The Role of Post-Hoc Tests in Repeated Measures ANOVA with Sphericity Issues

📚 Quick Study Guide

• 🧪 Repeated Measures ANOVA: A statistical test used to compare the means of related groups (i.e., same subjects tested multiple times).
• 🔄 Sphericity: An assumption of repeated measures ANOVA that the variances of the differences between all possible pairs of related groups are equal.
• ⚠️ Sphericity Violation: Often checked using Mauchly's Test. If significant (p < 0.05), sphericity is violated.
• ✔️ Corrections for Sphericity Violation: When sphericity is violated, adjustments like Greenhouse-Geisser or Huynh-Feldt are applied to the degrees of freedom, making the test more conservative.
• 🔍 Post-Hoc Tests: Used after a significant ANOVA result to determine which specific pairs of means are significantly different.
• 📊 Common Post-Hoc Tests (with Sphericity Correction):
  • 📌 Bonferroni: Controls familywise error rate by dividing alpha by the number of comparisons.
  • 📌 Tukey's HSD (Honestly Significant Difference): Controls familywise error rate when comparing all possible pairs of means. Not typically used *with* sphericity corrections, often used if sphericity is assumed.
  • 📌 Holm's Method: A step-down procedure that is more powerful than Bonferroni.
  • 📌 Sidák Correction: Similar to Bonferroni but slightly more powerful.
• 💡 Choosing the Right Post-Hoc Test: Consider the stringency needed (how much you want to control for Type I error) and the power (ability to detect true differences). The choice also depends on whether and how sphericity violations were corrected in the initial ANOVA.
• 📝 Important Note: After using Greenhouse-Geisser or Huynh-Feldt corrections in your ANOVA, you generally *do not* need to further adjust the p-values of your post-hoc tests, as the corrections already account for the inflated Type I error rate caused by sphericity violation. However, always consult statistical best practices and reporting guidelines.

Practice Quiz

1. What is the primary purpose of post-hoc tests in a repeated measures ANOVA?
   1. To determine if sphericity is violated.
   2. To identify which specific group means differ significantly after a significant ANOVA result.
   3. To correct for violations of normality.
   4. To increase the power of the ANOVA.
2. What assumption is tested by Mauchly's Test?
   1. Homogeneity of variance.
   2. Normality of data.
   3. Sphericity.
   4. Independence of observations.
3. If Mauchly's Test indicates a violation of sphericity, what is a common approach to address this issue in repeated measures ANOVA?
   1. Applying a logarithmic transformation to the data.
   2. Using a between-subjects ANOVA instead.
   3. Adjusting the degrees of freedom using corrections like Greenhouse-Geisser or Huynh-Feldt.
   4. Removing outliers from the data.
4. Which of the following is a common post-hoc test used after repeated measures ANOVA?
   1. Chi-Square test.
   2. Tukey's HSD.
   3. Independent Samples t-test.
   4. Mann-Whitney U test.
5. What is the Bonferroni correction primarily designed to control?
   1. Type II error rate.
   2. Familywise error rate (Type I error).
   3. Power of the test.
   4. Violation of normality.
6. After applying Greenhouse-Geisser correction to your repeated measures ANOVA due to sphericity violation, should you always apply further corrections to your post-hoc tests?
   1. Yes, it is always necessary.
   2. No, the Greenhouse-Geisser correction already accounts for the inflated Type I error rate.
   3. Only if the sample size is very small.
   4. Only if the p-value is close to the significance level.
7. Which of the following post-hoc methods adjusts p-values in a step-down manner, offering more power than the Bonferroni correction?
   1. Tukey's HSD
   2. Bonferroni
   3. Holm's Method
   4. Scheffe's Method