Longitudinal AnalysisLongitudinal QuestionsPhenotypic CholeskyCholesky DecompositionCommon Factor + ResidualsIn Matrix NotationSimplex ISimplex IIReciprocal CausationModel for X variablesSlide 11Simplex ExpressionSimplex ModelSimplex Model ISimplex Model IISimplex Model IIIPractical ExampleSummaryLongitudinal AnalysisHGEN619 class 2006HGEN61910/20/03Longitudinal QuestionsMultivariate Analysis: What are the contributions of genetic and environmental factors to the covariance between more than two traits?Longitudinal Analysis: What are the contributions of genetic and environmental factors to the covariance between a trait measured on several occasions?Phenotypic CholeskyF1 F4F3F2P1P4P3P2f11f41f31f21f22f33f42f32f43f4400000 0F1 F2P1t11 1P2t1F31P3t1F41P4t1Cholesky Decompositionf11f41f31f21f22f33f42f32f43f44000000F1 F4F3F2P1P4P3P2f11f41f31f21f22f33f42f32f43f4400000 0*F F'*Common Factor + Residualsf11P1t1P2t1f21P3t1P4t1f31f41F11E11E41E31E21e11e44e33e22E1 E4E3E2P1P4P3P2e11e22e33e4400000 0000000EF1P1P4P3P2f11f41f31f21FIn Matrix NotationE1 E4E3E2P1P4P3P2e11e22e33e4400000 0000000*e11e22e33e4400000 0000000*E E'F1P1P4P3P2f11f41f31f21*f11f21f31f41F*F'Simplex IP1t1P2t1b21P3t1P4t1b31b41E11E41E31E21e11e44e33e22E1 E4E3E2P1P4P3P2e11e22e33e4400000 0000000EP1 P4P3P2P1P4P3P2 b21b32b4300000 0000000B0Simplex IIf11P1t1P2t1b21P3t1P4t1b31b41F11E41E31E21e44e33e22E1 E4E3E2P1P4P3P2e11e22e33e4400000 0000000EP1 P4P3P2P1P4P3P2 b21b32b4300000 0000000B0Reciprocal CausationxzASImultivariate path diagramx variables are caused by a set of independent variables, z x variables may cause each other, hence the unidirectional arrow from x to itselfx = Ax + Iz = Ax + zModel for X variables x - Ax = z(I - A)x = z(I - A)-1(I - A)x = (I - A)-1z x = (I - A)-1zxx' = (I - A)-1z ((I - A)-1z)' = (I - A)-1zz'(I - A)-1'= (I - A)-1S(I - A)-1'In Matrix Notation-111100000 0000000-I~()P1 P4P3P2P1P4P3P2 b21b32b4300000 0000000B0Y=Simplex ExpressionE1 E4E3E2e11e22e33e4400000 0000000*e11e22e33e4400000 0000000*E E'Y & E* E'Y * * Y'Simplex ModelC11 1 1 1 111 1[X][Y][Z][T][P]P1t1P2t1z21P3t1P4t1z31z41E1 E4E3E21 111P1t1P2t1P3t1P4t1A1 A2 A3 A4 A1A2A3 A4[O]E11E41E31E21e11e44e33e221.0 / 0.5 1.0 / 0.5 1.0 / 0.51A11A21A31A41A11A21A31A4a11a44a33a22x21x31x41a11a44a33a22z21z31z41E1 E4E3E21 111E11E41E31E21e11e44e33e22x21x31x411.0 / 0.5Simplex Model IG1: female model parametersCalculation Begin Matrices; X Diag nvarm1 nvarm1 Free ! additive genetic transmission paths Y Diag nvarm1 nvarm1 Free ! common environmental transmission paths W Diag nvarm1 nvarm1 Fixed! dominance transmission paths Z Diag nvarm1 nvarm1 Free ! unique environmental transmission paths I Iden nvar nvar ! identity matrix F Zero nvarm1 1 ! right column of zeros to fill out trans U Zero 1 nvar ! top row of zeros to fill out trans O Diag nvar nvar Free ! genetic specifics P Diag nvar nvar Free ! common environmental specifics R Diag nvar nvar Fixed ! dominance specifics T Diag nvar nvar Free ! specific environmental specifics H Full 1 1 Q Full 1 1 End Matrices; Matrix H .5 Matrix Q .25Simplex Model IIBegin Algebra; G= (I-(U_X|F))~; K= (I-(U_Y|F))~; L= (I-(U_Z|F))~; J= (I-(U_W|F))~; A= G*O*O'*G'; C= K*P*P'*K'; E= L*T*T'*L'; D= J*R*R'*J'; End Algebra; Start .4 All Start .5 T 1 1 1 T 1 2 2 T 1 3 3 T 1 4 4 T 1 5 5 Start 0 X 1 1 1 - X 1 nvarm1 nvarm1 Y 1 1 1 - Y 1 nvarm1 nvarm1 Start 0 W 1 1 1 - W 1 nvarm1 nvarm1 Z 1 1 1 - Z 1 nvarm1 nvarm1EndSimplex Model IIIG4Calculation Matrices = Group 1 I Iden nvar nvar End Matrices; Begin Algebra; V= A+C+D+E ; S= \sqrt(I.(V)) ; M= \d2v(S~*O)_ \d2v(S~*P)_ \d2v(S~*R)_ \d2v(S~*T)_ S~*(U_X|F)_ S~*(U_Y|F)_ S~*(U_W|F)_ S~*(U_Z|F); End Algebra;EndPractical ExampleDataset: US-CVT StudyBMIvar1 = 11yvar2 = 12.5yvar3 = 14yvar4 = 15.5yvar5 = 17yN MZF: 108, DZF: 63SummaryCholesky DecompositionSaturatedIndependent Pathway ModelCommon FactorsResidualsSimplex
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