T-Test Essentials: Definition, Formula and Calculation

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

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

Visualization

Visualize the data using box plots. Plot weight by groups.

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

Computation

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

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:

stat.test <- genderweight %>%
  t_test(weight ~ group, var.equal = 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 F      M         20    20     -20.8    38 2.33e-22 ****

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.

Cohen’s d for Student t-test

This effect size is calculated by dividing the mean difference between the groups by the pooled standard deviation.

Cohen’s d formula:

d = (mean1 - mean2)/pooled.sd, where:

• pooled.sd is the common standard deviation of the two groups. pooled.sd = sqrt([var1*(n1-1) + var2*(n2-1)]/[n1 + n2 -2]);
• var1 and var2 are the variances (squared standard deviation) of group1 and 2, respectively.
• n1 and n2 are the sample counts for group 1 and 2, respectively.
• mean1 and mean2 are the means of each group, respectively.

Calculation:

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