Introduction
After conducting a one-way ANOVA and finding significant differences between group means, the next logical step is a post-hoc test. These tests help determine which specific groups differ significantly. There are several methods available—each with its strengths, assumptions, and use cases.
Why Use Post-Hoc Tests?
ANOVA tells us that at least one group differs, but not which ones. Post-hoc tests allow for detailed pairwise or contrast comparisons while controlling for Type I error. Choosing the right test is crucial to maintain statistical validity.
Tukey's Honestly Significant Difference (HSD)
Tukey HSD is a widely used post-hoc test designed for pairwise comparisons. It controls the family-wise error rate and is most reliable when sample sizes are equal and assumptions of normality and homogeneity of variances are met.
- Conservativeness: Moderate
- Assumptions: Equal variances, normal distribution
- Best for: All pairwise comparisons
Scheffé's Test
Scheffé’s test is more conservative but extremely versatile. Unlike Tukey, it allows for any linear combination of means—not just pairwise comparisons. It's particularly useful for exploring complex hypotheses.
- Conservativeness: High
- Assumptions: Same as ANOVA
- Best for: Complex or custom contrasts
Duncan's Multiple Range Test
Duncan’s test is less conservative than Tukey’s and focuses on detecting as many differences as possible. It uses a stepwise approach and has a higher chance of identifying significant differences, but also a higher risk of Type I error.
- Conservativeness: Low
- Error Control: Per-comparison, not family-wise
- Best for: When liberal comparisons are acceptable
Other Common Post-Hoc Tests
Bonferroni Correction
A simple and very conservative method that adjusts the significance level based on the number of comparisons. It’s easy to use but reduces statistical power.
Games-Howell Test
Ideal when assumptions of homogeneity of variance are violated. It accommodates unequal sample sizes and is a robust alternative to Tukey in such cases.
Comparison Table
| Test | Best For | Conservative? | Handles Unequal Variances? |
|---|---|---|---|
| Tukey HSD | All pairwise, equal sizes/vars | Moderate | No |
| Scheffé | Any contrast | Very | No |
| Duncan | Liberal pairwise | No | No |
| Bonferroni | Small # of comparisons | High | No |
| Games-Howell | Unequal variances/sizes | Moderate | Yes |
Try It Yourself: One-Way ANOVA Calculator
You can perform a one-way ANOVA easily using this free tool from statistical-calculators.site. After finding significant results, refer back here to choose the best post-hoc test.