1. Multivariate Genetic Analysis: Introduction
Frühling Rijsdijk & Shaun Purcell
Twin Workshop, Boulder
Wednesday March 3, 2004
In this second part we will talk about the Matrix specification of Bivariate genetic models.
Bivariate models are the most simple case of multivariate models: when we analyse two traits at a time.

2. Multivariate Twin Analyses
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

3. Univariate ACE Model for a Twin Pair1
1/.5 A
P x E C A A C E
X LOW 1  1 z y x x y z C
P y P1 P2
Y LOW 1  1 E
P z
RECAP
Before we’ll move on to a two-variable (bivariate) AE twin model, we’ll quickly review the univariate ACE model:
The effects of the latent genetic and environmental factors on the observed phenotypes are represented by the path coefficients small a, c, and e
These path coefficients are 1x1 matrices, so they are just single values.
The expected MZ and DZ matrices (the variances and covariances) that are formulated from these path coefficients are, therefore, 2x2 matrices.
2  2
Z LOW 1  1
2  2

4. x y z P2 1 P A2 A1 P2 1 P C2 C1 P2 1 P E2 E1 X LOWER 2  2
1/.5 A1 A2
y21
P11
P21
x22
x21 E1 E2
z11
z21
z22 C1 C2
y11
x11
P12
P22 1
When we have measured 2 variables for each twin, the A and E matrices are now 2x2 matrices which have a triangular or Cholesky decomposition. In the Cholesky there are as many factors as variables, i.e. A1 and A2, where A2 only influences the second variable and A1 influences both variables allowing for a correlation between the variables due to shared genetic effects.
The same counts for E. Show second half of diagram. Show second half of slide:
The complete variance covariance matrix will be a 4x4 matrix which can be divided into 4 quadrants:
The first Q represents the variances and covariances within an individual:
on the diagonals we have the variances of trait1 and trait 2 which can be obtained by path tracing rules : i.e for P1 this is x x112 and for P2 this is
x x z z212. On the off-diagonal we have the covariance between trait1-2, which is the sum of all chains or paths (via A and E) linking the two:
x11 * x z11 * z21.
Since the within twin variances and covariances are expected to be the same for Twin 1 and Twin 2 we can fill in the same expected 2x2 Var-Cov matrix for the 4th quadrant 
The second Q represents the covariances between Twin1 and 2 and can be either within trait (diagonals) or cross trait. Show path tracing.
The cross-twin within-trait covariances are in fact what we have already come across in the univariate analyses: the mz/dz ratios indicate the heritability of a trait. What is new here is the correlation between trait1-2. Whether this correlation is determined by shared genetic or shared environmental effects, will be indicated by the cross-twin cross-trait covariance.
In AE model the expected cross-twin cross-trait covariance ratio for MZ and DZ pairs will be 2:1, because MZ pairs share twice as many genetic effects as DZ. 22 21 11 P2 1 P A2 A1 x 22 21 11 P2 1 P C2 C1 y 22 21 11 P2 1 P E2 E1 z
X LOWER 2  2
Y LOWER 2  2
Z LOWER 2  2

5. 4  4 Twin1 p1 p2 Within-Twin Covariances Cross-Twin Covariances
Var P1
Cov P1-P2
Var P2
Twin2 p1 p2
Within Trait 1
Cross Traits
Within Trait 2
Twin1 p1
Twin2
To summarize the points made with the previous slides:
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 cross-twin cross-traits covariances
4  4

