2 Results of Model-Fitting

A summary of the adequacy of the models fitted to these data on younger children's internalizing problems is shown in Table 11.2. The illustrative program in Appendix  runs the analysis for the bias model with 34 degrees of freedom.

As can be seen from Table 11.2, all three types of model give excellent fits to the

Table 11.2: Summary of models fitted to younger children's internalizing problems.

	Fit statistics
Model$^*$	df	$\chi^2$	AIC
Restricted bias	36	30.07	-41.9
Bias	34	25.78	-42.2
Psychometric	32	20.71	-43.3
Biometric	32	20.95	-43.1

$^*$All models allow scalar sex limitation of individual parameters. The solution avoiding effects of opposite sign in boys and girls by allowing for sex-limitation of shared environmental influences specific to fathers' ratings gives $\chi^2_{31}=20.76$.

data for younger children, with the psychometric model being preferred by Akaike's criterion. Thus, our first conclusion would be that to a very good approximation, mothers and fathers can be assumed to be rating the same phenotype in their children when using the Child Behavior Checklist, at least as far as these internalizing behaviors are concerned. This may not be so for other behaviors or assessment instruments and in each particular case the assumption ought to be tested by a comparison of models of the kind we have described. Although there are numerous submodels or alternative models that may be considered, (for example: no sex limitation; non-scalar sex-limitation; and setting non-significant parameters to zero), only a subset will be presented here for illustration.

Table 11.3 shows the parameter

Table 11.3: Parameter estimates from the bias, psychometric, and biometric models for parental ratings of internalizing behaviors in younger children.

Bias model			Psychometric model			Biometric model
Path	Boys	Girls	Path	Boys	Girls	Path	Boys	Girls
$a$	.519	.163	$a$	.370	.145	$a_m$	.513	.134
$c$	.277	.363	$a_m$	.338	-.027	$a_{fm}$	.261	.132
$e$	.189	.156	$a_f$	-.069	.281	$a_f$	.265	.286
$a$	.671	1.416
$b_m$	.320	.545	$c$	.308	.449	$c_m$	.440	.659
$b_f$	.509	.473	$c_m$	.332	.479	$c_{fm}$	.225	.308
$r_m^2$	.074	.154	$c_f$	.437	.507	$c_f$	.490	.603
$r_f^2$	.175	.115
			e	.176	.200	$e_m$	.328	.423
			em	.278	.372	$e_{fm}$	.096	.097
			ef	.386	.333	$e_f$	.414	.377

estimates for the full bias and psychometric models allowing for scalar sex limitation and, in the case of the biometric model, we have allowed for non-scalar sex-limitation of the shared environmental influences specific to fathers' ratings ( $\chi^2_{31}=20.76$ for the model presented with the correlation between boys' and girls' effects of this kind estimated at 0.86 rather than unity). To show the relationship between the more parsimonious bias model and the full parameterization of the biometric model, in Table 11.4 we

Table 11.4: The contributions to the phenotypic variances and covariance of mothers' and fathers' ratings of young boys' internalizing behavior.

	Biometric model			Bias model
	Ratings		Cov (r)	Ratings		Cov (r)
Source	Mother	Father	M-F	Mother	Father	M-F
Additive
genetic (A)	.268	.138	.134 (.70)	.269	.121	.181 (1.0)
Shared
Env. (C)	.194	.291	.099 (.42)	.077	.035	.051 (1.0)
Bias	--	--	--	.102	.259	.000 (.00)
C + Bias	.194	.291	.099 (.42)	.179	.294	.051 (.22)
Random
Env. (E)	.108	.181	.031 (.22)	.036	.016	.024 (1.0)
Residual	--	--	--	.074	.175	.000 (.00)
E + Residual	.108	.181	.031 (.22)	.110	.191	.024 (.17)
Phenotypic
Total	.564	.609	.264 (.45)	.558	.606	.256 (.44)
Italicized numbers indicate parameters are fixed ex hypothesi in the rater bias
model.

present the expected contributions of A, C, and E to the variance of mothers' ratings, fathers' ratings, and the covariances between mothers' and fathers' ratings. What Table 11.4 shows is that, providing the rater bias model is adequate, we can partition the environmental variance of mothers' and fathers' ratings into variance attributable to those effects consistently rated by both parents and those effects which either represent rater bias or residual unreliable environmental variance. In this particular case, while a univariate consideration of maternal ratings would suggest a heritability of 47% [ $= .263/(.263+.194+.108)$], a shared environmental influence of 34%, and a non-shared environmental influence of 19%, it is clear that more than half of the shared environmental influence can be attributed to rater bias, and the major portion of the non-shared environmental influence to unreliability or inconsistency between ratings. The heritability of internalizing behaviors in young boys rated consistently by both parents may be as high as 70% [ $= .269/(.269+.077+.036)$].