October 21 2014 Elizabeth PromWormley amp Hermine Maes ecpromwormlevcuedu 8048288154 The Problems BMI may not be an appropriate measure for use in studying the genetics of obesity ID: 643129
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Slide1
Fitting Bivariate Models
October 21, 2014
Elizabeth Prom-Wormley &
Hermine
Maes
ecpromwormle@vcu.edu
804-828-8154Slide2
The Problem(s)
BMI may not be an appropriate measure for use in studying the genetics of obesity
Do height and weight share the same genetic/environmental influences?
Smoking is a risk factor for cardiovascular disease, but how ?
Does smoking
in adolescence lead to cardiovascular disease in adulthood?Slide3
A Solution
Bivariate Genetic Analysis
Two or more traits can be correlated because they share common genes or common environmental influences
e.g. Are the same genetic/environmental factors influencing the traits
?
With twin data on multiple traits it is possible to partition the
covariation
into its genetic and environmental components
Goal: to understand what factors make sets of variables correlate or co-varySlide4
Bivariate Analysis
A Roadmap
1- Use the data to test basic assumptions inherent to standard
genetic models
Saturated Bivariate
Model
2- Estimate the
contributions of genetic and environmental factors to the covariance between two
traits
Bivariate Genetic Models (Bivariate Genetic Analysis via
Cholesky
Decomposition)Slide5
Getting a Feel for the Data
Phenotypic and Twin Correlations
MZ
DZ
Height2
Weight2
Height2Weight2Height10.880.41 0.440.23Weight10.490.84 0.170.33
Open R script for today
r
Weight
/
Height
= 0.47Slide6
Getting a Feel for the Data
Phenotypic and Twin Correlations
MZ
DZ
Height2
Weight2
Height2Weight2Height10.880.41 0.440.23Weight10.490.84 0.170.33
r
Weight
/Height = 0.47
Expectations
1-
rMZ
>
rDZ
(cross-twin/cross-trait)
Genetic
effects contribute to the relationship between height and weight
2- Cross
-twin cross-variable correlations are not as big as the correlations between
twins
within
variables.
Variable specific
genetic
effects on height and/
or weightSlide7
So…How Can We be Sure?Slide8
Building the
Bivariate Genetic Model
Sources
of Information
Cross
-trait covariance
within
individuals
Within-Twin CovarianceCross-trait covariance between twinsCross-Twin CovarianceMZ:DZ ratio of cross-trait covariance between twinsSlide9
Basic Data Assumptions
MZ
and DZ twins are sampled from the same population, therefore we expect
Equal means/variances in Twin 1 and Twin 2
Equal means/variances in MZ and DZ
twins
Equal
covariances
