PCA followed by Wilcoxon-Mann-Whitney test on PC1: is it problematic? I have two groups of animals and 60 to 100 variables per animal (describing their behavior during 24h). I want to know whether the two groups of animals behave differently (I have 11 data points per group), and I do not care much about knowing what is different.
I started by doing a SVM followed by a permutation to assess the accuracy of the SVM statistically. Got a good accuracy (5/6 correct prediction) but it is far from getting to the significant threshold (p>0,2), due to the over-dispertion of the distribution of accuracy scores, probably linked to the the low sample size.
Now I am wondering about a most simpler approach: making a PCA and do a basic wilcox test on the first component (keeping p<0.05 as the threshold, since I am doing only one test). I think it is allowed: since the PCA does not use the grouping variable, it will treat inter- and between- groups variance similarly. The fact that the p-value with 11 animals per group is super super low make me wonder if I do not make any wrong assumption here.
 A: *

*Your PCA technique might miss the difference between the groups: That is because it focuses on the total variation rather than the variation between the groups.
If you are focused on exploring/picking-out a difference between the groups, then a linear discriminant analysis (LDA) might be a better (stronger) tool.
But since you started with SVM (support vector networks?) instead of MANOVA/LDA I wonder if you have some restrictions that made you choose a non-linear technique? PCA is not helping on  such issues that  much.
See the image below for a simple example that expresses these effects. The largest variations in the variables between groups (e.g. a total of var1 + var2) might be perpendicular to the differences within the groups (e.g. a distribution among var1 and var2). 


*Aside from this limitation in power your PCA technique is robust regarding type I errors. Given the null hypothesis that 'the two groups are the same', then you could sample from a single distribution, perform PCA, and assign groups afterwards. So the PCA can't have any effect, if the null hypothesis is true.
