IntroductiontoMultivariateGeneticAnalysis
MeikeBartels,HermineMaes,ElizabethProm-WormleyandMichelNivard
AimandRationale
Aim: toexaminethesourceoffactorsthatmaketraitscorrelateorco-vary
Rationale: Traitsmaybecorrelatedduetosharedgeneticfactors(A)orsharedenvironmentalfactors(CorE)
Canuseinformationonmultipletraitsfromtwinpairstopartitioncovariationintogeneticand
environmentalcomponents
Example1
Whydotraitscorrelate/covary?Howcanweexplaintheassociation?Additivegeneticfactors(rG)Sharedenvironment (rC)Non-sharedenvironment (rE)
ADHD
E1
E2
A1
A2
√σA112
IQ
rG
C1
C2
rC
1 1 1 1
1 1
rE
√σC112 √σA22
2 √σC222
√σE112 √σE22
2
Example2
• Associationsbetweenphenotypesovertimew Doesanxiety inchildhoodleadtodepression inadolescence?
• Howcanweexplaintheassociation?w Additivegeneticfactors(a21)w Sharedenvironment (c21)w Non-sharedenvironment (e21)w Howmuchisnotexplainedbyprior
anxiety?
Childhood anxiety
A1
A2
E1
E2
a11 a21 a22
e11 e21 e22
1
1
1
1
Adolescent depression
C1 C2
c11
c21
c22
11
SourcesofInformation
• Forexample:twotraitsmeasuredintwinpairs
• Interestedin:w Cross-traitcovariancewithin individualsw Cross-traitcovariancebetween twinswMZ:DZratioofcross-traitcovariancebetweentwins
ObservedCovarianceMatrix
Phenotype 1 Phenotype 2 Phenotype 1 Phenotype 2
Phenotype 1 Variance P1
Phenotype 2 Covariance P1-P2
Variance P2
Phenotype 1 Within-traitP1
Cross-trait Variance P1
Phenotype 2 Cross-trait Within-traitP2
Covariance P1-P2
Variance P2
Twin 1 Twin 2
ObservedCovarianceMatrix
Phenotype 1 Phenotype 2 Phenotype 1 Phenotype 2
Phenotype 1 Variance P1
Phenotype 2 Covariance P1-P2
Variance P2
Phenotype 1 Within-traitP1
Cross-trait Variance P1
Phenotype 2 Cross-trait Within-traitP2
Covariance P1-P2
Variance P2
Twin 1 Twin 2
Within-twin covariance
Within-twin covariance
ObservedCovarianceMatrix
Phenotype 1 Phenotype 2 Phenotype 1 Phenotype 2
Phenotype 1 Variance P1
Phenotype 2 Covariance P1-P2
Variance P2
Phenotype 1 Within-traitP1
Cross-trait Variance P1
Phenotype 2 Cross-trait Within-traitP2
Covariance P1-P2
Variance P2
Twin 1 Twin 2
Within-twin covariance
Within-twin covarianceCross-twin covariance
SEM:CholeskyDecomposition
Twin 1Phenotype 1
A1
A2
E1
E2
a11 a22
e11 e22
1
1
1
1
Twin 1Phenotype 2
C1 C2
c11 c22
11
SEM:CholeskyDecomposition
Twin 1Phenotype 1
A1
A2
E1
E2
a11 a21 a22
e11 e21 e22
1
1
1
1
Twin 1Phenotype 2
C1 C2
c11
c21
c22
11
SEM:CholeskyDecomposition
Twin 1Phenotype 1
A1
A2
E1
E2
a11 a21 a22
e11 e21 e22
1
1
1
1
Twin 1Phenotype 2
C1 C2
c11
c21
c22
11
Twin 2Phenotype 1
A1
A2
E1
E2
a11 a21 a22
e11 e21 e22
1
1
1
1
Twin 2Phenotype 2
C1 C2
c11
c21
c22
11
1/0.5 1/0.51 1
