2 Common Pathway Model for Atopy

In this section we focus on a much more stringent model which hypothesizes that the covariation between symptoms is determined by a single `phenotypic' latent variable called ``atopy.'' Atopy itself is determined by additive, dominance and individual environmental sources of variance. As in the independent pathway model, there are still specific genetic and environmental effects on each symptom. The path diagram for this model is shown in Figure 10.4. Because there is now a latent variable ATOPY which has direct phenotypic paths to each of the symptoms, this has been called the common pathway model (Kendler et al., 1987) or the psychometric factors model (McArdle and Goldsmith, 1990).

Figure 10.4: Common pathway model for four variables.

The Mx script corresponding to this path diagram, given in Appendix , contains several new features. Again, there are a number of alternative ways to specify this model in Mx. We use the same approach as in previous models and specify the genetic and environmental covariance matrices in a calculation group up front. In this example, matrices X, W and Z represent the additive and dominance genetic and specific environmental loadings on the latent phenotype. The factor loadings on the observed variables are estimated in $4\times1$ matrix S. The residual variances are decomposed in genetic and environmental diagonal matrices G and F. The data groups are identical to those of the independent pathway model. One final feature of the model is that since ATOPY is a latent variable whose scale (and hence variance) is not indexed to any measured variable, we must fix its residual variance term (EATOPY) to unity to make the model identified. This inevitably means that the estimates for the loadings contributing to ATOPY are arbitrary and hence so are the paths leading from ATOPY to the symptoms. It is thus particularly important to standardize the solution so that the total variance explained for each symptom is unity. The fixing of the loading on EATOPY clearly has implications for the calculation of degrees of freedom, as we shall see below. The condensed output for this model is presented below, showing the completely standardized estimates which give unit variance for each variable.

Table 10.12: Parameter estimates from the common pathway model for atopy

	$Atopy$	$E_{Atopy}$	$H_{Atopy}$	$D_{Atopy}$	$H_{spec}$	$E_{Spec}$
Asthma	.671	$-$	$-$	$-$	.531	.517
Hayfever	.814	$-$	$-$	$-$	.456	.358
Dust Allergy	.941	$-$	$-$	$-$	-.059	.334
Eczema	.301	$-$	$-$	$-$	.735	.608
Atopy	$-$	.686	.397	.610	$-$	$-$
$\chi^2=51.37$, 46 df, p=.238

Note that here NInput_vars=8 so there are 56 ( $2\times \mbox{\verb*+NI+}(\mbox{\verb*+NI+}-1)/2$) unique correlations. From the above table it appears that 15 parameters have been estimated, but in fact EATOPY was fixed and the four $E$ specifics are obtained by difference, so there are only 10 free parameters in the model, hence 46 degrees of freedom. The latent variable ATOPY has a broad heritability of over 0.6 ( $1-.610^2=.686^2+.397^2$) of which approximately a quarter is due to dominance, and this factor has an important phenotypic influence on all symptoms, particularly dust allergy (0.941) and hayfever (0.814). There are still sizeable specific genetic influences not accounted for by the ATOPY factor on all symptoms except dust allergy ($.059^2$). However, despite the appeal of this model, it does not fit as well as the independent pathway model and the imposition of constraints that covariation between symptoms arises purely from their phenotypic relation with the latent variable ATOPY has worsened fit by $\chi^2=12.93$ for 6 degrees of freedom, which is significant at the 5% level. We conclude that while there are common environmental, additive, and nonadditive genetic factors which influence all four symptoms of atopy, these have differential effects on the symptoms; the additive and non-additive factors, for example, having respectively greater and lesser proportional influence on hayfever than the other symptoms. While it is tempting to interpret this as evidence for at least two genes, or sets of genes, being responsible for the aggregation of symptoms we call atopy, this is simplistic as in fact such patterns could be consistent with the action of a single gene -- or indeed with polygenic effects. For a full discussion of this important point see Mather and Jinks (1982) and Carey (1988).