Mx, Matrix Algebra, Multivariate Genetic Analyses

1. Mx, • Matrix Algebra, • Multivariate Genetic Analyses Meike Bartels & Kate Morley Leuven 2008

2. 1/.5 1 A C E A C E IQ T1 IQ T2 The Univariate Model

3. 1 1/.5 C E A E C A a e c a c e P P + + + + + + + + + + 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 a a c c a a a a c c c c e e e e S S = = MZ DZ + + 2 2 2 2 . . 5 5 a a c c 2  2 2  2

5. GENES GENES GENES GENES ENVIRONMENTS ENVIRONMENT GENES ENVIRONMENTS Relationships Intelligence Brain Volume

6. Multivariate 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

7. Individual Within Between Within Measure Between Multivariate Twin Data Twin covariance Variance Cross-trait twin covariance Phenotypic covariance

8. X1 Y1 X2 Y2 X1 VX1 CX1Y1 CX1X2 CX1Y2 Y1 CX1Y1 VY1 CY1X2 CY1Y2 X2 CX2X1 CX2Y1 VX2 CX2Y2 Y2 CX1Y2 CY2Y1 CY2X2 VY2 Multivariate Twin Covariance Matrix

9. 1 / 0.5 1 / 0.5 rG rG A X A Y A X A Y aY aY aX aX X 1 Y1 X 2 Y2 Multivariate Model Parameters

10. rG A X A Y aX aY Y1 X 1 Correlated Factors • Genetic correlation rG • Chain of paths aXrGaY bivariate heritability • Component of phenotypic covariance rXY = aXrGaY +cXrCcY +eXrEeY

11. rG A X A Y A C A SY A SX A 1 A 2 aC aC aX aY aSX a21 a11 a22 aSY Y1 X 1 Y1 Y1 X 1 X 1 Cholesky Decomposition

12. A 1 A 2 a21 a11 a22 Y1 X 1 Cholesky Decomposition Tests of specificity • If a22 > 0 • genetic influences specific to Y

13. X1 Y1 X2 Y2 X1 VX1 CX1Y1 CX1X2 CX1Y2 Y1 CX1Y1 VY1 CY1X2 CY1Y2 X2 CX2X1 CX2Y1 VX2 CX2Y2 Y2 CX1Y2 CY2Y1 CY2X2 VY2 Multivariate Twin Covariance Matrix

14. 1/.5 1/.5 A1 A2 A1 A2 a11 a21 a22 a11 a21 a22 P11 P21 P12 P22 e11 e22 e11 e22 e21 e21 E1 E2 E2 E1 Twin1 Twin2 p1 p2 p1 p2 Twin1 p1 p2 Twin2 p1 p2 Within-Twin Covariances Cross-Twin Covariances a112 +e112 a11*a21+ e11*e21 1/.5*a112 1/.5*a11* a21 a21*a11+ e21*e11 a222 +a212+ e222 +e212 1/.5*a21* a11 1/.5*a222+1/.5*a212 Cross-Twin Covariances Within-Twin Covariances

15. 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

16. é ù 2 a a a 11 ê ú S = 11 21 ê ú A 2 2 a a a a + 21 22 ê ú ë û 21 11 A LOWER 2  2 A1 A2 A 1 A 2 a P 1 0 11 a a P2 21 22 a21 a11 a22 é ù a11a21+ 0*a22 a112+ 0*0 a a a é ù é ù 0 ê ú Y1 X 1 11 11 21 S = = = A * A ' * ê ú ê ú a212 + a222 a21a11 + a22*0 ê ú A a a a 0 ê ú ê ú ë û ë û ê ú 21 22 22 ë û Specification in Mx * or ‘Star’ Matrix Multiplication

17. A1 A2 é é ù ù 2 2 a e e a a e 11 11 ê ê ú ú S S = = 11 11 21 21 a11 a21 a22 ê ê ú ú E A 2 2 2 2 a e e a a e a e + + 21 21 22 22 ê ê ú ú ë ë û û 21 21 11 11 P1 P2 e11 e22 e21 a112 + e112 a11 a21 + e11 e21 SP = a21 a11 + e21 e11 a212 + a222 + a212 + a222 E1 E2 Within-Twin Covariances: Specification in Mx  P = A + E

18. 1 1 A1 A2 A1 A2 a11 a21 a22 a11 a21 a22 P11 P21 P12 P22 é ù 2 a a a 11 ê ú S = = 11 21 A * A ' ê ú A 2 2 a a a a + 21 22 ê ú ë û 21 11 Cross-Twin Covariances: Specification in Mx (MZ)

19. .5 .5 A1 A2 A1 A2 a11 a21 a22 a11 a21 a22 P11 P21 P12 P22 é ù 2 . 5 a . 5 a a 11 ê ú Ä S = Ä = 11 21 . 5 . 5 A * A ' ê ú A 2 2 . 5 a a . 5 a a + 21 22 ê ú ë û 21 11 Cross-Twin Covariances: Specification in Mx (DZ) Within-Traits (diagonals): P11-P12= a11  .5  a11 P21-P22= a22  .5  a22 + a21 .5  a21 Cross-Traits: P11-P22= a11  .5  a21 P21-P12= a21 .5  a11 Kronecker Product

