1 Rater Bias Model

Figure 11.1 shows a path model for the ratings of twins by their

Figure 11.1: Model for ratings of a pair of twins (1 and 2) by their parents. Maternal and paternal observed ratings ($MRT$ and $FRT$) are linear functions of the true phenotypes of the twins ($PT$), maternal and paternal rater bias ($B_M$ and $B_F$), and residual error ($R_{MRT}$ and $R_{FRT}$).

parents, in which the phenotypes of a pair of twins ($PT_1$ and $PT_2$) are functions of additive genetic influence ($A$), shared environments ($C$) and non-shared environments ($E$). The ratings by the mother ($MRT$) and father ($FRT$) are functions of the twin's phenotype, the maternal ($B_M$) or paternal ($B_F$) rater bias, and residual errors ($R_{1MRT}$, etc). If this model is correct, the following discriminations may be made:

1. the structural analysis of the latent phenotypes of the children can be considered independently of the rater biases and unreliability of the ratings;
2. the extent of rater biases and unreliability of ratings can be estimated;
3. the relative accuracy of maternal and paternal ratings can be assessed.

A simple implementation of the model in Mx is achieved by defining the model by the following matrix equations:

$$\left( \begin{array}{r} MRT_1\ MRT_2\ FRT_1\ FRT_2 \end... ...) \left(\begin{array}{r} PT_1\ PT_2 \end{array} \right)\\$$

$$+\left( \begin{array}{rrrr} r_{m1} & 0 & 0 & 0\ 0 & r_{m2} & 0 ... ...{array}{r} R_{1 MRT}\ R_{2 MRT}\ R_{1 FRT}\ R_{2 FRT} \end{array} \right)$$ (68)

or

$$\bf y = \bf B\bf b + \bf L\bf l + \bf R\bf r$$

and

$$\left( \begin{array}{r} PT_1\ PT_2 \end{array} \right) = \left( \begin{array}{cccccc} a& c& e& 0& 0& 0\ 0& 0& 0& a& c& e... ...begin{array}{r} A_1 \ C_1 \ E_1 \ A_2 \ C_2 \ E_2 \end{array} \right)$$ (69)

or

$$\bf l = \bf G\bf x$$

Thus

$$\bf y = \bf B\bf b + \bf L\bf G\bf x + \bf R\bf r$$

Then, the covariance matrix of the ratings is given by

$${\cal E}\{{\bf yy}'\} = {\cal E}\{\bf B\bf b + \bf L\bf G\bf x + \bf R\bf r\} \{\bf B\bf b + \bf L\bf G\bf x + \bf R\bf r\}'$$ (70)

$$= \bf B\bf B' + \bf R\bf R' + \bf L\bf G \cal E \{ \bf x\bf x'\} \bf G'\bf L'$$ (71)

The term $\bf G \cal E \{ \bf x\bf x'\} \bf G'$ generates the usual expectations for the ACE model. The expectations are filtered to the observed ratings through the factor structure L and are augmented by the contributions from rater bias ($\bf B$) and residual influences ($\bf R$). An Mx script for this model is listed in Appendix . In considering the rater bias model, and the other models discussed below, we should note that parameters need not be constrained to be equal when rating boys and girls and, as Neale and Stevenson (1988) pointed out, we need not necessarily assume that parental biases are equal for MZ and DZ twins' ratings. This latter relaxation of the parameter constraints allows us to consider the possibility that twin correlations differ across zygosities for reasons related to differential parental biases based on beliefs about their twins' zygosity.