CholeskyDecomposition
PathTracing
Within-TwinCovariances(A)
A1
A2
a11 a21 a22
11
Phenotype 1 Phenotype 2
Phenotype 1 a112+c11
2+e112
Phenotype 2 a11a21+c11c21+e11e21 a222+a21
2+c222+c21
2
+e222+e21
2
Twin 1Twin 1
Phenotype 1Twin 1
Phenotype 2
Within-TwinCovariances(A)
A1
A2
a11 a21 a22
11
Phenotype 1 Phenotype 2
Phenotype 1 a112+c11
2+e112
Phenotype 2 a11a21+c11c21+e11e21 a222+a21
2+c222+c21
2
+e222+e21
2
Twin 1Twin 1
Phenotype 1Twin 1
Phenotype 2
Within-TwinCovariances(A)
A1
A2
a11 a21 a22
11
Phenotype 1 Phenotype 2
Phenotype 1 a112+c11
2+e112
Phenotype 2 a11a21+c11c21+e11e21 a222+a21
2+c222+c21
2
+e222+e21
2
Twin 1Twin 1
Phenotype 1Twin 1
Phenotype 2
Within-TwinCovariances(A)
A1
A2
a11 a21 a22
11
Phenotype 1 Phenotype 2
Phenotype 1 a112+c11
2+e112
Phenotype 2 a11a21+c11c21+e11e21 a222+a21
2+c222+c21
2
+e222+e21
2
Twin 1Twin 1
Phenotype 1Twin 1
Phenotype 2
Within-TwinCovariances(C)
C1C2
c11 c21 c22
11
Phenotype 1 Phenotype 2
Phenotype 1 a112+c11
2+e112
Phenotype 2 a11a21+c11c21+e11e21 a222+a21
2+c222+c21
2
+e222+e21
2
Twin 1Twin 1
Phenotype 1Twin 1
Phenotype 2
Within-TwinCovariances(E)
Phenotype 1 Phenotype 2
Phenotype 1 a112+c11
2+e112
Phenotype 2 a11a21+c11c21+e11e21 a222+a21
2+c222+c21
2
+e222+e21
2
Twin 1
Twin 1Phenotype 1
E1
E2
e11 e21 e22
11
Twin 1Phenotype 2
Cross-TwinCovariances(A)
Twin 1Phenotype 1
A1
A2
a11 a21 a22
11
Twin 1Phenotype 2
Twin 2Phenotype 1
A1
A2
a11 a21 a22
11
Twin 2Phenotype 2
1/0.5 1/0.5
Phenotype 1 Phenotype 2
Phenotype 1
Phenotype 2 +c11c21 +c222+c21
2
Twin 1
Cross-TwinCovariances(A)
Twin 1Phenotype 1
A1
A2
a11 a21 a22
11
Twin 1Phenotype 2
Twin 2Phenotype 1
A1
A2
a11 a21 a22
11
Twin 2Phenotype 2
1/0.5 1/0.5
Phenotype 1 Phenotype 2
Phenotype 1 1/0.5a112+
Phenotype 2 +c11c21 +c222+c21
2
Twin 1
Cross-TwinCovariances(A)
Twin 1Phenotype 1
A1
A2
a11 a21 a22
11
Twin 1Phenotype 2
Twin 2Phenotype 1
A1
A2
a11 a21 a22
11
Twin 2Phenotype 2
1/0.5 1/0.5
Phenotype 1 Phenotype 2
Phenotype 1 1/0.5a112+c11
2
Phenotype 2 1/0.5a11a21+c11c21 +c222+c21
2
Twin 1
Cross-TwinCovariances(A)
Twin 1Phenotype 1
A1
A2
a11 a21 a22
11
Twin 1Phenotype 2
Twin 2Phenotype 1
A1
A2
a11 a21 a22
11
Twin 2Phenotype 2
1/0.5 1/0.5
Phenotype 1 Phenotype 2
Phenotype 1 1/0.5a112+c11
2
Phenotype 2 1/0.5a11a21+c11c21 1/0.5a222+1/0.5a21
2+c222+c21
2
Twin 1
Cross-TwinCovariances(C)
Twin 1Phenotype 1
Twin 1Phenotype 2
C1 C2
c11
c21
c22
11
Twin 2Phenotype 1
Twin 2Phenotype 2
C1 C2
c11
c21
c22
11
1 1
Phenotype 1 Phenotype 2
Phenotype 1 1/0.5a112+c11
2
Phenotype 2 1/0.5a11a21+c11c21 1/0.5a222+1/0.5a21
2+c222+c21
2
Twin 1
PredictedModel
Phenotype 1 Phenotype 2 Phenotype 1 Phenotype 2
