5 Mx Example: The Variance Components Model

We include the variance components parameterization of the basic structural equation model for completeness. It will not be developed and applied in as great detail as the path coefficients parameterization because (i) it is difficult to generalize to more complex pedigree structures or multivariate problems, and (ii) doing so would contribute much by weight but little by insight to this volume. Readers seeking an easy introduction to twin models in Mx may skip this section and focus their attention on Section 6.4, the path coefficients parameterization.

For MZ and DZ twin pairs reared in the same family, the variance components parameterization is presented in (Figure 5.4a). Under the simplifying assumptions of the present chapter, the $2\times 2$ expected covariance matrix of twin pairs (sigma) will be, in terms of variance components,

$$\left[ \begin{array}{ll} V_E + V_C + V_A + V_D & \omega_i V... ...V_A + \delta_i V_D & V_E + V_C + V_A + V_D \end{array}\right]$$

where $\omega_{i}$ is 1 for twins, full sibs or adoptees reared in the same household, but 0 for separated twins or other biological relatives reared apart; $a_{i}$ is 1 for MZ twin pairs, 0.5 for DZ pairs, full sibs, or parents and offspring, and 0 for genetically unrelated individuals; and $\delta_{i}$ is 1 for MZ pairs, 0.25 for DZ pairs or full sibs, and 0 for most other relationships. In terms of path coefficients, we need only substitute ${V_E} = e^{2}, {V_C} = c^{2}, {V_A} = h^{2}$, and ${V_D} = d^{2}$.

In data on twin pairs reared together the effects of shared environment and genetic dominance are confounded. If both additive genetic effects and shared environmental effects contribute to variation in a trait, the covariance of DZ twin pairs will be less than the MZ covariance, but greater than one-half the MZ covariance. If both additive genetic effects and dominance genetic effects contribute to variation in a trait, the covariance of DZ pairs will be less than one-half the MZ covariance. In terms of variance components, therefore, a substantial dominance genetic effect will lead to a negative estimate of the shared environmental variance component, if a model allowing for additive genetic and shared environmental variance components is fitted; while conversely a substantial shared environmental effect will lead to a negative estimate of the dominance genetic variance component, if a model allowing for additive and dominance genetic variance components is fitted (Martin et al., 1978). In terms of path coefficients, however, since we are estimating parameters $c$ or $d$, $c^{2}$ or $d^{2}$ can never take negative values, and so we will obtain an estimate of $c=0$ in the presence of dominance, or $d=0$ in the presence of shared environmental effects. Additional data on separated twin pairs (Jinks and Fulker, 1970) or on the parents or other relatives of twins (Fulker, 1982; Heath, 1983) are needed to resolve the effects of shared environment and genetic dominance when both are present.