What is the formula for Mauchly's W used for testing sphericity in univariate repeated measures anova?
Mauchley's test of sphericity is used to test the assumption in univariate repeated measures that the covariance matrix of the repeated measures has a particular form (see the Mixed Anovas talk in the [http://imaging.mrc-cbu.cam.ac.uk/statswiki/ Graduate Statistics Talk Series].)
After the W is computed, the “chi-square” (χ2) distribution may be used to assess whether the data show a statistically significant difference from sphericity.
This measure reports if there is a correlation between pairs of repeated measures. If the observed statistic for Mauchly’s test is statistically significant and the underlying distributions are normal, then the assumption of sphericity is rejected and an adjustment such as Greenhouse-Geisser or Huynh-Feldt will be necessary. It may be useful to switch to the multivariate form of repeated measures ANOVA in which sphericity is not assumed.
The test statistic, W, is compared to a chi-square distribution to assess the adequacy of the sphericity assumption. Full details are given [http://www.sagepub.com/upm-data/10927_Chapter9.pdf: here.]
The formula for W for a k x k covariance matrix, S, with determinant det(S), trace (or sum of the variances of S), tr(S), equals:
$$ W = det(S)( \frac{k+1}{tr(S)})^text{k+1}$$
It should be mentioned that although Mauchly’s test is a popular tool, it has been criticized for its inaccuracy when multivariate normality cannot be assured (Keselman, Rogan, Mendoza, & Breen, 1980; Rogan, Keselman, & Mendoza, 1979). Thus, prudent researchers need to be careful about mechanical use of this test. It makes sense to check the underlying normality of the distributions of data (and the transformed variables tracking the differences between each pair of repeated measures) before relying on the Mauchly test.