Phenotype 1a112+c112+e112
Phenotype 2 a11a21+c11c21+e11e21
a222+a212+c222+c212+e222+e212
Phenotype 11/.5a112+c112 a112+c112+e112
Phenotype 2 1/.5a11a21+c11c21
1/.5a222+1/.5 a212+c222+c212
a11a21+c11c21+e11e21
a222+a212+c222
+c212+e222+e212
Twin 1 Twin 2
Within-twin covariance
Within-twin covarianceCross-twin covariance
PredictedModel
Phenotype 1 Phenotype 2 Phenotype 1 Phenotype 2
Phenotype 1 Variance P1
Phenotype 2 Covariance P1-P2
Variance P2
Phenotype 1 Within-traitP1
Cross-trait Variance P1
Phenotype 2 Cross-trait Within-traitP2
Covariance P1-P2
Variance P2
Twin 1 Twin 2
Within-twin covariance
Within-twin covarianceCross-twin covariance
PredictedModel
Phenotype 1 Phenotype 2 Phenotype 1 Phenotype 2
Phenotype 1 Variance P1
Phenotype 2 Covariance P1-P2
Variance P2
Phenotype 1 Within-traitP1
Cross-trait Variance P1
Phenotype 2 Cross-trait Within-traitP2
Covariance P1-P2
Variance P2
Twin 1 Twin 2
Within-twin covariance
Within-twin covarianceCross-twin covariance
Variance of P1 and P2the same across twinsand zygosity groups
PredictedModel
Phenotype 1 Phenotype 2 Phenotype 1 Phenotype 2
Phenotype 1 Variance P1
Phenotype 2 Covariance P1-P2
Variance P2
Phenotype 1 Within-traitP1
Cross-trait Variance P1
Phenotype 2 Cross-trait Within-traitP2
Covariance P1-P2
Variance P2
Twin 1 Twin 2
Within-twin covariance
Within-twin covarianceCross-twin covariance
Covariance of P1 and P2 the same across twins and zygosity groups
PredictedModel
Phenotype 1 Phenotype 2 Phenotype 1 Phenotype 2
Phenotype 1 Variance P1
Phenotype 2 Covariance P1-P2
Variance P2
Phenotype 1 Within-traitP1
Cross-trait Variance P1
Phenotype 2 Cross-trait Within-traitP2
Covariance P1-P2
Variance P2
Twin 1 Twin 2
Within-twin covariance
Within-twin covarianceCross-twin covariance
Cross-twin covariancewithin each trait differsby zygosity
PredictedModel
Phenotype 1 Phenotype 2 Phenotype 1 Phenotype 2
Phenotype 1 Variance P1
Phenotype 2 Covariance P1-P2
Variance P2
Phenotype 1 Within-traitP1
Cross-trait Variance P1
Phenotype 2 Cross-trait Within-traitP2
Covariance P1-P2
Variance P2
Twin 1 Twin 2
Within-twin covariance
Within-twin covarianceCross-twin covariance
Cross-twin cross-traitcovariance differs byzygosity
ExampleCovarianceMatrix
P1 P2 P1 P2
P1 1
P2 .30 1
P1 0.79 0.49 1
P2 0.50 0.59 0.29 1
P1 P2 P1 P2
P1 1
P2 0.30 1
P1 0.39 0.25 1
P2 0.24 0.43 0.31 1
Twin 1
Twin 1
Twin 2
Twin 2MZ
DZ
Within-twin covariance
Within-twin covariance
Within-twin covariance
Within-twin covarianceCross-twin covariance
Cross-twin covariance
ExampleCovarianceMatrix
P1 P2 P1 P2
P1 1
P2 .30 1
P1 0.79 0.49 1
P2 0.50 0.59 0.29 1
P1 P2 P1 P2
P1 1
