Cholesky decomposition May 27th 2015 Helsinki, Finland E. Vuoksimaa

Univariate & multivariate approach Univariate models – A,C and E estimates Bivariate Cholesky – A,C and E estimates & covariance between two phenotypes – ACE more power compared to univariate scenario – two interpretations on the relationship between two phenotypes 1) how much of the variance is explained by A,C,E effect that are shared between phenotypes 2) decomposing phenotypic correlation into genetic and environmental correlations

Multivariate models – A, C and E estimates & covariance between phenotypes: trivariate & other multivariate Cholesky decompositions  extension of bivariate Cholesky; Independent (IP) (biometric) & common pathway (CP) models – testing against ACE Cholesky – Cholesky for genetic or environmental effects: e.g., Cholesky structure for C and CP for A and E

Example data Height (measured), weight (measured) also general cognitive ability (GCA, in-person neuropsychological testing, IQ based on two WAIS subtests) Residualized measures (age and sex) Standardized M=0, SD=1

Bivariate Vars <- c(’var1',’var2') nv <- 2 # number of variables ntv <- nv*2 # number of total variables selVars <- paste(Vars,c(rep(1,nv),rep(2,nv)),sep="") pathA <- mxMatrix( type="Lower", nrow=nv, ncol=nv, free=TRUE, values= ?, lbound=?, ubound=? name="a" ) pathC <- mxMatrix( type="Lower", nrow=nv, ncol=nv, free=TRUE, values= ?, lbound=?, ubound=? name=”c" ) pathE <- mxMatrix( type="Lower", nrow=nv, ncol=nv, free=TRUE, values= ?, lbound=?, ubound=? name=”c" )

Bivariate Cholesky AA HeightGCA AA Height GCA rg Correlated factors rg = genetic correlation

Bivariate Cholesky CC HeightGCA CC Height GCA rc Correlated factors rc = common enviromental correlation

Bivariate Cholesky EE HeightGCA EE Height GCA re Correlated factors re = unique environmental correlation

Cholesky decomposition A A HeightGCA A A Height GCA C C C C E E E E 1.0 MZ / 0.5 DZ 1.0 MZ / 1.0 DZ

Correlated factors A A HeightGCA A A Height GCA C C C C E E E E rc rg re

Additive genetic effects AA Height GCA a11 a21 a22 A1A2 Heighta11 GCAa21a22 omxSetParameters( CholAeModel_noAcor, labels=labLower("a",nv), free=c(TRUE,FALSE,TRUE)) Lower nvar x nvar matrix  parameters are estimated freely shared genetic effects can be constrained to zero (0): [nvar X (nvar+1)] / 2 = number of A paths  (2 X 3) / 2 = 3

Additive genetic effects CC Height GCA c11 c21 c22 C1C2 Heightc11 GCAc21c22 omxSetParameters( CholAeModel_noCcor, labels=labLower(”c",nv), free=c(TRUE,FALSE,TRUE)) Lower nvar x nvar matrix  parameters are estimated freely shared common environmental effects can be constrained to zero (0): [nvar X (nvar+1)] / 2 = number of C paths  (2 X 3) / 2 = 3

Additive genetic effects EE Height GCA e11 e21 e22 E1E2 Heighte11 GCAe21e22 omxSetParameters( CholAeModel_noEcor, labels=labLower(”e",nv), free=c(TRUE,FALSE,TRUE)) Lower nvar x nvar matrix  parameters are estimated freely shared unique environmental effects can be constrained to zero (0): [nvar X (nvar+1)] / 2 = number of E paths  (2 X 3) / 2 = 3

Number of parameters [nvar X (nvar+1)] / 2 = number of A paths  (2 X 3) / 2 = 3 [nvar X (nvar+1)] / 2 = number of C paths  (2 X 3) / 2 = 3 [nvar X (nvar+1)] / 2 = number of E paths  (2 X 3) / 2 = 3 means = 2 Bivariate number of parameters = 11 Number of parameters in AE-AE cholesky ? Number of parameters in AE-AE cholesky where rg = 0 ?

