Bivariate

1. Bivariateanalysis
Mxclass 2004

2. UnivariateACE model
T
2
AEAC
a ac e
T
1
1 111
E
e
1
C
c
1
1 or .5
1

3. Expected Covariance Matrices
a
2
+c
2
+e
2
.5a
2
+c
2
.5a
2
+c
2
a
2
+c
2
+e
2
E DZ =
a
2
+c
2
+e
2
a
2
+c
2
a
2
+c
2
a
2
+c
2
+e
2
E MZ = 2 x 2
2 x 2

4. BivariateQuestions I
nUnivariateAnalysis: What are the contributions
of additive genetic, dominance/shared
environmental and unique environmental factors
to the variance?
nBivariateAnalysis: What are the contributions of
genetic and environmental factors to the
covariance between two traits?

5. Two Traits
E
Y
E
X
X Y
A
X
A
Y
A
C
E
C

6. BivariateQuestions II
nTwo 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?
nWith twin data on multiple traits it is possible to
partition the covariationinto its genetic and
environmental components
nGoal: to understand what factors make sets of
variables correlate or co-vary

7. BivariateTwin Data
between
withintrait
betweenwithin
individual twin
variance twin covariance
trait covariancecross-trait
twin covariance

8. BivariateTwin Covariance Matrix
Y
2
X
2
Y
1
X
1
Y
2
X
2
Y
1
X
1
V
X1
C
X1X2
C
X2X1
V
X2
V
Y1
C
Y1Y2
C
Y2Y1
V
Y2
C
X1Y1
C
X2Y2
C
Y1X1
C
Y2X2
C
X1Y2
C
X2Y1
C
Y1X2
C
Y2X1

9. Genetic Correlation
Y
2
A
Y
A
X
A
Y
a
x
a
y
a
y
X
1
1 11
A
X
a
x
1
1 or .51 or .5
X
2
Y
1
r
g
r
g

10. Alternative Representations
A
X
A
Y
a
x
a
y
X
1
1 1
Y
1
r
g
A
SX A
SY
a
sx
a
sy
X
1
1 1
Y
1
A
C
1
a
c
a
c
A
1
A
2
a
11
a
22
X
1
1 1
Y
1
a
21

11. CholeskyDecomposition
A
1
A
2
a
11
a
22
X
1
1 1
Y
1
a
21
A
1
A
2
a
11
a
22
X
2
1 1
Y
2
a
21
1 or .51 or .5

12. More Variables
A
1
A
2
a
11
a
22
X
1
1 1
X
2
a
21
A
3
a
33
1
X
3
A
4
a
44
1
X
4
a
32
a
43
a
31
a
42
A
5
1
X
5

13. BivariateAE Model
A
1
A
2
a
11
a
22
X
1
1 1
Y
1
a
21
A
1
A
2
a
11
a
22
X
2
1 1
Y
2
a
21
1 or .51 or .5
E
1
E
2
e
11
e
22
1 1
e
21
E
1
E
2
e
11
e
22
1 1
e
21

14. MZ Twin Covariance Matrix
Y
2
X
2
Y
1
X
1
Y
2
X
2
Y
1
X
1
a
11
2
+
e
11
2
a
22
2
+a
21
2
+
e
22
2
+e
21
2
a
21
*a
11
+
e
21
*e
11
a
22
2
+a
21
2
a
11
2
a
21
*a
11

15. DZ Twin Covariance Matrix
Y
2
X
2
Y
1
X
1
Y
2
X
2
Y
1
X
1
a
11
2
+
e
11
2
a
22
2
+a
21
2
+
e
22
2
+e
21
2
a
21
*a
11
+
e
21
*e
11
.5a
22
2
+
.5a
21
2
.5a
11
2
.5a
21
*a
11

16. Within-Twin Covariances[Mx]
A
1
A
2
a
11
a
22
X
1
1 1
Y
1
a
21
a
11
a
22
0
a
21
A
1
A
2
X
1
Y
1
X Lower 2 2
a
11
2
a
22
2
+a
21
2
a
21
*a
11
a
11
*a
21
=
A=X*X'
a
11
a
22
0
a
21
a
11
a
22
0
a
21
*
EA=

17. Within-Twin Covariances
a
11
2
a
22
2
+a
21
2
a
21
*a
11
a
11
*a
21
EA=
e
11
2
e
22
2
+e
21
2
e
21
*e
11
e
11
*e
21
EE=
a
11
2
+ e
11
2
a
22
2
+a
21
2
+e
22
2
+e
21
2
a
11
*a
21
+ e
11
*e
21EP= EA+EE =
a
21
*a
11
+ e
21
*e
11

18. Cross-Twin Covariances
a
11
2
a
22
2
+a
21
2
a
21
*a
11
a
11
*a
21
EA=MZ
.5a
11
2
.5a
22
2
+.5a
21
2
.5a
21
*a
11
.5a
11
*a
21
.5@EA=DZ

19. Cross-Trait Covariances
nWithin-twin cross-trait covariancesimply
common etiological influences
nCross-twin cross-trait covariancesimply
familial common etiological influences
nMZ/DZ ratio of cross-twin cross-trait
covariancesreflects whether common
etiological influences are genetic or
environmental

20. UnivariateExpected Covariances
a
2
+c
2
+e
2
.5a
2
+c
2
.5a
2
+c
2
a
2
+c
2
+e
2
E DZ =
a
2
+c
2
+e
2
a
2
+c
2
a
2
+c
2
a
2
+c
2
+e
2
E MZ = 2 x 2
2 x 2

21. UnivariateExpected CovariancesII
E DZ =
EA+EC+EC .5@EA+EC
.5@EA+EC EA+EC+EC
EA+EC+EC EA+EC
EA+EC EA+EC+EC
E MZ = 2 x 2
2 x 2

22. BivariateExpected Covariances
E DZ =
EA+EC+EC .5@EA+EC
.5@EA+EC EA+EC+EC
EA+EC+EC EA+EC
EA+EC EA+EC+EC
E MZ = 4 x 4
4 x 4

23. Practical Example I
nDataset: MCV-CVT Study
n1983-1993
nBMI, skinfolds(bic,tri,calf,sil,ssc)
nLongitudinal: 11 years
nN MZFY: 107, DZF: 60

24. Practical Example II
nDataset: NL MRI Study
n1990’s
nWorking Memory, Gray & White Matter
nN MZFY: 68, DZF: 21