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Introduction to Multivariate Genetic Analysis Kate Morley and Frühling Rijsdijk 21st Twin and Family Methodology Workshop, March 2008
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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 from twin pairs to partition covariation into genetic and environmental components
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Example 1 Why do traits correlate/covary? How can we explain the association? Additive genetic factors (r G ) Shared environment (r C ) Non-shared environment (r E ) Kuntsi et al. (2004) Am J Med Genet B, 124:41 ADHD E1E1 E2E2 A1A1 A2A2 A11 2 IQ rGrG C1C1 C2C2 rCrC 1111 1 1 rErE C11 2 A22 2 C22 2 E11 2 E22 2
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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 (a 21 ) Shared environment (c 21 ) Non-shared environment (e 21 ) How much is not explained by prior anxiety? Rice et al. (2004) BMC Psychiatry 4:43 Childhood anxiety A1A1 A2A2 E1E1 E2E2 a 11 a 21 a 22 e 11 e 21 e 22 1 1 1 1 Adolescent depression C1C1 C2C2 c 11 c 21 c 22 1 1
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Sources of Information As an example: two traits measured in twin pairs Interested in: Cross-trait covariance within individuals Cross-trait covariance between twins MZ:DZ ratio of cross-trait covariance between twins
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Observed Covariance Matrix Phenotype 1Phenotype 2Phenotype 1Phenotype 2 Phenotype 1 Variance P1 Phenotype 2 Covariance P1-P2 Variance P2 Phenotype 1 Within-trait P1 Cross-traitVariance P1 Phenotype 2 Cross-traitWithin-trait P2 Covariance P1-P2 Variance P2 Twin 1 Twin 2
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Observed Covariance Matrix Phenotype 1Phenotype 2Phenotype 1Phenotype 2 Phenotype 1 Variance P1 Phenotype 2 Covariance P1-P2 Variance P2 Phenotype 1 Within-trait P1 Cross-traitVariance P1 Phenotype 2 Cross-traitWithin-trait P2 Covariance P1-P2 Variance P2 Twin 1 Twin 2 Within-twin covariance
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Observed Covariance Matrix Phenotype 1Phenotype 2Phenotype 1Phenotype 2 Phenotype 1 Variance P1 Phenotype 2 Covariance P1-P2 Variance P2 Phenotype 1 Within-trait P1 Cross-traitVariance P1 Phenotype 2 Cross-traitWithin-trait P2 Covariance P1-P2 Variance P2 Twin 1 Twin 2 Within-twin covariance Cross-twin covariance
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SEM: Cholesky Decomposition Twin 1 Phenotype 1 A1A1 A2A2 E1E1 E2E2 a 11 a 22 e 11 e 22 1 1 1 1 Twin 1 Phenotype 2 C1C1 C2C2 c 11 c 22 1 1
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SEM: Cholesky Decomposition Twin 1 Phenotype 1 A1A1 A2A2 E1E1 E2E2 a 11 a 21 a 22 e 11 e 21 e 22 1 1 1 1 Twin 1 Phenotype 2 C1C1 C2C2 c 11 c 21 c 22 1 1
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SEM: Cholesky Decomposition Twin 1 Phenotype 1 A1A1 A2A2 E1E1 E2E2 a 11 a 21 a 22 e 11 e 21 e 22 1 1 1 1 Twin 1 Phenotype 2 C1C1 C2C2 c 11 c 21 c 22 1 1 Twin 2 Phenotype 1 A1A1 A2A2 E1E1 E2E2 a 11 a 21 a 22 e 11 e 21 e 22 1 1 1 1 Twin 2 Phenotype 2 C1C1 C2C2 c 11 c 21 c 22 1 1 1/0.5 1 1
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Why 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: A = X*X’ Many other multivariate models possible Depends on data and hypotheses of interest
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Cholesky Decomposition Path Tracing
