Kepner and Wackerly summarized the literature in noting "by the late 1980s, the volume of literature on RT methods was rapidly expanding as new insights, both positive and negative, were gained regarding the utility of the method. Concerned that RT methods would be misused, Sawilowsky et al. (1989, p. 255) cautioned practitioners to avoid the use of these tests 'except in those specific situations where the characteristics of the tests are well understood'."  According to Hettmansperger and McKean,  "Sawilowsky (1990)  provides an excellent review of nonparametric approaches to testing for interaction" in ANOVA.
Mauchly’s test for sphericity can be run in the majority of statistical software, where it tends to be the default test for sphericity. Mauchly’s test is ideal for mid-size samples. It may fail to detect sphericity in small samples and it may over-detect in large samples.
If the test returns a small p-value (p ≤.05), this is an indication that your data has violated the assumption. The following picture of SPSS output for ANOVA shows that the significance “sig” attached to Mauchly’s is .274. This means that the assumption has not been violated for this set of data.