Multivariate Genetic Analysis: Introduction

1. Multivariate Genetic Analysis:Introduction Frühling Rijsdijk & Shaun Purcell Twin Workshop, Boulder Wednesday March 3, 2004

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

3. Univariate ACE Model for a Twin Pair 1 1/.5 A P x C E A E C A X LOW 1  1 C P y x z y x y z P1 P2 Y LOW 1  1 E P z 2  2 Z LOW 1  1 2  2

4. 1 1 1/.5 1/.5 C2 C1 A1 A2 C2 C1 A1 A2 y11 x11 y21 x22 y22 x21 y11 x11 y21 x22 y22 x21 P11 P21 P12 P22 z11 z22 z21 z11 z22 z21 E1 E2 E1 E2 E1 A1 C1 C2 A2 E2 z x y P P P 1 1 1 0 0 0 11 11 11 y z x z y x P2 P2 P2 21 21 21 22 22 22 X LOWER 2  2 Y LOWER 2  2 Z LOWER 2  2

5. Twin1 Twin2 p1 p2 p1 p2 Within-Twin Covariances Cross-Twin Covariances Twin1 p1 Within Trait 1 Var P1 Cross Traits Cross Traits Within Trait 2 Within Trait 2 Var P2 p2 Cov P1-P2 Cross-Twin Covariances Within-Twin Covariances Twin2 p1 Var P1 Within Trait 1 p2 Var P2 Cov P1-P2 4  4

6. 1/.5 1 1 1/.5 C2 C1 A1 A2 C2 C1 A1 A2 y11 x11 y21 x22 y22 x21 y11 x11 y21 x22 x21 P11 P21 P12 P22 z11 z22 z21 z11 z22 z21 E1 E2 E1 E2 p1 Twin 1 p2 y22 4  4 Twin1 Twin2 p1 p2 p1 p2 Within-Twin Covariances Cross-Twin Covariances x112+ y112 + z112 1/.5*x112 + 1/1 *y112 x21*x11+ y21*y11 + z21*z11 x222+ x212+ y222+ y212 + z222 +z212 1/.5*x21* x11 + 1/1 *y212 *y11 1/.5*x222+1/.5*x212 + 1/1*y222+1/1*y212 Rmz:Rdz will indicate whether A, C or E determine Rp1-p2 p1 Twin 2 p2

7. Twin1 MZ p1 p2 Within-Twin Covariances 1 p1 Twin 1 1 .30 p2 Cross-Twin Covariances Within-Twin Covariances . 79 .49 p1 1 Twin 1 .50 .59 . 29 1 p2 Twin1 DZ p1 p2 Within-Twin Covariances 1 p1 Twin 1 1 .30 p2 Cross-Twin Covariances Within-Twin Covariances .39 .25 p1 1 Twin 1 .24 .43 . 31 1 p2

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

9. Specification in Mx

10. A1 A2 x11 x21 x22 P1 P2 Within-Twin Covariances : A Path Tracing: ‘Star’ Matrix Multiplication (*)

11. Specification of C and E follow the same principals Begin Matrices; X LOW 2 2 FREE ! Additive Genetic PATHS Y LOW 2 2 FREE ! Common Env PATHS Z LOW 2 2 FREE ! Unique Env PATHS End Matrices; Begin Algebra; A=X*X’; ! Additive Genetic Cov matrix C=Y*Y’; ! Common Env Cov matrix E=Z*Z’; ! Unique Env Cov matrix P=A+C+E; End Algebra;

12. By rule of matrix addition:  P = A + C + E x211+ y211+z211 x11x21+ y11y21+z11z21 P = x21x11+ y21y11+z21z11 x221+x222 + y221+y222+ z221+z222

13. .5 .5 A1 A2 A1 A2 x11 x21 x22 x11 x21 x22 P11 P21 P12 P22 Twin 1 Twin 2 Cross-Twins Covariances (DZ): A Path Tracing: Within-Traits (diagonals): P11-P12= x11  .5 x11 P21-P22= (x22  .5  x22)+(x21 .5  x21) Cross-Traits: P11-P22= x11  .5  x21 P21-P12= x21 .5  x11 Kronecker Product 

14. Kronecker product  H FULL 1 1 MATRIX H .5  = @ (m  n)  (p  q) = (mp  nq) (1  1)  (2  2) = (2  2)

15. Cross-Twins Covariances (DZ): C Path Tracing: 1 1 Within-Traits (diagonals): P11-P12= y11y11 P21-P22= (y22  y22)+(y21 y21) Cross-Traits: P11-P22= y11  y21 P21-P12= y21 y11 C1 C2 C1 C2 y11 y21 y22 y11 y21 y22 P11 P21 P12 P22 Twin 1 Twin 2

16. .5x211+ y211 .5x21x11+ y21y11 .5A+C= .5x11x21+ y11y21 .5x222+x221 + y222+y221 Cross-Twin Covariances (DZ): (1/2 A + C) +

17. x211+ y211 .5x21x11+ y21y11 A+C= x11x21+ y11y21 x222+x221 + y222+y221 Cross-Twin Covariances (MZ): (A + C) +

18. Covariance matrix (MZ): specification in Mx Within-Twin Cov = A+C+E 2  2 Cross-Twin Cov = A+C 2  2 A+C+E | A+C _ A+C | A+C+E COV 4  4

19. Covariance matrix (DZ): specification in Mx Within-Twin Cov = A+C+E 2  2 Cross-Twin Cov = 1/2 A+C 2  2 A+C+E | H@A+C _ H@A+C | A+C+E COV 4  4

20. x211+ y211+ z211 x11x21+ y11y21+z11z21 P = x21x11+ y21y11+z21z11 x221+x222 + y221+y222+ z221+z222 The Within-Twin Covariance matrix describes how much of the phenotypic covariance between P1 and P2 is due to common A, common C and common E effects In order to get the Predicted Phenotypic correlation, we convert this Covariance matrix to a correlation matrix. In order to get the Genetic, Shared-environmental and Unique- environmental correlations (rg, rc, re), we convert the A, C and E Covariance matrices to correlation matrices.

21. Correlations • A correlation coefficient is a standardized covariance that lies between -1 and 1 so that it is easier to interpret • It is calculated by dividing the covariance by the square root of the product of the variances of the two variables

22. Covariances to Correlations In matrix form:

23. Correlations to covariances In matrix form:

24. How do we derive the Genetic Correlation? Matrix Function in Mx: \stnd(A)

25. . A11A12 A21A22 A11 0 0 A22 1 0 0 1 = I.A = Or......... R=\sqrt(I.A)~*A*\sqrt(I.A)~ ; Where I is an Identity Matrix of 2  2 and

26. Cholesky ACE for 3 variables A1 A2 A3 x22 x33 x32 x11 x21 x31 P1 P2 P3 Begin Matrices; X LOW 3 3 FREE ! Additive Genetic PATHS Y LOW 3 3 FREE ! Common Env PATHS Z LOW 3 3 FREE ! Unique Env PATHS End Matrices;

27. h2x h2y A1 A2 rg X Y c2x c2y C1 C2 rc rph due to A rph due to C rph due to E