Moderating the Covariance between Family Member’s Behavior

Moderating the Covariance between Family Member’s Behavior

Brad VerhulstSarah Medland

Motivation

Substance Use of Family Members is highly correlated

This may be a function of:

• Environmental Factors• Special Twin Environment• Sibling Interaction• Vertical Cultural Transmission

• Genetic Factors • Additive & Non-Additive

But if genes are expressed at different points in the lifespan, the differential expression will look like genetic non-additivity.

But twins may be a larger part of their co-twins environment not because of some magical special twin environment, but because they are the same age and are simply more likely to have the same interests, share the same friends, and have the same life experiences

So What’s the Alternative?• The correlation between behavior decays may decay as a

function of differences along a continuum.• In this case we will focus on age difference but realistically

several possible moderating variable could be used.

• The larger the age difference between relatives the lower the correlation

• Two reasons:• As siblings are more different in age, they are less likely to

share environments• Age specific gene expression

Background Methodology

Both Genetic and Environmental factors have cumulative effects

Some Effects are specific to the time of measurement

Related: Hopper and Matthews (1982, 1983)- The longer siblings live together, the more similar they are.

Relationship between the negative exponential function and a linear time series with discrete

time points

Existing Gene-Environment Interaction Models

Classical Twin Design

Pt1 Pt2

1

A

C

E

μ

E

C

A1

MZ=1

DZ = ½

1

1

1

1

1

1

a + βaM a + βaM

e+ βeM

c + βcM c + βcMe+ βeM

+ Mβm μ + Mβm

Purcell (2002)

Means Moderation Model

Basic Means and Variances

Our ModelPt1 Pt2

1

A

C

E

μ

E

C

A

1

1

1

1

1

1

a a

e

c c

e

+ Mβm μ + Mβm

T

t

1

MZ= e-|Δage| * αa

DZ = e-|Δage| * ½*αa

e-|Δage| * αc

T

t

1

Cov = Acov * e-|Δage|*αa + Ccov * e-|Δage|*αc + Tcov

Estimated Variance Components for the A, C, T and E Parameters

0.69 0.71 0.73

050

100

150

200

250

Additive Genetic

Males

0.42 0.44 0.46 0.48

050

100

150

200

Shared Environment

0.22 0.24 0.26 0.28

050

100

150

Twin Environment

0.490 0.500 0.510

050

100

150

200

Unique Environment

0.46 0.48 0.50 0.52

050

100

150

Females

0.47 0.49 0.51

050

100

150

200

250

0.26 0.28 0.30 0.32

050

100

150

0.625 0.635 0.645 0.655

050

100

150

200

Note: Red Line Indicates the Simulated Value

The Relationship between the Simulated and Estimated Moderation Parameters

0.0 0.2 0.4 0.6 0.8 1.0

-1.0

0.0

0.5

1.0

1.5

2.0

2.5

Genetic Moderation Parameter

n=10000 Families per Twin TypeSimulated Parameter Value

Est

imat

ed P

aram

eter

Slope 1.06Intercept 0.01

0.0 0.2 0.4 0.6 0.8 1.0

-1.0

0.0

0.5

1.0

1.5

2.0

2.5

Environmental Moderation Parameter

n=10000 Families per Twin TypeSimulated Parameter Value

Est

imat

ed P

aram

eter Slope 0.98

Intercept 0.01

What does our model predict about the correlations between relatives?

Exercise:

• Using ModGraph.R • Run the function• Adjust the parameter values to

fit your expectations about the decay in the correlations

• How does age related gene expression affect the model?

• What would happen if siblings shared less of a common environment as they were more discrepant in age?

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

Age Difference (in Years)

Cor

rela

tion

betw

een

Firs

t Deg

ree

Rel

ativ

es

DZ Correlation

MZ Correlation

A = 0.46C = 0.26T = 0.03E = 0.26

A Mod = 0.14C Mod = 0.18

The Decay in the Correlation between First Degree Relativesas a Function of Age Difference

Correlationdue to

Genetic Factors

Correlationdue to Shared Environmental

Factors

TotalCorrelation

ModGraph(a=1,c=1,t=1,e=1, AgeRange = 10, aM = 1, cM = 1)

Data Requirements

• 5 Types of Families (with twins) • Sex limitation model as the baseline model

• Each Family has up to 6 people:• 2 twins • 2 brothers • 2 sisters

• Missing data on the phenotypes is allowed• Code missing phenotypic observations NA

• Missing data on the definition variables is NOT allowed• Code missing definition variables some arbitrarily large

negative number: eg -9999

Reading the Data into R# ------------------------------------------------------------------------------# Program: TaSMod.R # Author: Brad Verhulst# Date: 10 22, 2012 ## Twin and Sibling Model with Genetic and Environmental Moderation Parameters# Matrix style model - Raw data - Continuous data# -------|---------|---------|---------|---------|---------|---------|---------|

# Call required packagesrequire(OpenMx)

# ------------------------------------------------------------------------------# PREPARE DATAsetwd("C:/Users/Brad Verhulst/Desktop/NIDA mod Scripts")

