The use of Cholesky decomposition in multivariate models

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The use of Cholesky decomposition in multivariate models of sex-limited genetic and environmental effects

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 )Univariate ok 2 2

The Problem )ACE model )Classical Twin Study )Sex limitation model )Univariate ok 2 2 5 rdzm =. 5 am + cm 2 5 rdzf =. 5 af + cf 2 5 rdzo =. 5 amaf + cmcf

Scalar sex-limitation DZ OS 0. 50 1. 00 A 1 M A 2 M

Scalar sex-limitation DZ OS 0. 50 1. 00 A 1 M A 2 M xm P 1 M ym 0. 50 A 1 F zm P 2 M 1. 00 xf P 1 F A 2 F yf zf P 2 F

Scalar sex-limitation DZ Females 0. 50 1. 00 A 1 F xf P 1

Scalar sex-limitation DZ Females 0. 50 1. 00 A 1 F xf P 1 F yf xf P 2 F P 1 F yf P 2 F

Scalar sex-limitation DZ Males 0. 50 1. 00 A 1 M xm P 1

Scalar sex-limitation DZ Males 0. 50 1. 00 A 1 M xm P 1 M 0. 50 1. 00 A 2 M A 1 M A 2 M xm zm P 2 M P 1 M zm P 2 M

Scalar sex-limitation Opposite sex 0. 50 1. 00 A 1 M xm P 1

Scalar sex-limitation Opposite sex 0. 50 1. 00 A 1 M xm P 1 M 0. 50 1. 00 A 2 M A 1 F A 2 F zm P 2 M xf P 1 F yf P 2 F

Algebraically Genetic covariances across twins P 1 P 2 rdzm = P 1. 5

Algebraically Genetic covariances across twins P 1 P 2 rdzm = P 1. 5 xm 2 0 P 2 0. 5 zm 2 rdzf = P 1. 5 xf 2. 5 xfyf P 2. 5 xfyf. 5 yf 2 P 1 M P 2 M rdzo = P 1 F. 5 xmxf 0 P 2 F. 5 xmyf 0

Conclusion )Whichever is second variable in males it cannot correlate with females )Whichever correlates

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

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

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

Solution )Same factors operate in males & females but have different sized effects )If

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

How to fix it )Re-parameterize model 5 Estimate correlations among factors 5 Constrain equal

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

Reparameterized Correlation Approach 0. 50 1. 00 rg A 1 M 1. 00 A

Reparameterized Correlation Approach 0. 50 1. 00 rg A 1 M 1. 00 A 2 1. 00 M xm P 1 M 0. 50 1. 00 A 2 F A 1 F zm P 2 M rg xf P 1 F zf P 2 F

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

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

Cholesky Approach A how-to guide ) Additive Genetic Loadings In Males 5 A =

Cholesky Approach A how-to guide ) 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

Cholesky approach Advantages Disadvantages Same old model Requires non-linear constraints Keeps positive definiteness Estimates

Cholesky approach Advantages Disadvantages Same old model Requires non-linear constraints Keeps positive definiteness Estimates more parameters

Final Answer )Use whichever you like )Need non-linear constraints either way )Problem is not

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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