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Latent variable Cholesky ACE model--Singular A and C matrices

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bwiernik's picture
Joined: 01/30/2014 - 19:39
Latent variable Cholesky ACE model--Singular A and C matrices

I'm running a model with a Cholesky decomposition on a set of latent variables for the Big Five trait. When I run the model, the estimated phenotypic covariance (A+C+E) matrix among the latent variables Big Five variables and the E matrix are fine, but the A and C matrices are both singular. The diagonal elements of C are all near zero, so I'm not surprised by that one. Several of the estimated genetic correlations for A are extremely high (.80-.90). I'm wondering if these results suggest some sort of misspecification or empirical under-identification, or if the issue is just sampling error making the genetic covariances spuriously high.

AdminNeale's picture
Joined: 03/01/2013 - 14:09
Singular A and C matrices are ok

Hi Brenton

As you note, a null C covariance matrix is reasonable if there is no evidence of C variation in the traits. The A matrix sounds to me as if a single genetic factor is operating. If so, the A matrix will not be of full rank (it would be rank 1 instead of the dimension of the matrix). You might want to try a single factor model instead of the Cholesky -- fixing all parameters not in the first column would do it.

I think you have figured out why (A+C+E) is positive definite, despite only the E matrix having this property. However, I'll explain in case others have a similar question. The criterion for positive definiteness of matrix M is that any (all) non-null real column vector L should yield a strictly positive result when pre- and post-multiplying the matrix by the vector, i.e., $$L' M L > 0$$ for all non-null vectors L. Since matrix multiplication is distributive over addition, we can write $$L' (A+C+E) L = L' A L + L' C L + L' E L > 0.$$ The positive definiteness you observe for E is sufficient to keep the A+C+E sum positive definite.