Causes of CoVariationNIDA WorkshopVIPBG, October 2010Monday, October 18, 2010•Univariate Analysis: What are the contributions of additive genetic, dominance/shared environmental and unique environmental factors to the variance?•Bi/Multivariate Analysis: What are the contributions of genetic and environmental factors to the covariance between two traits? What makes sets of variables correlate or co-vary, comorbid?QuestionsMonday, October 18, 2010Two Traits•Same Genes•Different Genes•Partially Different Genes•Same Environment•Different Environment•Partially Different Env.Monday, October 18, 2010Bivariate Twin DataMonday, October 18, 2010individual twinindividual twinwithinbetweentraitwithinwithin-twin within-trait co(variance)(cross-twin within-trait) covariancetraitbetween(within-twin cross-trait) covariancecross-twin cross-trait covarianceBivariate Twin DataMonday, October 18, 2010Bivariate Twin Covariance MatrixX1Y1X2Y2X1VX1CX1Y1CX1X2CX1Y2Y1CY1X1VY1CY1X2CY1Y2X2CX2X1CX2Y1VX2CX2Y2Y2CY2X1CY2Y1CY2X2VY2Monday, October 18, 2010Genetic CorrelationMonday, October 18, 2010Alternative RepresentationsMonday, October 18, 2010Cholesky DecompositionMonday, October 18, 2010Bivariate AE ModelMonday, October 18, 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 CovariancesMonday, October 18, 2010Bivariate•Saturated Model•equality of means/variances•Genetic Models (ACE)•bivariate -> Cholesky DecompositionMonday, October 18, 2010Multivariate•Saturated Model•equality of means/variances•Genetic Models (ACE)•multivariate -> Cholesky Decomposition•Independent Pathway •Common PathwayMonday, October 18, 2010Example Dataset•VATSPSUD•PI: K. Kendler•Adult Twins from VTR •FF3, MMMF2•smoking, Fagerstrom Questionnaire itemsMonday, October 18, 2010Smoking Indicators•Fagerstrom Questionnaire for Nicotine Dependence•Number of cigarettes smoked•How soon after waking do you smoke your first cigarette?•First cigarette most satisfying•Smoked more in morningMonday, October 18, 2010Example Dataset•CVT•PI: R. Schieken•Adolescent Twins from Virginia •11 year olds•body compositionMonday, October 18, 2010Body Composition•Skinfolds•Biceps, Triceps, Calf•Suprailiacal, Subscapular•BMI•Weight/ Height^2Monday, October 18, 2010Saturated ModelMonday, October 18, 2010Test Assumptions•Equality of means/thresholds across twin order and zygosity•Equality of variances across twin order and zygosityMonday, October 18, 2010Cholesky1μ2μ1P21F2P1f211F1f11P41F4P3F31μ4μ3f31f41TextText F1 F2 F3 F4P1 f11P2 f21P3 f31P4 f41Monday, October 18, 20101μ2μ1P21F2P11F1P41F4P3F31μ4μ3f42f32f22 F1 F2 F3 F4P1 f11 0P2 f21 f22P3 f31 f32P4 f41 f42Monday, October 18, 20101μ2μ1P21F2P11F1P41F4P3f43f33F31μ4μ3 F1 F2 F3 F4P1 f11 0 0P2 f21 f22 0P3 f31 f32 f33P4 f41 f42 f43Monday, October 18, 20101μ2μ1P21F2P11F1P41F4f44P3F31μ4μ3 F1 F2 F3 F4P1 f11 0 0 0P2 f21 f22 0 0P3 f31 f32 f33 0P4 f41 f42 f43 f44Monday, October 18, 2010Phenotypic1μ2μ1P21F2P1f211F1f11P41F4f44P3f43f33F31μ4μ3f31f42f41f32f22Monday, October 18, 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 covariance matrix, fully saturatedMonday, October 18, 2010Genetic1μ2μ1P21A2P1a211A1a11P41A4a44P3a43a33A31μ4μ3a31a42a41a32a22Monday, October 18, 2010& Environmental1μ2μ1P21A2P1a211A1a11P41A4a44P3a43a33A31μ4μ3a31a42a41a32a22P21C2P1c211C1c11P41C4c44P3c43c33C31c31c42c41c32c22P21E2P1e211E1e11P41E4e44P3e43e33E31e31e42e41e32e22Monday, October 18, 2010Triple CholeskyMonday, October 18, 2010model: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]MultiTwinRawOrd.RMonday, October 18, 2010Monday, October 18, 2010Latent Phenotype1μ2μ1P2es2P11Es1es1Es21P4P31Es3es3Es41μ4μ3es4LPfl2fl4fl1fl3Monday, October 18, 2010ACE Model1μ2μ1P2es2P11Es1es1Es21P4P31Es3es3Es41μ4μ3es4LPfl21Al1al1El11fl4el1fl1fl31Cl1cl1Monday, October 18, 2010F1 P1 f11 P2 f21 P3 f31 P4 f41 F1 F2 F3 F4P1 f11 f21 f31 f41%*% al11 %*% t(al11) %*%fl %*% al %*% t(al) %*% t(fl)fl %&% al %*% t(al)Monday, October 18, 2010Partition Residuals1μ2μ1P2es2P11Es1es1Es21P4P31Es3es3Es41μ4μ3es41As11Cs1cs1as1LPfl21Al1al1El11fl4el1fl1fl31Cl1cl1Monday, October 18, 2010Monday, October 18, 2010Common PathwayMonday, October 18, 2010model:ACE, matrix:al [,1][1,] [NA]model:ACE, matrix:cl [,1][1,] [NA]model:ACE, matrix:el [,1][1,] [NA]model:ACE, matrix:fl [,1] [1,] [fl11][2,] [fl21][3,] [fl31][4,] [fl41][5,] [fl51]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]
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