# Read in the DataTaSdata <- read.csv("TaSdata.csv")

Basic Model Parameters#Basic Parameters Required for the modelnv <- 1Vars <- c('tw','tw','br','br','si','si')selVars <- paste(Vars,c(rep(1,nv),rep(2,nv)),sep="") selAges <- paste('age',1:6, sep="")ntv <- nv*2npeople <-6

# Subset the data

MZMdata <- subset(TaSdata, zyg==1,c(selVars,selAges))MZFdata <- subset(TaSdata, zyg==2,c(selVars,selAges))DZMdata <- subset(TaSdata, zyg==3,c(selVars,selAges))DZFdata <- subset(TaSdata, zyg==4,c(selVars,selAges))DZOdata <- subset(TaSdata, zyg==5,c(selVars,selAges))

General Model Parameters### Specifying the Parameters for the model

# Matrices to store the Path Coefficients am <- mxMatrix( type="Lower", nrow=nv, ncol=nv, free=TRUE, values=.6, label="am11", lbound = .001, name="am")cm <- mxMatrix( type="Lower", nrow=nv, ncol=nv, free=TRUE, values=.6, label="cm11", lbound = .001, name="cm" )tm <- mxMatrix( type="Lower", nrow=nv, ncol=nv, free=TRUE, values=.6, label="tm11", lbound = .001, name="tm" )em <- mxMatrix( type="Lower", nrow=nv, ncol=nv, free=TRUE, values=.6, label="em11", lbound = .001, name="em" )af <- mxMatrix( type="Lower", nrow=nv, ncol=nv, free=TRUE, values=.6, label="af11", lbound = .001, name="af" )cf <- mxMatrix( type="Lower", nrow=nv, ncol=nv, free=TRUE, values=.6, label="cf11", lbound = .001, name="cf" )tf <- mxMatrix( type="Lower", nrow=nv, ncol=nv, free=TRUE, values=.6, label="tf11", lbound = .001, name="tf" )ef <- mxMatrix( type="Lower", nrow=nv, ncol=nv, free=TRUE, values=.6, label="ef11", lbound = .001, name="ef" )rg <- mxMatrix( type="Diag", nrow=1, ncol=1, free=FALSE, values=1, label="rg11", lbound = -1, ubound = 1, name="rg" )

paths <- c(am, cm, tm ,em, af, cf, tf, ef, rg) # Matrices A, C, and E compute variance componentsAm <- mxAlgebra( expression=am %*% t(am), name="Am" )Cm <- mxAlgebra( expression=cm %*% t(cm), name="Cm" )Tm <- mxAlgebra( expression=tm %*% t(tm), name="Tm" )Em <- mxAlgebra( expression=em %*% t(em), name="Em" )

Af <- mxAlgebra( expression=af %*% t(af), name="Af" )Cf <- mxAlgebra( expression=cf %*% t(cf), name="Cf" )Tf <- mxAlgebra( expression=tf %*% t(tf), name="Tf" )Ef <- mxAlgebra( expression=ef %*% t(ef), name="Ef" )

VCs <- c(Am, Cm, Tm, Em, Af, Cf, Tf, Ef, Amf, Cmf, Tmf, Emf, Afm, Cfm, Tfm, Efm)

Specifying the Genetic Correlation Matrix

mzmA <- mxAlgebra(expression= rbind(cbind(Am , Am , .5*Am , .5*Am , .5*Amf, .5*Amf), cbind(Am , Am , .5*Am , .5*Am , .5*Amf, .5*Amf), cbind(.5*Am , .5*Am , Am , .5*Am , .5*Amf, .5*Amf), cbind(.5*Am , .5*Am , .5*Am , Am , .5*Amf, .5*Amf), cbind(.5*Afm, .5*Afm, .5*Afm, .5*Afm, Af , .5*Af ), cbind(.5*Afm, .5*Afm, .5*Afm, .5*Afm, .5*Af , Af )), name = "mzmA")

mzfA <- mxAlgebra(expression= rbind(cbind(Af , Af , .5*Amf, .5*Amf, .5*Af , .5*Af ), cbind(Af , Af , .5*Amf, .5*Amf, .5*Af , .5*Af ), cbind(.5*Afm, .5*Afm, Am , .5*Am , .5*Amf, .5*Amf), cbind(.5*Afm, .5*Afm, .5*Am , Am , .5*Amf, .5*Amf), cbind(.5*Af , .5*Af , .5*Afm, .5*Afm, Af , .5*Af ), cbind(.5*Af , .5*Af , .5*Afm, .5*Afm, .5*Af , Af )), name = "mzfA")

dzmA <- mxAlgebra(expression= rbind(cbind(Am , .5*Am , .5*Am , .5*Am , .5*Amf, .5*Amf), cbind(.5*Am , Am , .5*Am , .5*Am , .5*Amf, .5*Amf), cbind(.5*Am , .5*Am , Am , .5*Am , .5*Amf, .5*Amf), cbind(.5*Am , .5*Am , .5*Am , Am , .5*Amf, .5*Amf), cbind(.5*Afm, .5*Afm, .5*Afm, .5*Afm, Af , .5*Af ), cbind(.5*Afm, .5*Afm, .5*Afm, .5*Afm, .5*Af , Af )), name = "dzmA")

