3 Fitting the Multivariate Genetic Model

To illustrate the genetic common factor model we fit it to the arithmetic computation data, but now using both members of the female twin pairs and specifying two groups for the MZ and DZ twins. The observed variances and correlations examined in this analysis are presented in Table 10.3. Appendix  shows the full Mx script for this model.

Table 10.3: Observed female MZ (above diagonal) and DZ (below diagonal) correlations and variances for arithmetic computation variables.

		Twin 1				Twin 2
		T0	T1	T2	T3	T0	T1	T2	T3
T1	T0	1.0	.81	.83	.87	.78	.65	.71	.68
	T1	.89	1.0	.87	.87	.74	.74	.74	.71
	T2	.85	.90	1.0	.90	.73	.66	.72	.70
	T3	.83	.86	.86	1.0	.74	.71	.74	.75
T2	T0	.23	.31	.36	.34	1.0	.73	.78	.79
	T1	.22	.32	.34	.38	.81	1.0	.86	.87
	T2	.16	.23	.27	.35	.79	.86	1.0	.87
	T3	.23	.31	.34	.37	.81	.86	.87	1.0
	MZ	297.9	229.4	247.4	274.9	281.9	359.7	326.9	281.1
	DZ	259.7	259.9	245.2	249.3	283.8	249.5	262.1	270.9

The results from this common factor model are shown in Table 10.3.3 The parameter estimates in the MX PARAMETER ESTIMATES section indicate a substantial genetic basis for the observed arithmetic covariances, as the genetic loadings are much higher than either the shared and non-shared environmental effects. The unique variances in F also appear substantial but these do not contribute to covariances among the measures, only to the variance of each observed variable. The $\chi^{2}_{56}$ value of 46.77 suggests that this single factor model provides a reasonable explanation of the data. (Note that the 56 degrees of freedom are obtained from $2\times 8(8+1)/2$ free statistics minus 16 estimated parameters).

Table 10.4: Parameter estimates from the full genetic common factor model

	$A_C$	$C_C$	$E_C$	$E_S$
Time 1	15.088	1.189	4.142	46.208
Time 2	13.416	5.119	6.250	39.171
Time 3	13.293	4.546	7.146	31.522
Time 4	13.553	5.230	5.765	34.684
$\chi^2=46.77$, 56 df, p=.806

Earlier in this chapter we alluded to the fact that confirmatory factor models allow one to statistically test the significance of model parameters. We can perform such a test on the present multivariate genetic model. The Mx output above shows that the shared environment factor loadings are much smaller than either the genetic or non-shared environment loadings. We can test whether these loadings are significantly different from zero by modifying slightly the Mx script to fix these parameters and then re-estimating the other model parameters. There are several possible ways in which one might modify the script to accomplish this task, but one of the easiest methods is simply to change the Y to have no free elements. Performing this modification in the first group effectively drops all $C$ loadings from all groups because the Matrices= Group 1 statement in the second and third group equates its loadings to those in the first. Thus, the modified script represents a model in which common factors are hypothesized for genetic and non-shared environmental effects to account for covariances among the observed variables, and unique effects are allowed to contribute to measurement variances. All shared environmental effects are omitted from the model. Since the modified multivariate model is a sub- or nested model of the full common factor specification, comparison of the goodness-of-fit chi-squared values provides a test of the significance of the deleted $C$ factor loadings (see Chapter ). The full model has 56 degrees of freedom and the reduced one: $2\times 8(8+1)/2 - 12 =60$ d.f. Thus, the difference chi-squared statistic for the test of $C$ loadings has $60 - 56 = 4$ degrees of freedom. As may be seen in the output fragment below, the $\chi^{2}_{60}$ of the reduced model is 51.08, and, therefore, the difference $\chi^{2}_{4}$ is $51.08 - 46.77 = 4.31$, which is non-significant at the .05 level. This non-significant chi-squared indicates that the shared environmental loadings can be dropped from the multivariate genetic model without significant loss of fit; that is, the arithmetic data are not influenced by environmental effects shared by twins. Parameter estimates from this reduced model are given below in Table 10.3.3

Table 10.5: Parameter estimates from the reduced genetic common factor model

	$A_C$	$C_C$	$E_C$	$E_S$
Time 1	14.756	$-$	3.559	59.502
Time 2	14.274	$-$	6.331	39.433
Time 3	14.081	$-$	7.047	30.843
Time 4	14.405	$-$	5.845	36.057
$\chi^2=51.08$, 60 df, p=.787

The estimates for the genetic and non-shared environment parameters differ somewhat between the reduced model and those estimated in the full common factor model. Such differences often appear when fitting nested models, and are not necessarily indicative of misspecification (of course, one would not expect the estimates to change in the case where parameters to be omitted are estimated as 0.0 in the full model). The fitting functions used in Mx (see Chapter ) are designed to produce parameter estimates that yield the closest match between the observed and estimated covariance matrices. Omission of selected parameters, for example, the $C$ loadings in the present model, generates a different model $\Sigma$ and thus may be expected to yield slightly different parameter estimates in order to best approximate the observed matrix.