1 General Model for Sex-limitation

The general sex-limitation model allows us to (1) estimate the magnitude of genetic and environmental effects on male and female phenotypes and (2) determine whether or not it is the same set of genes or shared environmental experiences that influence a trait in males and females. Although the first task may be achieved with data from like-sex twin pairs only, the second task requires that we have data from opposite-sex pairs (Eaves et al., 1978). Thus, the Mx script we describe will include model specifications for all 5 zygosity groups (MZ-male, MZ-female, DZ-male, DZ-female, DZ-opposite-sex). To introduce the general sex-limitation model, we consider a path diagram for opposite-sex pairs, shown in Figure 9.1. Included among the ultimate variables in the diagram are female and male additive genetic ($A_{f}$ and $A_{m}$), dominant genetic ($D_{f}$ and $D_{m}$), and unique environmental ($E_{f}$ and $E_{m}$) effects, which influence the latent phenotype of the female ($P_{f}$) or male ($P_{m}$) twin. The additive and dominant genetic effects are correlated within twin pairs ($\alpha$ = 0.50 for additive effects, and $\beta$ = 0.25 for dominant effects) as they are for DZ like-sex pairs in the simple univariate ACE model. This correlational structure implies that the genetic effects represent common sets of genes which influence the trait in both males and females; however, since $a_{m}$ and $a_{f}$ or $d_{m}$ and $d_{f}$ are not constrained to be equal, the common effects need not have the same magnitude across the sexes.

Figure 9.1: The general genotype $\times$ sex interaction model for twin data. Path diagram is shown for DZ opposite-sex twin pairs. $\alpha$ = 0.5 and $\beta$ = 0.25.

Figure 9.1 also includes ultimate variables for the male (or female) member of the opposite-sex twin pair ($A'_{m}$ and $D'_{m}$) which do not correlate with genetic effects on the female phenotype. For this reason, we refer to $A'_{m}$ and $D'_{m}$ as sex-specific variables. Significant estimates of their effects indicate that the set of genes which influences a trait in males is not identical to that which influences a trait in females. To determine the extent of male-female genetic similarity, one can calculate the male-female genetic correlation ($r_{g}$). As usual (see Chapter 2) the correlation is computed as the covariance of the two variables divided by the product of their respective standard deviations. Thus, for additive genetic effects we have

$$r_g=\frac{a_m a_f}{\sqrt{a_f^2(a_m^2+a_m^{\prime 2})}}$$

Alternatively, a similar estimate may be obtained for dominant genetic effects. However, the information available from twin pairs reared together precludes the estimation of both sex-specific parameters, $a'_{m}$ and $d'_{m}$ and, consequently, both additive and dominance genetic correlations. Instead, models including $A'_{m}$ or $D'_{m}$ may be fit to the data, and their fits compared using appropriate goodness-of-fit indices, such as Akaike's Information Criteria (AIC; Akaike, 1987; see Section ). This criterion may be used to compare the fit of an $ACE$ model to the fit of an $ADE$ model. AIC is one member of a class of indices that reflect both the goodness of fit of a model and its parsimony, or ability to account for the observed data with few parameters. To generalize the model specified in Figure 9.1 to other zygosity groups, the parameters associated with the female phenotype are equated to similar effects on the phenotypes of female same-sex MZ and DZ twin pairs. In the same manner, all parameters associated with the male phenotype (reflecting effects which are common to both sexes as well as those specific to males) are equated to effects on both members of male same-sex MZ and DZ pairs. As a result, the model predicts that variances will be equal for all female twins, and all male twins, regardless of zygosity group or twin status (i.e., twin 1 vs. twin 2). The model does not necessarily predict equality of variances across the sexes.