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Within-Twin Covariances (A) A1A1 A2A2 a 11 a 21 a 22 1 1 Phenotype 1Phenotype 2 Phenotype 1 a 11 2 +c 11 2 +e 11 2 Phenotype 2 a 11 a 21 +c 11 c 21 +e 11 e 21 a 22 2 +a 21 2 +c 22 2 +c 21 2 +e 22 2 +e 21 2 Twin 1 Phenotype 1 Twin 1 Phenotype 2
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Within-Twin Covariances (A) A1A1 A2A2 a 11 a 21 a 22 1 1 Phenotype 1Phenotype 2 Phenotype 1 a 11 2 +c 11 2 +e 11 2 Phenotype 2 a 11 a 21 +c 11 c 21 +e 11 e 21 a 22 2 +a 21 2 +c 22 2 +c 21 2 +e 22 2 +e 21 2 Twin 1 Phenotype 1 Twin 1 Phenotype 2
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Within-Twin Covariances (A) A1A1 A2A2 a 11 a 21 a 22 1 1 Phenotype 1Phenotype 2 Phenotype 1 a 11 2 +c 11 2 +e 11 2 Phenotype 2 a 11 a 21 +c 11 c 21 +e 11 e 21 a 22 2 +a 21 2 +c 22 2 +c 21 2 +e 22 2 +e 21 2 Twin 1 Phenotype 1 Twin 1 Phenotype 2
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Within-Twin Covariances (A) A1A1 A2A2 a 11 a 21 a 22 1 1 Phenotype 1Phenotype 2 Phenotype 1 a 11 2 +c 11 2 +e 11 2 Phenotype 2 a 11 a 21 +c 11 c 21 +e 11 e 21 a 22 2 +a 21 2 +c 22 2 +c 21 2 +e 22 2 +e 21 2 Twin 1 Phenotype 1 Twin 1 Phenotype 2
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Within-Twin Covariances (C) C1C1 C2C2 c 11 c 21 c 22 1 1 Phenotype 1Phenotype 2 Phenotype 1 a 11 2 +c 11 2 +e 11 2 Phenotype 2 a 11 a 21 +c 11 c 21 +e 11 e 21 a 22 2 +a 21 2 +c 22 2 +c 21 2 +e 22 2 +e 21 2 Twin 1 Phenotype 1 Twin 1 Phenotype 2
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Within-Twin Covariances (E) Phenotype 1Phenotype 2 Phenotype 1 a 11 2 +c 11 2 +e 11 2 Phenotype 2 a 11 a 21 +c 11 c 21 +e 11 e 21 a 22 2 +a 21 2 +c 22 2 +c 21 2 +e 22 2 +e 21 2 Twin 1 Phenotype 1 E1E1 E2E2 e 11 e 21 e 22 1 1 Twin 1 Phenotype 2
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Cross-Twin Covariances (A) Twin 1 Phenotype 1 A1A1 A2A2 a 11 a 21 a 22 1 1 Twin 1 Phenotype 2 Twin 2 Phenotype 1 A1A1 A2A2 a 11 a 21 a 22 1 1 Twin 2 Phenotype 2 1/0.5 Phenotype 1Phenotype 2 Phenotype 1 Phenotype 2 +c 11 c 21 +c 22 2 +c 21 2 Twin 1 Twin 2
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Cross-Twin Covariances (A) Twin 1 Phenotype 1 A1A1 A2A2 a 11 a 21 a 22 1 1 Twin 1 Phenotype 2 Twin 2 Phenotype 1 A1A1 A2A2 a 11 a 21 a 22 1 1 Twin 2 Phenotype 2 1/0.5 Phenotype 1Phenotype 2 Phenotype 1 1/0.5a 11 2 + Phenotype 2 +c 11 c 21 +c 22 2 +c 21 2 Twin 1 Twin 2
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Cross-Twin Covariances (A) Twin 1 Phenotype 1 A1A1 A2A2 a 11 a 21 a 22 1 1 Twin 1 Phenotype 2 Twin 2 Phenotype 1 A1A1 A2A2 a 11 a 21 a 22 1 1 Twin 2 Phenotype 2 1/0.5 Phenotype 1Phenotype 2 Phenotype 1 1/0.5a 11 2 +c 11 2 Phenotype 2 1/0.5a 11 a 21 +c 11 c 21 +c 22 2 +c 21 2 Twin 1 Twin 2
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Cross-Twin Covariances (A) Twin 1 Phenotype 1 A1A1 A2A2 a 11 a 21 a 22 1 1 Twin 1 Phenotype 2 Twin 2 Phenotype 1 A1A1 A2A2 a 11 a 21 a 22 1 1 Twin 2 Phenotype 2 1/0.5 Phenotype 1Phenotype 2 Phenotype 1 1/0.5a 11 2 +c 11 2 Phenotype 2 1/0.5a 11 a 21 +c 11 c 21 1/0.5a 22 2 +1/0.5a 21 2 +c 22 2 +c 21 2 Twin 1 Twin 2
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Cross-Twin Covariances (C) Twin 1 Phenotype 1 Twin 1 Phenotype 2 C1C1 C2C2 c 11 c 21 c 22 1 1 Twin 2 Phenotype 1 Twin 2 Phenotype 2 C1C1 C2C2 c 11 c 21 c 22 1 1 1 1 Phenotype 1Phenotype 2 Phenotype 1 1/0.5a 11 2 +c 11 2 Phenotype 2 1/0.5a 11 a 21 +c 11 c 21 1/0.5a 22 2 +1/0.5a 21 2 +c 22 2 +c 21 2 Twin 1 Twin 2