6. 1/.5 A1 A2
y21
P11
P21
x22
x21 E1 E2
z11
z21
z22 C1 C2
y11
y22
x11
P12
P22 1
y22
4  4
Twin1
Twin2
p p2
p p2
Within-Twin Covariances
Cross-Twin Covariances
Twin 1 p2 p1
When we have measured 2 variables for each twin, the A and E matrices are now 2x2 matrices which have a triangular or Cholesky decomposition. In the Cholesky there are as many factors as variables, i.e. A1 and A2, where A2 only influences the second variable and A1 influences both variables allowing for a correlation between the variables due to shared genetic effects.
The same counts for E. Show second half of diagram. Show second half of slide:
The complete variance covariance matrix will be a 4x4 matrix which can be divided into 4 quadrants:
The first Q represents the variances and covariances within an individual:
on the diagonals we have the variances of trait1 and trait 2 which can be obtained by path tracing rules : i.e for P1 this is x x112 and for P2 this is
x x z z212. On the off-diagonal we have the covariance between trait1-2, which is the sum of all chains or paths (via A and E) linking the two:
x11 * x z11 * z21.
Since the within twin variances and covariances are expected to be the same for Twin 1 and Twin 2 we can fill in the same expected 2x2 Var-Cov matrix for the 4th quadrant 
The second Q represents the covariances between Twin1 and 2 and can be either within trait (diagonals) or cross trait. Show path tracing.
The cross-twin within-trait covariances are in fact what we have already come across in the univariate analyses: the mz/dz ratios indicate the heritability of a trait. What is new here is the correlation between trait1-2. Whether this correlation is determined by shared genetic or shared environmental effects, will be indicated by the cross-twin cross-trait covariance.
In AE model the expected cross-twin cross-trait covariance ratio for MZ and DZ pairs will be 2:1, because MZ pairs share twice as many genetic effects as DZ.
x112 + y112
+ z112
1/.5*x112 +
1/1 * y112
x21*x11+ y21*y11 +
z21*z11
x x212+
y222 + y212 +
z222 +z212
1/.5*x21* x11 +
1/1 * y212 * y11
1/.5*x222+1/.5*x212 +
1/1*y222+1/1*y212
Rmz:Rdz will indicate
whether A, C or E
determine Rp1-p2 p1
Twin 2 p2

7. Twin1 MZ
p p2
Within-Twin Covariances p1 1
Twin 1 1 p2
.30
Cross-Twin Covariances
Within-Twin Covariances p1
. 79
.49 1
Twin 1
.50
.59
. 29 1 p2
Twin1 DZ
p p2
Within-Twin Covariances
When we have measured 2 variables for each twin, the A and E matrices are now 2x2 matrices which have a triangular or Cholesky decomposition. In the Cholesky there are as many factors as variables, i.e. A1 and A2, where A2 only influences the second variable and A1 influences both variables allowing for a correlation between the variables due to shared genetic effects.
The same counts for E. Show second half of diagram. Show second half of slide:
The complete variance covariance matrix will be a 4x4 matrix which can be divided into 4 quadrants:
The first Q represents the variances and covariances within an individual:
on the diagonals we have the variances of trait1 and trait 2 which can be obtained by path tracing rules : i.e for P1 this is x x112 and for P2 this is
x x z z212. On the off-diagonal we have the covariance between trait1-2, which is the sum of all chains or paths (via A and E) linking the two:
x11 * x z11 * z21.
Since the within twin variances and covariances are expected to be the same for Twin 1 and Twin 2 we can fill in the same expected 2x2 Var-Cov matrix for the 4th quadrant 
The second Q represents the covariances between Twin1 and 2 and can be either within trait (diagonals) or cross trait. Show path tracing.
The cross-twin within-trait covariances are in fact what we have already come across in the univariate analyses: the mz/dz ratios indicate the heritability of a trait. What is new here is the correlation between trait1-2. Whether this correlation is determined by shared genetic or shared environmental effects, will be indicated by the cross-twin cross-trait covariance.
In AE model the expected cross-twin cross-trait covariance ratio for MZ and DZ pairs will be 2:1, because MZ pairs share twice as many genetic effects as DZ. p1 1
Twin 1 p2
.30 1
Cross-Twin Covariances
Within-Twin Covariances p1
.39
.25 1
Twin 1
.24
.43
. 31 1 p2

8. Summary : Cross-traits covariances
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 cross-twin cross-traits covariances

9. Specification in Mx
We have seen how the within-twin variance-covariance matrix for the Additive genetic effects of the Cholesky decomposition can be obtained by path tracing.
The Cholesky decomposition of the A matrix is specified in Mx as a Lower matrix (which are always square) and in the bivariate case the dimension is 2x2. Show box.
The within-twin var-cov of A is given by the product of A and it’s tranpsose. In the transposed matrix the rows become the columns and vice-versa
We use the * or ordinary matrix multiplication : rows of A are multiplied by columns of A’ to form the elements of sigma A. If we work out the multiplication we see that we get exactly the same result as by path tracing.

