Introduction to Multivariate Genetic Analysis

1 Introduction to Multivariate Genetic AnalysisMeike Bartels, Hermine Maes, Elizabeth Prom-Wormley and Michel NivardAim and RationaleAim:to examine the source of factors that make traits correlate or co-varyRationale:Traits may be correlated due to sharedgenetic factors (A) or shared environmental factors (C or E)Can use information on multiple traits from twinpairs to partition covariation into Genetic and environmental componentsExample 1 Why do traits correlate/covary?How can we explain the association?Additive Genetic factors (rG)Shared environment (rC)Non-shared environment (rE)ADHDE1E2A1A2 A112 IQrGC1C2rC111111rE C112 A222 C222 E112 E222 Example 2 Associations between phenotypes over timewDoes anxiety in childhood lead to depression in adolescence?

2 How can we explain the association?wAdditive Genetic factors (a21)wShared environment (c21)wNon-shared environment (e21)wHow much is not explained by prior anxiety?Childhood anxietyA1A2E1E2a11a21a22e11e21e221111 Adolescent depressionC1C2c11c21c2211 Sources of Information For example: two traits measured in twin pairs Interested in:wCross-trait covariance withinindividualswCross-trait covariance betweentwinswMZ:DZ ratio of cross-trait covariance between twinsObserved Covariance MatrixPhenotype 1 Phenotype 2 Phenotype 1 Phenotype 2 Phenotype 1 Variance P1 Phenotype 2 Covariance P1-P2 Variance P2 Phenotype 1 Within-traitP1 Cross-traitVariance P1 Phenotype 2 Cross-traitWithin-traitP2 Covariance P1-P2 Variance P2 Twin 1 Twin 2 Observed Covariance MatrixPhenotype 1 Phenotype 2 Phenotype 1 Phenotype 2 Phenotype 1 Variance P1 Phenotype 2 Covariance P1-P2 Variance P2 Phenotype 1 Within-traitP1 Cross-traitVariance P1 Phenotype 2 Cross-traitWithin-traitP2 Covariance P1-P2 Variance P2 Twin

3 1 Twin 2 Within-twin covarianceWithin-twin covarianceObserved Covariance MatrixPhenotype 1 Phenotype 2 Phenotype 1 Phenotype 2 Phenotype 1 Variance P1 Phenotype 2 Covariance P1-P2 Variance P2 Phenotype 1 Within-traitP1 Cross-traitVariance P1 Phenotype 2 Cross-traitWithin-traitP2 Covariance P1-P2 Variance P2 Twin 1 Twin 2 Within-twin covarianceWithin-twin covarianceCross-twin covarianceSEM: Cholesky DecompositionTwin 1 Phenotype 1A1A2E1E2a11a22e11e221111 Twin 1 Phenotype 2C1C2c11c2211 SEM: Cholesky DecompositionTwin 1 Phenotype 1A1A2E1E2a11a21a22e11e21e221111 Twin 1 Phenotype 2C1C2c11c21c2211 SEM.

4 Cholesky DecompositionTwin 1 Phenotype 1A1A2E1E2a11a21a22e11e21e221111 Twin 1 Phenotype 2C1C2c11c21c2211 Twin 2 Phenotype 1A1A2E1E2a11a21a22e11e21e221111 Twin 2 Phenotype 2C1C2c11c21c22111 DecompositionPath TracingWithin-Twin Covariances (A)A1A2a11a21a2211 Phenotype 1 Phenotype 2 Phenotype 1a112+c112+e112 Phenotype 2a11a21+c11c21+e11e21a222+a212+c222+c212 +e222+e212 Twin 1 Twin 1 Phenotype 1 Twin 1 Phenotype 2 Within-Twin Covariances (A)A1A2a11a21a2211 Phenotype 1 Phenotype 2 Phenotype 1a112+c112+e112 Phenotype 2a11a21+c11c21+e11e21a222+a212+c222+c212 +e222+e212 Twin 1 Twin 1 Phenotype 1 Twin 1 Phenotype 2 Within-Twin Covariances (A)A1A2a11a21a2211 Phenotype 1 Phenotype 2 Phenotype 1a112+c112+e112 Phenotype 2a11a21+c11c21+e11e21a222+a212+c222+c212 +e222+e212 Twin 1 Twin 1 Phenotype 1 Twin 1 Phenotype 2 Within-Twin Covariances (A)