between Twin 1 and Twin 2 in MZ and DZ twins
9Slide10
Getting a Feel for the Data
Means
ht1
wt1
ht2
wt2
MZ
16.30
5.67
16.29
5.65
DZ
16.41
5.82
16.33
5.77Slide11
Getting a Feel for the Data
Variances-
Covariances
ht1
wt1
ht2
wt2
ht1wt1ht2wt2
ht1
wt1
ht2
wt2
ht1
wt1
ht2
wt2
MZ Twins
DZ Twins
Twin 1
Twin 2
Twin 1
Twin 2
Twin 1
Twin 2
Twin 1
Twin 2Slide12
Getting a Feel for the Data
Variances-
Covariances
Variances Slide13
Observed Variance-Covariance Matrix
P1
P
2
P1
P
2
P
1
Variance
Twin 1- P1
P2
Variance
Twin 1- P2
P
1
Variance
Twin2- P1
P
2
Variance
Twin 2- P2
Twin 1
Twin 2
Twin 1
Twin 2
VariancesSlide14
Getting a Feel for the Data
Variances-
Covariances
ht1
wt1
ht2
wt2
ht10.43wt10.73ht20.44
wt2
0.78
ht1
wt1
ht2
wt2
ht1
0.48
wt1
0.75
ht2
0.46
wt2
0.86
MZ Twins
DZ Twins
Twin 1
Twin 2
Twin 1
Twin 2
Twin 1
Twin 2
Twin 1
Twin 2Slide15
Getting a Feel for the Data
Variances-
Covariances
Cross-Trait / Within-Twin CovarianceSlide16
Observed Variance-Covariance Matrix
P1
P
2
P1
P
2
P
1
Variance
Twin 1- P1
Covariance
Twin 1
P1/P2
P2
Covariance
Twin 1
P1/P2
Variance
Twin 1- P2
P
1
Variance
Twin2- P1
Covariance
Twin 2
P1/P2
P
2
Covariance
Twin 2
P1/P2
Variance
Twin 2- P2
Twin 1
Twin 2
Twin 1
Twin 2
Variances
+
Cross-Trait
Within-Twin
CovariancesSlide17
Getting a Feel for the Data
Variances-
Covariances
ht1
wt1
ht2
wt2
ht10.430.28wt10.280.73ht20.44
0.26
wt2
0.26
0.78
ht1
wt1
ht2
wt2
ht1
0.48
0.26
wt1
0.26
0.75
ht2
0.46
0.28
wt2
0.28
0.86
MZ Twins
DZ Twins
Twin 1
Twin 2
Twin 1
Twin 2
Twin 1
Twin 2
Twin 1
Twin 2Slide18
Getting a Feel for the Data
Variances-
Covariances
Cross-Trait / Within-Twin CovarianceSlide19
Observed Variance-Covariance Matrix
P1
P
2
P1
P
2
P
1
Variance
Twin 1- P1
Covariance
Twin 1
P1/P2
P1 Within Trait
T1 /T2 Covariance
P2
Covariance
Twin 1
P1/P2
Variance
Twin 1- P2
P2 Within Trait
T1 /T2 Covariance
P
1
P1 Within Trait
T1 /T2
Covariance
Variance
Twin2- P1
Covariance
Twin 2
P1/P2
P
2
P2 Within Trait
T1 /T2
Covariance
Covariance
Twin 2
P1/P2
Variance
Twin 2- P2
Twin 1
Twin 2
Twin 1
Twin 2
Variances
+
Cross-Trait
Within-Twin
Covariances
+
Within-Trait
Cross-Twin
CovariancesSlide20
Getting a Feel for the Data
Variances-
Covariances
ht1
wt1
ht2
wt2
ht10.430.280.38wt10.280.730.64ht20.38
0.44
0.26
wt2
0.64
0.26
0.78
ht1
wt1
ht2
wt2
ht1
0.48
0.26
0.21
wt1
0.26
0.75
0.27
ht2
0.21
0.46
0.28
wt2
0.27
0.28
0.86
MZ Twins
DZ Twins
Twin 1
Twin 2
Twin 1
Twin 2
Twin 1
Twin 2
Twin 1Twin 2Slide21
Getting a Feel for the Data
Variances-
Covariances
Cross-Trait / Cross-Twin CovarianceSlide22
Observed Variance-Covariance Matrix
P1
P
2
P1
P
2
P
