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

Even when the factor model is correct, we observe some variants with specific effects. Variants in the APOE region are not generally associated with all cognitive tasks. Rather, they are associated with “fluid” tasks which decline with age, but not “crystallized” tasks.

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.

This is what has been observed for the “p factor,” i.e. the general factor of psychopathology (www.nature.com/articles/s41...), and more recently for impulsivity (www.medrxiv.org/content/10.1...).

However, if the common factor is an illusion- a statistical artifact of aggregation- then we should observe individual genetic variants that are associated with subsets of the phenotypes, but not variants that are associated with all phenotypes.

This is exactly what has been observed for constructs such as general intelligence (www.nature.com/articles/s41...) and externalizing psychopathology (www.nature.com/articles/s41...).

If the common factor model is correct, then we should observe individual genetic variants that are associated with all of the phenotypes composing that factor (the SNP effects should be proportional to the factor loadings).

Genome-wide association studies (GWAS) quantify associations between genetic variants and phenotypes. We can leverage GWAS data to resolve the factor indeterminacy problem and test the validity of latent variables.

However, the human genome contains elementary components that, due to the shuffling process associated with sexual reproduction, come to be naturally uncorrelated.

Reducing cognitive tasks into smaller components was unsuccessful – these narrow components still correlated with each other, leaving open the question of what model was responsible for those correlations.

Researchers have suggested breaking down cognitive tasks into “elementary processes” to identify components that correlate with the phenotypes but are uncorrelated with one another. Such a set of variables could be used to test whether a general factor was the true causal model.

That’s factor indeterminacy: just because a set of correlations is consistent with a common factor doesn’t mean that we can rule out other types of causal models. So what do we do about factor indeterminacy?

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.