Proportion of phenotypic correlation due to rg (√a 2 var1 X rg X √a 2 var2) / rp  (√ heritability of phenotype 1 X genetic correlation between phenotype 1 and phenotype 2 X √ heritability of phenotype 2) / phenotypic correlation

Trivariate Vars <- c(’var1',’var2’, ’var3’)# add 3rd variable, 4th, 5th, etc. nv <- 3 # number of variables# you need to change this, here 3 ntv <- nv*2 # number of total variables selVars <- paste(Vars,c(rep(1,nv),rep(2,nv)),sep="") pathA <- mxMatrix( type="Lower", nrow=nv, ncol=nv, free=TRUE, values= ?, lbound=?, ubound=? name="a" ) pathC <- mxMatrix( type="Lower", nrow=nv, ncol=nv, free=TRUE, values= ?, lbound=?, ubound=? name=”c" ) pathE <- mxMatrix( type="Lower", nrow=nv, ncol=nv, free=TRUE, values= ?, lbound=?, ubound=? name=”c" ) Matrices to get the path coefficients for a, c and e

Trivariate Cholesky decompostion A A Var1 Var2 a11 a21 a22 A1A2A3 Var1a11 Var2a21a22 Var3a31a32a33 Lower nvar x nvar matrix  parameters are estimated freely Var3 a31 A a32 a33 [nvar X (nvar+1)] / 2 = number of A paths  (3 X 4) / 2 = 6

Trivariate Cholesky decompostion C C Var1 Var2 c11 c21 c22 A1A2A3 Var1c11 Var2c21c22 Var3c31c32c33 Lower nvar x nvar matrix  parameters are estimated freely Var3 c31 C c32 c33 [nvar X (nvar+1)] / 2 = number of C paths  (3 X 4) / 2 = 6

Trivariate Cholesky decompostion E E Var1 Var2 e11 e21 e22 A1A2A3 Var1e11 Var2e21e22 Var3e31e32e33 Lower nvar x nvar matrix  parameters are estimated freely Var3 e31 E e32 e33 [nvar X (nvar+1)] / 2 = number of E paths  (3 X 4) / 2 = 6

Trivariate correlated factors A A Var1 Var2Var3 A

Trivariate correlated factors C C Var1 Var2Var3 C

Trivariate correlated factors E E Var1 Var2Var3 E

Number of parameters [nvar X (nvar+1)] / 2 = number of A paths [nvar X (nvar+1)] / 2 = number of C paths [nvar X (nvar+1)] / 2 = number of E paths means Trivariate number of parameters = ?

Included in the example script Saturated models are included in the script ACE-ACE Cholesky AE-AE Cholesky CE-CE Cholesky rg = 0 re = 0 no correlation between phenotypes

Bivariate with height & weight, also height & GCA and weight & GCA Calculate genetic and environmental correlations Can we set rg/re or both as zero? What is the proportion of phenotypic correlation due to rg?

Things to consider Do not automatically run AE-AE after ACE-ACE, e.g., consider if you want to keep C effects for one(/some) of the variables E.g., C effects of about 15% may be fixed to be zero, but you may still want to keep the C effects – less biased genetic correlation Cholesky in context of IP and CP models What is the question that you are asking

Suggested reading Carey G. (1988), Behavior Genetics, 18, Loehlin (1996). The Cholesky approach: a cautionary note. Behavior Genetics, 26, Carey G. (2005). Cholesky problems. Behavior Genetics, 35, Wu and Neale (2013). On the likelihood ratio tests in bivariate ACDE models. Psychometrika, 78, Panizzon et al. (2014). Genetic and environmental influences on general cognitive ability: is g a valid latent construct. Intelligence, 43,

Resources including presentations International Twin workshop, every March, Institute for behavioral genetics, University of Boulder Colorado QIMR, Workshop, Sarah Medland