Tukey's HSD

Statistics · Axiom Academy

Post-hoc analysis for pairwise comparisons after ANOVA When ANOVA shows a significant difference between groups, we need to identify which specific pairs of groups differ. Simply running multiple t-tests would inflate our Type I error rate. Example: With 4 groups, we have 6 pairwise comparisons. The family-wise error rate without correction would be approximately 1 - (0.95)^6 ≈ 0.265, much higher than 0.05! Tukey's HSD calculates the minimum difference between two group means needed to declare them significantly different: q = studentized range statistic (from Tukey's q-table) MSW = Mean Square Within groups (from ANOVA) n = sample size per group (assumed equal) 3. Understanding the q Distribution The studentized range statistic q is found using: k = number of groups being compared df = degrees of freedom for error (N - k) α = significance level (typically 0.05) For each pair of group means, we calculate the absolute difference and compare it to HSD: If |mean₁ - mean₂| > HSD → Groups are significantly different If |mean₁ - mean₂| ≤ HSD → No significant difference 5. Confidence Intervals for Mean Differences Tukey's HSD can also be expressed as confidence intervals for each pairwise difference: Scenario: A researcher tests 3 teaching methods (A, B, C) with n = 10 students each. ANOVA yields F = 8.45 (p < 0.01) with MSW = 25. Group Means: A = 75, B = 82, C = 88

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