Multivariate Genetic Analysis: Introduction(II) Frühling Rijsdijk & Shaun Purcell Wednesday March 6, 2002

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 genes or common environmental influences l 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 P 1 P A A C C E E 1/.5 a cecae 2  2

Within-Twin Covariances Twin1 p1 p2 Twin2 p1 p2 Within-Twin Covariances 4  4 a 21 *a 11 + e 21 *e 11 a e /.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 E 1 E 2 E 1 E 2 e 11 e 21 e 22 e 11 e 21 e 22 a a e e 21 2 Twin1 p1 p2 Twin2 p1 p2 Cross-Twin Covariances 1/.5*a /.5*a /.5*a /.5*a 21 * a 11 1/.5*a 11 * a 21 Cross-Twin Covariances a 11 *a 21 + e 11 *e 21

Twin1 p1 p2 Twin1 p1 p2 Within-Twin Covariances Twin2 p1 p2 Within-Twin Covariances 4  4 a 11 *a 21 + e 11 *e 21 a e /.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 E 1 E 2 E 1 E 2 e 11 e 21 e 22 e 11 e 21 e 22 a a e e 21 2 Twin2 p1 p2 Cross-Twin Covariances (1/.5)*a 11 2 (1/.5)*a (1/.5)*a 21 2 (1/.5)*a 21 * a 11 mz=.6 / dz=.3 Cross-Twin Covariances

Twin1 p1 p2 Within-Twin Covariances Cross-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 Within Trait 1 Cross TraitsWithin Trait 2 Cov P1-P2 Var P2 Var P1

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

Within-Twin Covariances: Specification in Mx Path Tracing * or ‘Star’ Matrix Multiplication P1P2 a 22 a 11 A 1 A 2 a 21 A LOWER 2  2

 P =  A +  E e 22 E 1 E 2 e 21 e 11 P1P2 a 22 a 11 A 1 A 2 a 21 By rule of matrix addition:

Within-Traits (diagonals): P 11 -P 12 = a 11 .5  a 11 P 21 -P 22 = a 22 .5  a 22 + a 21 .5  a 21 Cross-Traits: P 11 -P 22 = a 11 .5  a 21 P 21 -P 12 = a 21 .5  a 11 Cross-Twins Covariances (DZ): Specification in Mx.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 

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

Cross-Twins Covariances (MZ): Specification in Mx 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

Cross-Twins Covariances (MZ/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 C*C’ = c 11 2 c 22 2 c 11 c 21 c c 21 2

COV A+C+E| A+C_ A+C | A+C+E / MZ COV A+C+E| | A+C+E / DZ

Correlations l A correlation coefficient is a standardized covariance that lies between -1 and 1 so that it is easier to interpret l 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:

Genetic Correlations Matrix Function in Mx: \stnd(A*A’)