5 Application to Data from the Child Behavior Checklist

To illustrate the application of these models we consider an updated set of data first presented by Hewitt, et al., (1990) and now based on 983 families where both parents rated each of their twin children using Achenbach's Child Behavior Checklist (CBC; Achenbach and Edelbrock, 1983). For the full analysis, published in Hewitt et al. (1992), data from a population-based sample of 500 MZ twin pairs and 483 DZ twin pairs were considered and ratings were included irrespective of the biological or social relationship of the parent to the child. The children were Caucasian and ranged in age from 8 to 16 years. Ratings on 23 core items assessing children's internalizing behavior in both younger and older children and in either boys or girls were totalled to obtain an internalizing scale score for each child. The items contributing to this scale are listed in Appendix . For illustrative purposes in this chapter we just consider the ``prepubertal" subsample of younger children aged 8-11 years. More detailed analyses, including older children, may be found in Hewitt et al. (1992). The scale scores were log-transformed to approximate normality and adjusted for linear regression on age and sex within age cohorts. The observed variances, covariances, and correlations of the resulting scores are given in Table 11.1 by zygosity and sex group.

Table 11.1: Observed variance-covariance matrices (lower triangle) and twin correlations (above the diagonal) for parental ratings (mother (Mo); father (Fa)) of internalizing behavior problems in five zygosity-sex groups (MZ female, N=96; MZ male, N=102; DZ female, N=102; DZ male, N=97; DZ male-female, N=103). All twins were between 8 and 11 years at assessment.

Zygosity/sex		Male					Female
		Twin 1		Twin 2			Twin 1		Twin 2
		Mo	Fa	Mo	Fa		Mo	Fa	Mo	Fa
MZ	MoT1	.675	.40	.74	.43	MoT1	.694	.47	.84	.46
	FaT1	.265	.652	.35	.77	FaT1	.312	.638	.37	.72
	MoT2	.513	.237	.714	.51	MoT2	.569	.238	.666	.45
	FaT2	.292	.513	.354	.676	FaT2	.308	.461	.293	.647
DZ	MoT1	.621	.47	.70	.34	MoT1	.565	.41	.55	.29
	FaT1	.315	.719	.35	.73	FaT1	.241	.604	.25	.57
	MoT2	.434	.236	.623	.37	MoT2	.291	.137	.488	.52
	FaT2	.233	.531	.251	.743	FaT2	.171	.347	.285	.604
DZMF	MoT1	.538	.26	.49	.18
	FaT1	.162	.730	.17	.56
	MoT2	.243	.102	.465	.37
	FaT2	.103	.372	.191	.574

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: Model comparisons for internalizing problems analysis.

	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

data for younger children, with the psychometric model being preferred by Akaike's Information 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 fitting bias, psychometric, and biometric models for parental ratings of internalizing behaviors.

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
			$e_m$	.278	.372	$e_{fm}$	.096	.097
			$e_f$	.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: 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
A	.268	.138	.134 (.70)	.269	.121	.181 (1.0)
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)
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% [ $= .269/(.269+.077+.036)$].