4 Comparison of Models

We have considered four alternative models for parental ratings of children's behavior. Each model is for bivariate twin data where the two variables are the special case of mothers' ratings and fathers' ratings of the children's behavior. The least restrictive model, the biometric model, provides a baseline for comparison with the psychologically more informative psychometric and bias models. The most restricted bias model may be formally tested by likelihood ratio chi-square against either the psychometric or the unrestricted bias models. However, these latter two are not themselves nested. The relationships between these models, without taking into account sex limitations, are summarized in Figure 11.4. In this figure the solid arrows

Figure 11.4: Diagram of nesting of biometric, psychometric, and rater bias models.

represent the process of constraining a more general model to yield a more restrictive model; the model at the arrow head is nested within the model at the tail of the arrow and may be tested against it by a likelihood ratio chi square. The dashed arrows represent rotational constraints on the biometric model. The nine parameter psychometric model requires, for example, that the covariance between maternal and paternal ratings be no greater than the variance of either type of rating; in factor analytic terms this would require a constrained rotation of the biometric model solution. The ten parameter psychometric model, allowing $\alpha$ not equal to unity, still imposes the constraints that the contributions of the common influences to the variance of maternal ratings, the variance of paternal ratings, and the covariance between them be in the ratio $1:\alpha^2:\alpha$ for each source of influence. Thus, even though this model has 10 parameters (and hence is not identified for bivariate twin data) any of its solutions, arrived at by fixing one of the parameters to an arbitrary value, will again represent in factor analytic terms a constrained rotation of the biometric model.