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Multivariate AnalysisMultivariate Questions IPhenotypic Cholesky F1Phenotypic Cholesky F2Phenotypic CholeskySlide 6Cholesky DecompositionSaturated ModelPhenotypic Single FactorResidual VariancesFactor AnalysisTwin DataGenetic Single FactorSingle [Common] FactorCommon Environmental Single FactorSpecific Environmental Single FactorResiduals partitioned in ACEResidual FactorsIndependent Pathway ModelPath Diagram to MatricesIndependent Pathway IIndependent Pathway IIIndependent Pathway IIIIPSlide 25Latent PhenotypeSlide 27Factor on Latent PhenotypeCommon Pathway ModelSlide 30Common Pathway Model ICommon Pathway IICPPractical ExampleSlide 35WAIS-III IQPathway ModelTwo Common Pathway ModelTwo Independent CP ModelTwo Reduced Indep CP ModelCommon Pathway ModelIndependent Pathway ModelMultivariate AnalysisHGEN619 class 2006HGEN61910/20/03Multivariate Questions IBivariate Analysis: What are the contributions of genetic and environmental factors to the covariance between two traits?Multivariate Analysis: What are the contributions of genetic and environmental factors to the covariance between more than two traits?Phenotypic Cholesky F1F1 F4F3F2P1P4P3P2f11f41f31f21F1 F2f11P1t11 1P2t1f21F31P3t1F41P4t1f31f41Phenotypic Cholesky F2F1 F4F3F2P1P4P3P2f11f41f31f21f22f42f320F1 F2f22P1t11 1P2t1F31P3t1F41P4t1f32f42Phenotypic CholeskyF1 F4F3F2P1P4P3P2f11f41f31f21f22f33f42f32f43000F1 F2P1t11 1P2t1F3f331P3t1F41P4t1f43Phenotypic CholeskyF1 F4F3F2P1P4P3P2f11f41f31f21f22f33f42f32f43f4400000 0F1 F2P1t11 1P2t1F31P3t1F4f441P4t1Cholesky Decompositionf11f41f31f21f22f33f42f32f43f44000000F1 F4F3F2P1P4P3P2f11f41f31f21f22f33f42f32f43f4400000 0*F F'*Saturated ModelUse Cholesky decomposition to estimate covariance matrixFully saturatedModel: Cov P = F*F’F: Full nvar nvarPhenotypic Single FactorF1P1P4P3P2f11f41f31f21*f11f21f31f41F*F'F1f11P1t11P2t1f21P3t1P4t1f31f41Residual VariancesE1 E4E3E2P1P4P3P2e11e22e33e4400000 0000000*e11e22e33e4400000 0000000*E E'f11P1t1P2t1f21P3t1P4t1f31f41F11E11E41E31E21e11e44e33e22Factor AnalysisExplain covariance by limited number of factorsExploratory / ConfirmatoryModel: Cov P = F*F’ + E*E’F: Full nvar nfacE: Diag nvar nvarTwin Dataf11P1t1P2t1f21P3t1P4t1f31f41F11E11E41E31E21e11e44e33e22f11P1t2P2t2f21P3t2P4t2f31f41F11E11E41E31E21e11e44e33e22?Genetic Single Factora11P1t1P2t1a21P3t1P4t1a31a41A11E11E41E31E21e11e44e33e22a11P1t2P2t2a21P3t2P4t2a31a41A11E11E41E31E21e11e44e33e221.0 / 0.5Single [Common] FactorX: geneticFull 4 x 1Full nvar x nfacY: shared environmentalZ: specific environmentalA1P1P4P3P2a11a41a31a21*a11a21a31a41X*X'Common Environmental Single FactorP1t1P2t1P3t1P4t1A11E11E41E31E21e11e44e33e22P1t2P2t2P3t2P4t2A11E11E41E31E21e11e44e33e221.0 / 0.5c11c21c31c41C11c11c21c31c41C111.0Specific Environmental Single FactorP1t1P2t1P3t1P4t1A11E11E41E31E21e11e44e33e22P1t2P2t2P3t2P4t2A11E11E41E31E21e11e44e33e221.0 / 0.5C11C111.0e11e21e31e41E11e11e21e31e41E11Residuals partitioned in ACEP1t1P2t1P3t1P4t1A11E11E41E31E21e11e44e33e22P1t2P2t2P3t2P4t2A11E11E41E31E21e11e44e33e221.0 / 0.5C11C111.0E11E11C11c11A11a11Residual FactorsT: geneticU: shared environmentalV: specific environmentalDiag 4 x 4Diag nvar