dzfA <- mxAlgebra(expression= rbind(cbind(Af , .5*Af , .5*Amf, .5*Amf, .5*Af , .5*Af ), cbind(.5*Af , Af , .5*Amf, .5*Amf, .5*Af , .5*Af ), cbind(.5*Afm, .5*Afm, Am , .5*Am , .5*Amf, .5*Amf), cbind(.5*Afm, .5*Afm, .5*Am , Am , .5*Amf, .5*Amf), cbind(.5*Af , .5*Af , .5*Afm, .5*Afm, Af , .5*Af ), cbind(.5*Af , .5*Af , .5*Afm, .5*Afm, .5*Af , Af )), name = "dzfA")

dzoA <- mxAlgebra(expression= rbind(cbind(Am , .5*Amf, .5*Am , .5*Am , .5*Amf, .5*Amf), cbind(.5*Afm, Af , .5*Amf, .5*Amf, .5*Af , .5*Af ), cbind(.5*Am , .5*Afm, Am , .5*Am , .5*Amf, .5*Amf), cbind(.5*Am , .5*Afm, .5*Am , Am , .5*Amf, .5*Amf), cbind(.5*Afm, .5*Af , .5*Afm, .5*Afm, Af , .5*Af ), cbind(.5*Afm, .5*Af , .5*Afm, .5*Afm, .5*Af , Af )), name = "dzoA")

mzmA <- mxAlgebra(expression= rbind(cbind(Am , Am , .5*Am , .5*Am , .5*Amf, .5*Amf), cbind(Am , Am , .5*Am , .5*Am , .5*Amf, .5*Amf), cbind(.5*Am , .5*Am , Am , .5*Am , .5*Amf, .5*Amf), cbind(.5*Am , .5*Am , .5*Am , Am , .5*Amf, .5*Amf), cbind(.5*Afm, .5*Afm, .5*Afm, .5*Afm, Af , .5*Af ), cbind(.5*Afm, .5*Afm, .5*Afm, .5*Afm, .5*Af , Af )), name = "mzmA")

Specifying the Common Environmental Correlation Matrix

malC <- mxAlgebra(expression= rbind(cbind(Cm , Cm , Cm , Cm , Cmf, Cmf), cbind(Cm , Cm , Cm , Cm , Cmf, Cmf), cbind(Cm , Cm , Cm , Cm , Cmf, Cmf), cbind(Cm , Cm , Cm , Cm , Cmf, Cmf), cbind(Cfm , Cfm, Cfm, Cfm, Cf , Cf ), cbind(Cfm , Cfm, Cfm, Cfm, Cf , Cf )),name = "malC")

femC <- mxAlgebra(expression= rbind(cbind(Cf , Cf , Cmf, Cmf, Cf , Cf ), cbind(Cf , Cf , Cmf, Cmf, Cf , Cf ), cbind(Cfm , Cfm, Cm , Cm , Cmf, Cmf), cbind(Cfm , Cfm, Cm , Cm , Cmf, Cmf), cbind(Cf , Cf , Cfm, Cfm, Cf , Cf ), cbind(Cf , Cf , Cfm, Cfm, Cf , Cf )),name = "femC")

oppC <- mxAlgebra(expression= rbind(cbind(Cm , Cmf, Cm , Cm , Cmf, Cmf), cbind(Cfm , Cf , Cmf, Cmf, Cf , Cf ), cbind(Cm , Cfm, Cm , Cm , Cmf, Cmf), cbind(Cm , Cfm, Cm , Cm , Cmf, Cmf), cbind(Cfm , Cf , Cfm, Cfm, Cf , Cf ), cbind(Cfm , Cf , Cfm, Cfm, Cf , Cf )),name = "oppC")

Specifying the Special Twin Environment Correlation Matrix

malT <- mxAlgebra(expression= rbind(cbind(Tm, Tm, 0 , 0 , 0 , 0 ), cbind(Tm, Tm, 0 , 0 , 0 , 0 ), cbind(0 , 0 , Tm, 0 , 0 , 0 ), cbind(0 , 0 , 0 , Tm, 0 , 0 ), cbind(0 , 0 , 0 , 0 , Tf, 0 ), cbind(0 , 0 , 0 , 0 , 0 , Tf)),name = "malT")

femT <- mxAlgebra(expression= rbind(cbind(Tf, Tf, 0 , 0 , 0 , 0 ), cbind(Tf, Tf, 0 , 0 , 0 , 0 ), cbind(0 , 0 , Tm, 0 , 0 , 0 ), cbind(0 , 0 , 0 , Tm, 0 , 0 ), cbind(0 , 0 , 0 , 0 , Tf, 0 ), cbind(0 , 0 , 0 , 0 , 0 , Tf)),name = "femT")

oppT <- mxAlgebra(expression= rbind(cbind(Tm, Tmf, 0 , 0 , 0 , 0 ), cbind(Tfm, Tf, 0 , 0 , 0 , 0 ), cbind(0 , 0 , Tm, 0 , 0 , 0 ), cbind(0 , 0 , 0 , Tm, 0 , 0 ), cbind(0 , 0 , 0 , 0 , Tf, 0 ), cbind(0 , 0 , 0 , 0 , 0 , Tf)),name = "oppT")

