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The Problem

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. The Problem. ACE model Classical Twin Study Sex limitation model

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The Problem

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

  2. The Problem • ACE model • Classical Twin Study • Sex limitation model • Univariate ok • rdzm = .5am2 + cm2 • rdzf = .5af2 + cf2 • rdzo = .5amaf + cmcf

  3. Scalar sex-limitation DZ OS 0.50 0.50 1.00 1.00 1.00 1.00 A1 A2 A1 A1 A2 M M F F xm xm ym ym zm zm xf xf yf yf zf zf P1M P1M P2M P2M P1F P1F P2F P2F

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

  5. Scalar sex-limitation DZ Males 0.50 0.50 1.00 1.00 1.00 1.00 M A1 A2 A1 A2 M M M xm xm zm zm xm zm P1M P1M P1M P2M P2M P2M

  6. Scalar sex-limitation Opposite sex 0.50 0.50 1.00 1.00 1.00 1.00 A1 A2 A1 A2 M M F F xm xm zm zm xf xf yf yf P1M P1M P2M P2M P1F P1F P2F P2F

  7. Algebraically Genetic covariances across twins P1 P2 rdzm = P1 .5xm2 0 P2 0 .5zm2 rdzf = P1 .5xf2 .5xfyf P2 .5xfyf .5yf2 P1M P2M rdzo = P1F .5xmxf 0 P2F .5xmyf 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 • Estimate correlations among factors • Constrain equal across sexes • Linear constraints • Constrain Cholesky Model • Standardized covariance components should be equal • Non-linear constraints

  12. Reparameterized Correlation Approach 0.50 0.50 1.00 1.00 1.00 1.00 rg rg A2 A2 A1 A1 M M F F xm zm xf zf P1M P2M P1F P2F

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

  14. Cholesky Approach A how-to guide • Additive Genetic Loadings In Males • A = X*X' • Additive Genetic Loadings In Females • 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

  15. Cholesky approach Advantages Disadvantages Requires non-linear constraints Same old model 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

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