Bivariate analysis

HGEN619 class 2005

Univariate ACE model

T 2

AEA C

a ac e

T 1

1 111

E

e

1

C

c

1

1 or .5

1

Expected Covariance Matrices

a2+c 2+e 2 .5a 2+c 2

.5a 2+c 2 a 2+c 2+e 2 DZ =

a2+c 2+e 2 a2+c 2

a2+c 2 a 2+c 2+e 2 MZ = 2 x 2

2 x 2

Bivariate Questions I

Univariate Analysis: What are the contributions of additive genetic, dominance/shared environmental and unique environmental factors to the variance?

Bivariate Analysis: What are the contributions of genetic and environmental factors to the covariance between two traits?

Two Traits

E YE X

X Y

A X A YA C

E C

Bivariate Questions II

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? With twin data on multiple traits it is possible to

partition the covariation into its genetic and environmental components

Goal: to understand what factors make sets of variables correlate or co-vary

Bivariate Twin Data

individual twin

within between

trait within

between

(within-twin within-trait

co)variance(cross-twin within-trait)

covariance(cross-twin within-trait) covariance

cross-twin cross-trait covariance

Bivariate Twin Covariance Matrix

X1 Y1 X2 Y2

X1

Y1

X2

Y2

VX1 CX1X2

CX2X1 VX2

VY1 CY1Y2

CY2Y1 VY2

CX1Y1

CX2Y2

CY1X1

CY2X2

CX1Y2

CX2Y1

CY1X2

CY2X1

Genetic Correlation

Y 2

A YA X A Y

a x a ya y

X 1

1 11

A X

a x

1

1 or .5 1 or .5

X 2Y 1

rg rg

Alternative Representations

A X A Y

a x a y

X 1

1 1

Y 1

rg

A S X A S Y

a sx a sy

X 1

1 1

Y 1

A C

1

a c a c

A 1 A 2

a11 a22

X 1

1 1

Y 1

a21

Cholesky Decomposition

A 1 A 2

a11 a22

X 1

1 1

Y 1

a21

A 1 A 2

a11 a22

X 2

1 1

Y 2

a21

1 or .5 1 or .5

More Variables

A 1 A 2

a11 a22

X 1

1 1

X 2

a21

A 3

a33

1

X 3

A 4

a44

1

X 4

a32 a43a31 a42

A 5

1

X 5

Bivariate AE Model

A 1 A 2

a11 a22

X 1

1 1

Y 1

a21

A 1 A 2

a11 a22

X 2

1 1

Y 2

a21

1 or .5 1 or .5

E 1 E 2

e11 e22

1 1

e21

E 1 E 2

e11 e22

1 1

e21

MZ Twin Covariance Matrix

X1 Y1 X2 Y2

X1

Y1

X2

Y2

a112

+e112

a222+a21

2+e22

2+e212

a21*a11+e21*e11

a222+a21

2

a112

a21*a11

DZ Twin Covariance Matrix

X1 Y1 X2 Y2

X1

Y1

X2

Y2

a112

+e112

a222+a21

2+e22

2+e212

a21*a11+e21*e11

.5a222+

.5a212

.5a112

.5a21*a11

Within-Twin Covariances [Mx]

A 1 A 2

a11 a22

X 1

1 1

Y 1

a21a11

a22

0

a21

A 1 A 2

X 1

Y 1

X Lower 2 2

a112

a222+a 21

2a21 *a 11

a11 *a 21=

A=X*X'

a11

a22

0

a21

a11

a220

a21

* A=

Within-Twin Covariances

a112

a222+a 21

2a21 *a 11

a11 *a 21 A=

e112

e222+e 21

2e21 *e 11

e11 *e 21 E=

a112+ e 11

2

a222+a 21

2 +e 222+e 21

2

a11 *a 21 + e 11 *e 21 P= A+ E =a21 *a 11 + e 21 *e 11

Cross-Twin Covariances

a112

a222+a 21

2a21 *a 11

a11 *a 21 A=MZ

.5a 112

.5a 222+.5a 21

2.5a 21 *a 11

.5a 11 *a 21.5@ A=DZ

Cross-Trait Covariances

Within-twin cross-trait covariances imply common etiological influences

Cross-twin cross-trait covariances imply familial common etiological influences

MZ/DZ ratio of cross-twin cross-trait covariances reflects whether common etiological influences are genetic or environmental

Univariate Expected Covariances

a2+c 2+e 2 .5a 2+c 2

.5a 2+c 2 a 2+c 2+e 2 DZ =

a2+c 2+e 2 a2+c 2

a2+c 2 a 2+c 2+e 2 MZ = 2 x 2

2 x 2

Univariate Expected Covariances II

DZ = A+ C+ E .5@ A+ C.5@ A+ C A+ C+ E

A+ C+ E A+ C A+ C A+ C+ E

MZ = 2 x 2

2 x 2

Bivariate Expected Covariances

DZ = A+ C+ E .5@ A+ C.5@ A+ C A+ C+ E

A+ C+ E A+ C A+ C A+ C+ E

MZ = 4 x 4

4 x 4

Practical Example I

Dataset: MCV-CVT Study 1983-1993 BMI, skinfolds (bic,tri,calf,sil,ssc) Longitudinal: 11 years N MZF: 107, DZF: 60

Practical Example II

Dataset: NL MRI Study 1990’s Working Memory, Gray & White Matter

N MZFY: 68, DZF: 21