1 Application to CBC Data

By way of illustration we shall analyze data collected using the Achenbach Child Behavior Checklist (CBC; Achenbach & Edelbrock, 1983) on juvenile twins aged 8 through 16 years living in Virginia. Mothers were asked the extent to which a series of problem behaviors were characteristic of each of their twin children over the last six months. The 118 problem behaviors that were rated can be categorized, on the basis of empirical clustering, into two broad dimensions of internalizing and externalizing problems. The former are typified by fears, psychosomatic complaints, and symptoms of anxiety and depression. Externalizing behaviors are characterized by ``acting out'' -- delinquent and aggressive behaviors. The factor patterns vary somewhat with the age and sex of the child but there are core items which load on the broad factors in both boys and girls at younger (6-11 years) and older (12-16 year) ages. The 24 core items for the externalizing dimension analyzed by Silberg et al. (1992) and Hewitt et al. (1992) include among other things: arguing a lot, destructive behavior, disobedience, fighting, hanging around with children who get into trouble, running away from home, stealing, and bad language. For such behaviors we might suspect that siblings will influence each other in a cooperative manner through imitation or mutual reinforcement. The Mx script in Appendix  specifies the model for sibling interactions shown in Figure 8.2. By varying the script, the standard E, AE, CE, and ACE models may be fitted to the data to obtain the results shown in Table 8.1.

Table 8.1: Preliminary results of model fitting to externalizing behavior problems in Virginia boys from larger families.

				Parameter
	Fit statistics			Estimates
Model	df	$\chi^2$	AIC	a	c	e
AE	4	32.57	24.6	.78	--	.33
CE	4	29.80	21.8	--	.78	.43
ACE	3	4.95	-1.0	.50	.64	.34

Clearly the variation and co-aggregation of boys' behaviors problems cannot be explained either by a model which allows only for additive genetic effects (along with non-shared environmental influences), nor by a model which excludes genetic influences altogether. The ACE model fits very well ($p=.18$) and suggests a heritability of 33% with shared environmental factors accounting for 52% of the variance. But is the ACE model the best in this case? We observe that the pooled individual phenotypic variances of the MZ twins (0.915) are greater than those of the DZ twins (0.689) and, although this discrepancy is apparently not statistically significant with our sample sizes (171 MZ pairs and 194 DZ pairs), we might be motivated to consider sibling interactions. Fitting the model shown in Figure 8.2 yields results given in Table 8.2.

Table 8.2: Parameter estimates and goodness of fit statistics from fitting models of sibling interaction to CBC data.

	Fit statistics			Parameter estimates
Model	df	$\chi^2$	AIC	a	c	e	s
E+s	4	29.80	21.8	--	--	*	*
AE+s	3	1.80	-4.2	.611	--	.419	.230
CE+s	3	29.80	21.8	--	.882	.282	-.101
ACE+s	2	1.80	-2.2	.611	.000$^1$	.419	.230
* Indicates parameters out of bounds.
$^1$This parameter is fixed on the lower bound (0.0) by Mx

Our general conclusion is that while the evidence for social interactions is not unequivocal, a model including additive genetic effects, non-shared environments, and reciprocal sibling cooperation provides the best account of these data.