Specifying the Unique Environment Correlation Matrix

malE <- mxAlgebra(expression= rbind(cbind(Em, 0 , 0 , 0 , 0 , 0 ), cbind(0 , Em, 0 , 0 , 0 , 0 ), cbind(0 , 0 , Em, 0 , 0 , 0 ), cbind(0 , 0 , 0 , Em, 0 , 0 ), cbind(0 , 0 , 0 , 0 , Ef, 0 ), cbind(0 , 0 , 0 , 0 , 0 , Ef)),name = "malE")

femE <- mxAlgebra(expression= rbind(cbind(Ef, 0, 0 , 0 , 0 , 0 ), cbind(0 , Ef, 0 , 0 , 0 , 0 ), cbind(0 , 0 , Em, 0 , 0 , 0 ), cbind(0 , 0 , 0 , Em, 0 , 0 ), cbind(0 , 0 , 0 , 0 , Ef, 0 ), cbind(0 , 0 , 0 , 0 , 0 , Ef)),name = "femE")

oppE <- mxAlgebra(expression= rbind(cbind(Em, 0 , 0 , 0 , 0 , 0 ), cbind(0 , Ef, 0 , 0 , 0 , 0 ), cbind(0 , 0 , Em, 0 , 0 , 0 ), cbind(0 , 0 , 0 , Em, 0 , 0 ), cbind(0 , 0 , 0 , 0 , Ef, 0 ), cbind(0 , 0 , 0 , 0 , 0 , Ef)),name = "oppE")

These are the interesting matrices

## Variance - Covariance Matrices (Unmoderated)MaleVar <- mxAlgebra( expression=Am+Cm+Tm+Em, name="MaleVar" ) ## Male Total Variance FemaVar <- mxAlgebra( expression=Af+Cf+Tf+Ef, name="FemaVar" ) ### Female Total Variance

## Means (Unmoderated)Meanm <- mxMatrix( type="Full", nrow=1, ncol=nv, free=T, values= 1, label="meanm", name="Meanm" )Meanf <- mxMatrix( type="Full", nrow=1, ncol=nv, free=T, values= 1, label="meanf", name="Meanf" )

## Matrices to store the effect of Age on the Meanbm <- mxMatrix( type="Full", nrow=1, ncol=1, free=T, values=0, label=c("intM"), name="bm" )bf <- mxMatrix( type="Full", nrow=1, ncol=1, free=T, values=0, label=c("intF"), name="bf" )

# Matrices to store the moderation parameters aMOD <- mxMatrix( type="Full", nrow=nv, ncol=nv, free=TRUE, values=.2, label="aM11", name="aMOD" )cMOD <- mxMatrix( type="Full", nrow=nv, ncol=nv, free=TRUE, values=0 , label="cM11", name="cMOD" )

unit <- mxMatrix(type="Full",nrow=nv, ncol=npeople, free=FALSE, values=1, name="unit")

Matrices for the definition terms ### Matrices for the definition terms# MZMmzmDtw1 <- mxMatrix( type="Full", nrow=1, ncol=1, free=FALSE, labels=c("data.age1"), name="Dmt1") #twin1mzmDtw2 <- mxMatrix( type="Full", nrow=1, ncol=1, free=FALSE, labels=c("data.age2"), name="Dmt2") #twin2mzmDms1 <- mxMatrix( type="Full", nrow=1, ncol=1, free=FALSE, labels=c("data.age3"), name="Dms1") #Male Sib 1mzmDms2 <- mxMatrix( type="Full", nrow=1, ncol=1, free=FALSE, labels=c("data.age4"), name="Dms2") #Male Sib 2mzmDfs1 <- mxMatrix( type="Full", nrow=1, ncol=1, free=FALSE, labels=c("data.age5"), name="Dfs1") #Female Sib 1mzmDfs2 <- mxMatrix( type="Full", nrow=1, ncol=1, free=FALSE, labels=c("data.age6"), name="Dfs2") #Female Sib 2

# MZFmzfDtw1 <- mxMatrix( type="Full", nrow=1, ncol=1, free=FALSE, labels=c("data.age1"), name="Dft1") #twin1mzfDtw2 <- mxMatrix( type="Full", nrow=1, ncol=1, free=FALSE, labels=c("data.age2"), name="Dft2") #twin2mzfDms1 <- mxMatrix( type="Full", nrow=1, ncol=1, free=FALSE, labels=c("data.age3"), name="Dms1") #Male Sib 1mzfDms2 <- mxMatrix( type="Full", nrow=1, ncol=1, free=FALSE, labels=c("data.age4"), name="Dms2") #Male Sib 2mzfDfs1 <- mxMatrix( type="Full", nrow=1, ncol=1, free=FALSE, labels=c("data.age5"), name="Dfs1") #Female Sib 1mzfDfs2 <- mxMatrix( type="Full", nrow=1, ncol=1, free=FALSE, labels=c("data.age6"), name="Dfs2") #Female Sib 2

