The regular p-value calculations in the repeated
measures anova (
ranova) are accurate if the theoretical
distribution of the response variables have compound symmetry. This
means that all response variables have the same variance, and each
pair of response variables share a common correlation. That is,
If the compound symmetry assumption is false, then the degrees of freedom for the repeated measures anova test must be adjusted by a factor ε, and the p-value must be computed using the adjusted values.
Compound symmetry implies sphericity.
For a repeated measures model with responses y1, y2, ..., sphericity means that all pair-wise differences y1 – y2, y1 – y3, ... have the same theoretical variance. Mauchly's test is the most accepted test for sphericity.
Mauchly's W statistic is
M is a p-by-d orthogonal contrast matrix, Σ is the covariance matrix, p is the number of variables, and d = p – 1.
A chi-square test statistic assesses the significance of W. If n is the number of rows in the design matrix, and r is the rank of the design matrix, then the chi-square statistic is
The C test statistic has a chi-square distribution with (p(p – 1)/2) – 1 degrees of freedom. A small p-value for the Mauchly's test indicates that the sphericity assumption does not hold.
rmanova method computes the p-values
for the repeated measures anova based on the results of the Mauchly's
test and each epsilon value.
 Mauchly, J. W. "Significance Test for Sphericity of a Normal n-Variate Distribution. The Annals of Mathematical Statistics. Vol. 11, 1940, pp. 204–209.