5 A1A2a11a21a2211 Phenotype 1 Phenotype 2 Phenotype 1a112+c112+e112 Phenotype 2a11a21+c11c21+e11e21a222+a212+c222+c212 +e222+e212 Twin 1 Twin 1 Phenotype 1 Twin 1 Phenotype 2 Within-Twin Covariances (C)C1C2c11c21c2211 Phenotype 1 Phenotype 2 Phenotype 1a112+c112+e112 Phenotype 2a11a21+c11c21+e11e21a222+a212+c222+c212 +e222+e212 Twin 1 Twin 1 Phenotype 1 Twin 1 Phenotype 2 Within-Twin Covariances (E)Phenotype 1 Phenotype 2 Phenotype 1a112+c112+e112 Phenotype 2a11a21+c11c21+e11e21a222+a212+c222+c212 +e222+e212 Twin 1 Twin 1 Phenotype 1E1E2e11e21e2211 Twin 1 Phenotype 2 Cross-Twin Covariances (A)Twin 1 Phenotype 1A1A2a11a21a2211 Twin 1 Phenotype 2 Twin 2 Phenotype 1A1A2a11a21a2211 Twin 2 Phenotype 21 1 Phenotype 2 Phenotype 1 Phenotype 2+c11c21+c222+c212 Twin 1 Cross-Twin Covariances (A)

6 Twin 1 Phenotype 1A1A2a11a21a2211 Twin 1 Phenotype 2 Twin 2 Phenotype 1A1A2a11a21a2211 Twin 2 Phenotype 21 1 Phenotype 2 Phenotype 11 +Phenotype 2+c11c21+c222+c212 Twin 1 Cross-Twin Covariances (A)Twin 1 Phenotype 1A1A2a11a21a2211 Twin 1 Phenotype 2 Twin 2 Phenotype 1A1A2a11a21a2211 Twin 2 Phenotype 21 1 Phenotype 2 Phenotype 11 +c112 Phenotype 21 +c11c21+c222+c212 Twin 1 Cross-Twin Covariances (A)Twin 1 Phenotype 1A1A2a11a21a2211 Twin 1 Phenotype 2 Twin 2 Phenotype 1A1A2a11a21a2211 Twin 2 Phenotype 21 1 Phenotype 2 Phenotype 11 +c112 Phenotype 21 +c11c211 +1 +c222+c212 Twin 1 Cross-Twin Covariances (C)Twin 1 Phenotype 1 Twin 1 Phenotype 2C1C2c11c21c2211 Twin 2 Phenotype 1 Twin 2 Phenotype 2C1C2c11c21c221111 Phenotype 1 Phenotype 2 Phenotype 11 +c112 Phenotype 21 +c11c211 +1 +c222+c212 Twin 1 Predicted ModelPhenotype 1 Phenotype 2 Phenotype 1 Phenotype 2 Phenotype 1a112+c112+e112 Phenotype 2a11a21+c11c21+e11e21a222+a212+c222+c212 +e222+e212 Phenotype 11/.

7 5a112+c112a112+c112+e112 Phenotype 21/.5a11a21+c11c211/.5a222+1/.5 a212+c222+c212a11a21+c11c21+e11e21a222+a 212+c222+c212+e222+e212 Twin 1 Twin 2 Within-twin covarianceWithin-twin covarianceCross-twin covariancePredicted ModelPhenotype 1 Phenotype 2 Phenotype 1 Phenotype 2 Phenotype 1 Variance P1 Phenotype 2 Covariance P1-P2 Variance P2 Phenotype 1 Within-traitP1 Cross-traitVariance P1 Phenotype 2 Cross-traitWithin-traitP2 Covariance P1-P2 Variance P2 Twin 1 Twin 2 Within-twin covarianceWithin-twin covarianceCross-twin covariancePredicted ModelPhenotype 1 Phenotype 2 Phenotype 1 Phenotype 2 Phenotype 1 Variance P1 Phenotype 2 Covariance

