Sphericity

Assessing Departures from Sphericity with Mauchly's Test

Mauchly's test is a statistical procedure that might be used to determine whether the sphericity assumption has been met for a given data set. Repeated-measures ANOVA performed with the IBM¯ SPSS¯ (PASW) 18.0 statistical program (an IBM company) will automatically include Mauchley's test. If the Mauchley's test statistic is not significant (i.e., p > .05), then it might be assumed that the variances of the differences between treatment conditions are roughly equal, and the sphericity assumption has been met. However, if the Mauchley's test statistic is significant (i.e., p ¯ .5), then the variances are not equal and the assumption of sphericity has been violated.

Some caution is in order on using Mauchley's test to check for violations of sphericity. This test might be underpowered with small sample sizes, which might lead to violations of the sphericity assumption going undetected. Furthermore, Mauchley's test does not provide information regarding the degree to which sphericity has been violated.

Correcting for Violations of Sphericity

There are corrections that might be used when the sphericity assumption has been violated, although they result in the loss of some statistical power. The most commonly used are the Greenhouse—Geisser correction (developed by Samuel Greenhouse and Seymour Geisser), the Huynh-Feldt correction (developed by Huynh Huynh and Leonard Feldt), and a hybrid of the two. All three corrections decrease the degrees of freedom and thereby increase the p value needed to obtain significance.

These corrections first require generating a value known as epsilon. Epsilon varies between 1/(k − 1) (k = number of repeated treatments) and 1. The closer epsilon is to 1.0, the closer data are to meeting the sphericity assumption. The lower limit of epsilon naturally varies based on the number of repeated treatments. For example:

The Greenhouse—Geisser correction generates a conservative estimate of epsilon and thus might generate an overly conservative correction. The Huynh-Feldt correction generates a more liberal estimate of epsilon, and this correction might be more appropriate when the Greenhouse—Geisser estimate of epsilon is greater than 0.75. However, it has also been argued that the Huynh-Feldt estimate of epsilon is too liberal (overestimates sphericity), and thus it might be more appropriate to average the epsilon values obtained from both and use that averaged value for adjusting degrees of freedom.

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