2 Scalar Effects Sex-limitation Model

The scalar sex-limitation model is a sub-model of both the general model and the common effects model. In the scalar model, not only are the sex-specific effects removed, but the variance components for females are all constrained to be equal to a scalar multiple ($k^{2}$) of the male variance components, such that $a^{2}_{f}$ = $k^{2} a^{2}_{m}$, $d^{2}_{f}$ = $k^{2} d^{2}_{m}$, and $e^{2}_{f}$ = $k^{2} e^{2}_{m}$. As a result, the standardized variance components (e.g., heritability estimates) are equal across sexes, even though the unstandardized components differ. Figure 9.2 shows a path diagram for DZ opposite-sex under the scalar sex-limitation model, and Appendix  provides the Mx specification. Unlike the model in Figure , the scalar model does not include separate parameters for genetic and environmental effects on males and females -- instead, these effects are equated across the sexes. Because of this equality, negative estimates of male-female genetic covariance cannot result. To introduce a scaling factor for the male (or female) variance components, we can pre and postmultiply the expected variances by a scalar.

Figure 9.2: The scalar genotype $\times$ sex interaction model for twin data. Path diagram is shown for DZ opposite-sex twin pairs. The $\alpha$ = 0.5 and $\beta$ = 0.25.

The full scalar sex-limitation model may be compared to the full common effects model using a $\chi^{2}$ difference test with 2 degrees of freedom. Similarly, the scalar sex-limitation model may be compared to the model with no sex differences (that is, one which fixes k to 1.0) using a $\chi^{2}$ difference test with a single degree of freedom. The restricted sex-limitation models described in this section are not an exhaustive list of the sub-models of the general sex-limitation model. Within either of these restricted models (as within the general model), one can test hypotheses regarding the significance of genetic or environmental effects. Also, within the common effects sex-limitation model, one may test whether specific components of variance are equal across the sexes (e.g., $a_{m}$ may be equated to $a_{f}$, or $e_{m}$ to $e_{f}$). Again, sub-models may be compared to more saturated ones through $\chi^{2}$ difference tests, or to models with the same number of parameters with Akaike's Information Criteria.