Interpreting Mauchly's Test Results: What to Do When Sphericity is Violated

📚 Quick Study Guide

• 📐 Mauchly's Test: Tests the assumption of sphericity in repeated measures ANOVA. Sphericity means the variances of the differences between all possible pairs of related groups are equal.
• 📉 Violation of Sphericity: Occurs when the variances of the differences are not equal.
• ⚠️ Consequences of Violation: Inflated Type I error rate (i.e., falsely rejecting the null hypothesis).
• ✅ Corrections: When sphericity is violated, use corrections like Greenhouse-Geisser or Huynh-Feldt.
• 🧪 Greenhouse-Geisser: A more conservative correction; use when sphericity is severely violated (epsilon < 0.75).
• 📈 Huynh-Feldt: A less conservative correction; use when sphericity is not severely violated (epsilon > 0.75).
• 📊 Epsilon ($\epsilon$): Represents the degree to which sphericity is violated. Ranges from 1 (perfect sphericity) to 1/(k-1) (maximum violation), where k is the number of repeated measures.
• 🔎 Reporting: Report the corrected F-statistic, degrees of freedom (adjusted), and p-value.

Practice Quiz

1. Which assumption does Mauchly's test assess?

   1. Equal variances between groups
   2. Normality of data
   3. Sphericity
   4. Homogeneity of regression slopes

2. What does it mean when sphericity is violated?

   1. The means of all groups are equal.
   2. The variances of the differences between all possible pairs of related groups are unequal.
   3. The data is normally distributed.
   4. There are no significant differences between groups.

3. What is a consequence of violating the assumption of sphericity?

   1. Deflated Type I error rate
   2. Inflated Type I error rate
   3. Reduced statistical power
   4. Increased Type II error rate

4. Which correction is generally more conservative?

   1. Huynh-Feldt
   2. Greenhouse-Geisser
   3. Bonferroni
   4. Sidak

5. When is the Greenhouse-Geisser correction typically preferred?

   1. When sphericity is not violated.
   2. When sphericity is mildly violated.
   3. When sphericity is severely violated (epsilon < 0.75).
   4. When sphericity is moderately violated (epsilon = 0.75).

6. What does epsilon ($\epsilon$) represent in the context of sphericity?

   1. The sample size.
   2. The degree to which sphericity is violated.
   3. The p-value of Mauchly's test.
   4. The F-statistic.

7. If Mauchly's test is significant, what should you do?

   1. Ignore the result and proceed with the uncorrected ANOVA.
   2. Apply a correction for violation of sphericity.
   3. Transform the data to achieve sphericity.
   4. Remove outliers from the data.