Introduction to Multivariate Genetic Analysis

1 Introduction to Multivariate Genetic AnalysisKate Morley and Fr hling Rijsdijk21st Twin and Family Methodology Workshop, March 2008 Aim and Rationale Aim: to examine the source of factors that make traits correlate or co-vary Rationale: Traits may be correlated due to shared Genetic factors (A) or shared environmental factors (C or E) Can use information on multiple traits fromtwin pairs 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) Kuntsi et al.

2 (2004) Am J Med Genet B, 124:41 ADHDE1E2A1A2 A112 IQrGC1C2rC111111rE C112 A222 C222 E112 E222 Example 2 Associations between phenotypes over time Does anxiety in childhood lead to depression in adolescence? How can we explain the association? Additive Genetic factors (a21) Shared environment (c21) Non-shared environment (e21) How much is not explained by prioranxiety? Rice et al. (2004) BMC Psychiatry 4:43 Childhood anxietyA1A2E1E2a11a21a22e11e21e221111 Adolescent depressionC1C2c11c21c2211 Sources of Information As an example: two traits measured in twin pairs Interested in: Cross-trait covariance withinindividuals Cross-trait covariance betweentwins MZ.

3 DZ ratio of cross-trait covariance between twinsObserved Covariance MatrixVarianceP2 Covariance P1-P2 Within-traitP2 Cross-traitPhenotype 2 VarianceP1 Cross-traitWithin-traitP1 Phenotype 1 VarianceP2 Covariance P1-P2 Phenotype 2 VarianceP1 Phenotype 1 Phenotype 2 Phenotype 1 Phenotype 2 Phenotype 1 Twin 1 Twin 1 Twin2 Twin 2 Observed Covariance MatrixVarianceP2 Covariance P1-P2 Within-traitP2 Cross-traitPhenotype 2 VarianceP1 Cross-traitWithin-traitP1 Phenotype 1 VarianceP2 Covariance P1-P2 Phenotype 2 VarianceP1 Phenotype 1 Phenotype 2 Phenotype 1 Phenotype 2 Phenotype 1 Twin 1 Twin 1 Twin2 Twin 2 Within-twin covarianceWithin-twin covarianceObserved Covariance MatrixVarianceP2 Covariance P1-P2 Within-traitP2 Cross-traitPhenotype 2 VarianceP1 Cross-traitWithin-traitP1 Phenotype 1 VarianceP2 Covariance P1-P2 Phenotype 2 VarianceP1 Phenotype 1 Phenotype 2 Phenotype 1 Phenotype 2 Phenotype 1 Twin 1 Twin 1 Twin2 Twin 2 Within-twin covarianceWithin-twin covarianceCross-twin covarianceSEM.

4 Cholesky DecompositionTwin 1 Phenotype 1A1A2E1E2a11a22e11e221111 Twin 1 Phenotype 2C1C2c11c2211 SEM: Cholesky DecompositionTwin 1 Phenotype 1A1A2E1E2a11a21a22e11e21e221111 Twin 1 Phenotype 2C1C2c11c21c2211 SEM: Cholesky DecompositionTwin 1 Phenotype 1A1A2E1E2a11a21a22e11e21e221111 Twin 1 Phenotype 2C1C2c11c21c2211 Twin 2 Phenotype 1A1A2E1E2a11a21a22e11e21e221111 Twin 2 Phenotype 2C1C2c11c21c22111 Fit This Model? Covariance matrices must be positive definite If a matrix is positive definite, it can be decomposed into the product of a triangular matrix and its transpose.

5 A = X*X Many other Multivariate models possible Depends on data and hypotheses of interestCholesky DecompositionPath TracingWithin-Twin Covariances (A)A1A2a11a21a2211a222+a212+c222+c212+e2 22+e212a11a21+c11c21+e11e21 Phenotype 2a112+c112+e112 Phenotype 1 Phenotype 2 Phenotype 1 Twin 1 Twin 1 Twin 1 Phenotype 1 Twin 1 Phenotype 2 Within-Twin Covariances (A)A1A2a11a21a2211a222+a212+c222+c212+e2 22+e212a11a21+c11c21+e11e21 Phenotype 2a112+c112+e112 Phenotype 1 Phenotype 2 Phenotype 1 Twin 1 Twin 1 Twin 1 Phenotype 1 Twin 1 Phenotype 2 Within-Twin Covariances (A)A1A2a11a21a2211a222+a212+c222+c212+e2 22+e212a11a21+c11c21+e11e21 Phenotype 2a112+c112+e112 Phenotype 1 Phenotype 2 Phenotype 1 Twin 1 Twin 1 Twin 1 Phenotype 1 Twin 1 Phenotype 2 Within-Twin Covariances (A)

6 A1A2a11a21a2211a222+a212+c222+c212+e222+ e212a11a21+c11c21+e11e21 Phenotype 2a112+c112+e112 Phenotype 1 Phenotype 2 Phenotype 1 Twin 1 Twin 1 Twin 1 Phenotype 1 Twin 1 Phenotype 2 Within-Twin Covariances (C)C1C2c11c21c2211a222+a212+c222+c212+e2 22+e212a11a21+c11c21+e11e21 Phenotype 2a112+c112+e112 Phenotype 1 Phenotype 2 Phenotype 1 Twin 1 Twin 1 Twin 1 Phenotype 1 Twin 1 Phenotype 2 Within-Twin Covariances (E)a222+a212+c222+c212+e222+e212a11a21+c 11c21+e11e21 Phenotype 2a112+c112+e112 Phenotype 1 Phenotype 2 Phenotype 1 Twin 1 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 +c222+c212+c11c21 Phenotype 2 Phenotype 1 Phenotype 2 Phenotype 1 Twin 1 Twin2 Cross-Twin Covariances (A)

