Is it okay to perform PerMANOVA on PCA values? Is it acceptable to first perform principal components analysis on a dataset, and then use permutational MANOVA on those principal components values, rather than on the original values in the dataset? For example, in a dataset looking at sex differences in 11 variables, I don't get a significant p-value when performing perMANOVA on the dataset values, but do when the perMANOVA is performed on the PC values.
I have a feeling that this is probably not statistically valid, but I wanted to check. Thanks!
 A: I will leave it to others to decide if a permutational multivariate analysis of variance on principal components is 'acceptable' to them. But there isn't any logical error in computing such a statistic. It is simply composing functions of random variables. The more difficult question for you to ask yourself is whether this statistical procedure answers your research question.
Recall that PCA involves performing a linear transformation that optimally, in a certain linear sense, decorrelates your variables. Of considerable importance here is that the principal components are ordered by how much variation they linearly explain.
Also recall that the permutational analysis of variance tests a null that location and dispersion remain unchanged when values are exchanged among groups.
If your perMANOVA was not significant on the original variables, but was on the principal components of those variables, that suggests to me that you have successfully detected that your PCA did something. Namely, it unequalized the location or dispersion among groups. This makes sense given that PCA puts an aforementioned ordering on its components. If that is what you wanted to test with this procedure, then great! If not, then do something else.
