I'm reviewing a paper in which the authors conducted a PCA on some data that were a split among some a priori groups, and then looked for differences between those groups by running a separate ANOVA for each of the first four principal component axes. I'm not quite sure if that approach is statistically valid. Are principal component scores expressed in a sufficiently metric space such that they can be treated in downstream analysis like any other continuous numeric variable? Does the answer depend to some degree on how the PCA was rotated?
I do not think the interactions are valid to run ANOVA here. Try to check the significance of the model first then the significance of the other factors in the model before determining the validity of the ANOVA.
Separate ANOVAs don't make sense anyway for a multivariate problem. What would be the meaning of conflicting test results for the different PC scores (sometimes null hypothesis is declined and sometimes not)? ... probably the author would come to the wrong conclusion that separability between a priori groups has been shown to be significant.
(The space, however, should be similar, e.g. distance/position of data points relative to each other should not change.)
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