Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Slide 36Slide 37Slide 38Slide 39Slide 40Slide 41Slide 42Slide 43Slide 44Slide 45Slide 46Slide 47Slide 48Slide 49Slide 50Slide 51Slide 52Slide 53Slide 54Slide 55Slide 56Slide 57Slide 58Twin Analysis 104October 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 ITwo Traits•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 IIindividual twinwithin betweentraitwithinwithin-twin within-trait co(variance)(cross-twin within-trait) covariancebetween(within-twin cross-trait) covariancecross-twin cross-trait covarianceBivariate Twin DataBivariate Twin Covariance MatrixX1Y1X2Y2X1VX1CX1Y1CX1X2CX1Y2Y1CY1X1VY1CY1X2CY1Y2X2CX2X1CX2Y1VX2CX2Y2Y2CY2X1CY2Y1CY2X2VY2Genetic CorrelationAlternative RepresentationsCholesky DecompositionBivariate AE ModelX1Y1X2Y2X1Y1X2Y2a112+e112a222+a212+e222+e212a21*a11+e21*e11a222+a212a112a21*a11MZ Twin Covariance MatrixX1Y1X2Y2X1Y1X2Y2a112+e112a222+a212+e222+e212a21*a11+e21*e11.5a222+.5a212.5a112.5a21*a11DZ Twin Covariance MatrixWithin-Twin Covariances [Mx]Within-Twin CovariancesCross-Twin Covariances•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 Covariances•Dataset: MCV-CVT Study•1983-1993•BMI, skinfolds (bic,tri,calf,sil,ssc)•Longitudinal: 11 years•N MZF: 107, DZF: 60Practical Example IBivariate•Saturated Model•equality of means/variances•Genetic Models (ACE)•bivariate -> Cholesky Decomposition> 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]Multivariate•Saturated Model•equality of means/variances•Genetic Models (ACE)•multivariate -> Cholesky Decomposition•Independent Pathway •Common PathwayScientific 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 traitsCholeskyTextText F1 F2 F3 F4P1 f11P2 f21P3 f31P4 f41F1 F2 F3 F4P1 f11 0P2 f21 f22P3 f31 f32P4 f41 f42F1 F2 F3 F4P1 f11 0 0P2 f21 f22 0P3 f31 f32 f33P4 f41 f42 f43F1 F2 F3 F4P1 f11 0 0 0P2 f21 f22 0 0P3 f31 f32 f33 0P4 f41 f42 f43 f44PhenotypicCholesky 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 covariance matrix, fully saturatedGenetic& Environmentalmodel:ACE, matrix:a [,1] [,2] [,3] [,4] [,5] [1,] [a11] 0 0 0 0 [2,] [a21] [a22] 0 0 0 [3,] [a31] [a32] [a33] 0 0 [4,] [a41] [a42] [a43] [a44] 0 [5,] [a51] [a52] [a53] [a54] [a55]model:ACE, matrix:c [,1] [,2] [,3] [,4] [,5] [1,] [c11] 0 0 0 0 [2,] [c21] [c22] 0 0 0 [3,] [c31] [c32] [c33] 0 0 [4,] [c41] [c42] [c43] [c44] 0 [5,] [c51] [c52] [c53] [c54] [c55]model:ACE, matrix:e [,1] [,2] [,3] [,4] [,5] [1,] [e11] 0 0 0 0 [2,] [e21] [e22] 0 0 0 [3,] [e31] [e32] [e33] 0 0 [4,] [e41] [e42] [e43] [e44] 0 [5,] [e51] [e52] [e53] [e54] [e55]model:ACE, matrix:Mean [,1] [,2] [,3] [,4] [,5][1,] [NA] [NA] [NA] [NA] [NA]•IPCommon FactorF1 P1 f11 P2 f21 P3 f31 P4 f41 F1 F2 F3 F4P1 f11 f21 f31 f41%*%F %*% t(F)ResidualsE1 E2 E3 E4P1 f11 0 0 0P2 0 f22 0 0P3 0 0 f33 0P4 0 0 0 f44%*%E %*% t(E) E1 E2 E3 E4P1 f11 0 0 0P2 0 f22 0 0P3 0 0 f33 0P4 0 0 0 f44F %*% t(F) + E %*% t(E)with Twin Datatwin 1GeneticCommon FactorShared environmentalCommon FactorUnique environmental Common FactorGenetic Residuals& C & E ResidualsA1 P1 a11 P2 a21 P3 a31 P4 a41 P1 P2 P3 P4A1 a11 a21 a31 a41%*%ac %*% t(ac)A1 A2 A3 A4P1 a11 0 0 0P2 0 a22 0 0P3 0 0 a33 0P4 0 0 0 a44%*%as %*% t(as) P1 P2 P3 P4A1 a11 0 0 0A2 0 a22 0 0A3 0 0 a33 0A4 0 0 0 a44model:ACE, matrix:ac [,1] [1,] [ac11][2,] [ac21][3,] [ac31][4,] [ac41][5,] [ac51]model:ACE, matrix:cc [,1] [1,] [cc11][2,] [cc21][3,] [cc31][4,] [cc41][5,] [cc51]model:ACE, matrix:ec [,1] [1,] [ec11][2,] [ec21][3,] [ec31][4,] [ec41][5,] [ec51]model:ACE, matrix:as [,1] [,2] [,3] [,4] [,5] [1,] [as11] 0 0 0 0 [2,] 0 [as21] 0 0 0 [3,] 0 0 [as31] 0 0 [4,] 0 0 0 [as41] 0 [5,] 0 0 0 0 [as51]model:ACE, matrix:cs [,1] [,2] [,3] [,4] [,5] [1,] [cs11] 0 0 0 0 [2,] 0 [cs21] 0 0 0 [3,] 0 0 [cs31] 0 0 [4,] 0 0 0 [cs41] 0 [5,] 0 0 0 0 [cs51]model:ACE, matrix:es [,1] [,2] [,3] [,4] [,5] [1,] [es11] 0 0 0 0 [2,] 0 [es21] 0 0 0 [3,] 0 0 [es31] 0 0 [4,] 0 0 0 [es41] 0 [5,] 0 0 0 0 [es51]model:ACE, matrix:M [,1] [,2] [,3] [,4] [,5][1,] [NA] [NA] [NA] [NA] [NA]Independent Pathway•Biometric model•Different covariance structure for
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