1
Variance
Twin 1- P1
Covariance
Twin 1
P1/P2
P1 Within Trait
T1 /T2 Covariance
P1/P2
Cross-Trait
T1 /T2
Covariance
P2
Covariance
Twin 1
P1/P2
Variance
Twin 1- P2
P2/P1
Cross-Trait
T1 /T2
Covariance
P2 Within Trait
T1 /T2 Covariance
P
1
P1 Within Trait
T1 /T2
Covariance
P2/P1
Cross-Trait
T1 /T2
Covariance
Variance
Twin2- P1
Covariance
Twin 2 P1/P2P 2
P1/P2
Cross-Trait
T1 /T2
Covariance
P2 Within Trait
T1 /T2
Covariance
Covariance
Twin 2
P1/P2
Variance
Twin 2- P2
Twin 1
Twin 2
Twin 1
Twin 2
Variances
+
Cross-Trait
Within-Twin
Covariances
+
Within-Trait
Cross-Twin
CovariancesSlide23
Observed Variance-Covariance Matrix
P1
P
2
P1
P
2
P
1
Variance
Twin 1- P1
Covariance
Twin 1
P1/P2
P1 Within Trait
T1 /T2 Covariance
P1/P2
Cross-Trait
T1 /T2
Covariance
P2
Covariance
Twin 1
P1/P2
Variance
Twin 1- P2
P2/P1
Cross-Trait
T1 /T2
Covariance
P2 Within Trait
T1 /T2 Covariance
P
1
P1 Within Trait
T1 /T2
Covariance
P2/P1
Cross-Trait
T1 /T2
Covariance
Variance
Twin2- P1
Covariance
Twin 2 P1/P2P 2
P1/P2
Cross-Trait
T1 /T2
Covariance
P2 Within Trait
T1 /T2
Covariance
Covariance
Twin 2
P1/P2
Variance
Twin 2- P2
Twin 1
Twin 2
Twin 1
Twin 2
Variances
+
Cross-Trait
Within-Twin
Covariances
+
Within-Trait
Cross-Twin
Covariances
+
Cross-Twin
Cross-Trait
CovariancesSlide24
Observed Variance-Covariance Matrix
P1
P
2
P1
P
2
P
1
Variance
P1
P2
Covariance P1-P2
Variance
P2
P
1
Within-trait
P1
Cross-trait
Variance
P1
P
2
Cross-trait
Within-trait
P2
Covariance P1-P2
Variance
P2
Twin 1
Twin 2
Twin 1
Twin 2
Within-twin
covariance
Within-twin
covariance
Cross-twin
covarianceSlide25
Getting a Feel for the Data
Variances-
Covariances
ht1
wt1
ht2
wt2
ht10.430.280.380.24wt10.280.730.280.64ht20.380.28
0.44
0.26
wt2
0.24
0.64
0.26
0.78
ht1
wt1
ht2
wt2
ht1
0.48
0.26
0.21
0.15
wt1
0.26
0.75
0.110.27ht2
0.21
0.11
0.46
0.28
wt2
0.15
0.27
0.28
0.86
MZ Twins
DZ Twins
Twin 1
Twin 2
Twin 1
Twin 2Twin 1Twin 2Twin 1Twin 2Slide26
Within-twin cross-trait
covariances
imply common etiological influences
Cross
-twin cross-trait
covariances
imply familial common etiological influences
MZ
/DZ ratio of cross-twin cross-trait
covariances reflects whether common etiological influences are genetic or environmentalCross-Trait CovariancesSlide27
Bivariate Twin Covariance Matrix
X
1
Y
1
X
2
Y
2
X
1
V
X1
C
X1
Y1
C
X1X2
C
X1
Y2
Y
1
C
Y1
X1
V
Y1
C
Y1
X2
C
Y1Y2
X
2
C
X2X1
C
X2
Y1
V
X2
C
X2
Y2Y2CY2X1CY2Y1CY2X2VY2twin 1twin 1twin 2twin 2Variances of X & Y same across twins and zygosity groupsSlide28
Bivariate Twin Covariance Matrix
X
1
Y
1
X
2
Y
2
X
1
V
X1
C
X1
Y1
C
X1X2
C
X1
Y2
Y
1
C
Y1
X1
V
Y1
C
