Twin Analysis 104October 2010Friday, October 15, 2010•Univariate Analysis: What are the contributions of additive genetic, dominance/shared environmental and unique environmental factors to the variance?•Bivariate Analysis: What are the contributions of genetic and environmental factors to the covariance between two traits?Bivariate Questions IFriday, October 15, 2010Two TraitsFriday, October 15, 2010•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?•With twin data on multiple traits it is possible to partition the covariation into its genetic and environmental components•Goal: to understand what factors make sets of variables correlate or co-varyBivariate Questions IIFriday, October 15, 2010Bivariate Twin DataFriday, October 15, 2010individual twinindividual twinwithinbetweentraitwithinwithin-twin within-trait co(variance)(cross-twin within-trait) covariancetraitbetween(within-twin cross-trait) covariancecross-twin cross-trait covarianceBivariate Twin DataFriday, October 15, 2010Bivariate Twin Covariance MatrixX1Y1X2Y2X1VX1CX1Y1CX1X2CX1Y2Y1CY1X1VY1CY1X2CY1Y2X2CX2X1CX2Y1VX2CX2Y2Y2CY2X1CY2Y1CY2X2VY2Friday, October 15, 2010Genetic CorrelationFriday, October 15, 2010Alternative RepresentationsFriday, October 15, 2010Cholesky DecompositionFriday, October 15, 2010Bivariate AE ModelFriday, October 15, 2010MZ Twin Covariance MatrixFriday, October 15, 2010X1Y1X2Y2X1Y1X2Y2MZ Twin Covariance MatrixFriday, October 15, 2010X1Y1X2Y2X1Y1X2Y2a112+e112MZ Twin Covariance MatrixFriday, October 15, 2010X1Y1X2Y2X1Y1X2Y2a112+e112a222+a212+e222+e212MZ Twin Covariance MatrixFriday, October 15, 2010X1Y1X2Y2X1Y1X2Y2a112+e112a222+a212+e222+e212a21*a11+e21*e11MZ Twin Covariance MatrixFriday, October 15, 2010X1Y1X2Y2X1Y1X2Y2a112+e112a222+a212+e222+e212a21*a11+e21*e11a112MZ Twin Covariance MatrixFriday, October 15, 2010X1Y1X2Y2X1Y1X2Y2a112+e112a222+a212+e222+e212a21*a11+e21*e11a222+a212a112MZ Twin Covariance MatrixFriday, October 15, 2010X1Y1X2Y2X1Y1X2Y2a112+e112a222+a212+e222+e212a21*a11+e21*e11a222+a212a112a21*a11MZ Twin Covariance MatrixFriday, October 15, 2010X1Y1X2Y2X1Y1X2Y2a112+e112a222+a212+e222+e212a21*a11+e21*e11a222+a212a112a21*a11MZ Twin Covariance MatrixFriday, October 15, 2010X1Y1X2Y2X1Y1X2Y2a112+e112a222+a212+e222+e212a21*a11+e21*e11a222+a212a112a21*a11MZ Twin Covariance MatrixFriday, October 15, 2010X1Y1X2Y2X1Y1X2Y2a112+e112a222+a212+e222+e212a21*a11+e21*e11a222+a212a112a21*a11MZ Twin Covariance MatrixFriday, October 15, 2010X1Y1X2Y2X1Y1X2Y2a112+e112a222+a212+e222+e212a21*a11+e21*e11a222+a212a112a21*a11MZ Twin Covariance MatrixFriday, October 15, 2010DZ Twin Covariance MatrixFriday, October 15, 2010X1Y1X2Y2X1Y1X2Y2DZ Twin Covariance MatrixFriday, October 15, 2010X1Y1X2Y2X1Y1X2Y2a112+e112DZ Twin Covariance MatrixFriday, October 15, 2010X1Y1X2Y2X1Y1X2Y2a112+e112a222+a212+e222+e212DZ Twin Covariance MatrixFriday, October 15, 2010X1Y1X2Y2X1Y1X2Y2a112+e112a222+a212+e222+e212a21*a11+e21*e11DZ Twin Covariance MatrixFriday, October 15, 2010X1Y1X2Y2X1Y1X2Y2a112+e112a222+a212+e222+e212a21*a11+e21*e11.5a112DZ Twin Covariance MatrixFriday, October 15, 