1. Multivariate Genetic Analysis (Introduction) Frühling Rijsdijk Wednesday March 8, 2006

2. Multivariate Twin Analyses l Goal: to understand what factors make sets of variables correlate or co-vary l Two or more traits can be correlated because they share common genetic or common environmental influences (C or E) l With twin data on multiple traits it’s possible to partition the covariation into it’s genetic and environmental components

3. Example 1 How can we explain the association? - Additive Genetic effects (r g ) - Shared environment (r c ) - Non-shared environment (r e ) Kuntsi et al. Neuropsychiatric Genetics, Interested in reason for covariance / correlation between phenotypes, e.g. IQ and ADHD

4. Observed Cov matrices: 4×4 Twin1 p1 p2 Within-Twin Covariances Var P1 Cov P1- P2 Var P2 Twin2 p1 p2 Within-Twin CovariancesCross-Twin Covariances Within P1 Cross TraitsWithin P2 Twin1 p1 p2 Twin2 p1 p2 Cov P1- P2 Var P2 Var P1 Cross Traits

5. Twin 1 Phenotype 1 Twin 1 Phenotype 2 A1 A2 C1 C2 E1 E2 a 11 a 22 a 21 c 11 c 22 c 21 e 11 e 21 e 22

6. Twin 1 Phenotype 1 Twin 1 Phenotype 2 A1 A2 C1 C2 E1 E2 a 11 a 22 a 21 c 11 c 22 c 21 e 11 e 21 e 22 Twin 2 Phenotype 1 Twin 2 Phenotype 2 A1 A2 C1 C2 E1 E2 a 11 a 22 a 21 c 11 c 22 c 21 e 11 e 21 e 22 mz=1; dz=.5 1 1

7. Cholesky Decomposition: Path Tracing

8. Within-Twin Covariances Twin1 P2 Twin1 p1 p2 A 1 A 2 a 11 P1 1 P2 1 a 22 a 21 P1 a 11 2 a 22 2 + a 21 2 a 11 a 21

9. Within-Twin Covariances Twin1 P2 Twin1 p1 p2 P1 C 1 C 2 c 11 P1 1 P2 1 c 22 c 21 a 11 2 a 22 2 + a 21 2 a 11 a 21 + c 11 2 + c 22 2 + c 21 2 + c 11 c 21

10. Within-Twin Covariances Twin1 P2 a 11 2 a 22 2 + a 21 2 Twin1 p1 p2 a 11 a 21 P1 P1 1 P2 1 E 1 E 2 e 11 e 21 e 22 + c 11 2 + e 11 2 + e 11 e 21 + c 11 c 21 + e 22 2 + e 21 2 + c 22 2 + c 21 2

11. Cross-Twin Covariances Twin2 P2 1/.5a 11 2 1/.5a 22 2 + 1/.5a 21 2 Twin1 p1 p2 1/.5a 11 a 21 P1 1/.5 A 1 A 2 a 11 P1 1 P2 1 a 22 a 21 A 1 A 2 a 11 P1 2 P2 2 a 22 a 21

12. Cross-Twin Covariances Twin2 P2 1/.5a 11 2 1/.5a 22 2 + 1/.5a 21 2 Twin1 p1 p2 1/.5a 11 a 21 P1 + c 11 2 + c 11 c 21 + c 22 2 + c 21 2 11 C 1 C 2 c 11 P1 1 P2 1 c 22 c 21 C 1 C 2 c 11 P1 2 P2 2 c 22 c 21

13. Predicted Model Twin1 p1 p2 Within-Twin Covariances Var P1 Cov P1- P2 Var P2 Twin2 p1 p2 Within-Twin CovariancesCross-Twin Covariances Within P1 Cross TraitsWithin P2 Twin1 p1 p2 Twin2 p1 p2 Cov P1- P2 Var P2 Var P1 Var of P1 and P2 same across twins and zygosity groups

