T-Test Essentials: Definition, Formula and Calculation

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 compting the Cohen’s d.

ToothGrowth %>% cohens_d(len ~ supp, var.equal = TRUE)

## # A tibble: 1 x 7
##   .y.   group1 group2 effsize    n1    n2 magnitude
## * <chr> <chr>  <chr>    <dbl> <int> <int> <ord>    
## 1 len   OJ     VC       0.495    30    30 small

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).

ToothGrowth %>% cohens_d(
  len ~ supp, var.equal = TRUE, 
  hedges.correction = TRUE
  )

## # A tibble: 1 x 7
##   .y.   group1 group2 effsize    n1    n2 magnitude
## * <chr> <chr>  <chr>    <dbl> <int> <int> <ord>    
## 1 len   OJ     VC       0.488    30    30 small

Cohen’s d for Welch 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:

ToothGrowth %>% cohens_d(len ~ supp, var.equal = FALSE)

## # A tibble: 1 x 7
##   .y.   group1 group2 effsize    n1    n2 magnitude
## * <chr> <chr>  <chr>    <dbl> <int> <int> <ord>    
## 1 len   OJ     VC       0.495    30    30 small