Y1
X2
C
Y1Y2
X
2
C
X2X1
C
X2
Y1
V
X2
C
X2
Y2Y2CY2X1CY2Y1CY2X2VY2twin 1twin 1twin 2twin 2Covariances of X & Y same across twins and zygosity groupsSlide29
Bivariate Twin Covariance Matrix
X
1
Y
1
X
2
Y
2
X
1
V
X1
C
X1
Y1
C
X1X2
C
X1
Y2
Y
1
C
Y1
X1
V
Y1
C
Y1
X2
C
Y1Y2
X
2
C
X2X1
C
X2
Y1
V
X2
C
X2
Y2Y2CY2X1CY2Y1CY2X2VY2twin 1twin 1twin 2twin 2Cross-Twin Within-Trait Covariances differ by zygositySlide30
Bivariate Twin Covariance Matrix
X
1
Y
1
X
2
Y
2
X
1
V
X1
C
X1
Y1
C
X1X2
C
X1
Y2
Y
1
C
Y1
X1
V
Y1
C
Y1
X2
C
Y1Y2
X
2
C
X2X1
C
X2
Y1
V
X2
C
X2
Y2Y2CY2X1CY2Y1CY2X2VY2twin 1twin 1twin 2twin 2Cross-Twin Cross-Trait Covariances differ by zygositySlide31
Saturated Model TestingSlide32
Bivariate Analysis
A Roadmap
1- Use the data to test basic assumptions inherent to standard
genetic models
Saturated Bivariate
Model
2- Estimate the
contributions of genetic and environmental factors to the covariance between two
traits
Bivariate Genetic Models (Bivariate Genetic Analysis via Cholesky Decomposition)Slide33
Genetic Modeling with Twin Data
a
c
a
c
MZ = 1
DZ = 0.5
1
A
C
Twin
1
Height
Twin
2
Height
e
e
E
E
C
ASlide34
a
11
a
21
c
11
c
22
c
21
e
11
e
21
e
22
a
11
a
22
a
21
c
11
c
22
c
21
e
11
e
21
e
22
MZ = 1
DZ = 0.5
MZ = 1
DZ = 0.5
1
1
C1
C2
C1
C2
Twin 1HeightTwin 1 WeightTwin 2HeightTwin 2WeightE1E2E1E2a22Bivariate Genetic ModelingA1A2A1A2Slide35
Building the
Bivariate Genetic Model
Sources
of Information
Cross
-trait covariance
within
individuals
Within-Twin CovarianceCross-trait covariance between twinsCross-Twin CovarianceMZ:DZ ratio of cross-trait covariance between twinsSlide36
Alternative RepresentationsSlide37Slide38
Bivariate Twin Covariance Matrix
X
1
Y
1
X
1
V
X1
C
X1
Y1
Y
1
C
Y1
X1
V
Y1
twin 1
twin 1Slide39
Bivariate Twin Covariance Matrix
X
1
Y
1
X
1
a
11
2
C
X1
Y1
Y
1
C
Y1
X1
V
Y1
twin 1
twin 1Slide40
Bivariate Twin Covariance Matrix
X
1
Y
1
X
1
a
11
2
C
X1
Y1
Y
1
a
21
*a
11
V
Y1
twin 1
twin 1Slide41
Bivariate Twin Covariance Matrix
X
1
Y
1
X
1
a
11
2
a
21
*a
11
Y
1
a
21
*a
11
V
Y1
twin 1
twin 1Slide42
Bivariate Twin Covariance Matrix
X
1
Y
1
X
1
a
11
2
a
21
*a
11
Y
1
a
21
*a
11
a
22
2
+a
21
2
twin 1
twin 1Slide43
Bivariate Twin Covariance Matrix
X
1
Y
1
X
1
a
11
2
+
e
11
2
a
21
*a
11
+
e
21
*e
11
Y
1
a
21
*a
11+ e
21
*e
11
a
22
2
+
a
21
2
+
e
22
2
+e212twin 1twin 1Slide44
Bivariate Twin Covariance Matrix
X
1
Y
1
X
2
C
X2X1
C
X2
Y1
Y
2
C
Y2
X1
C
Y2Y1
twin 1
twin 2Slide45
Bivariate Twin Covariance Matrix
X
1
Y
1
X
2
1/0.5 * a
11
2
C
X2
Y1
Y
2
C
Y2
X1
C
Y2Y1
twin 1
twin 2Slide46
Bivariate Twin Covariance Matrix
X
1
Y
1
X
2
