Comparing PCAs using the distances between the centroids of groups For example, I have a data for N countries where quantitative values for men and women are given. I ran PCA separately for the data from each country. PC1 and PC2 explain most of the variance, so I use them for further steps. I want to test how similar or different the groups - i.e. men and women - are, in respect to their separation or closeness on the PCA scatterplot (PC1 vs PC2). To do this, I calculated a quantitative proxy which is the (Euclidean) distance between the centroids of the two groups, as shown in the schematic (top). The two plots are some two countries.
I have a few questions about this approach to compare, in the context of the PCAs:

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*Could the (Euclidean) distance between the centroids of the groups be used as an appropriate proxy to measure how similar/different the groups are?

*Can such a proxy be comparable across different PCAs? In the context of the example, because the distance for a country x is higher than country y, can I state that that men and women from country x are relatively different compared to country y?

*Finally, extending the pairwise comparisons, based on the distribution of the distances, can I make a statement that a country has higher or lower than average difference between men and women?



Have you looked into techniques such as Permanova or ANOSIM? I think that could help you in your goal of comparing groups. It is usually used in the context of ecology, to compare distance matrices based on a variable of interest (such as men Vs women in your case).
As far as I know, vegan have some interesting functions for this purpose:
https://cran.r-project.org/web/packages/vegan/index.html