P2 0.30 1
P1 0.39 0.25 1
P2 0.24 0.43 0.31 1
Twin 1
Twin 1
Twin 2
Twin 2MZ
DZ
Within-twin covariance
Within-twin covariance
Within-twin covariance
Within-twin covarianceCross-twin covariance
Cross-twin covariance
ExampleCovarianceMatrix
P1 P2 P1 P2
P1 1
P2 .30 1
P1 0.79 0.49 1
P2 0.50 0.59 0.29 1
P1 P2 P1 P2
P1 1
P2 0.30 1
P1 0.39 0.25 1
P2 0.24 0.43 0.31 1
Twin 1
Twin 1
Twin 2
Twin 2MZ
DZ
Within-twin covariance
Within-twin covariance
Within-twin covariance
Within-twin covarianceCross-twin covariance
Cross-twin covariance
Summary
• Within-individualcross-traitcovarianceimpliescommonaetiologicalinfluences
• Cross-twincross-traitcovarianceimpliescommonaetiologicalinfluencesarefamilial
• WhetherfamilialinfluencesgeneticorenvironmentalshownbyMZ:DZratioofcross-twincross-traitcovariances
CholeskyDecompositionBivariateGeneticanalyses
SpecificationinOpenMx
Within-TwinCovariance
P1 P2
A1
A2
a11 a21 a22
11 Path Tracing:
Within-TwinCovariance
P1 P2
A1
A2
a11 a21 a22
11 Path Tracing:
a Lower 2 x 2:P1
P2
a1 a2
Starting values and labels
Within-TwinCovariance
P1 P2
A1
A2
a11 a21 a22
11 Path Tracing:
a Lower 2 x 2:P1
P2
a1 a2
a11 a21 a22
€
ΣA = a* aT
ΣA = a% *%t(a)
Within-TwinCovariance
€
ΣA = a* aT
ΣA = a%*%t(a)
TotalWithin-TwinCovar.
Using matrix addition, the total within-twin covariancefor the phenotypes is defined as:
€
ΣA = a %*% t(a)
€
ΣC = c %*% t(c)
€
ΣE = e %*% t(e)
€
ΣV
€
ΣV
OpenMxMatrices&Algebra
AdditiveGeneticCross-TwinCovariance(DZ)
P11 P21
A1
A2
a11 a21 a22
11
P12 P22
A1
A2
a11 a21 a22
11
0.5 0.5
Twin 1 Twin 2
Path Tracing:Within-traitsP11-P12 = 0.5a11
2
P21-P22 = 0.5a222+0.5a21
2
Cross-traitsP11-P22 = 0.5a11a21P21-P12 = 0.5a21a11
€
0.5⊗ ΣA = 0.5%x%(a%*%t(a))
AdditiveGeneticCross-TwinCovariance(MZ)
P11 P21
A1
A2
a11 a21 a22
11
P12 P22
A1
A2
a11 a21 a22
11
1 1
Twin 1 Twin 2
€
1⊗ ΣA =1%x%(a%*%t(a))
CommonEnvironmentCross-TwinCovariance
P11 P21
C1
C2
c11 c21 c22
11
P12 P22
C1
C2
c11 c21 c22
11
1 1
Twin 1 Twin 2
€
1⊗ ΣC =1%x%(c%*%t(c))
CovarianceModelforTwinPairs
ObtainingStandardizedEstimates
ThreeImportantResults1. VarianceDecomposition->Heritability,(Shared)environmental influences2. CovarianceDecomposition->Theinfluencesofgenesandenvironmentonthe
covariancebetweenthetwovariables
“howmuchofthephenotypiccorrelationisaccountedforbygeneticandenvironmental influences”
3. GeneticandEnvironmental correlations ->theoverlapingenesandenvironmentaleffects
“istherealargeoverlapingene/environmental sets”
OpenMxOutput1. VarianceDecomposition->Heritability,(Shared)environmental influences2. CovarianceDecomposition->Theinfluencesofgenesandenvironmentonthe
covariancebetweenthetwovariables