# DZMdzmDtw1 <- mxMatrix( type="Full", nrow=1, ncol=1, free=FALSE, labels=c("data.age1"), name="Dtw1") #twin1dzmDtw2 <- mxMatrix( type="Full", nrow=1, ncol=1, free=FALSE, labels=c("data.age2"), name="Dtw2") #twin2dzmDms1 <- mxMatrix( type="Full", nrow=1, ncol=1, free=FALSE, labels=c("data.age3"), name="Dms1") #Male Sib 1dzmDms2 <- mxMatrix( type="Full", nrow=1, ncol=1, free=FALSE, labels=c("data.age4"), name="Dms2") #Male Sib 2dzmDfs1 <- mxMatrix( type="Full", nrow=1, ncol=1, free=FALSE, labels=c("data.age5"), name="Dfs1") #Female Sib 1dzmDfs2 <- mxMatrix( type="Full", nrow=1, ncol=1, free=FALSE, labels=c("data.age6"), name="Dfs2") #Female Sib 2

# DZFdzfDtw1 <- mxMatrix( type="Full", nrow=1, ncol=1, free=FALSE, labels=c("data.age1"), name="Dtw1") #twin1dzfDtw2 <- mxMatrix( type="Full", nrow=1, ncol=1, free=FALSE, labels=c("data.age2"), name="Dtw2") #twin2dzfDms1 <- mxMatrix( type="Full", nrow=1, ncol=1, free=FALSE, labels=c("data.age3"), name="Dms1") #Male Sib 1dzfDms2 <- mxMatrix( type="Full", nrow=1, ncol=1, free=FALSE, labels=c("data.age4"), name="Dms2") #Male Sib 2dzfDfs1 <- mxMatrix( type="Full", nrow=1, ncol=1, free=FALSE, labels=c("data.age5"), name="Dfs1") #Female Sib 1dzfDfs2 <- mxMatrix( type="Full", nrow=1, ncol=1, free=FALSE, labels=c("data.age6"), name="Dfs2") #Female Sib 2

# DZOdzoDtw1 <- mxMatrix( type="Full", nrow=1, ncol=1, free=FALSE, labels=c("data.age1"), name="Dtw1") #twin1dzoDtw2 <- mxMatrix( type="Full", nrow=1, ncol=1, free=FALSE, labels=c("data.age2"), name="Dtw2") #twin2dzoDms1 <- mxMatrix( type="Full", nrow=1, ncol=1, free=FALSE, labels=c("data.age3"), name="Dms1") #Male Sib 1dzoDms2 <- mxMatrix( type="Full", nrow=1, ncol=1, free=FALSE, labels=c("data.age4"), name="Dms2") #Male Sib 2dzoDfs1 <- mxMatrix( type="Full", nrow=1, ncol=1, free=FALSE, labels=c("data.age5"), name="Dfs1") #Female Sib 1dzoDfs2 <- mxMatrix( type="Full", nrow=1, ncol=1, free=FALSE, labels=c("data.age6"), name="Dfs2") #Female Sib 2

Algebra for the definition terms ### Algebra for effect of the definition term on the mean# MZMmzmdefmt1 <- mxAlgebra( expression= bm %*% Dmt1, name="defmt1")mzmdefmt2 <- mxAlgebra( expression= bm %*% Dmt2, name="defmt2")mzmdefms1 <- mxAlgebra( expression= bm %*% Dms1, name="defms1")mzmdefms2 <- mxAlgebra( expression= bm %*% Dms2, name="defms2")mzmdeffs1 <- mxAlgebra( expression= bf %*% Dfs1, name="deffs1")mzmdeffs2 <- mxAlgebra( expression= bf %*% Dfs2, name="deffs2")

# MZFmzfdefmt1 <- mxAlgebra( expression= bf %*% Dft1, name="defft1")mzfdefmt2 <- mxAlgebra( expression= bf %*% Dft2, name="defft2")mzfdefms1 <- mxAlgebra( expression= bm %*% Dms1, name="defms1")mzfdefms2 <- mxAlgebra( expression= bm %*% Dms2, name="defms2")mzfdeffs1 <- mxAlgebra( expression= bf %*% Dfs1, name="deffs1")mzfdeffs2 <- mxAlgebra( expression= bf %*% Dfs2, name="deffs2")

# DZMdzmdefmt1 <- mxAlgebra( expression= bm %*% Dtw1, name="defmt1")dzmdefmt2 <- mxAlgebra( expression= bm %*% Dtw2, name="defmt2")dzmdefms1 <- mxAlgebra( expression= bm %*% Dms1, name="defms1")dzmdefms2 <- mxAlgebra( expression= bm %*% Dms2, name="defms2")dzmdeffs1 <- mxAlgebra( expression= bf %*% Dfs1, name="deffs1")dzmdeffs2 <- mxAlgebra( expression= bf %*% Dfs2, name="deffs2")

