# 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.

1. Could the (Euclidean) distance between the centroids of the groups be used as an appropriate proxy to measure how similar/different the groups are?
2. 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?
3. 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?

• First of all, one may ask: are you sure the pair of PCs skim the same amount of valuable or relevant information from each country's data. If use the original p variables (which I surmise are the same in the countries), it is one story; but you chose to reduce dimensionality differently in different countries, it is another story. What if the crucial individual or gender differences can reside in a single PC in one country but demand to extract 3 PCs in another country? Are you sure you need independent PCA analyses for the countries? Commented Aug 24, 2022 at 18:47
• @ttnphns Great point about the need for independent PCA analyses. I would need to get an expert opinion on this to be sure, but if the the distances are calculated on from the same PCA, I would guess they would be more likely to be comparable. In this case, at least the features represented by the PCs would be uniform across all countries which would avoid the possible scenario of for example, comparing PCs representing height in one country with PCs representing number of toes in other the country. Commented Aug 24, 2022 at 19:31

I have concerns with this methodology. Note that there's no guarantee that PC1 and PC2 have anything to do with gender differences. Seeing that men and women are no different in PC1 and PC2 does not indicate that men and women are identical, it just indicates there is no difference in the most variant dimensions. You may find very large differences in PC3, PC4, and beyond, despite the fact that those dimensions don't capture the most variance in the population as a whole.

The fact that you're computing a different PCA for every country makes distances in PCA space incomparable, since you are likely comparing men and women in different ways for every country. PC1 in country X may be a proxy for height in which case you would find differences between men and women, but PC1 in country Y may be a proxy for something like "number of toes", which probably doesn't vary much by gender. Seeing a greater difference in PC1 for country X doesn't mean that men and women are more different in country X, it just means they are more different in whatever dimension happened to have the most variance.

You'd probably be best off computing a single principal component decomposition and applying it consistently to all countries. With that you, can evaluate the size of male vs. female differences along a particular dimension, and fairly compare those differences between countries.

So, to answer your specific questions: 1) Yes, distance in PCA space can be a proxy of distance in feature space, but note that your top PCAs may or may not have anything to do with gender differences. 2) No, comparing between different PCAs probably isn't meaningful since you are comparing differences on different dimensions. Seeing that men and women are different in the "apples" dimension in country X, and that they are not different in the "oranges" dimension in country Y allows you to make no statement about whether they are more or less alike overall in X or Y. 3) With consistent PCs across countries, you could evaluate which countries have the largest/smallest gender differences along a particular dimension.

• thanks for the answer/s which clarify things a lot. Firstly, I agree that the interpretation should be that 'there is some/no difference in the most variant dimensions' instead of groups are different/similar. I had intentionally used a little bit loose language in my question to keep things simple. Thanks for pointing that out. Commented Aug 24, 2022 at 18:39
• (1) Good to know that distance in PCA space can be a proxy. (2) My intention behind deriving the proxy was to get the general "sense" of the differences regardless of the specific feature (e.g. height or toe) driving the difference. In that case, I wonder if I could make this statement: 'men and women in country x have more/less difference in the most variant dimensions compared to those country y'(?). Commented Aug 24, 2022 at 18:45
• @User8i4y389 I suppose that would be a true statement, but it's badly confounded by what the dimensions mean in different countries to the point that it may not be a useful statement. It may be the case that men and women have identical differences in both countries, but that one dimension is more variant in one country than another. The statement in that case is saying far more about differences in the most variant dimensions by country than anything about men vs. women differences by country. You'd need more context to avoid that true statement being misleading. Commented Aug 24, 2022 at 18:53
• Ok, on the hindsight, I should have been more specific with my objective in the question. I was aiming for the differences in the most variant dimensions, which based on your comment the proxies i.e. the distances might be capturing, right? Commented Aug 24, 2022 at 18:59
• most variant dimensions should be somehow comparable between countries - quantitavely or qualitatively, in order to compare something "in them". Your pair of chief PCs in one country may account for 70% variance and have one "meaning" in one country and 77% variance plus another "meaning" in another country. Commented Aug 24, 2022 at 19:08

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

• (Andre) Thanks for the suggestion. I am not from ecology background. So I am not familiar with these methods. Could you please let me know if they are alternatives to PCA for dimensionality reduction purpose, or methods that can be applied to analyze the distances between the PCA clusters? Commented Aug 25, 2022 at 15:51
• I don't think this is exclusive to ecology, I know it is standard in the field though. If you want, there is an example using iris dataset in here: stackoverflow.com/a/20267537 Commented Aug 25, 2022 at 16:25
• Another good resource: archetypalecology.wordpress.com/2018/02/21/… Commented Aug 25, 2022 at 16:26
• (Andre) Thanks for the links. Appreciate a lot. So, these methods seem to be applicable for analyzing the distances between the PCA clusters. Commented Aug 25, 2022 at 19:49
• Yes, I would say so. In fact, I've used it in some omics data unrelated with ecology. Commented Aug 26, 2022 at 7:53