Cholesky decomposition

May 27th 2015Helsinki, 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

phenotypes1) how much of the variance is explained by A,C,E effect that are shared between phenotypes2) 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 variablesntv <- nv*2 # number of total variablesselVars <- 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

A A

Height GCA

A A

Height GCA

rg

Correlated factors

rg = genetic correlation

Bivariate Cholesky

C C

Height GCA

C C

Height GCA

rc

Correlated factors

rc = common enviromental correlation

Bivariate Cholesky

E E

Height GCA

E E

Height GCA

re

Correlated factors

re = unique environmental correlation

Cholesky decomposition

A A

Height GCA

A A

Height GCA

C C C C

E EE E

1.0 MZ / 0.5 DZ 1.0 MZ / 0.5 DZ

1.0 MZ / 1.0 DZ 1.0 MZ / 1.0 DZ

Correlated factors

A A

Height GCA

A A

Height GCA

C C C C

E EE E

rcrc rgrg

rere

Additive genetic effects

A A

Height GCA

a11a21

a22

A1 A2

Height a11

GCA a21 a22

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

C C

Height GCA

c11c21

c22

C1 C2

Height c11

GCA c21 c22

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

E E

Height GCA

e11e21

e22

E1 E2

Height e11

GCA e21 e22

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

• (√a2var1 X rg X √a2var2) / 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 3ntv <- nv*2 # number of total variablesselVars <- 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 a21a22

A1 A2 A3

Var1 a11

Var2 a21 a22

Var3 a31 a32 a33

Lower nvar x nvar matrix parameters are estimated freely

Var3

a31 Aa32

a33

[nvar X (nvar+1)] / 2 = number of A paths (3 X 4) / 2 = 6

Trivariate Cholesky decompostion

C C

Var1 Var2

c11 c21c22

A1 A2 A3

Var1 c11

Var2 c21 c22

Var3 c31 c32 c33

Lower nvar x nvar matrix parameters are estimated freely

Var3

c31 Cc32

c33

[nvar X (nvar+1)] / 2 = number of C paths (3 X 4) / 2 = 6

Trivariate Cholesky decompostion

E E

Var1 Var2

e11 e21e22

A1 A2 A3

Var1 e11

Var2 e21 e22

Var3 e31 e32 e33

Lower nvar x nvar matrix parameters are estimated freely

Var3

e31 Ee32

e33

[nvar X (nvar+1)] / 2 = number of E paths (3 X 4) / 2 = 6

Trivariate correlated factors

A A

Var1 Var2 Var3

A

Trivariate correlated factors

C C

Var1 Var2 Var3

C

Trivariate correlated factors

E E

Var1 Var2 Var3

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, 329-338.• Loehlin (1996). The Cholesky approach: a cautionary

note. Behavior Genetics, 26, 65-69. • Carey G. (2005). Cholesky problems. Behavior Genetics,

35, 653-665.• Wu and Neale (2013). On the likelihood ratio tests in

bivariate ACDE models. Psychometrika, 78, 441-463.• Panizzon et al. (2014). Genetic and environmental

influences on general cognitive ability: is g a valid latent construct. Intelligence, 43, 65-76.

Resources including presentations

• International Twin workshop, every March, Institute for behavioral genetics, University of Boulder Colorado

• QIMR, Workshop, Sarah Medland