This document discusses bivariate analysis of twin data to examine the genetic and environmental contributions to the covariance between two traits. It introduces bivariate twin models that partition the covariance into additive genetic (A), shared environmental (C), and unique environmental (E) components. The expected covariance matrices for MZ and DZ twins are presented. The goal of bivariate analysis is to understand what factors make sets of variables correlate through examining within-twin and cross-twin covariances. Examples of applications to real twin datasets are given.

1. Bivariate analysis
Mx class 2004

2. Univariate ACE model
1 or .5
1
1 1 1 1 1 1
A C E E C A
a c e e c a
T1 T2

3. Expected Covariance Matrices
a2+c2+e2 a2+c2
E MZ = 2x2
a2+c2 a2+c2+e2
a2+c2+e2 .5a2+c2
E DZ = 2x2
.5a2+c2 a2+c2+e2

4. Bivariate Questions I
n Univariate Analysis: What are the contributions
of additive genetic, dominance/shared
environmental and unique environmental factors
to the variance?
n Bivariate Analysis: What are the contributions of
genetic and environmental factors to the
covariance between two traits?

5. Two Traits
AX AC AY
X Y
EX EC EY

6. Bivariate Questions II
n Two or more traits can be correlated because
they share common genes or common
environmental influences
¨ e.g. Are the same genetic/environmental factors
influencing the traits?
n With twin data on multiple traits it is possible to
partition the covariation into its genetic and
environmental components
n Goal: to understand what factors make sets of
variables correlate or co-vary

7. Bivariate Twin Data
individual twin
within between
trait within variance twin covariance
between trait covariance cross-trait
twin covariance

8. Bivariate Twin Covariance Matrix
X1 Y1 X2 Y2
X1 VX1 CX1Y1 CX1X2 CX1Y2
Y1 CY1X1 VY1 CY1X2 CY1Y2
X2 CX2X1 CX2Y1 VX2 CX2Y2
Y2 CY2X1 CY2Y1 CY2X2 VY2

9. Genetic Correlation
1 or .5 1 or .5
rg rg
1 1 1 1
AX AY AX AY
ax ay ax ay
X1 Y1 X2 Y2

10. Alternative Representations
1
rg AC
1 1 1 1 1 1
AX AY ASX A SY A1 A2
ac ac
ax ay asx asy a11 a21 a22
X1 Y1 X1 Y1 X1 Y1

11. Cholesky Decomposition
1 or .5 1 or .5
1 1 1 1
A1 A2 A1 A2
a11 a21 a22 a11 a21 a22
X1 Y1 X2 Y2

12. More Variables
1 1 1 1 1
A1 A2 A3 A4 A5
a11 a21 aa22 a32 a4233 a43
31 a a44
X1 X2 X3 X4 X5

13. Bivariate AE Model
1 or .5 1 or .5
1 1 1 1
A1 A2 A1 A2
a11 a21 a22 a11 a21 a22
X1 Y1 X2 Y2
e11 e21 e22 e11 e21 e22
E1 E2 E1 E2
1 1 1 1

14. MZ Twin Covariance Matrix
X1 Y1 X2 Y2
X1 a112+e112 a112
Y1 a21*a11+ a222+a212+ a21*a11 a222+a212
e21*e11 e222+e212
X2
Y2

15. DZ Twin Covariance Matrix
X1 Y1 X2 Y2
X1 a112+e112 .5a112
Y1 a21*a11+ a222+a212+ .5a21*a11 .5a222+
e21*e11 e222+e212 .5a212
X2
Y2

16. Within-Twin Covariances [Mx]
1 1
A1 A2 X Lower 2 2
A1 A2
a11 a21 a22
X1 a11 0
X1 Y1 Y1 a21 a22
A=X*X'
a11 0 a11 a21 a112 a11*a21
EA= * =
a21 a22 0 a22 a21*a11 a222+a212

17. Within-Twin Covariances
a112 a11*a21
EA=
a21*a11 a222+a212
e112 e11*e21
EE=
e21*e11 e222+e212
a112+ e112 a11*a21 + e11*e21
EP= EA+EE =
a21*a11 + e21*e11 a222+a212 +e222+e212

18. Cross-Twin Covariances
a112 a11*a21
MZ EA=
a21*a11 a222+a212
.5a112 .5a11*a21
DZ .5@EA=
.5a21*a11 .5a222+.5a212

19. Cross-Trait Covariances
n Within-twin cross-trait covariances imply
common etiological influences
n Cross-twin cross-trait covariances imply
familial common etiological influences
n MZ/DZ ratio of cross-twin cross-trait
covariances reflects whether common
etiological influences are genetic or
environmental

20. Univariate Expected Covariances
a2+c2+e2 a2+c2
E MZ = 2x2
a2+c2 a2+c2+e2
a2+c2+e2 .5a2+c2
E DZ = 2x2
.5a2+c2 a2+c2+e2

21. Univariate Expected Covariances II
EA+EC+EC EA+EC
E MZ = 2x2
EA+EC EA+EC+EC
EA+EC+EC .5@EA+EC
E DZ = 2x2
.5@EA+EC EA+EC+EC

22. Bivariate Expected Covariances
EA+EC+EC EA+EC
E MZ = 4x4
EA+EC EA+EC+EC
EA+EC+EC .5@EA+EC
E DZ = 4x4
.5@EA+EC EA+EC+EC

23. Practical Example I
n Dataset: MCV-CVT Study
n 1983-1993
n BMI, skinfolds (bic,tri,calf,sil,ssc)
n Longitudinal: 11 years
n N MZFY: 107, DZF: 60

24. Practical Example II
n Dataset: NL MRI Study
n 1990’s
n Working Memory, Gray & White Matter
n N MZFY: 68, DZF: 21