2 Alternate Representation of the Multivariate Genetic Factor Model

One of the features of Mx is its flexibility for specifying the same or very similar models in different ways. Frequently the choice of model specification is simply a matter of individual preference, convenience, or familiarity with Mx notation, particularly when a model can be written in several different ways with no change in the substantive or numerical outcome. However, at other times very subtle changes in the Mx formulation of a model translate into a completely different substantive question. While it may be true that flexibility imparts confusion, it is important to recognize and distinguish alternative representations of genetic models in Mx. While the approach discussed above may be fairly intuitive, the B matrix may become relatively big, therefore increasing the chance of errors in editing. An alternative approach is to specify the common factors and residual variances for genetic, shared and specific environmental factors in separate matrices. One advantage of this approach is that the model can be easily adapted for a different number of common factors or observed variables. For example, if we use a $4\times1$ matrix X for the genetic common factor, a $4\times1$ matrix Y for the shared environmental common factor, a $4\times1$ matrix Z for the specific environmental common factor and a $4\times 4$ diagonal matrix F for the unique variances, the matrices section in Mx would be

 X Full 4 1 Free ! genetic common factor Y Full 4 1
Free ! shared environmental common factor Z Full 4 1 Free ! specific
environmental common factor F Diag 4 4 Free ! specific environmental
unique variances

We can then pre-calculate the genetic, shared and specific environmental variance components in the algebra section:

 A= X*X';  C= Y*Y';  E= Z*Z' +F*F';

and these matrices can be used to specify the expected covariance matrices for MZ and DZ twins in a similar fashion as the univariate models. Note that by using a Kronecker product for the genetic variance component in DZ twins (H@A) every element of the A matrix is multiplied by one half. One additional feature in Mx that allows for flexible model specification is the #define statement. One possible use is to define the number of variables up front, e.g.

#define nvar 4

and use the 'defined' variables in the matrices section:

X Full nvar 1 Free       ! genetic common factor
Y Full nvar 1 Free       ! shared environmental common factor
Z Full nvar 1 Free       ! specific environmental common factor
F Diag nvar nvar Free    ! specific environmental unique variances

If we wanted to do an analysis with just three variables, the only change to be made, besides the NInput_vars and Select statements, is the #define statement.