20. é ù 2 . 5 a . 5 a a 11 ê ú Ä S = @ = 11 21 . 5 H A * A ' ê ú A 2 2 . 5 a a . 5 a a + 21 22 ê ú ë û 21 11 Kronecker Product H FULL 1 1 MATRIX H .5  = @

21. 1 1 C1 C2 C1 C2 a11 a21 a22 a11 a21 a22 P11 P21 P12 P22 é ù 2 c c c 11 ê ú Ä S = = 11 21 1 C * C ' ê ú C 2 2 c c c c + 21 22 ê ú ë û 21 11 Cross-Twin Covariances: (MZ/DZ)

22. MX MZ COV A+C+E | A+C_ A+C | A+C+E / DZ COV A+C+E | H@A+C_ H@A+C | A+C+E /

23. Practical Session MRI-IQ dataset WM = working memory BBGM = gray matter volume of the cerebrum BBWM = white matter volume of the cerebrum (Variables have already been corrected for the effects of age and sex) Mx job: bivariate_WM+GM.mx (F:\meike\multivariate) Data file: mri.dat (F:\meike\multivariate) Posthuma et al. Nature Neuroscience, Feb 2002

24. Practical Session MRI-IQ dataset Phenotypic correlations Working memory – gray matter volume 0.27 Working memory – white matter volume 0.28

25. Practical Session MRI-IQ dataset Twin correlations WM BBGM BBWM MZ 0.72 0.86 0.89 DZ 0.27 0.45 0.34

26. Practical Session • Run Bivariate Model • Modify Mx Job to a trivariate model

27. Practical Session • Do the genes that influence WM also influence Brain Volume? • Are there genes that are unique to Brain Volume? • Is the phenotypic correlation caused by genetic correlation? • What is the genetic correlation? • The same questions apply to environmental (shared and unique) influences.

28. A1 A2 A3 A4 A5 V1 V2 V3 V4 V5 C1 C2 C3 C4 C5 More than two variables

29. More than two variables Phenotypic Single Factor

30. Residual Variances

31. Twin Data

32. Genetic Single Factor

33. ACE all Single Factors

34. Independent Pathway Model

35. Independent Pathway Model Mx script • G1: Define matrices • Calculation • Begin Matrices; • X full nvar nfac Free ! common factor genetic path coefficients • Y full nvar nfac Free ! common factor shared environment paths • Z full nvar nfac Free ! common factor unique environment paths • T diag nvar nvar Free ! variable specific genetic paths • U diag nvar nvar Free ! variable specific shared env paths • V diag nvar nvar Free ! variable specific residual paths • M full 1 nvar Free ! means • End Matrices; • Start … • Begin Algebra; • A= X*X' + T*T'; ! additive genetic variance components • C= Y*Y' + U*U'; ! shared environment variance components • E= Z*Z' + V*V'; ! nonshared environment variance components • End Algebra; • End

36. Independent Pathway Model Mx script • G2: MZ twins • #include iqnlmz.dat • Begin Matrices = Group 1; • Means M | M ; • Covariance A+C+E | A+C _ • A+C | A+C+E ; • Option Rsiduals • End • G3: DZ twins • #include iqnldz.dat • Begin Matrices= Group 1; • H full 1 1 • End Matrices; • Matrix H .5 • Means M | M ; • Covariance A+C+E | H@A+C _ • H@A+C | A+C+E ; • Option Rsiduals • End

37. Phenotypic Single Factor

38. Latent Phenotype

39. Twin Data

40. Factor on Latent Phenotype

41. Common Pathway Model

42. Common Pathway Model • G1: Define matrices • Calculation • Begin Matrices; • X full nfac nfac Free ! latent factor genetic path coefficient • Y full nfac nfac Free ! latent factor shared environment path • Z full nfac nfac Free ! latent factor unique environment path • T diag nvar nvar Free ! variable specific genetic paths • U diag nvar nvar Free ! variable specific shared env paths • V diag nvar nvar Free ! variable specific residual paths • F full nvar nfac Free ! loadings of variables on latent factor • I Iden 2 2 • M full 1 nvar Free ! means • End Matrices; • Start .. • Begin Algebra; • A= F&(X*X') + T*T'; ! genetic variance components • C= F&(Y*Y') + U*U'; ! shared environment variance components • E= F&(Z*Z') + V*V'; ! nonshared environment variance components • L= X*X' + Y*Y' + Z*Z'; ! variance of latent factor • End Algebra; • End

43. Common Pathway Model • G4: Constrain variance of latent factor to 1 • Constraint • Begin Matrices; • L computed =L1 • I unit 1 1 • End Matrices; • Constraint L = I ; • End • G5: Calculate Standardised Solution • Calculation • Matrices = Group 1 • D Iden nvar nvar • End Matrices; • Begin Algebra; • R=A+C+E; ! total variance • S=(\sqrt(D.R))~; ! diagonal matrix of standard deviations • P=S*F; ! standardized estimates for loadings on F • Q=S*T_ S*U_ S*V; ! standardized estimates for specific factors • End Algebra; • Options NDecimals=4 • End

44. Mx Scripts