1/0.5 * a
11
2
C
X2
Y1
Y
2
1/0.5 * a
21
*a
11
C
Y2Y1
twin 2
twin 1Slide47
Bivariate Twin Covariance Matrix
X
1
Y
1
X
2
1/0.5 * a
11
2
1/0.5 * a
21
*a
11
Y
2
1/0.5 * a
21
*a
11
C
Y2Y1
twin 2
twin 1Slide48
Bivariate Twin Covariance Matrix
X
1
Y
1
X
2
1/0.5
*
a
11
2
1/0.5
*
a
21
*a
11
Y
2
1/0.5
*
a
21
*a
11
1/0.5
*
a
22
2
+
1/0.5
*
a
21
2
twin 2
twin 1Slide49
Bivariate Twin Covariance Matrix
X
1
Y
1
X
2
Y
2
X
1
V
X1
C
X1
Y1
C
X1X2
C
X1
Y2
Y
1
C
Y1
X1
V
Y1
C
Y1
X2
C
Y1Y2
X
2
C
X2X1
C
X2
Y1
V
X2
C
X2
Y2Y2CY2X1CY2Y1CY2X2VY2twin 1twin 1twin 2twin 2Slide50
Predicted Twin Covariance Matrix
X
1
Y
1
X
2
Y
2
X
1
a
11
2
+
e
11
2
a
21
*a
11
+
e
21
*e
11
1/0.5
*
a
11
2
1/0.5
*
a
21
*a
11
Y
1
a
21
*a
11+ e21*e11a222+a212+ e222+e2121/0.5 * a21*a111/0.5*a222+1/0.5*a212X21/0.5* a1121/0.5 * a21*a11a112+e112 a21*a11+ e21*e11Y21/0.5 * a21*a11
1/0.5
*
a
22
2
+
1/0.5
*
a
21
2
a
21
*a
11
+
e
21
*e
11
a
22
2
+
a
21
2
+
e
22
2
+
e
21
2
twin 1
twin 1
twin 2
twin 2Slide51
Predicted MZ Twin Covariance
X
1
Y
1
X
2
Y
2
X
1
a
11
2
+
e
11
2
a
21
*a
11
+
e
21
*e
11
a
11
2
a
21
*a
11
Y
1
a
21
*a
11
+
e
21
*e11a222+a212+ e222+e212 a21*a11a222+a212X2a112a21*a11a112+e112 a21*a11+ e21*e11Y2 a21*a11a222+a212a21
*a
11
+
e21
*e
11
a
22
2
+
a
21
2
+
e
22
2
+
e
21
2
twin 1
twin 1
twin 2
twin 2Slide52
Predicted DZ Twin Covariance
X
1
Y
1
X
2
Y
2
X
1
a
11
2
+
e
11
2
a
21
*a
11
+
e
21
*e
11
0.5*a
11
2
0.5*a
21
*a
11
Y
1
a
21
*a
11
+
e
21
*e11a222+a212+ e222+e2120.5*a21*a110.5*a222+ 0.5* a212X20.5*a1120.5*a21*a11a112+e112 a21*a11+ e21*e11Y20.5*a21*a110.5*a222+ 0.5* a212a21
*a
11
+
e
21
*e
11
a
22
2
+
a
21
2
+
e
22
2
+
e
21
2
twin 1
twin 1
twin 2
twin 2Slide53
Predicted Covariance Matrix
X
1
Y
1
X
2
Y
2
X
1
V
X1
C
X1
Y1
C
X1X2
C
X1
Y2
Y
1
C
Y1
X1
V
Y1
C
Y1
X2
C
Y1Y2
X
2
C
X2X1
C
X2
Y1
V
X2
C
X2
Y2Y2CY2X1CY2Y1CY2X2VY2twin 1twin 1twin 2twin 2Variances of X & Y same across twins and zygosity groupsSlide54
Predicted Covariance Matrix
X
1
Y
1
X
2
Y
2
X
1
V
X1
C
X1
Y1
C
X1X2
C
X1
Y2
Y
1
C
Y1
X1
V
Y1
C
Y1
X2
C
Y1Y2
X
2
C
X2X1
C
X2
Y1
V
X2
C
X2
Y2Y2CY2X1CY2Y1CY2X2VY2twin 1twin 1twin 2twin 2Covariances of X & Y same across twins and zygosity groupsSlide55
Predicted Covariance Matrix
X
1
Y
1
X
2
Y
2
X
1
V
X1
C
X1
Y1
C
X1X2
C
X1
Y2
Y
1
C
Y1
X1
V
Y1
C
Y1
X2
C
Y1Y2
X
2
C
X2X1
C
X2
Y1
V
X2
C
X2
Y2Y2CY2X1CY2Y1CY2X2VY2twin 1twin 1twin 2twin 2Cross-Twin Within-Trait Covariances differ by zygositySlide56
Predicted Covariance Matrix
X
1
Y
1
X
2
Y
2
X
1
V
X1
C
X1
Y1
C
X1X2
C
X1
Y2
Y
1
C
Y1
X1
V
Y1
C
Y1
X2
C
Y1Y2
X
2
C
X2X1
C
X2
Y1
V
X2
C
X2
Y2Y2CY2X1CY2Y1CY2X2VY2twin 1twin 1twin 2twin 2Cross-Twin Cross-Trait Covariances differ by zygositySlide57
