3 Biometric Model

The final model to be considered is the biometric model shown in Figure 11.3, and

Figure 11.3: Biometric or independent 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 general (subscript $M$) and restricted (subscript $F$) genetic and environmental factors.

again may be readily implemented using the procedure described in Chapter 10. In this model there are two factors for each source of variance ($A$, $C$, and $E$). One factor is subscripted M, e.g., $A_M$, and loads on the maternal rating ($MRT$) and on the paternal rating ($FRT$). The other factor subscripted F, e.g., $A_F$, loads only on the paternal rating. Thus, for each source of influence we estimate three factor loadings which enable us to reconstruct estimates of the contribution of this influence to the variance of maternal ratings, the variance of paternal ratings and the covariance between them. Which factor loads on both types of rating and which on only one is arbitrary. This type of model is referred to as a Cholesky model or decomposition or a triangular model and provides a standard general approach to multivariate biometrical analysis (see Chapter 10). This biometric model is a saturated unconstrained model for the nine unique expected variances and covariances (in the absence of sibling interactions or other influences giving rise to heterogeneity of variances across zygosities, cf. Heath et al., 1989) and provides the most general approach to estimating the genetic, shared environmental and non-shared environmental components of variance and covariance. However, the absence of theoretically motivated constraints lessens the psychological informativeness of the model for the analysis of parental ratings. In this context, we may use the biometric model first to test the adequacy of the assumption that of the 20 observed variances and covariances for bivariate twin data of a given sex, 11 represent replicate estimates of the 9 unique structural expectations. Once again, sex differences in factor loadings (scalar sex limitation) may in principle lead to model failure for opposite sex data even though the biometric model is adequate for a given sex. In this case the non-scalar sex limitation model described in Heath et al. (1989) and Chapter 9 would be required. The bivariate biometric model provides a baseline for comparison of the adequacy of the psychometric and bias models. This comparison alerts us to the important possibility that mothers and fathers are assessing different (but possibly correlated) phenotypes as, for example, they might be if mothers and fathers were reporting on behaviors observed in different situations or without a common understanding of the behavioral descriptions used in the assessment protocol.