Games-Howell Test Calculator

Games-Howell Post-Hoc Test

Group 1 Data

Group 2 Data

Group 3 Data

Family-Wise Significance Level (α) Default threshold is 0.05 (5% family-wise error rate)

Significance level α must be strictly between 0 and 1 (e.g. 0.05).

Post-Hoc Unequal Variances

What Is the Games-Howell Test?

The Games-Howell Post-Hoc Test is a non-parametric-like multiple comparison procedure used following a one-way ANOVA when the assumption of homogeneity of variances (equal variances) is violated (e.g., Levene's test is significant) and sample sizes may be unequal.

While Tukey's HSD assumes equal variances across all groups using a pooled mean square error ($MS_{\text{within}}$), Games-Howell uses separate sample variances ($s_i^2$) and the **Welch-Satterthwaite adjusted degrees of freedom ($df'_{ij}$)** for each pairwise comparison.

When to Use It

Heteroscedasticity Post-Hoc: When Levene's or Bartlett's test rejects equal population variances ($p \le 0.05$).
Unequal Sample Sizes & Variances: Where standard Tukey HSD or Tukey-Kramer becomes either overly liberal or conservative.
Following Welch's ANOVA: The ideal post-hoc pairing when Welch's omnibus $F$-test was used instead of standard ANOVA.
Conservative Family-Wise Control: Controls Type I error across all $k(k-1)/2$ pairwise comparisons.

The Mathematical Formulas

$$SE_{ij} = \sqrt{\frac{s_i^2}{n_i} + \frac{s_j^2}{n_j}}, \quad q = \frac{|\bar{x}_i - \bar{x}_j|}{\sqrt{\frac{1}{2}\left(\frac{s_i^2}{n_i} + \frac{s_j^2}{n_j}\right)}}$$ $$df'_{ij} = \frac{\left( \frac{s_i^2}{n_i} + \frac{s_j^2}{n_j} \right)^2}{\frac{(s_i^2 / n_i)^2}{n_i - 1} + \frac{(s_j^2 / n_j)^2}{n_j - 1}}$$ $$\text{Simultaneous } (1-\alpha)\text{ CI} = (\bar{x}_i - \bar{x}_j) \pm \frac{q_{\text{crit}}(k, df'_{ij}, \alpha)}{\sqrt{2}} SE_{ij}$$

Where $k$ is the total number of groups compared, $s_i^2$ and $n_i$ are group sample variances and sample sizes, and $q_{\text{crit}}$ is derived from the Studentized Range distribution.

Worked Example

Scenario: Group 1 ($n_1=5, \bar{x}_1=15.0, s_1^2=5.0$), Group 2 ($n_2=4, \bar{x}_2=26.0, s_2^2=26.67$), Group 3 ($n_3=6, \bar{x}_3=40.0, s_3^2=52.4$) at $\alpha=0.05$.

Step 1: Pairwise Standard Error & Welch's $df'$ (Group 1 vs 2):
$SE_{12} = \sqrt{\frac{5.0}{5} + \frac{26.67}{4}} = \sqrt{1.0 + 6.6667} = 2.7689$
$df'_{12} = \frac{(7.6667)^2}{\frac{1.0^2}{4} + \frac{6.6667^2}{3}} = \frac{58.7778}{0.25 + 14.8148} = 3.90$.

Step 2: Calculate Studentized Range Statistic ($q$):
$q = \frac{\sqrt{2} |15.0 - 26.0|}{2.7689} = \frac{1.4142 \times 11.0}{2.7689} = 5.6179$.

Step 3: Evaluate P-Value & Significance:
$q_{\text{crit}}(k=3, df'=3.90, \alpha=0.05) \approx 5.62 \implies p = 0.0242 \le 0.05$.
Group 1 differs significantly from Group 2.

Unequal-Variance Post-Hoc Complete

Games-Howell Test Results

Overall Pairwise Findings

Significant Differences Detected

Pairwise comparisons adjusted for unequal group variances and sample sizes (α = 0.05).

Total Groups (k) 3

Comparisons 3

Total Sample (N) 15

Sig Comparisons 3 of 3

Variance Assumption Unequal (Welch)

Family Alpha (α) 0.05

Pairwise Comparisons & Simultaneous 95% Confidence Intervals

Comparison	Mean Diff (Mean_i - Mean_j)	Std Error (SE)	Welch df'	Studentized q	Simultaneous 95% CI	p-value	Significant?

Games-Howell Test Calculator

Games-Howell Post-Hoc Test

Games-Howell Test Results

Pairwise Comparisons & Simultaneous 95% Confidence Intervals

Step-by-Step Mathematical Calculations

Interpretation & Reporting

Simultaneous Confidence Intervals Plot (vs. Zero Difference)

Perfect for students, researchers, and professionals looking to build real statistical skills.