8 P1-P2 Variance P2 Phenotype 1 Within-traitP1 Cross-traitVariance P1 Phenotype 2 Cross-traitWithin-traitP2 Covariance P1-P2 Variance P2 Twin 1 Twin 2 Within-twin covarianceWithin-twin covarianceCross-twin covarianceVariance of P1 and P2the same across twinsand zygosity groupsPredicted ModelPhenotype 1 Phenotype 2 Phenotype 1 Phenotype 2 Phenotype 1 Variance P1 Phenotype 2 Covariance P1-P2 Variance P2 Phenotype 1 Within-traitP1 Cross-traitVariance P1 Phenotype 2 Cross-traitWithin-traitP2 Covariance P1-P2 Variance P2 Twin 1 Twin 2 Within-twin covarianceWithin-twin covarianceCross-twin covarianceCovariance of P1 and P2 the same across twins and zygosity groupsPredicted ModelPhenotype 1 Phenotype 2 Phenotype 1 Phenotype 2 Phenotype 1 Variance P1 Phenotype 2 Covariance P1-P2 Variance P2 Phenotype 1 Within-traitP1 Cross-traitVariance P1 Phenotype 2 Cross-traitWithin-traitP2 Covariance P1-P2 Variance P2 Twin 1 Twin 2 Within-twin covarianceWithin-twin covarianceCross-twin covarianceCross-twin covariancewithin each trait differsby zygosityPredicted ModelPhenotype 1 Phenotype 2 Phenotype 1 Phenotype 2 Phenotype 1 Variance P1 Phenotype 2

9 Covariance P1-P2 Variance P2 Phenotype 1 Within-traitP1 Cross-traitVariance P1 Phenotype 2 Cross-traitWithin-traitP2 Covariance P1-P2 Variance P2 Twin 1 Twin 2 Within-twin covarianceWithin-twin covarianceCross-twin covarianceCross-twin cross-traitcovariance differs byzygosityExample Covariance MatrixP1P2P1P2P11 1 Twin 1 Twin 2 Twin 2 MZDZW ithin-twin covarianceWithin-twin covarianceWithin-twin covarianceWithin-twin covarianceCross-twin covarianceCross-twin covarianceExample Covariance MatrixP1P2P1P2P11 1 Twin 1 Twin 2 Twin 2 MZDZW ithin-twin covarianceWithin-twin covarianceWithin-twin covarianceWithin-twin covarianceCross-twin covarianceCross-twin covarianceExample Covariance MatrixP1P2P1P2P11 1 Twin 1 Twin 2 Twin 2 MZDZW ithin-twin covarianceWithin-twin covarianceWithin-twin covarianceWithin-twin covarianceCross-twin covarianceCross-twin covarianceSummary Within-individual cross-trait covariance implies common aetiological influences Cross-twin cross-trait covariance implies common aetiological influences are familial Whether familial influences Genetic or environmental shown by MZ.

10 DZ ratio of cross-twin cross-trait covariancesCholesky Decomposition Bivariate Genetic analysesSpecification in OpenMxWithin-Twin CovarianceP1P2A1A2a11a21a2211 Path Tracing:Within-Twin CovarianceP1P2A1A2a11a21a2211 Path Tracing:a Lower 2 x 2:P1P2a1a2 Starting values and labelsWithin-Twin CovarianceP1P2A1A2a11a21a2211 Path Tracing:a Lower 2 x 2:P1P2a1a2a11a21a22 A=a*aT A= a%*%t(a)Within-Twin Covariance A=a*aT A=a%*%t(a)Total Within-Twin matrix addition, the total within-twin covariancefor the phenotypes is defined as: A=a %*% t(a) C=c %*% t(c) E=e %*% t(e) V VOpenMx Matrices & AlgebraAdditive Genetic Cross-Twin Covariance (DZ) 1 Twin 2 Path Tracing:Within-traitsP11-P12 = = + = = A= (a%*%t(a))Additive Genetic Cross-Twin Covariance (MZ)P11P21A1A2a11a21a2211P12P22A1A2a