2010X1Y1X2Y2X1Y1X2Y2a112+e112a222+a212+e222+e212a21*a11+e21*e11.5a222+.5a212.5a112DZ Twin Covariance MatrixFriday, October 15, 2010X1Y1X2Y2X1Y1X2Y2a112+e112a222+a212+e222+e212a21*a11+e21*e11.5a222+.5a212.5a112.5a21*a11DZ Twin Covariance MatrixFriday, October 15, 2010X1Y1X2Y2X1Y1X2Y2a112+e112a222+a212+e222+e212a21*a11+e21*e11.5a222+.5a212.5a112.5a21*a11DZ Twin Covariance MatrixFriday, October 15, 2010X1Y1X2Y2X1Y1X2Y2a112+e112a222+a212+e222+e212a21*a11+e21*e11.5a222+.5a212.5a112.5a21*a11DZ Twin Covariance MatrixFriday, October 15, 2010X1Y1X2Y2X1Y1X2Y2a112+e112a222+a212+e222+e212a21*a11+e21*e11.5a222+.5a212.5a112.5a21*a11DZ Twin Covariance MatrixFriday, October 15, 2010X1Y1X2Y2X1Y1X2Y2a112+e112a222+a212+e222+e212a21*a11+e21*e11.5a222+.5a212.5a112.5a21*a11DZ Twin Covariance MatrixFriday, October 15, 2010Within-Twin Covariances [Mx]Friday, October 15, 2010Within-Twin CovariancesFriday, October 15, 2010Cross-Twin CovariancesFriday, October 15, 2010•Within-twin cross-trait covariances imply common etiological influences•Cross-twin cross-trait covariances imply familial common etiological influences•MZ/DZ ratio of cross-twin cross-trait covariances reflects whether common etiological influences are genetic or environmentalCross-Trait CovariancesFriday, October 15, 2010•Dataset: MCV-CVT Study•1983-1993•BMI, skinfolds (bic,tri,calf,sil,ssc)•Longitudinal: 11 years•N MZF: 107, DZF: 60Practical Example IFriday, October 15, 2010Bivariate•Saturated Model•equality of means/variances•Genetic Models (ACE)•bivariate -> Cholesky DecompositionFriday, October 15, 2010> parameterSpecifications(bivACEFit)model:ACE, matrix:a [,1] [,2] [1,] [a11] 0 [2,] [a21] [a22]model:ACE, matrix:c [,1] [,2] [1,] [c11] 0 [2,] [c21] [c22]model:ACE, matrix:e [,1] [,2] [1,] [e11] 0 [2,] [e21] [e22]model:ACE, matrix:Mean [,1] [,2][1,] [NA] [NA]Friday, October 15, 2010Multivariate•Saturated Model•equality of means/variances•Genetic Models (ACE)•multivariate -> Cholesky Decomposition•Independent Pathway •Common PathwayFriday, October 15, 2010Scientific Questions•Are these measures influenced by the same genes (single common factor)?•Is there more than one factor (overall fat- fat distribution)?•What is the structure of C and E?•Contribution of A, C, E factors to covariance between traitsFriday, October 15, 2010Cholesky1μ2μ1P21F2P1f211F1f11P41F4P3F31μ4μ3f31f41TextText F1 F2 F3 F4P1 f11P2 f21P3 f31P4 f41Friday, October 15, 20101μ2μ1P21F2P11F1P41F4P3F31μ4μ3f42f32f22 F1 F2 F3 F4P1 f11 0P2 f21 f22P3 f31 f32P4 f41 f42Friday, October 15, 20101μ2μ1P21F2P11F1P41F4P3f43f33F31μ4μ3 F1 F2 F3 F4P1 f11 0 0P2 f21 f22 0P3 f31 f32 f33P4 f41 f42 f43Friday, October 15, 20101μ2μ1P21F2P11F1P41F4f44P3F31μ4μ3 F1 F2 F3 F4P1 f11 0 0 0P2 f21 f22 0 0P3 f31 f32 f33 0P4 f41 f42 f43 f44Friday, October 15, 2010Phenotypic1μ2μ1P21F2P1f211F1f11P41F4f44P3f43f33F31μ4μ3f31f42f41f32f22Friday, October 15, 2010Cholesky Decomposition F1 F2 F3 F4P1 f11 0 0 0P2 f21 f22 0 0P3 f31 f32 f33 0P4 f41 f42 f43 f44 F1 F2 F3 F4P1 f11 f21 f31 f41P2 0 f22 f32 f42P3 0 0 f33 f43P4 0 0 0 f44%*%F %*% t(F)Estimate
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