10. Within-Twin Covariances : AP1 P2
x22
x11 A1 A2
x21
Path Tracing:
‘Star’ Matrix Multiplication (*)
We have seen how the within-twin variance-covariance matrix for the Additive genetic effects of the Cholesky decomposition can be obtained by path tracing.
The Cholesky decomposition of the A matrix is specified in Mx as a Lower matrix (which are always square) and in the bivariate case the dimension is 2x2. Show box.
The within-twin var-cov of A is given by the product of A and it’s tranpsose. In the transposed matrix the rows become the columns and vice-versa
We use the * or ordinary matrix multiplication : rows of A are multiplied by columns of A’ to form the elements of sigma A. If we work out the multiplication we see that we get exactly the same result as by path tracing.

11. Specification of C and E follow the same principals
Begin Matrices;
X LOW 2 2 FREE ! Additive Genetic PATHS
Y LOW 2 2 FREE ! Common Env PATHS
Z LOW 2 2 FREE ! Unique Env PATHS
End Matrices;
Begin Algebra;
A=X*X’; ! Additive Genetic Cov matrix
C=Y*Y’; ! Common Env Cov matrix
E=Z*Z’; ! Unique Env Cov matrix
P=A+C+E;
End Algebra;
For the simplest, two-variable multivariate AE model the total phenotypic variance is P = A + E , where A, and E are 2  2 matrices rather than single values. P can be written as: the sum of the additive genetic and E covariance matrices. An extension to an ACE model is straightforward.

12. P = By rule of matrix addition:  P =  A +  C +  E
For the simplest, two-variable multivariate AE model the total phenotypic variance is P = A + E , where A, and E are 2  2 matrices rather than single values. P can be written as: the sum of the additive genetic and E covariance matrices. An extention to an ACE model is straightforward.
x211 + y211 +z211
x11x21 + y11y21+ z11z21
P =
x21x11 + y21y11+ z21z11
x221+x222 + y221+y222 + z221+z222

13. Cross-Twins Covariances (DZ): A
Path Tracing: .5 A1 A2
x11
P11
P21
x22
x21
P12
P22
Twin 1
Twin 2
Within-Traits (diagonals):
P11-P12= x11  .5  x11
P21-P22= (x22  .5  x22)+(x21 .5  x21)
Cross-Traits:
P11-P22= x11  .5  x21
P21-P12= x21  .5  x11
Kronecker Product 

14. Kronecker product  H FULL 1 1  = @ MATRIX H .5
 = @
(m  n)  (p  q) = (mp  nq)
(1  1)  (2  2) = (2  2)
Kronecker Product : Specification in MX
The right Kronecker product of two matrices is formed by multiplying each element of the first matrix by the second matrix.
So, the result matrix in which all elements of the matrix A*A’ are multiplied by .5 is the Kronecker product of a 1x1 matrix i.e. H which we assign the value .5
There are no conformability criteria for this type of product, the matrices can be of different order.
In Mx, the Kronecker product is denoted woth symbol.

