Multivariate Genetic Analysis Boulder 2004

Phenotypic Cholesky T1 P2 F1 F4 P1 P3 P4 F2 F3 1.00

Phenotypic Cholesky T1 P2 F1 F4 P1 P3 P4 F2 F3 1.00

Phenotypic Cholesky T1 P2 F1 F4 P1 P3 P4 F2 F3 1.00

Phenotypic Cholesky T1 P2 F1 F4 P1 P3 P4 F2 F3 1.00

Cholesky Decomposition

Cholesky Example Script: f:\hmaes\a17\chol.mx 3 MRI measures – 2 IQ subtests: var1 = cerebellum var2 = grey matter var3 = white matter var4 = calculation var5 = letters and numbers Data: f:\hmaes\a17\mri-iqfactor.rec T:\hmaes/a17\mri-iq-mz.dat T:\hmaes\a17\mri-iq-dz.dat

Cholesky #define nvar 5 Begin Matrices; X lower nvar nvar Free ! genetic structure Y lower nvar nvar Free ! shared environment structure Z lower nvar nvar Free ! specific environment structure M full 1 nvar Free ! grand means End Matrices; Begin Algebra; A= X*X'; ! additive genetic covariance C= Y*Y'; ! shared environment covariance E= Z*Z'; ! nonshared environment covariance End Algebra;

Standardization 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 End Algebra;

Phenotypic Single Factor 1.00

Residual Variances T1 P2 F1 P1 P3 P4 1.00 E1 E2 E3 E4

Twin Data T1 P2 F1 P1 P3 P4 T2 E1 E2 E3 E4 1.00 ?

Genetic Single Factor T1 P2 A1 P1 P3 P4 T2 E1 E2 E3 E4 1.00 1.0 / 0.5

Single [Common] Factor X: genetic Full 4 x 1 Full nvar x nfac Y: shared environmental Z: specific environmental

Common Environmental Single Factor P2 A1 P1 P3 P4 T2 E1 E2 E3 E4 1.00 1.0 / 0.5 C1

Specific Environmental Single Factor 1.00 1.0 / 0.5 C1

Residuals partitioned in ACE 1.00 1.0 / 0.5

Residual Factors T: genetic U: shared environmental V: specific environmental Diag 4 x 4 Diag nvar x nvar

Independent Pathway Model C A E P1 P3 P4 T2 A1 A2 A3 A4 E1 E2 E3 E4 C1 1.00 1.00 / 0.50 [X] [Y] [Z] [T] [V] [U]

Independent Pathway Example Script: f:\hmaes\a17\ind3f.mx 3 MRI measures – 2 IQ subtests: var1 = cerebellum var2 = grey matter var3 = white matter var4 = calculation var5 = letters and numbers Data: f:\hmaes\a17\mri-iqfactor.rec T:\hmaes/a17\mri-iq-mz.dat T:\hmaes\a17\mri-iq-dz.dat

Independent Pathway #define nvar 5 #define nfac 1 G1: Define matrices Calculation Begin Matrices; X full nvar 3 ! common factor genetic paths Y full nvar nfac ! common factor shared env paths Z full nvar nfac Free ! common factor specific env paths T diag nvar nvar Free ! variable specific genetic paths U diag nvar nvar ! variable specific shared env paths V diag nvar nvar Free ! variable specific residual paths M full 1 nvar Free ! grand means End Matrices;

Three Genetic Factors Specify X ! declare free parameters for 3 genetic common factors ! G1 G2 G3 100 110 0 101 111 0 102 112 0 103 0 120 104 0 121 Drop T 1 5 5 ! fix genetic specific for last variable for identification

Three Genetic Factors CB GM WM RK CL 1.00 A1 A2 A3 AS1 AS2 AS3 AS4 AS5

Independent II Begin Algebra; A= X*X' + T*T'; ! additive genetic covariance C= Y*Y' + U*U'; ! shared environment covariance E= Z*Z' + V*V'; ! specific environment covariance End Algebra; R=A+C+E; ! total variance S=(\sqrt(I.R))~; ! diagonal matrix of standard deviations P=S*X| S*Y| S*Z| \d2v(S*T)'| \d2v(S*U)'| \d2v(S*V)'; ! standardized estimates for common and specific factors \d2v : diagonal to vector

Path Diagram to Matrices Variance Component a2 c2 e2 Common Factors [X]full 5 x 1 [Y]full [Z]full Residual Factors [T]diag 5 x 5 [U]diag [V]diag #define nvar 5 #define nfac 1

Path Diagram to Matrices Variance Component a2 c2 e2 Common Factors [X]full 5 x 3 [Y] [Z]full 5 x 1 Residual Factors [T]diag 5 x 5 [U] [V]diag #define nvar 5 #define nfac 1

Phenotypic Single Factor 1.00

Latent Phenotype T1 P2 A P1 P3 P4 T2 C F1 1.00 1.0 / 0.5 E

Factor on Latent Phenotype

Common Pathway Model T1 P2 C A E P1 P3 P4 T2 A1 A2 A3 A4 E1 E2 E3 E4 L 1.00 1.00 / 0.50 [T] [V] constraint [X] [Y] [Z] [F] [U]

Common Pathway Example Script: f:\hmaes\a17\comp.mx 3 MRI measures – 2 IQ subtests: var1 = cerebellum var2 = grey matter var3 = white matter var4 = calculation var5 = letters and numbers Data: f:\hmaes\a17\mri-iqfactor.rec T:\hmaes/a17\mri-iq-mz.dat T:\hmaes\a17\mri-iq-dz.dat

Compath Pathway #define nvar 5 #define nfac 1 Begin Matrices; X full nfac nfac Free ! latent factor genetic paths Y full nfac nfac ! latent factor shared env paths Z full nfac nfac Free ! latent factor specific env paths T diag nvar nvar Free ! variable specific genetic paths U diag nvar nvar ! variable specific shared env paths V diag nvar nvar Free ! variable specific residual paths F full nvar nfac Free ! loadings on latent factor M full 1 nvar Free ! grand means End Matrices;

Compath II Begin Algebra; A= F&(X*X') + T*T'; ! genetic variance components C= F&(Y*Y') + U*U'; ! shared environment covariance E= F&(Z*Z') + V*V'; ! specific environment covariance L= X*X' + Y*Y' + Z*Z'; ! variance of latent factor End Algebra; R=A+C+E; ! total variance S=(\sqrt(D.R))~; ! diagonal matrix of standard deviations P=S*F| \d2v(S*T)'| \d2v(S*U)'| \d2v(S*V)'; ! standardized estimates for loadings and specific factors W= X*X' | Y*Y'| Z*Z'; ! standardized estimates for latent phenotype

Path Diagram to Matrices Variance Component a2 c2 e2 Common Factor [X]full 1 x 1 [Y]full [Z]full [F]full 5 x 1 Residual Factors [T]diag 5 x 5 [U]diag [V]diag #define nvar 5 #define nfac 1

Path Diagram to Matrices Variance Component a2 c2 e2 Common Factor [X]full 1 x 1 [Y] [Z]full [F]full 5 x 1 Residual Factors [T]diag 5 x 5 [U] [V]diag #define nvar 5 #define nfac 1

Summary Independent Pathway Model Common Pathway Model Biometric Factors Model Loadings differ for genetic and environmental common factors Common Pathway Model Psychometric Factors Model Loadings equal for genetic and environmental common factor