3 Sibling Interaction Model

Suppose that we are considering a phenotype like number of cigarettes smoked. For the sake of exposition we will set aside questions about the appropriate scale of measurement, what to do about non-smokers and so on, and assume that there is a well-behaved quantitative variable, which we can call `smoking' for short. What we want to specify is the influence of one sibling's (twin's) smoking on the other sibling's (cotwin's) smoking. Figure 8.2 shows a path diagram which extends the basic univariate model for twins to

Figure 8.2: Path diagram for univariate twin data, incorporating sibling interaction.

include a path of magnitude $s$ from each twin's smoking to the cotwin. If the path $s$ is positive then the sibling interaction is essentially cooperative, i.e., the more (less) one twin smokes the more (less) the cotwin will smoke as a consequence of this direct influence. We can easily conceive of a highly plausible mechanism for this kind of influence when twins are cohabiting; as a twin lights up she offers her cotwin a cigarette. If the path $s$ is negative then the sibling interaction is essentially competitive. The more (less) one twin smokes the less (more) the cotwin smokes. Although such competition contributes negatively to the covariance between twins, it may well not override the positive covariance resulting from shared familial factors. Thus, even in the presence of competition the observed phenotypic covariation may still be positive. If interactions are cooperative in some situations and competitive in others, our analyses will reveal the predominant mode. But before considering the detail of our expectations, let us look more closely at how the model is specified. The linear model is now:

$$P_1 = sP_2 + aA_1 + cC_1 + eE_1$$ (45)

$$P_2 = sP_1+aA_2+cC_2+eE_2$$ (46)

In matrix form we have

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

or

$$\bf y = {\bf B} \bf y + \bf G \bf x$$

In this form the B matrix is a square matrix with the number of rows and columns equal to the number of dependent variables. The leading diagonal of the B matrix contains zeros. The element in row $i$ and column $j$ represents the path from the $j^{th}$ dependent variable to the $i^{th}$ dependent variable. From this equation we can deduce, as shown in more detail below, that:

$$\bf y (\bf I - \bf B) = \bf G \bf x$$

$$\bf y = {(\bf I - \bf B)}^{-1} \bf G \bf x$$