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Predicted Model Phenotype 1Phenotype 2Phenotype 1Phenotype 2 Phenotype 1 a 11 2 +c 11 2 +e 11 2 Phenotype 2 a 11 a 21 +c 11 c 21 + e 11 e 21 a 22 2 +a 21 2 +c 22 2 + c 21 2 +e 22 2 +e 21 2 Phenotype 1 1/.5a 11 2 +c 11 2 a 11 2 +c 11 2 +e 11 2 Phenotype 2 1/.5a 11 a 21 + c 11 c 21 1/.5a 22 2 +1/.5 a 21 2 +c 22 2 +c 21 2 a 11 a 21 +c 11 c 21 + e 11 e 21 a 22 2 +a 21 2 +c 22 2 +c 21 2 +e 22 2 +e 21 2 Twin 1 Twin 2 Within-twin covariance Cross-twin covariance
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Predicted Model Phenotype 1Phenotype 2Phenotype 1Phenotype 2 Phenotype 1 Variance P1 Phenotype 2 Covariance P1-P2 Variance P2 Phenotype 1 Within-trait P1 Cross-traitVariance P1 Phenotype 2 Cross-traitWithin-trait P2 Covariance P1-P2 Variance P2 Twin 1 Twin 2 Within-twin covariance Cross-twin covariance
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Predicted Model Phenotype 1Phenotype 2Phenotype 1Phenotype 2 Phenotype 1 Variance P1 Phenotype 2 Covariance P1-P2 Variance P2 Phenotype 1 Within-trait P1 Cross-traitVariance P1 Phenotype 2 Cross-traitWithin-trait P2 Covariance P1-P2 Variance P2 Twin 1 Twin 2 Within-twin covariance Cross-twin covariance Variance of P1 and P2 the same across twins and zygosity groups
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Predicted Model Phenotype 1Phenotype 2Phenotype 1Phenotype 2 Phenotype 1 Variance P1 Phenotype 2 Covariance P1-P2 Variance P2 Phenotype 1 Within-trait P1 Cross-traitVariance P1 Phenotype 2 Cross-traitWithin-trait P2 Covariance P1-P2 Variance P2 Twin 1 Twin 2 Within-twin covariance Cross-twin covariance Covariance of P1 and P2 the same across twins and zygosity groups
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Predicted Model Phenotype 1Phenotype 2Phenotype 1Phenotype 2 Phenotype 1 Variance P1 Phenotype 2 Covariance P1-P2 Variance P2 Phenotype 1 Within-trait P1 Cross-traitVariance P1 Phenotype 2 Cross-traitWithin-trait P2 Covariance P1-P2 Variance P2 Twin 1 Twin 2 Within-twin covariance Cross-twin covariance within each trait differs by zygosity
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Predicted Model Phenotype 1Phenotype 2Phenotype 1Phenotype 2 Phenotype 1 Variance P1 Phenotype 2 Covariance P1-P2 Variance P2 Phenotype 1 Within-trait P1 Cross-traitVariance P1 Phenotype 2 Cross-traitWithin-trait P2 Covariance P1-P2 Variance P2 Twin 1 Twin 2 Within-twin covariance Cross-twin covariance Cross-twin cross-trait covariance differs by zygosity
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Example Covariance Matrix P1P2P1P2 P1 1 P2.301 P1 0.790.491 P2 0.500.590.291 P1P2P1P2 P1 1 P2 0.301 P1 0.390.251 P2 0.240.430.311 Twin 1 Twin 2 Twin 1 Twin 2 MZ DZ Within-twin covariance Cross-twin covariance
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Example Covariance Matrix P1P2P1P2 P1 1 P2.301 P1 0.790.491 P2 0.500.590.291 P1P2P1P2 P1 1 P2 0.301 P1 0.390.251 P2 0.240.430.311 Twin 1 Twin 2 Twin 1 Twin 2 MZ DZ Within-twin covariance Cross-twin covariance
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Example Covariance Matrix P1P2P1P2 P1 1 P2.301 P1 0.790.491 P2 0.500.590.291 P1P2P1P2 P1 1 P2 0.301 P1 0.390.251 P2 0.240.430.311 Twin 1 Twin 2 Twin 1 Twin 2 MZ DZ Within-twin covariance Cross-twin covariance
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Example Covariance Matrix P1P2P1P2 P1 1 P2.301 P1 0.790.251 P2 0.240.590.291 P1P2P1P2 P1 1 P2 0.301 P1 0.390.251 P2 0.240.430.311 Twin 1 Twin 2 Twin 1 Twin 2 MZ DZ Within-twin covariance Cross-twin covariance
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Example Covariance Matrix P1P2P1P2 P1 1 P2.301 P1 0.790.011 P2 0.010.590.291 P1P2P1P2 P1 1 P2 0.301 P1 0.390.011 P2 0.010.430.311 Twin 1 Twin 2 Twin 1 Twin 2 MZ DZ Within-twin covariance Cross-twin covariance
