1Bivariate analysisHGEN619 class 2006Univariate ACE modelT2AEA Ca ac eT11 111Ee1Cc11 or .512Expected Covariance Matricesa2+c2+e2 .5a2+c2.5a2+c2 a2+c2+e2E DZ =a2+c2+e2 a2+c2a2+c2 a2+c2+e2E MZ = 2 x 22 x 2Bivariate Questions In Univariate Analysis: What are the contributions of additive genetic, dominance/shared environmental and unique environmental factors to the variance?n Bivariate Analysis: What are the contributions of genetic and environmental factors to the covariance between two traits?3Two TraitsEYEXX YAXAYACECBivariate Questions IIn Two or more traits can be correlated because they share common genes or common environmental influences¨ e.g. Are the same genetic/environmental factors influencing the traits?n With twin data on multiple traits it is possible to partition the covariationinto its genetic and environmental componentsn Goal: to understand what factors make sets of variables correlate or co-vary4Bivariate Twin Databetweenwithintraitbetweenwithinindividual twin(within-twin within-trait co)variance(cross-twinwithin-trait) covariance(cross-twin within-trait) covariancecross-twin cross-trait covarianceBivariate Twin Covariance MatrixY2X2Y1X1Y2X2Y1X1VX1CX1X2CX2X1VX2VY1CY1Y2CY2Y1VY2CX1Y1CX2Y2CY1X1CY2X2CX1Y2CX2Y1CY1X2CY2X15Genetic CorrelationY2AYAXAYaxayayX11 11AXax11 or .5 1 or .5X2Y1rgrgAlternative RepresentationsAXAYaxayX11 1Y1rgASXASYasxasyX11 1Y1AC1acacA1A2a11a22X11 1Y1a216Cholesky DecompositionA1A2a11a22X11 1Y1a21A1A2a11a22X21 1Y2a211 or .5 1 or .5More VariablesA1A2a11a22X11 1X2a21A3a331X3A4a441X4a32a43a31a42A51X57Bivariate AE ModelA1A2a11a22X11 1Y1a21A1A2a11a22X21 1Y2a211 or .5 1 or .5E1E2e11e221 1e21E1E2e11e221 1e21MZ Twin Covariance MatrixY2X2Y1X1Y2X2Y1X1a112+e112a222+a212+e222+e212a21*a11+e21*e11a222+a212a112a21*a118DZ Twin Covariance MatrixY2X2Y1X1Y2X2Y1X1a112+e112a222+a212+e222+e212a21*a11+e21*e11.5a222+.5a212.5a112.5a21*a11Within-Twin Covariances [Mx]A1A2a11a22X11 1Y1a21a11a220a21A1A2X1Y1X Lower 2 2a112a222+a212a21*a11a11*a21=A=X*X'a11a220a21a11a220a21*EA=9Within-Twin Covariancesa112a222+a212a21*a11a11*a21EA=e112e222+e212e21*e11e11*e21EE=a112+ e112a222+a212 +e222+e212a11*a21 + e11*e21EP= EA+EE =a21*a11 + e21*e11Cross-Twin Covariancesa112a222+a212a21*a11a11*a21EA=MZ.5a112.5a222+.5a212.5a21*a11.5a11*a21.5@EA=DZ10Cross-Trait Covariancesn Within-twin cross-trait covariances imply common etiological influencesn Cross-twin cross-trait covariances imply familial common etiological influencesn MZ/DZ ratio of cross-twin cross-trait covariances reflects whether common etiological influences are genetic or environmentalUnivariate Expected Covariancesa2+c2+e2 .5a2+c2.5a2+c2 a2+c2+e2E DZ =a2+c2+e2 a2+c2a2+c2 a2+c2+e2E MZ = 2 x 22 x 211Univariate Expected Covariances IIE DZ =EA+EC+EE .5@EA+EC.5@EA+EC EA+EC+EEEA+EC+EE EA+ECEA+EC EA+EC+EEE MZ = 2 x 22 x 2Bivariate Expected CovariancesE DZ =EA+EC+EE .5@EA+EC.5@EA+EC EA+EC+EEEA+EC+EE EA+ECEA+EC EA+EC+EEE MZ = 4 x 44 x 412Practical Example In Dataset: MCV-CVT Studyn 1983-1993n BMI, skinfolds (bic,tri,calf,sil,ssc)n Longitudinal: 11 yearsn N MZF: 107, DZF: 60Practical Example IIn Dataset: NL MRI Studyn 1990’sn Working Memory, Gray & White Mattern N MZFY: 68, DZF:
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