Data Assumption: Sphericity

Very brief description

The assumption of sphericity refers to the equality of variances of the differences between treatment levels.  In Repeated Measures ANOVA it is a measure of the homogeneity of the variances of the differences between levels so it is quite similar to homogeneity of variance in between-groups in the univariate ANOVA. It is denoted by ε and sometimes referred to as “circularity”.
Who cares
Sphericity applies to repeated measures ANOVA and MANOVA. While technically not an assumption of Factor Analysis, “Bartlett’s test of sphericity” is applied to test the hypothesis that variables are uncorrelated with each other – so they only correlate with themselves (referred to as an “identity matrix”). Here we want the test of sphericity to be significant (we reject the null hypothesis that our factor analysed data equals an “identity matrix” with no inter-variable correlations), so factorability is assumed.

Why it is important
The violation of sphericity may cause our ANOVA or MANOVA significance test to become too “liberal” which increases the likelihood of a Type I error (incorrectly rejecting a true null hypothesis).
How to Test
Mauchly’s sphericity test of the residual covariance matrix: If the Mauchly’s sphericity test is significant (e.g. p<0.05) then we can conclude that there are significant differences between the “variance of differences” so the condition of sphericity has not been met and we should use the Greenhouse-Geisser or the Huynh-Feldt corrected F-value. However, if the Mauchly’s test statistic is not significant then we can use the sphericity assumed F-value. 
How to fix the problem
If you are lucky, your statistics software will provide you with the necessary remedy. SPSS provides an F-value for when sphericity is assumed, as well as options for corrections applied to produce a valid F-value. Among them are the Greenhouse-Geisser corrected F-value and the Huynh-Feldt F-value. The heuristic is that when Mauchly’s sphericity (Mauchly’s W) is greater than 0.75, then use Huynh-Feldt, and when less than 0.75 (or unknown), use Greenhouse-Geisser corrected F-value.