OpenMx Specification
X
1
Y
1
X
2
Y
2
X
1
V
X1
C
X1
Y1
C
X1X2
C
X1
Y2
Y
1
C
Y1
X1
V
Y1
C
Y1
X2
C
Y1Y2
X
2
C
X2X1
C
X2
Y1
V
X2
C
X2
Y2Y2CY2X1CY2Y1CY2X2VY2OpenMx scriptSlide58
Read in and Transform Variable(s)
transform variables to make variances with similar order of magnitudes
# Load Data
data(
twinData
)
describe(
twinData
)
twinData[,'ht1'] <- twinData[,'ht1']*10twinData[,'ht2'] <- twinData[,'ht2'
]
*
10
twinData
[,
'wt1'
]
<-
twinData
[,
'wt1'
]
/
10
twinData
[,
'wt2'
]
<- twinData[,'wt2']/10# Select Variables for AnalysisVars <- c('ht','wt')nv <- 2 # number of variablesntv
<-
nv
*2
# number of total variablesselVars
<- paste(
Vars,c(rep(
1
,nv
),rep(
2
,
nv
)),sep="") #c('ht1','wt1,'ht2','wt2')# Select Data for AnalysismzData <- subset(twinData, zyg==1, selVars)dzData <- subset(twinData, zyg==3, selVars)Slide59
# Set Starting Values
svMe
<-
c(
15
,
5
)
# start value for meanslaMe <- paste(selVars,"Mean",sep="_")svPa <- .6 # start value for parameterssvPas <- diag(svPa,nv,nv)
laA
<-
paste(
"a"
,rev(
nv
+
1
-
sequence(
1
:
nv
)),rep(
1
:
nv
,
nv:1),sep="") # c("a11","a21","a22")laD <- paste("d",rev(nv+1-sequence(1:nv)),rep(1:nv,nv:1),sep
=""
)laC
<- paste(
"c"
,rev(nv
+1
-sequence(
1
:nv
)),rep(
1
:
nv
,nv:1),sep="")laE <- paste("e",rev(nv+1-sequence(1:nv)),rep(1:nv,nv:1),sep="")Start Values, LabelsMeans & Parameterscreate numbered labels to fill the lower triangular matrices with the first number corresponding to the variable being pointed to and the second number corresponding to the factor Slide60
Within-Twin Covariance [A]
Path Tracing:
Matrix Algebra: Lower 2x2
A
1
A
2
P
1
P2Slide61
# Matrices declared to store a, c, and e Path Coefficients
pathA
<-
mxMatrix( type
=
"Lower"
, nrow
=
nv, ncol=nv, free=T, values=svPas, label=laA, name="a" ) pathD <- mxMatrix( type="Lower", nrow=nv, ncol
=
nv
, free
=
T, values
=
svPas
, label
=
laD
, name
=
"d"
)
pathC
<-
mxMatrix( type
=
"Lower", nrow=nv, ncol=nv, free=F, values=0, label=laC, name="c" )pathE <- mxMatrix( type="Lower", nrow=nv, ncol=nv
, free
=T, values
=svPas
, label
=laE
, name=
"e"
)
# Matrices generated to hold A, C, and E computed Variance Components
covA
<-
mxAlgebra( expression
=
a %*% t(a), name="A" )covD <- mxAlgebra( expression=d %*% t(d), name="D" )covC <- mxAlgebra( expression=c %*% t(c), name="C" )covE <- mxAlgebra( expression=e %*% t(e), name="E" )Path CoefficientsVariance Componentsregular multiplication of lower triangular matrix and its transposeA1 A2P1P2a %*% t(a)Slide62
Within-Twin Covariance [A]+[E]