14. Predicted Model Twin1 p1 p2 Within-Twin Covariances Var P1 Cov P1- P2 Var P2 Twin2 p1 p2 Within-Twin CovariancesCross-Twin Covariances Within Trait 1 Cross TraitsWithin Trait 2 Twin1 p1 p2 Twin2 p1 p2 Cov P1- P2 Var P2 Var P1 Cov P1 - P2 same across twins and zygosity groups

15. Predicted Model Twin1 p1 p2 Within-Twin Covariances Var P1 Cov P1- P2 Var P2 Twin2 p1 p2 Within-Twin CovariancesCross-Twin Covariances Within P1 Cross TraitsWithin P2 Twin1 p1 p2 Twin2 p1 p2 Cov P1- P2 Var P2 Var P1 Cross Twin Cov within each trait, different for MZ and DZ

16. Predicted Model Twin1 p1 p2 Within-Twin Covariances Var P1 Cov P1- P2 Var P2 Twin2 p1 p2 Within-Twin CovariancesCross-Twin Covariances Within P1 Cross TraitsWithin P2 Twin1 p1 p2 Twin2 p1 p2 Cov P1- P2 Var P2 Var P1 Cross Twin - Cross trait, different for MZ and DZ

17. Within-Twin Covariances.30 1 1 Twin1 p1 p2 Twin2 p1 p2 Twin 1 p2 p1 Twin 2 p2 p1.59. 79.50.49. 29 1 1 MZ Within-Twin CovariancesCross-Twin Covariances Within-Twin Covariances.30 1 1 Twin1 p1 p2 Twin2 p1 p2 Twin 1 p2 p1 Twin 2 p2 p1.43.39.24.25. 31 1 1 DZ Within-Twin CovariancesCross-Twin Covariances

18. Within-Twin Covariances.30 1 1 Twin1 p1 p2 Twin2 p1 p2 Twin 1 p2 p1 Twin 2 p2 p1.59. 79.25.24. 29 1 1 MZ Within-Twin CovariancesCross-Twin Covariances Within-Twin Covariances.30 1 1 Twin1 p1 p2 Twin2 p1 p2 Twin 1 p2 p1 Twin 2 p2 p1.43.39.24.23. 31 1 1 DZ Within-Twin CovariancesCross-Twin Covariances

19. Within-Twin Covariances.30 1 1 Twin1 p1 p2 Twin2 p1 p2 Twin 1 p2 p1 Twin 2 p2 p1.59. 79.01. 29 1 1 MZ Within-Twin CovariancesCross-Twin Covariances Within-Twin Covariances.30 1 1 Twin1 p1 p2 Twin2 p1 p2 Twin 1 p2 p1 Twin 2 p2 p1.43.39.01. 31 1 1 DZ Within-Twin CovariancesCross-Twin Covariances

20. Summary Within-individual cross-trait covariance implies common etiological influences Cross-twin cross-trait covariance implies that these common etiological influences are familial Whether these common familial influences are genetic or environmental, is reflected in the MZ/DZ ratio of the cross-twin cross-traits covariances

21. Cholesky Decomposition: Specification in Mx

22. Mx: Parameter Matrices #define nvar 2 Begin Matrices; X lower nvar nvar free! Genetic coefficients Y lower nvar nvar free! C coefficients Z lower nvar nvar free! E coefficients G Full 1 nvar free! means End Matrices; Begin Algebra; A=X*X’;! Gen var/cov C=Y*Y’;! C var/cov E=Z*Z’; ! E var/cov P=A+C+E End Algebra;

23. Within-Twin Covariances Path Tracing * or ‘Star’ Matrix Multiplication P1P2 a 22 a 11 A 1 A 2 a 21 X LOWER 2  2 P1 P2 A1 A2 a 11 0 a 21 a 22

24.  P =  A +  C +  E By rule of matrix addition:

25. Within-Traits (diagonals): P 11 -P 12 =.5 a 11 2 P 21 -P 22 =.5 a 22 2 +.5 a 21 2 Cross-Traits: P 11 -P 22 =.5 a 11 a 21 P 21 -P 12 =.5 a 21 a 11 Cross-Twins Covariances, Genetic effects (DZ).5 A 1 A 2 a 11 P11P21 a 22 a 21 A 1 A 2 a 11 P12P22 a 22 a 21 Twin 1Twin 2 Path Tracing Kronecker Product  @

