1Multivariate AnalysisHGEN619 class 2005Multivariate Questions In Bivariate Analysis: What are the contributions of genetic and environmental factors to the covariance between two traits?n 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 1P2t1F31P3t1F4f441P4t12Cholesky Decompositionf11f41f31f21f22f33f42f32f43f44000000F1 F4F3F2P1P4P3P2f11f41f31f21f22f33f42f32f43f4400000 0*F F'*Saturated Modeln Use Cholesky decomposition to estimate covariance matrixn Fully saturatedn Model: Cov P = F*F’¨F: Full nvar nvarPhenotypic Single FactorF1P1P4P3P2f11f41f31f21*f11f21f31f41F*F'F1f11P1t11P2t1f21P3t1P4t1f31f41Residual VariancesE1 E4E3E2P1P4P3P2e11e22e33e4400000 0000000*e11e22e33e4400000 0000000*E E'f11P1t1P2t1f21P3t1P4t1f31f41F11E11E41E31E21e11e44e33e22Factor Analysisn Explain covariance by limited number of factorsn Exploratory / Confirmatoryn Model: Cov P = F*F’ + E*E’¨F: Full nvar nfac¨E: Diag nvar nvarTwin Dataf11P1t1P2t1f21P3t1P4t1f31f41F11E11E41E31E21e11e44e33e22f11P1t2P2t2f21P3t2P4t2f31f41F11E11E41E31E21e11e44e33e22?3Genetic Single Factora11P1t1P2t1a21P3t1P4t1a31a41A11E11E41E31E21e11e44e33e22a11P1t2P2t2a21P3t2P4t2a31a41A11E11E41E31E21e11e44e33e221.0 / 0.5Single [Common] Factorn X: genetic¨ Full 4 x 1¨ Full nvar x nfacn Y: shared environmentaln 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 Factorsn T: geneticn U: shared environmentaln V: specific environmental¨ Diag 4 x 4¨ Diag nvar x nvarE1 E4E3E2P1P4P3P2e11e22e33e4400000 0000000*e11e22e33e4400000 0000000*V V'4Independent 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 Matrices#define nvar 6#define nfac 1[V]6 x 6[U]6 x 6[T]6 x 6Residual Factors[Z]6 x 1[Y]6 x 1[X]6 x 1Common Factorse2c2a2Variance ComponentIndependent Pathway In G1: Define matricesn Calculationn Begin Matrices;n X full nvar nfac Free ! common factor genetic path coefficientsn Y full nvar nfac Free ! common factor shared environment pathsn Z full nvar nfac Free ! common factor unique environment pathsn T diag nvar nvar Free ! variable specific genetic pathsn U diag nvar nvar Free ! variable specific shared env pathsn V diag nvar nvar Free ! variable specific residual pathsn M full 1 nvar Free ! meansn End Matrices;n Start …n Begin Algebra;n A= X*X' + T*T'; ! additive genetic variance componentsn C= Y*Y' + U*U'; ! shared environment variance componentsn E= Z*Z' + V*V'; ! nonshared environment variance componentsn End Algebra;n Endindpath.mxIndependent Pathway IIn G2: MZ twinsn #include iqnlmz.datn Begin Matrices = Group 1;n Means M | M ;n Covariance A+C+E | A+C _n A+C | A+C+E ;n Option Rsidualsn Endn G3: DZ twinsn #include iqnldz.datn Begin Matrices= Group 1;n H full 1 1n End Matrices;n Matrix H .5n Means M | M ;n Covariance A+C+E | H@A+C _n H@A+C | A+C+E ;n Option Rsidualsn EndIndependent Pathway IIIn G4: Calculate Standardised Solutionn Calculationn Matrices = Group 1n I Iden nvar nvarn End Matrices;n Begin Algebra;n R=A+C+E; ! total variancen S=(\sqrt(I.R ))~; ! diagonal matrix of standard deviationsn P=S*X_ S*Y_ S*Z; ! standardized estimates for common factorsn Q=S*T_ S*U_ S*V; ! standardized estimates for spec factorsn End Algebra;n Labels Row P a1 a2 a3 a4 a5 a6 c1 c2 c3 c4 c5 c6 e1 e2 e3 e4 e5 e6n Labels Col P var1 var2 var3 var4 var5 var6n Labels Row Q as1 as2 as3 as4 as5 as6 cs1 cs2 cs3 cs4 cs5 cs6 es1 es2 es3 es4 es5 es6n Labels Col Q var1 var2 var3 var4 var5 var6n Options NDecimals=4n EndIPn Independent pathways¨Biometric model¨Different covariance structure for A, C and E5Phenotypic Single FactorF1P1P4P3P2f11f41f31f21*f11f21f31f41F*F'F1f11P1t11P2t1f21P3t1P4t1f31f41Latent PhenotypeF1f11P1t1P2t1f21P3t1P4t1f31f41E1C1A1ecaTwin DataF1f11P1t1P2t1f21P3t1P4t1f31f41E1C1A1ecaF1f11P1t1P2t1f21P3t1P4t1f31f41E1C1A1eca1.0 / 0.5 1.0Factor on Latent PhenotypeF1P1P4P3P2f11f41f31f21*f11f21f31f41F*F'a a* *X X'* *F&X X'*( )=Common Pathway ModelF1f11P1t1P2t1f21P3t1P4t1f31f41E1C1A1ecaF1f11P1t1P2t1f21P3t1P4t1f31f41E1C1A1eca1.0 / 0.5 1.0C11E11A11E21A21E31A31E41A41E11A11E21A21E31A31E41A411.0 / 0.5 1.0 / 0.5 1.0 / 0.5 1.0 / 0.5[T][V][U][X][Y][Z][F]Path Diagram to Matrices#define nvar 6#define nfac 1[V]6 x 6[U]6 x 6[T]6 x 6Residual Factors[F]6 x 1[Z]1 x 1[Y]1 x 1[X]1 x 1Common Factore2c2a2Variance Component6Common Pathway Model In G1: Define matricesn Calculationn Begin Matrices;n X full nfac nfac Free ! latent factor genetic path coefficientn Y full nfac nfac Free ! latent factor shared environment pathn Z full nfac nfac Free ! latent factor unique environment pathn T diag nvar nvar Free ! variable specific genetic pathsn U diag nvar nvar Free ! variable specific shared env pathsn V diag nvar nvar Free ! variable specific residual pathsn F full nvar nfac Free ! loadings of variables on latent factorn I Iden 2 2n M full 1 nvar Free ! meansn End Matrices;n Start ..n Begin Algebra;n A= F&(X*X') + T*T'; ! genetic variance componentsn C= F&(Y*Y') + U*U'; ! shared environment variance componentsn E= F&(Z*Z') + V*V'; ! nonshared environment variance componentsn L= X*X' + Y*Y' + Z*Z'; ! variance of latent factorn End Algebra;n EndCommon Pathway IIn G4: Constrain variance of latent factor to 1n Constraintn Begin Matrices;n L computed =L1n I unit 1 1n End Matrices;n Constraint L = I ;n
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