1 Independent Pathway Model for Atopy

Inspection of the correlation matrices in Table 10.6 reveals that the presence of any one of the symptoms is associated with an increased risk of the others within an individual (hence the concept of ``atopy''). All four symptoms show higher MZ correlations ($0.592$, $0.593$, $0.518$, $0.589$) than DZ correlations in liability ($0.262$, $0.318$, $0.214$, $0.313$) and there is a hint of genetic dominance (or epistasis) for asthma and dust allergy (DZ correlations less than half their MZ counterparts). Preliminary multivariate analysis suggests that dominance is acting at the level of a common factor influencing all symptoms, rather than as specific dominance contributions to individual symptoms. Our first model for covariation of these symptoms is shown in the path diagram of Figure 10.1

Figure 10.1: Independent pathway model for asthma, hayfever, dust allergy, and eczema. All labels for path-coefficients have been omitted. All four correlations at the bottom of the figure are fixed at 1 for MZ and .5 for DZ twins.

Because each of the three common factors () has its own paths to each of the four variables, this has been called the independent pathway model (Kendler et al., 1987) or the biometric factors model (McArdle and Goldsmith, 1990). This is translated into Mx in the Appendix  script. The specification of this example is very similar to the multivariate genetic factor model described earlier in this chapter. The three common factors are specified in nvar$\times$1 matrices X, W and Z, where nvar is defined as 4, representing the four atopy measures. The genetic and environmental specifics are estimated in nvar$\times$nvar matrices G and F. The genetic, dominance and specific environmental covariance matrices are then calculated in the algebra section. The rest of the script is virtually identical to that for the univariate model.

One important new feature of the model shown in Figure 10.1 is the treatment of variance specific to each variable. Such residual variance does not generally receive much attention in regular non-genetic factor analysis, for at least two reasons. First, the primary goal of factor analysis (and of many multivariate methods) is to understand the covariance between variables in terms of reduced number of factors. Thus the residual, variable specific, components are not the focus. A second reason is that with phenotypic factor analysis, there is simply no information very similar to further decompose the variable specific variance. However, in the case of data on groups of relatives, we have two parallel goals of understanding not only the within-person covariance for different variables, but also the across-relatives covariance structure both within and across variables. The genetic and environmental factor structure at the top of Figure  addresses the genetic and environmental components of variance common to the different variables. However, there remains information to discriminate between genetic and environmental components of the residuals, which in essence answers the question of whether family members correlate for the variable specific portions of variance.

A second important difference in this example -- using correlation matrices in which diagonal variance elements are standardized to one -- is that the degrees of freedom available for model testing are different from the case of fitting to covariance matrices in which all $k(k+1)/2$ elements are available, where $k$ is the number of input variables. We encountered this difference in the univariate case in Section 6.10.1, but it is slightly more complex in multivariate analysis. For correlation matrices, since the $k$ diagonal elements are fixed to one, we apparently have $g \times k$ fewer degrees of freedom than if we were fitting to covariances, where $g$ is the number of data groups. However, since for a given variable the sum of squared estimates always equals unity (within rounding error), it is apparent that not all the parameters are free, and we may conceptualize the unique environment specific standard deviations (i.e., the $e_i$'s) as being obtained as the square roots of one minus the sum of squares of all the other estimates. Since there are $v$ (number of variables) such constrained estimates, we actually have $v$ more degrees of freedom than the above discussion indicates, the correct adjustment to the degrees of freedom when fitting genetic multivariate models to correlation matrices is $-(g \times k-v)$. Since in most applications $k=2v$, the adjustment is usually $-3v$. In our example $v=4$ and the adjustment is indicated by the option DFreedom=-12. (Note that the DFreedom adjustment applies for the goodness-of-fit chi-squared for the whole problem, not just the adjustment for that group).

Edited highlights of the Mx output are shown below and the goodness-of-fit chi-squared indicates an acceptable fit to the data. The adjustment of $-12$ to the degrees of freedom which would be available were we working with covariance matrices (72) leaves 60 statistics. We have to estimate $3\times 4$ factor loadings and $2\times 4$ specific loadings (20 parameters in all), so there are $60-20=40$ d.f. It is a wise precaution always to go through this calculation of degrees of freedom -- not because Mx is likely to get them wrong, but as a further check that the model has been specified correctly.

Table 10.7: Mx parameter estimates from the independent pathway model

	$E_{Atopy}$	$H_{Atopy}$	$D_{Atopy}$	$H_{spec}$	$E_{Spec}$
Asthma	.320	.431	.466	.441	.548
Hayfever	.494	.772	.095	.000	.388
Dust Allergy	.660	.516	.431	.297	-.159
Eczema	.092	.221	.260	.712	.606
$\chi^2=38.44$, 40 df, p=.540

We can test variations of the above model by dropping the common factors one at a time, or by setting additive genetic specifics to zero. This is easily done by dropping the appropriate elements. Note that fixing $E$ specifics to zero usually results in model failure since it generates singular expected covariance matrices ($\Sigma$). Neither does it make biological sense since it is tantamount to saying that a variable can be measured without error; it is hard to think of a single example of this in nature! We could also elaborate the model by specifying a third source of specific variance components, or by substituting shared environment for dominance, either as a general factor or as specific variance components.