26. Cross-Twins Covariances, Genetic effects (MZ) 11 A 1 A 2 a 11 P11P21 a 22 a 21 A 1 A 2 a 11 P12P22 a 22 a 21 Twin 1Twin 2

27. Cross-Twins Covariances, C effects (MZ and DZ) 11 C 1 C 2 c 11 P11P21 c 22 c 21 C 1 C 2 c 11 P12P22 c 22 c 21 Twin 1Twin 2

28. Covariance Model for Twin pairs CovarianceA+C+E| A+C _ A+C| A+C+E / CovarianceA+C+E| H@A+C _ H@A+C| A+C+E / MZ DZ

29. Standardized Estimates

30. Correlated Factors Solution

31. Covariances to Correlations In matrix form:

32. Genetic Correlations

33. Specification in Mx Matrix Function in Mx: R = \stnd (A); R = \sqrt( I. A )˜ * A * \sqrt( I. A )˜; Where I is an Identity matrix : 1 2 1 0 0 1  2 A11 0 0  2 A22 And I.A =

34. Interpretation §High Genetic correlation: large overlap in genetic effects on the two traits §Does it mean that the phenotypic correlation between the traits is largely due to genetic effects? l No, the substantive importance of a particular r g depends both on the value of the correlation and the value of the A paths, i.e. the heritabilities of both traits.

35. A 1 A 2 h2h2 P1P2 h2h2 rgrg (  h 2 p1 * r g *  h 2 p2 ) / r PH ( .63 * -0.525 * .33) / -0.29 =.8357

36. Interpretation For Example, consider a phenotypic correlation of 0.40 With  h 2 p1 =.7 and  h 2 p2 =.6 with rg =.3.19 (49%) of the phenotypic correlation can be attributed to additive genetic effects. With  h 2 p1 =.2 and  h 2 p2 =.4 with rg =.8.20 (49%) of the phenotypic correlation can be attributed to additive genetic effects. Weakly heritable traits can still have a large portion of their correlation attributable to genetic effects.

37. More variables…… Twin 1 Phenotype 1 Twin 1 Phenotype 2 A1 A2 C1 C2 E1 E2 a 11 a 22 a 21 c 11 c 22 c 21 e 11 e 21 e 22 Twin 1 Phenotype 3 A3C3 a 33 E3 e 33 c 33 c 31 a 23 c 23 a 31 e 31 e 23

38. More variables…… Twin 1 Phenotype 1 Twin 1 Phenotype 2 A1A1 A2A2 C1C1 C2C2 E1 E2 a 11 a 22 a 21 c 11 c 22 c 21 e 11 e 21 e 22 Twin 1 Phenotype 3 A3A3 C3C3 a 33 E3 e 33 c 33 c 31 a 23 c 23 a 31 e 31 e 23 e 22 Twin 2 Phenotype 1 Twin 2 Phenotype 2 A1A1 A2A2 C1C1 C2C2 E1 E2 a 11 a 22 a 21 c 11 c 22 c 21 e 11 e 21 e 22 Twin 2 Phenotype 3 A3A3 C3C3 a 33 E3 e 33 c 33 c 31 a 23 c 23 a 31 e 31 e 23 e 22

39. Mx: Parameter Matrices #define nvar 3 Begin Matrices; X lower nvar nvar free! Genetic coefficients Y lower nvar nvar free! C coefficients Z lower nvar nvar free! E coefficients G Full 1 nvar free! means End Matrices; Begin Algebra; A=X*X’;! Gen var/cov C=Y*Y’;! C var/cov E=Z*Z’; ! E var/cov P=A+C+E End Algebra;

40. X LOWER 3  3 A1 A2 A3 a 11 0 0 a 21 a 22 0 a 31 a 32 a 33 Y LOWER 3  3 C1 C2 C3 c 11 0 0 c 21 c 22 0 c 31 c 32 c 33 Z LOWER 3  3 E1 E2 E3 e 11 0 0 e 21 e 22 0 e 31 e 32 e 33