# DZFdzfdefmt1 <- mxAlgebra( expression= bf %*% Dtw1, name="defft1")dzfdefmt2 <- mxAlgebra( expression= bf %*% Dtw2, name="defft2")dzfdefms1 <- mxAlgebra( expression= bm %*% Dms1, name="defms1")dzfdefms2 <- mxAlgebra( expression= bm %*% Dms2, name="defms2")dzfdeffs1 <- mxAlgebra( expression= bf %*% Dfs1, name="deffs1") dzfdeffs2 <- mxAlgebra( expression= bf %*% Dfs2, name="deffs2")

# DZOdzodefmt1 <- mxAlgebra( expression= bm %*% Dtw1, name="defmt1")dzodefmt2 <- mxAlgebra( expression= bf %*% Dtw2, name="defft2")dzodefms1 <- mxAlgebra( expression= bm %*% Dms1, name="defms1")dzodefms2 <- mxAlgebra( expression= bm %*% Dms2, name="defms2")dzodeffs1 <- mxAlgebra( expression= bf %*% Dfs1, name="deffs1")dzodeffs2 <- mxAlgebra( expression= bf %*% Dfs2, name="deffs2")

Calculating the Mean

## Algebra to calculate the Expected Mean expMeanMZm <- mxAlgebra( expression= cbind((Meanm + defmt1),(Meanm + defmt2),(Meanm + defms1),(Meanm + defms2),(Meanf + deffs1),(Meanf + deffs2)), name="expMeanMZm")expMeanMZf <- mxAlgebra( expression= cbind((Meanf + defft1),(Meanf + defft2),(Meanm + defms1),(Meanm + defms2),(Meanf + deffs1),(Meanf + deffs2)), name="expMeanMZf")expMeanDZm <- mxAlgebra( expression= cbind((Meanm + defmt1),(Meanm + defmt2),(Meanm + defms1),(Meanm + defms2),(Meanf + deffs1),(Meanf + deffs2)), name="expMeanDZm")expMeanDZf <- mxAlgebra( expression= cbind((Meanf + defft1),(Meanf + defft2),(Meanm + defms1),(Meanm + defms2),(Meanf + deffs1),(Meanf + deffs2)), name="expMeanDZf")expMeanDZmf <- mxAlgebra( expression= cbind((Meanm + defmt1),(Meanf + defft2),(Meanm + defms1),(Meanm + defms2),(Meanf + deffs1),(Meanf + deffs2)), name="expMeanDZmf")

Setting up the Moderation Matrix### Setting up the moderation matrix# Data for Ages AgeMZm <- mxMatrix(type="Full",nrow=npeople, ncol=nv, free=FALSE, labels=c("data.age1","data.age2","data.age3","data.age4","data.age5","data.age6"), name="AgeMZm")AgeMZf <- mxMatrix(type="Full",nrow=npeople, ncol=nv, free=FALSE, labels=c("data.age1","data.age2","data.age3","data.age4","data.age5","data.age6"), name="AgeMZf")AgeDZm <- mxMatrix(type="Full",nrow=npeople, ncol=nv, free=FALSE, labels=c("data.age1","data.age2","data.age3","data.age4","data.age5","data.age6"), name="AgeDZm")AgeDZf <- mxMatrix(type="Full",nrow=npeople, ncol=nv, free=FALSE, labels=c("data.age1","data.age2","data.age3","data.age4","data.age5","data.age6"), name="AgeDZf")AgeDZmf <- mxMatrix(type="Full",nrow=npeople, ncol=nv, free=FALSE, labels=c("data.age1","data.age2","data.age3","data.age4","data.age5","data.age6"), name="AgeDZmf")

# Age matrix (pre)ageMatMZm <- mxAlgebra(AgeMZm %x% unit, name = "ageMatMZm")ageMatMZf <- mxAlgebra(AgeMZf %x% unit, name = "ageMatMZf")ageMatDZm <- mxAlgebra(AgeDZm %x% unit, name = "ageMatDZm")ageMatDZf <- mxAlgebra(AgeDZf %x% unit, name = "ageMatDZf")ageMatDZmf <- mxAlgebra(AgeDZmf %x% unit, name = "ageMatDZmf")

# Age Difference MatrixdeltaMZm <- mxAlgebra(abs(ageMatMZm - t(ageMatMZm)), name = "deltaMZm" )deltaMZf <- mxAlgebra(abs(ageMatMZf - t(ageMatMZf)), name = "deltaMZf" )deltaDZm <- mxAlgebra(abs(ageMatDZm - t(ageMatDZm)), name = "deltaDZm" )deltaDZf <- mxAlgebra(abs(ageMatDZf - t(ageMatDZf)), name = "deltaDZf" )deltaDZmf <- mxAlgebra(abs(ageMatDZmf - t(ageMatDZmf)), name = "deltaDZmf" )

1 1 1 1 1 1

252530322927

%x%

25 25 25 25 25 2525 25 25 25 25 2530 30 30 30 30 3032 32 32 32 32 3229 29 29 29 29 2927 27 27 27 27 27

