# Multilevel Principal Component Analysis

I needed to run a PCA on a dataset with a multilevel structure. My question is similar to the one asked here: Principal components analysis on nested data

In my case, however, the two levels are crossed rather than nested. Is there an R package for this? Would it be better to just report by-item and by-subject results?

A bit of context on the study: the data results from experiments where proofreaders (subjects) checked a number of written sentences (items). All proofreaders saw all sentences. I want to investigate correlations across multiple measures of proofreading behaviour (proofreading time, number of editing operations, etc.)

I'd be happy to just go with by-item and by-subject analyses, but results are quite different and I'm not sure about how to interpret this. Biplots show that the proofreading behaviour variables are much more correlated with each other in the by-item analysis. Would this be due to high between-subject variability, making the by-item analysis more reliable?

• You can always run PCA on the mutlivariate data and then use an LME on the scores to account for the levels. It should work fine. Otherwise you have a strong identifiability issue. – usεr11852 May 6 '16 at 19:06
• @usεr11852 but LME doesn't do what I want... I don't have an outcome variable. The idea is inspecting how close together/far apart all variables are, so it's more of an explorative approach rather than hypothesis testing. Biplots would be the best way to visualise this I guess, so what I'd need is a method that'd allow me to account for the levels graphically in a biplot. Not sure if that exists... – user3744206 May 6 '16 at 22:11
• OK, I understand what you mean now. How about controlling in an LME each proofreading behaviour for subject and item effect first individually and then using the residuals for your biplots? That way you would have factored out the multilevel structure of your behaviour data. – usεr11852 May 7 '16 at 1:57
• @usεr11852 hadn't thought of that... Makes sense to me and seems like quite a simple solution. Have you seen this anywhere or would you know of any source I could cite on residuralization being used as a way to get round the independence assumption? – user3744206 May 8 '16 at 0:01
• I have seen this many times in Paediatrics studies where people say "we examine $X$ after controlling for variable $Y$ and $Y^*$. I do not work on item-response theory unfortunately. I have not seen it cited - I think just writing it out is enough as it is a straight-forward procedure stemming from a basic regression application. – usεr11852 May 8 '16 at 5:35