What to read to catch up on multivariate statistics?

Henry Harpending writes:

I am writing to ask you for a recommendation of something I can read to catch up on multivariate statistics. I am happy with random processes and linear algebra since they are important in population genetics. My last encounter with real statistics was several decades ago.

Recently I have had to dip my toes into real multivariate statistics again and I am completely lost. I can’t, for example, figure out how a random effects model is different from what we used to call “partialing out” nuisance covariates. I have a hard time concentrating on exactly what a “BLURP” model is because the name is so silly.

Can you recommend something accessible to me that would put me on track?

My reply: if you’re interested particularly in random effects models, I will (parochially) refer you to my own book with Jennifer Hill. You can jump straight to the chapters on multilevel modeling.

If the question is about traditional multivariate methods such as factor analysis, principal components, etc., that I don’t really know! But I think my book would be a good start.

Do readers have any suggestions for a good book, preferably model-based, on multivariate methods such as factor analysis, principal components, etc.?