0 0 -5 -7 -4 -20 0 -5 -7 -4 -25 5 0 -2 1 37 7 -2 0 3 54 4 1 -3 0 22 2 3 -5 -2 0

Algebras for the Moderation Matrices

# Algebra for the Genetic Moderation ParameteraMODmzm <- mxAlgebra(exp(-deltaMZm %x% aMOD), name = "aMODmzm")aMODmzf <- mxAlgebra(exp(-deltaMZf %x% aMOD), name = "aMODmzf")aMODdzm <- mxAlgebra(exp(-deltaDZm %x% aMOD), name = "aMODdzm")aMODdzf <- mxAlgebra(exp(-deltaDZf %x% aMOD), name = "aMODdzf")aMODdzmf <- mxAlgebra(exp(-deltaDZmf %x% aMOD), name = "aMODdzmf")

# Algebra for the Common Environmental Moderation ParametercMODmzm <- mxAlgebra(exp(-deltaMZm %x% cMOD), name = "cMODmzm")cMODmzf <- mxAlgebra(exp(-deltaMZf %x% cMOD), name = "cMODmzf")cMODdzm <- mxAlgebra(exp(-deltaDZm %x% cMOD), name = "cMODdzm")cMODdzf <- mxAlgebra(exp(-deltaDZf %x% cMOD), name = "cMODdzf")cMODdzmf <- mxAlgebra(exp(-deltaDZmf %x% cMOD), name = "cMODdzmf")

Specifying the Expected Covariance Matrices

## Expected Covariance MatricesexpCovMZm <- mxAlgebra(mzmA * aMODmzm + malC* cMODmzm + malT + malE, name="expCovMZm")expCovMZf <- mxAlgebra(mzfA * aMODmzf + femC* cMODmzf + femT + femE , name="expCovMZf")expCovDZm <- mxAlgebra(dzmA * aMODdzm + malC* cMODdzm + malT + malE, name="expCovDZm")expCovDZf <- mxAlgebra(dzfA * aMODdzf + femC* cMODdzf + femT + femE, name="expCovDZf")expCovDZmf <- mxAlgebra(dzoA * aMODdzmf+ oppC* cMODdzmf+ oppT + oppE, name="expCovDZmf")

Specifying the Data and Objective Functions

# Specifying the Data FramesdataMZM <- mxData( observed=MZMdata, type="raw" )dataMZF <- mxData( observed=MZFdata, type="raw")dataDZM <- mxData( observed=DZMdata, type="raw" )dataDZF <- mxData( observed=DZFdata, type="raw" )dataDZO <- mxData( observed=DZOdata, type="raw" )

# Specifying the Objective FunctionsMZMobj <- mxFIMLObjective( covariance="expCovMZm", means="expMeanMZm", dimnames=selVars )MZFobj <- mxFIMLObjective( covariance="expCovMZf", means="expMeanMZf", dimnames=selVars )DZMobj <- mxFIMLObjective( covariance="expCovDZm", means="expMeanDZm", dimnames=selVars )DZFobj <- mxFIMLObjective( covariance="expCovDZf", means="expMeanDZf", dimnames=selVars )DZOobj <- mxFIMLObjective( covariance="expCovDZmf", means="expMeanDZmf", dimnames=selVars )

Specifying the Submodels MZm <- mxModel("MZm",paths, VCs,aMOD,cMOD,bm,bf,MaleVar,FemaVar,mzmA,aMODmzm,malC,cMODmzm,malT,malE,mzmDtw1,mzmDtw2,mzmDms1,mzmDms2,mzmDfs1,mzmDfs2,mzmdefmt1,mzmdefmt2,mzmdefms1,mzmdefms2,mzmdeffs1,mzmdeffs2,Meanm,Meanf,expMeanMZm,unit,AgeMZm,ageMatMZm,deltaMZm,aMODmzm,cMODmzm,expCovMZm,dataMZM,MZMobj )

MZf <- mxModel("MZf",paths, VCs,aMOD,cMOD,bm,bf,MaleVar,FemaVar,mzfA,aMODmzf,femC,cMODmzf,femT,femE,mzfDtw1,mzfDtw2,mzfDms1,mzfDms2,mzfDfs1,mzfDfs2,mzfdefmt1,mzfdefmt2,mzfdefms1,mzfdefms2,mzfdeffs1,mzfdeffs2,Meanm,Meanf,expMeanMZf,unit,AgeMZf,ageMatMZf,deltaMZf,aMODmzf,cMODmzf,expCovMZf,dataMZF,MZFobj )

DZm <- mxModel("DZm", paths, VCs,aMOD,cMOD,bm,bf,MaleVar,FemaVar,dzmA,aMODdzm,malC,cMODdzm,malT,malE,dzmDtw1,dzmDtw2,dzmDms1,dzmDms2,dzmDfs1,dzmDfs2,dzmdefmt1,dzmdefmt2,dzmdefms1,dzmdefms2,dzmdeffs1,dzmdeffs2,Meanm,Meanf,expMeanDZm,unit,AgeDZm,ageMatDZm,deltaDZm,aMODdzm,cMODdzm,expCovDZm,dataDZM,DZMobj )

