2 Numerical Illustration

To illustrate these effects numerically, let us consider a simplified situation in which $a^2=.5, d^2=0, c^2=0, e^2=.5$ in the absence of social interaction (i.e., $s=0$); in the presence of strong cooperation, $s=.5$; and in the presence of strong competition, $s=-.5$. Table 8.4 gives the numerical values for MZ and DZ twins and unrelated pairs of individuals reared together (e.g., adoptive siblings).

Table 8.4: Effects of strong sibling interaction on the variance and covariance between MZ, DZ, and unrelated individuals reared together. The interaction parameter $s$ takes the values $0$, $.5$, and $-.5$ for no sibling interaction, cooperation, and competition, respectively.

	MZ twins			DZ twins			Unrelated
Interaction	Var	Cov	$r$	Var	Cov	$r$	Var	Cov	$r$
None	1.00	.50	.50	1.00	.25	.25	1.00	.00	.00
Cooperation	3.11	2.89	.93	2.67	2.33	.88	2.22	1.78	.80
Competition	1.33	.44	.33	1.78	-.67	-.38	2.22	-1.78	-.80

In terms of correlations, phenotypic cooperation mimics the effects of shared environment while phenotypic competition may mimic the effects of non-additive genetic variance. However, the effects can be distinguished because social interactions result in different total phenotypic variances for differently related pairs of individuals. All of the other kinds of models we have considered predict that the population variance of individuals is not affected by the presence or absence of relatives. However, cooperative interactions increase the variance of more closely related individuals the most, while competitive interactions increase them the least and under some circumstances may decrease them. Thus, in twin data, cooperation is distinguished from shared environmental effects because cooperation results in greater total phenotypic variance in MZ than in DZ twins. Competition is distinguished from non-additive genetic effects because it results in lower total phenotypic variance in MZ than in DZ twins. This is the bottom line: social interactions cause the variance of a phenotype to depend on the degree of relationship of the social actors. There are three observations we should make about this result. First, a test of the contrary assumption, i.e., that the total observed variance is independent of zygosity in twins, was set out by Jinks and Fulker (1970) as a preliminary requirement of their analyses and, as has been noted, is implicitly provided whenever we fit models without social interactions to covariance matrices. For I.Q., educational attainment, psychometric assessments of personality, social attitudes, body mass index, heart rate reactivity, and so on, the behavior genetic literature is replete with evidence for the absence of the effects of social interaction. Second, analyses of family correlations (rather than variances and covariances) effectively standardize the variances of different groups of individuals and throw away the very information we need to distinguish social interactions from other influences. Third, if we are working with categorical data and adopting a threshold model (see Chapter 2), we can make predictions about the standardized thresholds in different groups. Higher quantitative variances lead to smaller (i.e., less deviant) thresholds and therefore higher prevalence for the extreme categories. Thus, for example, if abstinence vs. drinking status is influenced by sibling cooperation on a latent underlying phenotype, and abstinence has a frequency of 10% in DZ twins, we should expect a higher frequency of abstinence in MZ twins. These models are relatively simple to implement in Mx (Neale, 1997).