AACCEESASASC SCSESEVC0.48850.34460.09340.05400.43400.08310.4809 0.71540.09190.11210.42720.1725VC0.34460.49840.05400.03120.08310.71800.7154 0.39950.11210.02500.17250.5755
1/.5 1/.5
A1 A2
a11
P11 P21
a22a21
A1 A2a11
P12 P22
a22a21
Twin 1 Twin 2
€
rg =a21a11
a112 * (a21
2 + a222 )
Geneticcorrelation
OpenMxOutputØ fitACE$rAØ [,1][,2]Ø [1,]1.000000000.69847252Ø [2,]0.698472521.00000000
Ø fitACE$rCØ [,1][,2]Ø [1,]1.000000000.99999984Ø [2,]0.999999841.00000000
Ø fitACE$rEØ [,1][,2]Ø [1,]1.000000000.14881543Ø [2,]0.148815431.00000000
Geneticcorrelation&contributiontoobservedcorrelation
A1 A2
a11
P11 P21
a22
Twin 1
rg
Iftherg=1,thetwosetsofgenesoverlapcompletely
Ifhowevera11anda22areneartozero,genesdonotcontributetotheobservedcorrelation
InterpretingResults
• Highgeneticcorrelation=largeoverlapingeneticeffectsonthetwophenotypes
• Doesitmeanthatthephenotypiccorrelationbetweenthetraitsislargelyduetogeneticeffects?w No:thesubstantive importanceofaparticularrG dependsthevalueofthecorrelationand thevalueofthe√σA
2 pathsi.e.importanceisalsodetermined bytheheritabilityofeachphenotype
MoreVariables…
Twin 1Phenotype 1
A1
A2
E1
E2
a11
a21
a22
e11e21
e22
1
1
1
1
Twin 1Phenotype 2
C1 C2
c11
c21 c22
11
Twin 1Phenotype 3
E3
e33
e31 e32
C3
c33
1A3
a33
c32a31
c31
a32
1
1
MoreVariables…
Twin 1Phenotype 1
A1
A2
E1
E2
a11
a21
a22
e11e21
e22
1
1
1
1
Twin 1Phenotype 2
C1 C2
c11
c21 c22
11
1/0.5 1/0.51
1
Twin 1Phenotype 3
E3
e33
e31 e32
C3
c33
1A3
a33
c32a31
c31
a32
1
1
Twin 2Phenotype 1
A1
A2
E1
E2
a11
a21
a22
e11e21
e22
1
1
1
1
Twin 2Phenotype 2
C1 C2
c11
c21 c22
11
Twin 2Phenotype 3
E3
e33
e31 e32
C3
c33
1A3
a33
c32a31
c31
a32
1
1
1/0.5
1
ExpandedMatrices
a Lower 3 x 3
c Lower 3 x 3
e Lower 3 x 3
OpenMxParameterMatrices
vars <- c(’varx', ’vary’, ‘varz’)
nv <- 3
OpenMx
Practical
SCRIPT:F:\meike\2016\Multivariate
DATA:DHBQ_bs.dat
DATA
- GeneralFamilyFunctioning,SubjectiveHappiness
- T1,T2,brother,sister
- missing-999
Observed Cross-twin Cross-trait Correlations
colMeans(mzData,na.rm=TRUE)
colMeans(dzData,na.rm=TRUE)
cov(mzData,use="complete")
cov(dzData,use="complete")
cor(mzData,use="complete")
cor(dzData,use="complete”)
Observed Cross-twin Cross-trait Correlations>cor(mzData,use="complete")
family1happy1 family2happy2
family1 1.00 0.40 0.58 0.37
happy1 0.40 1.00 0.35 0.46
family2 0.58 0.35 1.00 0.41
happy2 0.37 0.46 0.41 1.00
>cor(dzData,use="complete")
family1happy1family2happy2
family1 1.00 0.47 0.30 0.06
happy1 0.47 1.00 0.30 0.21
family2 0.30 0.30 1.00 0.37
happy2 0.06 0.21 0.37 1.00
PRACI.TheACEmodelanditsestimates
1.RuntheACEmodel
2.WhatistheheritabilityofFAMandHAP?
