Although the four standard multivariate tests often give similar results, there are situations where they differ, and one might have advantages over another. Unfortunately, there is no clear winner. In general, here is the order of preference in terms of power:
1. Pillai’s Trace
2. Wilks’ Lambda
3. Hotelling-Lawley Trace
4. Roy’s Maximum Root
When there is a large deviation from the null hypothesis and the eigenvalues differ widely, the order of preference is the reverse (Seber 1984).