Mauchly test

TL;DR

Tests whether the variances of the differences between all pairs of levels of a within-subjects factor are equal.
A significant result indicates a violation of sphericity, which can bias effect estimates and p-values in repeated measures ANOVA.
If sphericity is violated, use correction methods such as Greenhouse-Geisser or Huynh-Feldt to obtain accurate results.

Definition

The Mauchly test is a statistical test used to determine if the assumption of sphericity has been violated in a repeated measures analysis of variance (ANOVA). The sphericity assumption states that the variances of the differences between all pairs of levels of the within-subjects factor are equal. Violation of this assumption can result in biased and inaccurate estimates of the effects of the within-subjects factor, as well as incorrect p-values and Type I error rates.

Explanation

To perform the Mauchly test, the researcher calculates the variances of the differences between every pair of levels of the within-subjects factor. If these variances are not equal, the assumption of sphericity is violated and the Mauchly test will be significant. A significant Mauchly test indicates that repeated measures ANOVA results (effect estimates, p-values, and Type I error rates) may be unreliable without adjustment. Common adjustment methods mentioned are Greenhouse-Geisser and Huynh-Feldt corrections.

Examples

Exercise intervention and weight loss

A study examines three different exercise interventions (aerobic, resistance, and combination) and measures weight loss at three different time points (baseline, 6 weeks, and 12 weeks). The within-subjects factor is the time point, and the between-subjects factor is the exercise intervention.

To conduct the Mauchly test, the researcher calculates the variances of the differences between all pairs of levels of the within-subjects factor (the differences between baseline, 6 weeks, and 12 weeks). If these variances are not equal, the Mauchly test is significant and the assumption of sphericity is violated. Unequal variances of differences in this study could be due to individual differences in weight loss rates or the effectiveness of the exercise interventions. In that case, the researcher would use a correction method, such as the Greenhouse-Geisser or Huynh-Feldt, to obtain accurate estimates of the effects of time point and exercise intervention on weight loss.

Teaching methods and student achievement

A study examines three different teaching methods (lecture, discussion, and cooperative learning) and measures student achievement at three different grades (3rd, 4th, and 5th). The within-subjects factor is the grade, and the between-subjects factor is the teaching method.

The researcher calculates the variances of the differences between all pairs of levels of the within-subjects factor (the differences between 3rd, 4th, and 5th grades). If these variances are not equal, the Mauchly test is significant and sphericity is violated. Unequal variances of differences in this study could be due to individual differences in learning abilities or the effectiveness of the teaching methods. If sphericity is violated, the researcher would apply a correction method, such as the Greenhouse-Geisser or Huynh-Feldt, to obtain accurate estimates of the effects of grade and teaching method on student achievement.

Notes or pitfalls

Violation of sphericity can produce biased and inaccurate estimates of within-subject effects and lead to incorrect p-values and inflated Type I error rates.
When the Mauchly test is significant, apply a correction method (for example, Greenhouse-Geisser or Huynh-Feldt) to adjust degrees of freedom and obtain more accurate inference.

Sphericity
Repeated measures ANOVA
Within-subjects factor
Between-subjects factor
Greenhouse-Geisser
Huynh-Feldt