1. The use of Cholesky decomposition in multivariate models of sex-limited genetic and environmental effects Michael C. Neale Virginia Institute for Psychiatric and Behavioral Genetics Virginia Commonwealth University

2. The Problem ) ACE model ) Classical Twin Study ) Sex limitation model ) Univariate ok 5 rdzm=.5a m 2 + c m 2 5 rdzf=.5a f 2 + c f 2 5 rdzo =.5a m a f + c m c f

3. Scalar sex-limitation DZ OS zmzf 1.00 M A1A2 M P1M P2M xmymxmymzm A1 F A2 F P1F P2F xfyfxfyf 1.00 zf 1.00 0.50 1.00 0.50

4. Scalar sex-limitation DZ Females 1.00 F A1 P2F P1F xf yf A1 F P1F P2F 1.00 xfyfxfyf 0.50

5. Scalar sex-limitation DZ Males xmzm 1.00 0.50 1.00 M A1A2 M P1M P2M xmzm A1A2 M P2M xm zm 1.00 M P1M

6. Scalar sex-limitation Opposite sex xmzm 1.00 M A1A2 M P1M P2M xmzm A1 F A2 F P1F P2F xfyfxfyf 1.00 0.50 1.00 0.50

7. Algebraically Genetic covariances across twins P1 P2 rdzm = P1.5x m 2 0 P2 0.5z m 2 rdzf = P1.5x f 2.5x f y f P2.5x f y f.5y f 2 P1M P2M rdzo = P1F.5x m x f 0 P2F.5x m y f 0

8. Conclusion ) Whichever is second variable in males it cannot correlate with females ) Whichever correlates less empirically will fit better ) Something's screwy

9. Questions ) What does scalar sex-limitation mean ) Why does Cholesky not obey?

10. Solution ) Same factors operate in males & females but have different sized effects ) If they are the same factors, they should correlate the same ) Cholesky allows different covariance structure among factors

11. How to fix it ) Re-parameterize model 5 Estimate correlations among factors 5 Constrain equal across sexes 5 Linear constraints ) Constrain Cholesky Model 5 Standardized covariance components should be equal 5 Non-linear constraints

12. Reparameterized Correlation Approach A1 M A2 M P1MP2M xmzm A1 F A2 F P1FP2F xf 1.00 zf 1.00 0.50 1.00 0.50 rg

13. Correlation approach AdvantagesDisadvantages Conceptually Elegant Non-positive definiteness may occur Linear constraints

14. Cholesky Approach ) Additive Genetic Loadings In Males 5 A = X*X' ) Additive Genetic Loadings In Females 5 G = K*K' ) Declare F Izero nvar-1 nvar ) Constraint \vech(F&\stnd(A)) = \vech(F&\stnd(G)) ) Do Same for C/D and E matrices A how-to guide

15. Cholesky approach AdvantagesDisadvantages Same old model Requires non-linear constraints Keeps positive definiteness Estimates more parameters

16. Final Answer ) Use whichever you like ) Need non-linear constraints either way ) Problem is not limited to Cholesky Model ) Fix models with > 1 factor