

# 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))

## Paired t-test

### 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

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

In the result above :

• t is the t-test statistic value (t = -20.79),
• df is the degrees of freedom (df= 26.872),
• p-value is the significance level of the t-test (p-value = 4.29810^{-18}).
• conf.int is the confidence interval of the mean of the differences at 95% (conf.int = [-24.5314, -20.1235]);
• sample estimates is the mean of the differences (mean = 63.499, 85.826).

#### Using the rstatix 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 conduct a 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.

## References

Cohen, J. 1998. Statistical Power Analysis for the Behavioral Sciences. 2nd ed. Hillsdale, NJ: Lawrence Erlbaum Associates.

Hedges, Larry, and Ingram Olkin. 1985. “Statistical Methods in Meta-Analysis.” In Stat Med. Vol. 20. doi:10.2307/1164953.

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