2 Basic Univariate Model without Interaction

Up to this point, we have been concerned primarily with decomposing observed phenotypic variation into its genetic and environmental components. This has been accomplished by estimating the paths from latent or independent variables to dependent variables. A basic univariate path diagram is set out in Figure 8.1.

Figure 8.1: Basic path diagram for univariate twin data.

This path diagram shows the deviation phenotypes $P_1$ and $P_2$, of a pair of twins. Here we refer to the phenotypes as deviation phenotypes to emphasize the point that the model assumes variables to be measured as deviations from the means, which is the case whenever we fit models to covariance matrices and do not include means. The deviation phenotypes $P_1$ and $P_2$ result from their respective additive genetic deviations, $A_1$ and $A_2$, their shared environment deviations, $C_1$ and $C_2$, and their non-shared environmental deviations, $E_1$ and $E_2$. The linear model corresponding to the path diagram is:

$$\begin{array}{lllll} P_1&=&aA_1+cC_1+eE_1\\ P_2&=&aA_2+cC_2+eE_2\end{array}$$

In matrix form we can write:

$$\left( \begin{array}{r} P_1 \ P_2 \end{array} \right) = \... ...ay}{r} A_1\ C_1\ E_1\ A_2\ C_2\ E_2 \end{array} \right)$$

or as a matrix expression

$$\bf y = \bf G \bf x$$

Details of specifying and estimating this basic univariate model are given in Chapter 6. One of the interesting assumptions of this basic ACE model is that the siblings' or twins' phenotypes have no influence on each other. This assumption may well be true of height or finger print ridge count, but is it necessarily true for a behavior like smoking, a psychiatric condition like depression, delinquent behavior in children or even an anthropometric measure like the body mass index? We should not, in general, assume a priori that a source of variation is absent, especially when an empirical test of the assumption may be readily performed. However, we may as well recognize from the onset that evidence for social interactions or sibling effects is pretty scarce. The fact is that usually one form or another of the basic univariate model adequately describes a twin or family data set, within the power of the study. This tells us that there will not be evidence of significant social interactions since, were such effects substantial, they would lead to failure of basic univariate models. Nevertheless, this extension of the basic models is of considerable theoretical interest and studying its outcome on the expectations derived from the models can provide insight into the nature and results of social influences. The applications to bivariate and multivariate causal modeling are perhaps even more intriguing and will be taken up in chapter .