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 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.