+
using matrix addition to generate total within-twin covarianceSlide63
# Algebra to compute total variances and standard deviations (diagonal only)
covP
<-
mxAlgebra( expression
=
A
+
D
+C+E, name="V" )matI <- mxMatrix( type="Iden", nrow=nv, ncol=nv, name="I"
)
invSD
<-
mxAlgebra( expression
=
solve(sqrt(
I
*
V
)), name
=
"iSD"
)
# Algebras generated to hold Parameter Estimates and Derived Variance Components
rowVars
<-
rep(
'vars',nv)colVars <- rep(c('A','D','C','E','SA','SD','SC','SE'),each=nv)estVars
<-
mxAlgebra( cbind(
A,
D,
C
,E
,A
/
V,
D/
V
,
C
/
V,E/V), name="Vars", dimnames=list(rowVars,colVars))Total VariancesVariance Componentseach of covariance matrices is of size nv x nvSlide64
Cross-Twin Covariances [A] & 0.5[A]
+
using Kronecker product to multiple every element of matrix by scalarSlide65
# Algebra for expected Mean and Variance/Covariance Matrices in MZ & DZ twins
meanG
<-
mxMatrix( type
=
"Full"
, nrow
=
1, ncol=nv, free=T, values=svMe, labels=laMe, name="Mean" )meanT
<-
mxAlgebra( cbind(
Mean
,
Mean
), name
=
"expMean"
)
covMZ
<-
mxAlgebra( rbind( cbind(
V
,
A
+
D
+
C), cbind(A+D+C , V )), name="expCovMZ" )covDZ <- mxAlgebra( rbind( cbind(V , 0.5%x%A+0.25%x%D+C),
cbind(
0.5%x%
A+
0.25
%x%D
+C
,
V )), name
="expCovDZ"
)
Expected Means
& Covariances
cbind creates two nv x ntv row matrices
rbind turns them into to ntv x ntv matrixSlide66
# Data objects for Multiple Groups
dataMZ
<-
mxData( observed
=
mzData
, type
=
"raw" )dataDZ <- mxData( observed=dzData, type="raw" )# Objective objects for Multiple GroupsobjMZ <- mxFIMLObjective( covariance="expCovMZ", means="expMean", dimnames=
selVars
)
objDZ
<-
mxFIMLObjective( covariance
=
"expCovDZ"
, means
=
"expMean"
, dimnames
=
selVars
)
# Combine Groups
pars
<-
list( pathA, pathD, pathC, pathE, covA, covD, covC, covE, covP, matI, invSD, estVars, meanG
, meanT
)
modelMZ
<-
mxModel( pars
, covMZ
,
dataMZ,
objMZ, name
=
"MZ"
)
modelDZ
<- mxModel( pars, covDZ, dataDZ, objDZ, name="DZ" )minus2ll <- mxAlgebra( expression=MZ.objective + DZ.objective, name="m2LL" )obj <- mxAlgebraObjective( "m2LL" )BivAceModel <- mxModel( "BivACE", pars, modelMZ, modelDZ, minus2ll, obj )Data, Objectives & Model Objectsexpected covariances and meansobserved dataSlide67
# Run Bivariate ACE model
BivAceFit
<-
mxRun(
BivAceModel
)
BivAceSum
<- summary(BivAceFit)BivAceSum$paround(BivAceFit@output$estimate,4)round(BivAceFit$Vars@result,4
)
BivAceSum
$
Mi
# Generate Output with Functions
source(
"GenEpiHelperFunctions.R"
)
parameterSpecifications(
BivAceFit
)
expectedMeansCovariances(
BivAceFit
