Why is Welch's t-test preferred over Student's t-test?

Welch's t-test does not assume equal variances between groups (homoscedasticity). Empirical studies (e.g., Delacre et al., 2017) demonstrate that Welch's t-test maintains robust Type I error rates even when variances and sample sizes are unequal.

How is Cohen's d interpreted?

According to Cohen (1988), d = 0.2 represents a small effect, d = 0.5 represents a medium effect, and d = 0.8 or higher indicates a large effect size.

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Theory & Interpretation of the t-Test

The two-sample $t$-test assesses whether the empirical means of two populations differ significantly from each other. Originally developed by William Sealy Gosset in 1908 under the pseudonym "Student", the test remains the foundational inferential tool across clinical medicine, pharmacology, and behavioral epidemiology.

Why Welch's t-Test is Recommended by Default

Standard Student's $t$-test requires the assumption of homoscedasticity (equal variances $\sigma_1^2 = \sigma_2^2$). In biomedical practice, unequal variances between experimental arms are standard rather than anomalous. When sample sizes and variances differ simultaneously, the classical Student's $t$-test produces heavily distorted Type I error rates.

Welch's $t$-test (Welch, 1947) corrects for heteroscedasticity without requiring preliminary variance tests (such as Levene's test, which inflate family-wise error rates). Leading biometricians (Delacre et al., 2017) recommend using Welch's test as the default standard for all two-sample continuous comparisons.

Mathematical Formulas

The Welch $t$-statistic is defined as:

$$t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$$

The degrees of freedom are approximated via the Welch-Satterthwaite equation:

$$\nu \approx \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{(s_1^2/n_1)^2}{n_1 - 1} + \frac{(s_2^2/n_2)^2}{n_2 - 1}}$$

Effect Size: Cohen's d

Statistical significance ($p < 0.05$) indicates that an observed difference is unlikely due to chance, but does not measure clinical importance. Cohen's $d$ quantifies the magnitude of the difference in standardized standard deviation units:

$$d = \frac{|\bar{x}_1 - \bar{x}_2|}{s_{\text{pooled}}}, \quad \text{where } s_{\text{pooled}} = \sqrt{\frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1 + n_2 - 2}}$$

$d = 0.20$: Small effect size (subtle clinical difference).
$d = 0.50$: Medium effect size (readily noticeable to an experienced practitioner).
$d \ge 0.80$: Large effect size (substantial clinical impact).

Assumptions of the Test

Continuous Metric: The dependent variable is measured on an interval or ratio scale.
Independence of Observations: Data points within each group and between groups are mutually independent (for independent samples).
Normality: The distributions within each group approximate a Gaussian curve. Note that by the Central Limit Theorem (CLT), with $n \ge 30$ per arm, the $t$-test is robust to moderate violations of normality.

Academic References

Student. (1908). The probable error of a mean. Biometrika, 6(1), 1–25.
Welch, B. L. (1947). The generalization of 'Student's' problem when several different population variances are involved. Biometrika, 34(1/2), 28–35.
Delacre, M., Lakens, D., & Leys, C. (2017). Why psychologists should by default use Welch's t-test instead of Student's t-test. International Review of Social Psychology, 30(1), 92–101.
Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Lawrence Erlbaum Associates.

Welch Degrees of Freedom (df)	62.84
One-Tailed p-Value	0.0045
95% Confidence Interval	[-15.32, -2.28]
Effect Size (Cohen's d)	0.66

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