Mauchly's sphericity test

If a univariate method is selected, the repeated-measures ANOVA must be appropriately corrected depending on the degree to which sphericity has been violated.

[2] To further illustrate the concept of sphericity, consider a matrix representing data from patients who receive three different types of drug treatments in Figure 1.

[1] Statistical software should not provide output for a test of sphericity for two levels of a repeated measure factor; however, some versions of SPSS produce an output table with degrees of freedom equal to 0, and a period in place of a numeric p value.

However, if Mauchly's test is significant then the F-ratios produced must be interpreted with caution as the violations of this assumption can result in an increase in the Type I error rate, and influence the conclusions drawn from your analysis.

[4] In instances where Mauchly's test is significant, modifications need to be made to the degrees of freedom so that a valid F-ratio can be obtained.

Each of these corrections have been developed to alter the degrees of freedom and produce an F-ratio where the Type I error rate is reduced.

The actual F-ratio does not change as a result of applying the corrections; only the degrees of freedom.

When epsilon is > .75, the Greenhouse–Geisser correction is believed to be too conservative, and would result in incorrectly rejecting the null hypothesis that sphericity holds.

Another alternative procedure is using the multivariate test statistics (MANOVA) since they do not require the assumption of sphericity.

[9] However, this procedure can be less powerful than using a repeated measures ANOVA, especially when sphericity violation is not large or sample sizes are small.

[10] O’Brien and Kaiser[11] suggested that when you have a large violation of sphericity (i.e., epsilon < .70) and your sample size is greater than k + 10 (i.e., the number of levels of the repeated measures factor + 10), then a MANOVA is more powerful; in other cases, repeated measures design should be selected.

[5] Additionally, the power of MANOVA is contingent upon the correlations between the dependent variables, so the relationship between the different conditions must also be considered.