15. Cross-Twins Covariances (DZ): C
Path Tracing: 1 1
Within-Traits (diagonals):
P11-P12= y11  y11
P21-P22= (y22  y22)+(y21  y21)
Cross-Traits:
P11-P22= y11  y21
P21-P12= y21  y11 C1 C2 C1 C2
y11
y21
y22
y11
y21
y22
P11
P21
P12
P22
Twin 1
Twin 2

16. Cross-Twin Covariances (DZ): (1/2 A + C)
.5x211 + y211
.5x222+x221 + y222+y221
.5x11x21 + y11y21
.5x21x11 + y21y11

17. Cross-Twin Covariances (MZ): (A + C)
x211 + y211
x222+x221 + y222+y221
x11x21 + y11y21
.5x21x11 + y21y11

18. A+C+E | A+C _ A+C | A+C+E 4  4 Within-Twin Cov = A+C+E 2  2
Covariance matrix (MZ): specification in Mx
Within-Twin Cov = A+C+E 2  2
Cross-Twin Cov = A+C 2  2
A+C+E | A+C _
A+C | A+C+E
COV
We now have all relevant bits to specify the predicted Var-Cov matrix of the MZ twin pairs.
The usefulness of the vertical and horizontal adhesion operators can now be shown.
Since the predicted Var-Cov matrix is a 2x2 matrix with 4 elements, but some of the elements are the same according to the model i.e the Var of twin 1 = Var of twin 2 (diagonal elements) and the cov between Twin 1 and Twin 2 = the cov between Twin 2 and Twin1, we use | and _ to combine the two information units to get the predicted 2x2 var-cov matrix.
4  4

19. A+C+E | H@A+C _ H@A+C | A+C+E 4  4 Within-Twin Cov = A+C+E 2  2
Covariance matrix (DZ): specification in Mx
Within-Twin Cov = A+C+E 2  2
Cross-Twin Cov = 1/2 A+C 2  2
A+C+E | _
| A+C+E
COV
We now have all relevant bits to specify the predicted Var-Cov matrix of the MZ twin pairs.
The usefulness of the vertical and horizontal adhesion operators can now be shown.
Since the predicted Var-Cov matrix is a 2x2 matrix with 4 elements, but some of the elements are the same according to the model i.e the Var of twin 1 = Var of twin 2 (diagonal elements) and the cov between Twin 1 and Twin 2 = the cov between Twin 2 and Twin1, we use | and _ to combine the two information units to get the predicted 2x2 var-cov matrix.
4  4

20. The Within-Twin Covariance matrix describes how much of the phenotypic covariance between P1 and P2 is due to common A, common C and common E effects
P =
x211 + y211 + z211
x221+x222 + y221+y222 + z221+z222
x21x11 + y21y11+ z21z11
x11x21 + y11y21+ z11z21
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 Unique-
environmental correlations (rg, rc, re), we convert the A, C and E
Covariance matrices to correlation matrices.
For the simplest, two-variable multivariate AE model the total phenotypic variance is P = A + E , where A, and E are 2  2 matrices rather than single values. P can be written as: the sum of the additive genetic and E covariance matrices. An extention to an ACE model is straightforward.

21. Correlations
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

22. Covariances to Correlations
In matrix form:

23. Correlations to covariances
In matrix form:

24. How do we derive the Genetic Correlation?
The Standardize operation converts a covariance matrix into a correlation matrix by dividing the covariance between two variables 1 and 2 by the square root of the product of the variances of variable 1 and variable 2.
Matrix Function in Mx: \stnd(A)

25. R=\sqrt(I.A)~*A*\sqrt(I.A)~ ;Or
R=\sqrt(I.A)~*A*\sqrt(I.A)~ ;
The Standardize operation converts a covariance matrix into a correlation matrix by dividing the covariance between two variables 1 and 2 by the square root of the product of the variances of variable 1 and variable 2.
Where I is an Identity Matrix of 2  2 and
I.A = .
1 0
0 1
A11 A12
A21 A22 = A
0 A22

26. Cholesky ACE for 3 variables
x22
x32
x33
x11
x21
x31 P1 P2 P3
Begin Matrices;
X LOW 3 3 FREE ! Additive Genetic PATHS
Y LOW 3 3 FREE ! Common Env PATHS
Z LOW 3 3 FREE ! Unique Env PATHS
End Matrices;

27. rph due to A rph due to C rph due to E X Y h2x h2y c2x c2y rg A1rc
c2x
c2y
rph due to C
rph due to E