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Summary 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:DZ ratio of cross-twin cross-trait covariances
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Cholesky Decomposition Specification in Mx
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Mx Parameter Matrices #define nvar 2 Begin Matrices; X lower nvar nvar free! Genetic coefficients Y lower nvar nvar free! C coefficients Z lower nvar nvar free! E coefficients G Full 1 nvar free! Means H Full 1 1 fix! 0.5 for DZ A covar End Matrices; Begin Algebra; A=X*X’;! A var/cov C=Y*Y’;! C var/cov E=Z*Z’; ! E var/cov P=A+C+E End Algebra;
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Within-Twin Covariance P1 P2 A1A1 A2A2 a 11 a 21 a 22 1 1 Path Tracing:
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Within-Twin Covariance P1 P2 A1A1 A2A2 a 11 a 21 a 22 1 1 Path Tracing: X Lower 2 x 2: P1 P2 A1A2
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Within-Twin Covariance P1 P2 A1A1 A2A2 a 11 a 21 a 22 1 1 Path Tracing: X Lower 2 x 2: P1 P2 A1A2 a 11 a 21 a 22
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Total Within-Twin Covar. Using matrix addition, the total within-twin covariance for the phenotypes is defined as:
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Cross-Twin Covariance (DZ) P11 P21 A1A1 A2A2 a 11 a 21 a 22 1 1 P12 P22 A1A1 A2A2 a 11 a 21 a 22 1 1 0.5 Twin 1Twin 2
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Cross-Twin Covariance (DZ) P11 P21 A1A1 A2A2 a 11 a 21 a 22 1 1 P12 P22 A1A1 A2A2 a 11 a 21 a 22 1 1 0.5 Twin 1Twin 2 Within-traits P11-P12 = 0.5a 11 2 P21-P22 = 0.5a 22 2 +0.5a 21 2
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Cross-Twin Covariance (DZ) P11 P21 A1A1 A2A2 a 11 a 21 a 22 1 1 P12 P22 A1A1 A2A2 a 11 a 21 a 22 1 1 0.5 Twin 1Twin 2 Path Tracing: Within-traits P11-P12 = 0.5a 11 2 P21-P22 = 0.5a 22 2 +0.5a 21 2 Cross-traits P11-P22 = 0.5a 11 a 21 P21-P12 = 0.5a 21 a 11
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Additive Genetic Cross-Twin Covariance (DZ) P11 P21 A1A1 A2A2 a 11 a 21 a 22 1 1 P12 P22 A1A1 A2A2 a 11 a 21 a 22 1 1 0.5 Twin 1Twin 2 Path Tracing: Within-traits P11-P12 = 0.5a 11 2 P21-P22 = 0.5a 22 2 +0.5a 21 2 Cross-traits P11-P22 = 0.5a 11 a 21 P21-P12 = 0.5a 21 a 11 N.B. in Mx = @
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Additive Genetic Cross-Twin Covariance (MZ) P11 P21 A1A1 A2A2 a 11 a 21 a 22 1 1 P12 P22 A1A1 A2A2 a 11 a 21 a 22 1 1 1 1 Twin 1Twin 2
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Common Environment Cross- Twin Covariance P11 P21 C1C1 C2C2 c 11 c 21 c 22 1 1 P12 P22 C1C1 C2C2 c 11 c 21 c 22 1 1 1 1 Twin 1Twin 2
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Covariance Model for Twin Pairs MZ: CovarianceA+C+E | A+C_ A+C | A+C+E / DZ: CovarianceA+C+E | H@A+C_ H@A+C | A+C+E / N.B. H Full 1 1 Fixed = 0.5
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Obtaining Standardised Estimates
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Correlated Factors Solution Phenotype 1 E1E1 E2E2 A1A1 A2A2 A11 2 Phenotype 2 rGrG C1C1 C2C2 rCrC 1111 1 1 rErE C11 2 A22 2 C22 2 E11 2 E22 2 Each variable decomposed into genetic/environmental components Correlations across variables estimated Results from Cholesky can be converted to this model
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Using matrix algebra notation: Covariance to Correlation
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Genetic Correlations