)
tableFitStatistics(
BivAceFit
)
Parameter Estimates
Variance Components
two ways to get parameter estimatesprint pre-calculated unstandardized variance components and standardized variance componentsSlide68
Three Important
Results from Bivariate Genetic Analysis
Variance
Decomposition -> Heritability, (Shared) environmental influences
Covariance Decomposition -> The influences of genes and environment on the covariance between the two
variables
“How
much of the phenotypic correlation is accounted for by genetic
and environmental influences?”
Genetic and Environmental correlations -> the overlap in genes and environmental effects “Is there a large overlap in genetic/ environmental factors?Slide69
From Cholesky to Genetic Correlation
standardized solution = correlated factors solutionSlide70
Genetic Covariance to Genetic Correlation
calculated by dividing genetic covariance by
square root of product of genetic variances of two variablesSlide71
# Calculate genetic and environmental correlations
corA
<-
mxAlgebra( expression
=
solve(sqrt(
I
*
A))%&%A), name ="rA" )corD <- mxAlgebra( expression=solve(sqrt(I*D))%&%E), name ="rD" )
corC
<-
mxAlgebra( expression
=
solve(sqrt(
I
*
C
))%
&
%
C
), name
=
"rC"
)
corE
<- mxAlgebra( expression=solve(sqrt(I*E))%&%E), name ="rE" )Genetic CorrelationAlgebraSlide72
72
Contribution to Phenotypic Correlation
if rg=1, then two sets of genes overlap completely
if however, a11 and a22 are near to zero, genes do not contribute much to phenotypic correlation
contribution to phenotypic correlation is
function of both heritabilities and rgSlide73
Interpreting Results
High genetic correlation = large overlap in genetic effects on the two phenotypes
Does it mean that the phenotypic correlation between the traits is largely due to genetic effects?
No:
the substantive importance of a particular r
G
depends the value of the correlation
and
the value of the
A2 paths i.e. importance is also determined by the heritability of each phenotypeSlide74
Interpretation of Correlations
Consider two traits with a phenotypic correlation
(
r
P
) of
0.40 :
h
2
P1 = 0.7 and h2P2 = 0.6 with rG = .3Correlation due to additive genetic effects = ? Proportion of phenotypic correlation attributable to additive genetic effects = ? h2P1 = 0.2 and h2
P2
= 0.3 with
r
G
= 0.8
Correlation due to additive genetic effects =
?
Proportion of phenotypic correlation attributable to additive genetic effects =
?
Correlation due to A:
Divide by r
P
to find proportion of phenotypic correlation.Slide75
Bivariate Cholesky
Multivariate Cholesky