Finally, I couldn’t end without mentioning that we were able to work in this fantastic representation of factor analysis from Buzz Hunt.

We can see this in the results of a simple simulation. Here, although the square correlations matrices (intercorrelations among phenotypes) are indistinguishable, the rectangular matrices (associations with individual genetic variants) differ starkly.

Here are correlations generated using a different model, where different combinations of 6 phenotypes share a cause, but never all 9 (so there is no common factor). It looks the same as the other matrix!

To make this a little more salient, let’s look at some correlation matrices. These matrices show the correlation among 9 phenotypes. Here’s one that was generated by a common factor model, where one shared cause influences all of the phenotypes in the correlation matrix.

Many social scientists use latent variables to capture variables that cannot be directly observed (extraversion, intelligence). However, the correlations we use to infer latent factors can result from alternative data generating mechanisms. We refer to this problem as factor indeterminacy.