Post-Hoc Procedures & Repeated-Measures Sphericity
Choosing between Tukey HSD, Bonferroni, Dunnett, and Games-Howell, and applying Greenhouse-Geisser corrections when Mauchly's test fails.
In this lesson you will learn
- 1Match each post-hoc procedure (Tukey HSD, Bonferroni, Dunnett, Games-Howell) to its specific trial structure and variance profile.
- 2Define Sphericity in Repeated Measures ANOVA as equality of variances of all pairwise timepoint difference scores.
- 3Apply the Greenhouse-Geisser or Huynh-Feldt epsilon (ε) multiplier to degrees of freedom when Mauchly's test yields p < .05.
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Post-Hoc Decision Matrix & Sphericity Corrections
This section compares the four primary post-hoc tests and explains how Mauchly's sphericity test and Greenhouse-Geisser ε work in longitudinal studies.
Reporting unadjusted Repeated Measures ANOVA degrees of freedom when Mauchly's p < .05 inflates Type I error and is routinely flagged by peer reviewers.
Once a significant omnibus ANOVA establishes that group means differ, select the post-hoc procedure that matches your contrast plan and variance diagnostics:
| Procedure | Optimal Scenario | Key Property |
|---|---|---|
| Tukey HSD | All pairwise comparisons across equal-variance groups. | Uses studentized range distribution; higher power than Bonferroni for all pairs. |
| Dunnett's Test | Comparing (k − 1) active treatments against a single reference Control only. | Saves alpha by skipping treatment-vs-treatment tests. |
| Bonferroni / Holm | A small number of pre-planned contrasts (or after Kruskal-Wallis Dunn test). | Conservative when number of comparisons is large (α / c). |
| Games-Howell | All pairwise comparisons when Levene's test fails (p < .05) or group sizes differ markedly. | Uses Welch-adjusted df per pair without pooling variances. |
Repeated Measures & Sphericity: When the same cohort is measured across k ≥ 3 timepoints (e.g. Baseline, Month 1, Month 3, Month 6), Repeated Measures ANOVA requires Sphericity—the variances of the differences between all pairs of timepoints must be equal. If Mauchly's Test of Sphericity yields p < .05, apply the Greenhouse-Geisser (ε) adjustment to shrink the degrees of freedom.
Repeated-Measures Sphericity & Greenhouse-Geisser Calculator
See how epsilon (ε) scales degrees of freedom when Mauchly's sphericity test is significant
Clinical Example
A cohort of 25 asthma patients has FEV1 measured across 4 timepoints (Baseline, 1h, 3h, 6h). Mauchly's test is significant (W = 0.48, p = .003), with Greenhouse-Geisser ε = 0.68.
Interpretation: Multiplying the unadjusted degrees of freedom (3 and 72) by ε = 0.68 yields decimal degrees of freedom F(2.04, 48.96), which immediately signals to reviewers that sphericity was properly corrected.
Common Pitfall
Using Bonferroni across 6 or more groups (15+ pairwise tests), which lowers the per-test alpha so drastically that genuine treatment effects are missed (Type II error).
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