2 Psychometric Model

Figure 11.2 shows a bivariate psychometric or `common pathway' model. Implementation of this model in Mx can be achieved by the

Figure 11.2: Psychometric or common pathway 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 latent phenotypes of the twins ($PT$), and rater specific variance (e.g., $A_M$, $C_M$ and $E_M$).

approaches illustrated in Chapter 10. The psychometric model estimates, for each source of influence ($A$, $C$, and $E$) the variance for mothers' ratings, the variance for fathers' ratings and the covariance between these ratings. These estimates are subject to the constraints that the covariances are positive and neither individual rating variance can be less than the covariance between the ratings. The psychological implication of this psychometric model is that the mothers' and fathers' ratings are composed of consistent assessments of reliable trait variance, together with assessments of specific phenotypes uncorrelated between the parents. There are some technical points to note with this model. First, bivariate data for MZ and DZ twins (of a given sex) yield 20 observed variances and covariances. However, only 9 of these have unique expectations under the classes of model we are considering, the remaining 11 being replicate estimates of particular expectations (e.g., the variance of maternal ratings of MZ twin 1, of MZ twin 2, of DZ twin 1 and of DZ twin 2 are four replicate estimates of the variance of maternal ratings in the population). Given this, we might expect our 9 parameter psychometric model to fit as well as any other 9 parameter model for bivariate twin data. However, there are some implicit constraints in our psychometric model. For example, the phenotypic covariance of mothers' and fathers' ratings cannot be greater than the variance of either type of rating. Such constraints may cause the model to fail in some circumstances even though the 9 parameter biometric model discussed below (Figure 11.3) may fit adequately. The second technical point is that if we do not constrain the loadings of the common factor to be equal on the mothers' ratings and on the fathers' ratings, and assume that there is no specific genetic variance for either mothers' ratings or for fathers' ratings, then this variant of the psychometric model is formally equivalent to our version in the Neale and Stevenson bias model described above. In this case the ``shared environmental'' specific variances for the mothers' and fathers' ratings are formally equivalent to the maternal and paternal biases in the earlier model, while the ``non-shared'' specific variances are equal to the unreliability variance of the earlier parameterization. Thus, although the 9 parameter psychometric model and the bias model do not form a nested pair (Mulaik et al., 1989), they represent alternative sets of constraints on a more general 10 parameter model (which is not identified with two-rater twin data) and these constrained models may be compared in terms of parsimony and goodness of fit. Furthermore, we may consider a restricted bias model in which the scaling factor in Figure 11.1 is set to unity and which, therefore, has 7 free parameters and is nested within both the psychometric model and the unrestricted bias model. This restricted bias model may therefore be tested directly against either the psychometric or the unrestricted bias models by a likelihood ratio chi-square.