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Multivariate Genetic Analysis

Multivariate Genetic Analysis. Boulder 2004. 1.00. 1.00. 1.00. 1.00. F1. F2. F3. F4. P1. P2. P3. P4. T1. T1. T1. T1. Phenotypic Cholesky. 1.00. 1.00. 1.00. 1.00. F1. F2. F3. F4. P1. P2. P3. P4. T1. T1. T1. T1. Phenotypic Cholesky. 1.00. 1.00. 1.00. 1.00. F1.

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Multivariate Genetic Analysis

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  1. Multivariate Genetic Analysis Boulder 2004

  2. 1.00 1.00 1.00 1.00 F1 F2 F3 F4 P1 P2 P3 P4 T1 T1 T1 T1 Phenotypic Cholesky

  3. 1.00 1.00 1.00 1.00 F1 F2 F3 F4 P1 P2 P3 P4 T1 T1 T1 T1 Phenotypic Cholesky

  4. 1.00 1.00 1.00 1.00 F1 F2 F3 F4 P1 P2 P3 P4 T1 T1 T1 T1 Phenotypic Cholesky

  5. 1.00 1.00 1.00 1.00 F1 F2 F3 F4 P1 P2 P3 P4 T1 T1 T1 T1 Phenotypic Cholesky

  6. Cholesky Decomposition

  7. 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

  8. 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;

  9. 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;

  10. 1.00 F1 P1 P2 P3 P4 T1 T1 T1 T1 Phenotypic Single Factor

  11. 1.00 F1 P1 P2 P3 P4 T1 T1 T1 T1 E1 E2 E3 E4 1.00 1.00 1.00 1.00 Residual Variances

  12. 1.00 ? 1.00 F1 F1 P1 P2 P3 P4 P1 P2 P3 P4 T1 T1 T1 T1 T2 T2 T2 T2 E1 E2 E3 E4 E1 E2 E3 E4 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 Twin Data

  13. 1.0 / 0.5 1.00 1.00 A1 A1 P1 P2 P3 P4 P1 P2 P3 P4 T1 T1 T1 T1 T2 T2 T2 T2 E1 E2 E3 E4 E1 E2 E3 E4 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 Genetic Single Factor

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

  15. 1.00 1.0 / 0.5 1.00 1.00 1.00 1.00 A1 C1 A1 C1 P1 P2 P3 P4 P1 P2 P3 P4 T1 T1 T1 T1 T2 T2 T2 T2 E1 E2 E3 E4 E1 E2 E3 E4 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 Common Environmental Single Factor

  16. 1.00 1.0 / 0.5 1.00 1.00 1.00 1.00 1.00 1.00 A1 C1 E1 A1 C1 E1 P1 P2 P3 P4 P1 P2 P3 P4 T1 T1 T1 T1 T2 T2 T2 T2 E1 E2 E3 E4 E1 E2 E3 E4 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 Specific Environmental Single Factor

  17. 1.00 1.0 / 0.5 1.00 1.00 1.00 1.00 1.00 1.00 A1 C1 E1 A1 C1 E1 P1 P2 P3 P4 P1 P2 P3 P4 T1 T1 T1 T1 T2 T2 T2 T2 A1 C1 E1 E2 E3 E4 E1 E2 E3 E4 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 Residuals partitioned in ACE

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

  19. 1.00 / 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 A C E A C E C C C C C C [Y] [X] [Z] P1 P2 P3 P4 P1 P2 P3 P4 T1 T1 T1 T1 T2 T2 T2 T2 [V] [U] [T] E1 E2 E3 E4 E1 E2 E3 E4 C1 A1 1.00 A2 1.00 A3 1.00 A4 1.00 A1 1.00 A2 1.00 A3 1.00 A4 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 / 0.50 1.00 / 0.50 1.00 / 0.50 1.00 / 0.50 Independent Pathway Model

  20. 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

  21. 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;

  22. 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

  23. 1.00 1.00 1.00 A1 A2 A3 CB GM WM RK CL AS1 AS2 AS3 AS4 AS5 1.00 1.00 1.00 1.00 1.00 Three Genetic Factors

  24. 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; Begin 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 End Algebra; \d2v : diagonal to vector

  25. Path Diagram to Matrices

  26. Path Diagram to Matrices

  27. 1.00 F1 P1 P2 P3 P4 T1 T1 T1 T1 Phenotypic Single Factor

  28. 1.00 1.0 / 0.5 1.00 1.00 1.00 1.00 1.00 1.00 A C E A C E F1 F1 P1 P2 P3 P4 P1 P2 P3 P4 T1 T1 T1 T1 T2 T2 T2 T2 Latent Phenotype

  29. Factor on Latent Phenotype

  30. 1.00 / 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 A C E A C E C C C C C C [X] [Y] [Z] constraint L L constraint [F] P1 P2 P3 P4 P1 P2 P3 P4 T1 T1 T1 T1 T2 T2 T2 T2 [V] [U] [T] E1 E2 E3 E4 E1 E2 E3 E4 C1 A1 1.00 A2 1.00 A3 1.00 A4 1.00 A1 1.00 A2 1.00 A3 1.00 A4 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 / 0.50 1.00 / 0.50 1.00 / 0.50 1.00 / 0.50 Common Pathway Model

  31. 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

  32. 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;

  33. 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; Begin 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 End Algebra;

  34. Path Diagram to Matrices

  35. Path Diagram to Matrices

  36. Summary • Independent 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

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