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Welch's T-Test

Welch's T-Test

Definition

Core Statement

Welch's T-Test is a modification of the Student's t-test that is reliable even when the two groups have unequal variances and/or unequal sample sizes. It does not pool variances, making it robust to heteroscedasticity.

Purpose

  1. Compare means of two independent groups when the homogeneity of variance assumption is violated.
  2. Serve as a safer default for two-sample mean comparisons.

When to Use

Always Use Welch's T-Test

Modern statistical advice recommends using Welch's test by default. It performs as well as Student's t-test when variances are equal, and much better when they are not.

R Default

The t.test() function in R uses Welch's t-test by default (var.equal = FALSE).

Theoretical Background

Differences from Student's T

Feature Student's T Welch's T
Variance Pooling Yes (sp) No (Separate s1, s2)
Degrees of Freedom n1+n2−2 Welch-Satterthwaite (Non-integer)
Assumption Equal variances No assumption about variances

Welch-Satterthwaite Degrees of Freedom

df=(s12n1+s22n2)2(s12/n1)2n1−1+(s22/n2)2n2−1

This formula yields a non-integer df (e.g., 23.7), which penalizes for higher uncertainty when variances differ.

Assumptions

Limitations

Pitfalls

  1. Still Sensitive to Outliers: The mean is affected by outliers. For robust comparisons, consider Mann-Whitney U Test or bootstrapping.
  2. Requires Normality: For severely non-normal data with small n, use non-parametric tests.

Python Implementation

from scipy import stats
import numpy as np

group_A = np.array([10, 12, 11, 14, 15])
group_B = np.array([18, 19, 21, 18, 20, 17, 19, 22, 25])  # Different n, different var

# Welch's T-Test (equal_var=False)
t_stat, p_val = stats.ttest_ind(group_A, group_B, equal_var=False)

print(f"Welch t-statistic: {t_stat:.3f}")
print(f"p-value: {p_val:.4f}")

R Implementation

# Welch's T-Test (default in R)
t.test(group_A, group_B)  # var.equal = FALSE is default

# Note: df will be a non-integer (e.g., 12.34)

Worked Numerical Example

CEO vs Intern Salaries (Extreme Variance)

Data:

  • Group A (Interns): Mean = 40k, SD = 5k, n=50
  • Group B (CEOs): Mean = 5M, SD = 2M, n=20

Student's t-test (Wrong): Assumes same variance. Finds pooled SD. Might fail to detect diff or exaggerate significance depending on N.

Welch's t-test (Correct):

  • Accounts for SD=5k vs SD=2,000k difference.
  • df calculation penalizes the small N of the high-variance group.
  • Result: t=−11.5,p<0.001.
  • Conclusion: Differences are real despite noise.

Interpretation Guide

Output Interpretation Edge Case Notes
Non-integer df (e.g., 23.45) Welch-Satterthwaite correction applied. The closer df is to (N1+N2-2), the more similar variances are.
p<0.05 Means are significantly different. Valid even if SD1 = 1 and SD2 = 1000.
Larger SE than Student's Welch is being conservatives. The price of robustness.
df drops drastically Severe heteroscedasticity. E.g., N=100 total, but df=15. Means variance is driven by small group.

Common Pitfall Example

"But my Textbook says check Variances first..."

Old School Workflow:

  1. Run Levene's Test.
  2. If significant → Welch.
  3. If not → Student's.

Modern Best Practice:

  • Just use Welch's.
  • Why? Testing for variance equality first is a "conditional procedure" that affects error rates.
  • Welch's test loses very little power even if variances are equal.
  • Recommendation: Set equal_var=False (Python) or rely on t.test default (R) and forget about it.