Why the Default results of RMANOVA are corrected by GG? The textbook and wikipedia told us that when the results of Mauchly Tests for Sphericity is not significant, there ‘s no need for correction. When the results of Mauchly Tests for Sphericity is significant, we should choice the the correction method according to epsilon.
The reason I have not implemented the approach you suggest is because I believe it is not statistically fully appropriate. Specifically, the Mauchly test, as any significance test, has a specific power and Type I error rate (the latter of which is controlled by the alpha/significance level). I do not believe it is worthwhile to incorporate these additional error probabilities into an analysis. Instead, I believe that the small loss in power by always using Greenhouse-Geisser is overall a more appropriate strategy and more conservative.
Consider the case in which the Mauchly test fails to correctly detect a violation because the power is too low in a specific case. If this happens, you get inflated Type I error rates. I consider this a potentially larger drawback than occasionally committing a Type II error in case Greenhouse-Geisser is too strict.