x nvarE1 E4E3E2P1P4P3P2e11e22e33e4400000 0000000*e11e22e33e4400000 0000000*V V'Independent Pathway ModelP1t1P2t1P3t1P4t1AC1P1t2P2t2P3t2P4t2AC11.0 / 0.5CC1CC11.0EC1EC1C11E11A11E21A21E31A31E41A41E11A11E21A21E31A31E41A411.0 / 0.5 1.0 / 0.5 1.0 / 0.5 1.0 / 0.5[X] [Y] [Z][T][V][U]Path Diagram to MatricesVariance Componenta2c2e2Common Factors[X]6 x 1[Y]6 x 1[Z]6 x 1Residual Factors[T]6 x 6[U]6 x 6[V]6 x 6#define nvar 6#define nfac 1Independent Pathway IG1: Define matrices Calculation Begin Matrices; X full nvar nfac Free ! common factor genetic path coefficients Y full nvar nfac Free ! common factor shared environment paths Z full nvar nfac Free ! common factor unique environment paths T diag nvar nvar Free ! variable specific genetic paths U diag nvar nvar Free ! variable specific shared env paths V diag nvar nvar Free ! variable specific residual paths M full 1 nvar Free ! means End Matrices; Start … Begin Algebra; A= X*X' + T*T'; ! additive genetic variance components C= Y*Y' + U*U'; ! shared environment variance components E= Z*Z' + V*V'; ! nonshared environment variance components End Algebra;Endindpath.mxIndependent Pathway IIG2: MZ twins#include iqnlmz.dat Begin Matrices = Group 1; Means M | M ; Covariance A+C+E | A+C _ A+C | A+C+E ; Option RsidualsEndG3: DZ twins#include iqnldz.dat Begin Matrices= Group 1; H full 1 1 End Matrices; Matrix H .5 Means M | M ; Covariance A+C+E | H@A+C _ H@A+C | A+C+E ; Option RsidualsEndIndependent Pathway IIIG4: Calculate Standardised Solution Calculation Matrices = Group 1 I Iden nvar nvar End Matrices; Begin Algebra; R=A+C+E; ! total variance S=(\sqrt(I.R))~; ! diagonal matrix of standard deviations P=S*X_ S*Y_ S*Z; ! standardized estimates for common factors Q=S*T_ S*U_ S*V; ! standardized estimates for spec factors End Algebra; Labels Row P a1 a2 a3 a4 a5 a6 c1 c2 c3 c4 c5 c6 e1 e2 e3 e4 e5 e6 Labels Col P var1 var2 var3 var4 var5 var6 Labels Row Q as1 as2 as3 as4 as5 as6 cs1 cs2 cs3 cs4 cs5 cs6 es1 es2 es3 es4 es5 es6 Labels Col Q var1 var2 var3 var4 var5 var6 Options NDecimals=4EndIPIndependent pathwaysBiometric modelDifferent covariance structure for A, C and EPhenotypic Single FactorF1P1P4P3P2f11f41f31f21*f11f21f31f41F*F'F1f11P1t11P2t1f21P3t1P4t1f31f41Latent PhenotypeF1f11P1t1P2t1f21P3t1P4t1f31f41E1C1A1ecaTwin DataF1f11P1t1P2t1f21P3t1P4t1f31f41E1C1A1ecaF1f11P1t1P2t1f21P3t1P4t1f31f41E1C1A1eca1.0 / 0.51.0Factor on Latent PhenotypeF1P1P4P3P2f11f41f31f21*f11f21f31f41F*F 'aa* *X X'* *F&X X'*( )=Common Pathway ModelF1f11P1t1P2t1f21P3t1P4t1f31f41E1C1A1ecaF1f11P1t1P2t1f21P3t1P4t1f31f41E1C1A1eca1.0 / 0.51.0C11E11A11E21A21E31A31E41A41E11A11E21A21E31A31E41A411.0 / 0.5 1.0 / 0.5 1.0 / 0.5 1.0 / 0.5[T][V][U][X][Y][Z][F]Path Diagram to MatricesVariance Componenta2c2e2Common Factor[X]1 x 1[Y]1 x 1[Z]1 x 1[F]6 x 1Residual Factors[T]6 x 6[U]6 x 6[V]6 x 6#define nvar 6#define nfac 1Common Pathway Model IG1: Define matrices Calculation Begin Matrices; X full nfac nfac Free ! latent factor genetic path coefficient Y full nfac nfac Free ! latent factor shared


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VCU HGEN 619 - Multivariate Analysis

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