DZf <- mxModel("DZf", paths, VCs,aMOD,cMOD,bm,bf,MaleVar,FemaVar,dzfA,aMODdzf,femC,cMODdzf,femT,femE,dzfDtw1,dzfDtw2,dzfDms1,dzfDms2,dzfDfs1,dzfDfs2,dzfdefmt1,dzfdefmt2,dzfdefms1,dzfdefms2,dzfdeffs1,dzfdeffs2,Meanm,Meanf,expMeanDZf,unit,AgeDZf,ageMatDZf,deltaDZf,aMODdzf,cMODdzf,expCovDZf,dataDZF,DZFobj )

DZmf <- mxModel("DZmf", paths, VCs,aMOD,cMOD,bm,bf,MaleVar,FemaVar,dzoA,aMODdzmf,oppC,cMODdzmf,oppT,oppE,dzoDtw1,dzoDtw2,dzoDms1,dzoDms2,dzoDfs1,dzoDfs2,dzodefmt1,dzodefmt2,dzodefms1,dzodefms2,dzodeffs1,dzodeffs2, Meanm,Meanf,expMeanDZmf,unit,AgeDZmf,ageMatDZmf,deltaDZmf,aMODdzmf,cMODdzmf,expCovDZmf,dataDZO,DZOobj )

Running the ModelTaSModModel <- mxModel("TaSHetACE",MZm, MZf, DZm, DZf, DZmf,mxAlgebra( expression=MZm.objective + MZf.objective +DZm.objective + DZf.objective + DZmf.objective, name="minus2sumloglikelihood" ),mxAlgebraObjective("minus2sumloglikelihood"))

TaSModFit <- mxRun(TaSModModel )TaSModSumm <- summary(TaSModFit )TaSModSumm

Looking at the Outputfree parameters: name matrix row col Estimate Std.Error lbound ubound1 am11 MZm.am 1 1 0.69356237 0.020621014 0.001 2 cm11 MZm.cm 1 1 0.45026574 0.017106106 0.001 3 tm11 MZm.tm 1 1 0.27076106 0.035208626 0.001 4 em11 MZm.em 1 1 0.49112887 0.010067007 0.001 5 af11 MZm.af 1 1 0.44093396 0.025880749 0.001 6 cf11 MZm.cf 1 1 0.70389297 0.014464905 0.001 7 tf11 MZm.tf 1 1 0.03834467 0.061631613 0.001 8 ef11 MZm.ef 1 1 0.50301198 0.009813318 0.001 9 aM11 MZm.aMOD 1 1 0.35600616 0.684446181 10 cM11 MZm.cMOD 1 1 0.31349403 0.253406803 11 intM MZm.bm 1 1 -0.03724208 0.156987583 12 intF MZm.bf 1 1 0.14033247 0.106558294 13 meanm MZm.Meanm 1 1 0.02990390 0.084356156 14 meanf MZm.Meanf 1 1 -0.07092350 0.059893376

observed statistics: 30000 estimated parameters: 14 degrees of freedom: 29986 -2 log likelihood: 71950.61 saturated -2 log likelihood: NA number of observations: 5000 chi-square: NA p: NA Information Criteria: df Penalty Parameters Penalty Sample-Size AdjustedAIC: 11978.61 71978.61 NABIC: -183445.95 72069.85 72025.36CFI: NA TLI: NA RMSEA: NA timestamp: 2012-10-18 11:36:44 frontend time: 3.249828 secs backend time: 1.830538 mins independent submodels time: 0 secs wall clock time: 113.0821 secs cpu time: 113.0821 secs openmx version number: 1.3.0-2169

Looking at the Output-2 log likelihood: 71950.61 saturated -2 log likelihood: NA number of observations: 5000 chi-square: NA p: NA Information Criteria: df Penalty Parameters Penalty Sample-Size AdjustedAIC: 11978.61 71978.61 NABIC: -183445.95 72069.85 72025.36CFI: NA TLI: NA RMSEA: NA timestamp: 2012-10-18 11:36:44 frontend time: 3.249828 secs backend time: 1.830538 mins independent submodels time: 0 secs wall clock time: 113.0821 secs cpu time: 113.0821 secs openmx version number: 1.3.0-2169

Communicating the Results

Males Females

0 2 4 6 8 10

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0.4

0.6

0.8

1.0

Age Difference (in Years)

Cor

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t Deg

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DZ Correlation

MZ CorrelationA = 0.48C = 0.2T = 0.07E = 0.24

A Mod = 0.35C Mod = 0.31

The Decay in the Correlation between First Degree Relativesas a Function of Age Difference

Correlationdue to

Genetic Factors

Correlationdue to Shared Environmental

Factors

TotalCorrelation

0 2 4 6 8 10

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Age Difference (in Years)

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DZ Correlation

MZ Correlation

A = 0.21C = 0.53T = 0E = 0.26

A Mod = 0.39C Mod = 0.34

The Decay in the Correlation between First Degree Relativesas a Function of Age Difference

Correlationdue to

Genetic Factors

Correlationdue to Shared Environmental

Factors

TotalCorrelation

Thank You

Special Acknowledgements

R25DA026119 (PI: Neale)

Mike Neale

Hermine Maes

Lindon Eaves