7 Twin 1 Phenotype 1A1A2a11a21a2211 Twin 1 Phenotype 2 Twin 2 Phenotype 1A1A2a11a21a2211 Twin 2 Phenotype 21 +c222+c212+c11c21 Phenotype 21 +Phenotype 1 Phenotype 2 Phenotype 1 Twin 1 Twin2 Cross-Twin Covariances (A)Twin 1 Phenotype 1A1A2a11a21a2211 Twin 1 Phenotype 2 Twin 2 Phenotype 1A1A2a11a21a2211 Twin 2 Phenotype 21 +c222+c2121 +c11c21 Phenotype 21 +c112 Phenotype 1 Phenotype 2 Phenotype 1 Twin 1 Twin2 Cross-Twin Covariances (A)Twin 1 Phenotype 1A1A2a11a21a2211 Twin 1 Phenotype 2 Twin 2 Phenotype 1A1A2a11a21a2211 Twin 2 Phenotype 21 +1 +c222+c2121 +c11c21 Phenotype 21 +c112 Phenotype 1 Phenotype 2 Phenotype 1 Twin 1 Twin2 Cross-Twin Covariances (C)Twin 1 Phenotype 1 Twin 1 Phenotype 2C1C2c11c21c2211 Twin 2 Phenotype 1 Twin 2 Phenotype 2C1C2c11c21c2211111 +1 +c222+c2121 +c11c21 Phenotype 21 +c112 Phenotype 1 Phenotype 2 Phenotype 1 Twin 1 Twin2 Predicted Modela222+a212+c222+c212+e222+e212a11a21 +c11c21+e11e211/.

8 5a222+1/.5 a212+c222+c2121/.5a11a21+c11c21 Phenotype 2a112+c112+e1121/.5a112+c112 Phenotype 1a222+a212+c222+c212+e222+e212a11a21+c11 c21+e11e21 Phenotype 2a112+c112+e112 Phenotype 1 Phenotype 2 Phenotype 1 Phenotype 2 Phenotype 1 Twin 1 Twin 1 Twin2 Twin 2 Within-twin covarianceWithin-twin covarianceCross-twin covariancePredicted ModelVarianceP2 Covariance P1-P2 Within-traitP2 Cross-traitPhenotype 2 VarianceP1 Cross-traitWithin-traitP1 Phenotype 1 VarianceP2 Covariance P1-P2 Phenotype 2 VarianceP1 Phenotype 1 Phenotype 2 Phenotype 1 Phenotype 2 Phenotype 1 Twin 1 Twin 1 Twin2 Twin 2 Within-twin

9 CovarianceWithin-twin covarianceCross-twin covariancePredicted ModelVarianceP2 Covariance P1-P2 Within-traitP2 Cross-traitPhenotype 2 VarianceP1 Cross-traitWithin-traitP1 Phenotype 1 VarianceP2 Covariance P1-P2 Phenotype 2 VarianceP1 Phenotype 1 Phenotype 2 Phenotype 1 Phenotype 2 Phenotype 1 Twin 1 Twin 1 Twin2 Twin 2 Within-twin covarianceWithin-twin covarianceCross-twin covarianceVariance of P1 and P2the same across twinsand zygosity groupsPredicted ModelVarianceP2 Covariance P1-P2 Within-traitP2 Cross-traitPhenotype 2 VarianceP1 Cross-traitWithin-traitP1 Phenotype 1 VarianceP2 Covariance P1-P2 Phenotype 2 VarianceP1 Phenotype 1 Phenotype 2 Phenotype 1 Phenotype 2 Phenotype 1 Twin 1 Twin 1 Twin2 Twin 2 Within-twin covarianceWithin-twin covarianceCross-twin covarianceCovariance of P1 andP2 the same acrosstwins and zygosity groupsPredicted ModelVarianceP2 Covariance P1-P2 Within-traitP2 Cross-traitPhenotype 2 VarianceP1 Cross-traitWithin-traitP1 Phenotype 1 VarianceP2 Covariance P1-P2 Phenotype 2 VarianceP1 Phenotype 1 Phenotype 2 Phenotype 1 Phenotype 2

10 Phenotype 1 Twin 1 Twin 1 Twin2 Twin 2 Within-twin covarianceWithin-twin covarianceCross-twin covarianceCross-twin covariancewithin each trait differsby zygosityPredicted ModelVarianceP2 Covariance P1-P2 Within-traitP2 Cross-traitPhenotype 2 VarianceP1 Cross-traitWithin-traitP1 Phenotype 1 VarianceP2 Covariance P1-P2 Phenotype 2 VarianceP1 Phenotype 1 Phenotype 2 Phenotype 1 Phenotype 2 Phenotype 1 Twin 1 Twin 1 Twin2 Twin 2 Within-twin covarianceWithin-twin covarianceCross-twin covarianceCross-twin cross-traitcovariance differs byzygosityExample Covariance Matrix1 1 Twin 1 Twin 2 Twin 2 Twin 1 Twin 1 Twin2 Twin2 MZDZW ithin-twin covarianceWithin-twin covarianceWithin-twin covarianceWithin-twin covarianceCross-twin covarianceCross-twin covarianceExample Covariance Matrix1 1 Twin 1 Twin 2 Twin 2 Twin 1 Twin 1 Twin2 Twin2 MZDZW ithin-twin covarianceWithin-twin covarianceWithin-twin covarianceWithin-twin covarianceCross-twin covarianceCross-twin covarianceExample Covariance Matrix1