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Specification in Mx 1.Matrix function in Mx: R = \stnd(A); 2.R = \sqrt(I.A)˜ * A * \sqrt(I.A)˜; Where I is an identity matrix: and I.A =
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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 A 2 paths i.e. importance is also determined by the heritability of each phenotype
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Example ADHD A1A1 A2A2 h P1 2 IQ rGrG 11 h P2 2 N.B. h P1 2 = A11 2 Proportion of r P due to additive genetic factors:
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Standardised Results Begin algebra; K = A%P|C%P|E%P; End algebra; % is the Mx operator for element division Proportion of the phenotypic correlation due to genetic effects
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Example Mx Output Matrix K: Additive Genetic Component of ADHD = 63%, for IQ = 33% % of covariance between ADHD and IQ due to A = 84% Matrix K Estimates [A%P|C%P|E%P] 1 2 3 4 5 6 1 0.6286 0.8360 0.0000 0.0000 0.3714 0.1640 2 0.8360 0.3338 0.0000 0.3666 0.1640 0.2996 [S=\STND(P)] [R=\STND(A)] 1 21 2 1 1.0000 -0.2865 1 1.0000 -0.2865 2 -0.2865 1.0000 2 -0.5248 1.0000 Phenotypic correlation Genetic correlation
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Interpretation of Correlations Consider two traits with a phenotypic correlation of 0.40 : h 2 P1 = 0.7 and h 2 P2 = 0.6 with r G =.3 Correlation due to additive genetic effects = ? Proportion of phenotypic correlation attributable to additive genetic effects = ? h 2 P1 = 0.2 and h 2 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.
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Interpretation of Correlations Consider two traits with a phenotypic correlation of 0.40 : h 2 P1 = 0.7 and h 2 P2 = 0.6 with r G =.3 Correlation due to additive genetic effects = 0.19 Proportion of phenotypic correlation attributable to additive genetic effects = 0.49 h 2 P1 = 0.2 and h 2 P2 = 0.3 with r G = 0.8 Correlation due to additive genetic effects = 0.20 Proportion of phenotypic correlation attributable to additive genetic effects = 0.49 Weakly heritable traits can still have a large portion of their correlation attributable to genetic effects.
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More Variables… Twin 1 Phenotype 1 A1A1 A2A2 E1E1 E2E2 a 11 a 21 a 22 e 11 e 21 e 22 1 1 1 1 Twin 1 Phenotype 2 C1C1 C2C2 c 11 c 21 c 22 1 1 Twin 1 Phenotype 3 E3E3 e 33 e 31 e 32 C3C3 c 33 1 A3A3 a 33 c 32 a 31 c 31 a 32 1 1
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More Variables… Twin 1 Phenotype 1 A1A1 A2A2 E1E1 E2E2 a 11 a 21 a 22 e 11 e 21 e 22 1 1 1 1 Twin 1 Phenotype 2 C1C1 C2C2 c 11 c 21 c 22 1 1 1/0.5 1 1 Twin 1 Phenotype 3 E3E3 e 33 e 31 e 32 C3C3 c 33 1 A3A3 a 33 c 32 a 31 c 31 a 32 1 1 Twin 2 Phenotype 1 A1A1 A2A2 E1E1 E2E2 a 11 a 21 a 22 e 11 e 21 e 22 1 1 1 1 Twin 2 Phenotype 2 C1C1 C2C2 c 11 c 21 c 22 1 1 Twin 2 Phenotype 3 E3E3 e 33 e 31 e 32 C3C3 c 33 1 A3A3 a 33 c 32 a 31 c 31 a 32 1 1 1/0.5 1
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Mx Parameter Matrices #define nvar 3 Begin Matrices; X lower nvar nvar free! Genetic coefficients Y lower nvar nvar free! C coefficients Z lower nvar nvar free! E coefficients G Full 1 nvar free! Means H Full 1 1 fix! 0.5 for DZ A covar End Matrices; Begin Algebra; A=X*X’;! Gen var/cov C=Y*Y’;! C var/cov E=Z*Z’; ! E var/cov P=A+C+E End Algebra;
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Expanded Matrices X Lower 3 x 3 Y Lower 3 x 3 Z Lower 3 x 3
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