Introduction to Multivariate Genetic Analysisibg.colorado.edu/cdrom2016/bartels/Multivariate/Multivar...Introduction to Multivariate Genetic Analysis Meike Bartels, Hermine Maes, Elizabeth

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





Variance P2




Covariance P1-P2

Variance P2

Twin 1 Twin 2

Within-twin covariance






Variance P2




Covariance P1-P2

Variance P2

Twin 1 Twin 2


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


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 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


A1

A2

a11 a21 a22

11



2+e112


2+c222+c21

2

+e222+e21

2

Twin 1Twin 1

Phenotype 1Twin 1

Phenotype 2


A1

A2

a11 a21 a22

11



2+e112


2+c222+c21

2

+e222+e21

2

Twin 1Twin 1

Phenotype 1Twin 1

Phenotype 2


A1

A2

a11 a21 a22

11



2+e112


2+c222+c21

2

+e222+e21

2

Twin 1Twin 1

Phenotype 1Twin 1

Phenotype 2

Within-TwinCovariances(C)

C1C2

c11 c21 c22

11



2+e112


2+c222+c21

2

+e222+e21

2

Twin 1Twin 1

Phenotype 1Twin 1

Phenotype 2

Within-TwinCovariances(E)



2+e112


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 +c11c21 +c222+c21

2

Twin 1


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 1/0.5a112+

Phenotype 2 +c11c21 +c222+c21

2

Twin 1


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 1/0.5a112+c11

2

Phenotype 2 1/0.5a11a21+c11c21 +c222+c21

2

Twin 1


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



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



2

Phenotype 2 1/0.5a11a21+c11c21 1/0.5a222+1/0.5a21

2+c222+c21

2

Twin 1

PredictedModel


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



PredictedModel




Variance P2




Covariance P1-P2

Variance P2

Twin 1 Twin 2



PredictedModel




Variance P2




Covariance P1-P2

Variance P2

Twin 1 Twin 2



Variance of P1 and P2the same across twinsand zygosity groups

PredictedModel




Variance P2




Covariance P1-P2

Variance P2

Twin 1 Twin 2



Covariance of P1 and P2 the same across twins and zygosity groups

PredictedModel




Variance P2




Covariance P1-P2

Variance P2

Twin 1 Twin 2



Cross-twin covariancewithin each trait differsby zygosity

PredictedModel




Variance P2




Covariance P1-P2

Variance P2

Twin 1 Twin 2



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





Cross-twin covariance


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







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






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:


P1 P2

A1

A2

a11 a21 a22

11 Path Tracing:

a Lower 2 x 2:P1

P2

a1 a2

Starting values and labels


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)


€

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


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


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


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


[1]"Matrixsolve(sqrt(I*E))%&%E"FamilyHappyLife


GeneticModels

Geneticandenvironmentalfactoranalysisexamples

CommonPathwayModel IndependentPathwayModel

Introduction to Multivariate Genetic Analysisibg.colorado.edu/cdrom2016/bartels/Multivariate/Multivar...Introduction to Multivariate Genetic Analysis Meike Bartels, Hermine Maes, Elizabeth

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