cholesky decomposition may 27th 2015 helsinki, finland e. vuoksimaa
TRANSCRIPT
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