I don't see a reason why you could drop the sphericity assumption. Greenhouse-Geisser is a typical solution, and some authors recommend to use it anyways, because you cannot rely on the results from Mauchly test (e.g. Maxwell & Delaney,

2004).

Other alternatives could be an MANOVA approach to repeated measures (Tabachnick & Fidell, 2007) or multilevel modeling. Both have pros and cons, but the latter is apparently the most flexible tool to incorporate different variance-covariance structures.

I think you could, assume sphericity, report the violation and the correction approach. Indeed, "repeated measures ANOVAs are ...susceptible to violating the assumption of sphericity, which causes the test to become too liberal, ['inflating' ]"

Some suggestions are:

- When ε ≤ 0.75 (or you don’t know what the value for the statistic is), use the Greenhouse-Geisser correction. {This is a conservative correction that increases the risk of Type II error.}

- When ε > 0.75, use the Huynh-Feldt correction. Abdi (2010) believed this correction was more efficient, and powerful than the Greenhouse-Geisser correction.

I don't really go for the idea of multilevel modelling, just be sure there is no significant outlier. Goodluck.

Thank you very much both for your recommendation. I understand from your answers that I should assume sphericity and conduct the two way repeated measures ANOVA using this method.

Yes, but with GG-correction.