4 Fitting a Second Genetic Factor

The genetic common factor model we introduced in Sections 10.3.3 and 10.3.2 may be extended to address more specific questions about the data. In the arithmetic computation measures, for example, it is reasonable to hypothesize two genetic factors: one general factor contributing to all measurements of arithmetic computation, and a second ``alcohol'' factor which influences the measures taken after the challenge dose of alcohol. The most parsimonious extension of our common factor model may involve the addition of only 1 free parameter which represents each of the factor loadings on the alcohol factor (that is, the alcohol loadings may be equated for all alcohol measurements). The Mx script corresponds very closely to that used in section 10.3.2, using the X for the genetic common factors We add the latent alcohol factors for twins 1 and 2 as a second column with the following specification statement:

Specify X
1 0 
2 5
3 5
4 5

The addition of the single parameter for all alcohol loadings reflects a model having 13 parameters and $2\times 8(8+1)/2 - 13 = 59$ degrees of freedom. We can, therefore, test the significance of the alcohol factor by comparing the goodness-of-fit chi-squared value for this model with that obtained from the model of Section 10.3.2 for a $60-59=1$ d.f. test. Table 10.3.4 shows the results of the two-factor multivariate genetic model.

Table 10.6: Parameter estimates from the two genetic factors model

	$A_C1$	$A_C2$	$E_C$	$E_S$
Time 1	15.067	0.000	4.408	6.674
Time 2	13.701	4.270	6.091	6.277
Time 3	13.518	4.270	6.800	5.644
Time 4	13.832	4.270	5.695	5.928
$\chi^2=47.52$, 59 df, p=.858

The estimated genetic factor loading for the alcohol variables ($4.27$) is reasonably large, but much smaller than the loadings on the general genetic factor. This difference is more apparent when we consider proportions of genetic variance accounted for by these two factors, being ${4.27^2}/( 13.70^2+4.27)$ or 9% for the alcohol factor, and $100-9=91\%$ for the general genetic factor. The model yields a $\chi^{2}_{59} = 47.52$ ($p$ = .86), indicating a good fit to the data. The chi-squared test for the significance of the alcohol factor loadings is $51.08 - 47.52 = 3.56$, which is not quite significant at the .05 level. Thus, while the hypothesis of there being genetic effects on the alcohol measures additional to those influencing arithmetic skills fits the observed data better, the increase in fit obtained by adding the alcohol factor does not reach statistical significance.