3.Whatisthegeneticinfluenceonthecovariance?
4.Whatisthegeneticcorrelation?
OpenMxOutput1. VarianceDecomposition->Heritability,(Shared)environmental influences2. CovarianceDecomposition->Theinfluencesofgenesandenvironmentonthe
covariancebetweenthetwovariables
AACCEESASASC SCSESEVC0.48850.34460.09340.05400.43400.08310.4809 0.71540.09190.11210.42720.1725VC0.34460.49840.05400.03120.08310.71800.7154 0.39950.11210.02500.17250.5755
OpenMxOutputØ fitACE$rAØ [,1][,2]Ø [1,]1.000000000.69847252Ø [2,]0.698472521.00000000
Ø fitACE$rCØ [,1][,2]Ø [1,]1.000000000.99999984Ø [2,]0.999999841.00000000
Ø fitACE$rEØ [,1][,2]Ø [1,]1.000000000.14881543Ø [2,]0.148815431.00000000
PRACII.TrivariateModel
1. Addathirdvariable(SatisfactionwithLife)tothemodel
2. Runmodel
3. Whataretheparameterestimates?
4. Whatisthegeneticcorrelation?
Changesthathadtobemade
#SelectVariablesforAnalysis
Vars<- c('family','happy','life')
nv<- 3#numberofvariables
ntv<- nv*2#numberoftotalvariables
selVars<- paste(Vars,c(rep(1,nv),rep(2,nv)),sep="")
svMe<- c(7,5,5)
GeneticandEnvironmentalInfluences[1]"MatrixA/V"
stCovA1 stCovA2 stCovA3family 0.4117 0.6549 0.5697happy 0.6549 0.3297 0.3224life 0.5697 0.3224 0.2246
[1]"MatrixC/V"stCovC1 stCovC2 stCovC3
Family 0.1517 0.1709 0.3004happy 0.1709 0.0681 0.1275life 0.3004 0.1275 0.1406
[1]"MatrixE/V"stCovE1 stCovE2 stCovE3
Family 0.4366 0.1742 0.1299happy 0.1742 0.6022 0.5501life 0.1299 0.5501 0.6348
GeneticandEnvironmentalCorrelations[1]"Matrixsolve(sqrt(I*A))%&%A"
FamilyHappyLifeFamily 1.00 0.76 0.78Happy 0.76 1.00 0.88life 0.78 0.88 1.00
[1]"Matrixsolve(sqrt(I*C))%&%C"FamilyHappyLife
Family 1.00 0.72 0.85happy 0.72 1.00 0.97life 0.85 0.97 1.00
[1]"Matrixsolve(sqrt(I*E))%&%E"FamilyHappyLife
Family 1.00 0.14 0.10happy 0.14 1.00 0.66life 0.10 0.66 1.00
GeneticModels
Geneticandenvironmentalfactoranalysisexamples
CommonPathwayModel IndependentPathwayModel
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