Multivariate Genetic Analysis Introduction Frhling Rijsdijk Shaun Purcell

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

Multivariate Twin Analyses 1. 2. 3. 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

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

1 1 1/. 5 C 1 A 2 y 11 x 11 y 21 x 21 P 11 z 11 C 2 C 1 y 22 y 11 x 11 y 21 A 1 P 21 z 21 E 1 A 1 x 22 1/. 5 z 22 z 11 X LOWER 2 2 C 1 x 22 y 11 0 P 2 y 21 y 22 Y LOWER 2 2 y 22 z 22 E 2 C 2 P 1 C 2 P 22 z 21 E 1 A 2 x 11 0 P 2 x 21 x 22 x 21 P 12 E 2 P 1 A 2 E 1 E 2 z 11 0 P 2 z 21 z 22 P 1 Z LOWER 2 2

Twin 1 p 1 p 2 Twin 2 p 1 p 2 Within-Twin Covariances Cross-Twin Covariances Var P 1 Within Trait 1 Cov P 1 -P 2 Var P 2 Cross Traits Within Trait 2 Cross-Twin Covariances Within Trait 1 Var P 1 Cross Traits Within Trait 2 Cov P 1 -P 2 4 4 Var P 2

1 A 1 C 1 y 11 x 11 y 21 A 2 x 21 P 11 1/. 5 C 2 C 1 y 22 y 11 x 11 y 21 P 21 z 11 E 2 Twin 1 p 2 p 1 p 2 A 2 x 21 p 2 x 22 4 4 E 2 Twin 2 p 1 p 2 Cross-Twin Covariances x 112 + y 112 + z 112 1/. 5*x 112 + 1/1 * y 112 x 222 + x 212+ y 222 + y 212 + z 222 +z 212 y 22 z 21 Within-Twin Covariances x 21*x 11+ y 21*y 11 + z 21*z 11 C 2 P 22 E 1 Twin 1 p 1 A 1 P 12 z 21 1 1/. 5*x 21* x 11 + 1/1 * y 212 * y 11 1/. 5*x 222+1/. 5*x 212 + 1/1*y 222+1/1*y 212 Rmz: Rdz will indicate whether A, C or E determine Rp 1 -p 2

Twin 1 MZ p 1 p 2 Within-Twin Covariances p 1 1 p 2 . 30 1 Twin 1 Cross-Twin Covariances Within-Twin Covariances p 1 . 79 . 49 1 p 2 . 50 . 59 . 29 1 Twin 1 DZ p 1 p 2 Within-Twin Covariances p 1 1 p 2 . 30 1 Twin 1 Cross-Twin Covariances Within-Twin Covariances p 1 . 39 . 25 1 p 2 . 24 . 43 . 31 1

Summary : Cross-traits covariances l l l 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 crosstwin cross-traits covariances

Specification in Mx

Within-Twin Covariances : A A 1 x 11 x 21 P 1 Path Tracing: A 2 x 22 P 2 ‘Star’ Matrix Multiplication (*)

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

By rule of matrix addition: P = A + C + E P = x 211 + y 211 +z 211 x 11 x 21 + y 11 y 21+ z 11 z 21 x 21 x 11 + y 21 y 11+ z 21 z 11 x 221+x 222 + y 221+y 222 + z 221+z 222

Cross-Twins Covariances (DZ): A A 1 x 11 x 21 P 11 Path Tracing: . 5 A 2 A 1 x 22 P 21 x 11 x 21 P 12 Twin 1 A 2 x 22 Within-Traits (diagonals): P 11 -P 12= x 11 . 5 x 11 P 21 -P 22= (x 22 . 5 x 22)+(x 21 . 5 x 21) P 22 Twin 2 Cross-Traits: P 11 -P 22= x 11 . 5 x 21 P 21 -P 12= x 21 . 5 x 11 Kronecker Product

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

Cross-Twins Covariances (DZ): C C 1 y 11 y 21 P 11 1 C 2 C 1 y 22 P 21 Twin 1 Path Tracing: 1 y 11 C 2 y 21 P 12 y 22 Within-Traits (diagonals): P 11 -P 12= y 11 P 21 -P 22= (y 22 y 22)+(y 21 y 21) P 22 Twin 2 Cross-Traits: P 11 -P 22= y 11 y 21 P 21 -P 12= y 21 y 11

Cross-Twin Covariances (DZ): (1/2 A + C) + . 5 A+C = . 5 x 211 + y 211 . 5 x 21 x 11 + y 21 y 11 . 5 x 11 x 21 + y 11 y 21 . 5 x 222+x 221 + y 222+y 221

Cross-Twin Covariances (MZ): (A + C) + A+C = x 211 + y 211 . 5 x 21 x 11 + y 21 y 11 x 11 x 21 + y 11 y 21 x 222+x 221 + y 222+y 221

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

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

The Within-Twin Covariance matrix describes how much of the phenotypic covariance between P 1 and P 2 is due to common A, common C and common E effects P = x 211 + y 211 + z 211 x 11 x 21 + y 11 y 21+ z 11 z 21 x 21 x 11 + y 21 y 11+ z 21 z 11 x 221+x 222 + y 221+y 222 + z 221+z 222 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 Uniqueenvironmental correlations (rg, rc, re), we convert the A, C and E Covariance matrices to correlation matrices.

Correlations l l 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

Covariances to Correlations In matrix form:

Correlations to covariances In matrix form:

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

Or. . R=sqrt(I. A)~*A*sqrt(I. A)~ ; Where I is an Identity Matrix of 2 2 and 1 0 I. A = 0 1 . A 11 A 12 A 21 A 22 A 11 0 = 0 A 22

Cholesky ACE for 3 variables A 1 x 11 A 2 x 21 P 1 Begin Matrices; X LOW 3 3 FREE Y LOW 3 3 FREE Z LOW 3 3 FREE End Matrices; x 31 A 3 x 22 P 2 x 33 P 3 ! Additive Genetic PATHS ! Common Env PATHS ! Unique Env PATHS

rph due to A rg A 1 A 2 h 2 x h 2 y X Y c 2 x rph due to C c 2 y C 1 rc C 2 rph due to E