6 Interpreting Univariate Results

In model-fitting to univariate twin data, whether we use a variance components or a path coefficients model, we are essentially testing the following hypotheses:

1. No family resemblance (``E'' model: $e>0$: $a=c=d=0$)
2. Family resemblance solely due to additive genetic effects (``AE'' model: $a>0, e>0, c=d=0$)
3. Family resemblance solely due to shared environmental effects (``CE'' model: $e>0, c > 0, a=d=0$)
4. Family resemblance due to additive genetic plus dominance genetic effects (``ADE'' model: $a>0, d>0, e > 0, c=0$)
5. Family resemblance due to additive genetic plus shared environmental effects (``ACE'' model: $a>0,c>0,e>0, d=0$).

Note that we never fit a model that excludes random environmental effects, because it predicts perfect MZ twin pair correlations, which in turn generate a singular expected covariance matrix. From inspection of the twin pair correlations for BMI, we noted that they were most consistent with a model allowing for additive genetic, dominance genetic, and random environmental effects. Model-fitting gives three important advantages at this stage:

1. An overall test of the goodness of fit of the model
2. A test of the relative goodness of fit of different models, as assessed by likelihood-ratio $\chi^2$. For example, we can test whether the fit is significantly worse if we omit genetic dominance for BMI
3. Maximum-likelihood parameter estimates under the best-fitting model.

Table 6.4 tabulates goodness-of-fit chi-squares obtained in four

Table 6.4: Results of fitting models to twin pairs covariance matrices for Body Mass Index: Two-group analyses, complete pairs only.

	Females				Males
	Young		Older		Young		Older
Model (d.f.)	$\chi^2$	$p$	$\chi^2$	$p$	$\chi^2$	$p$	$\chi^2$	$p$
CE (4)	160.72	$<$.001	87.36	$<$.001	97.20	$<$.001	37.14	$<$.001
AE (4)	8.06	.09	2.38	.67	10.88	.03	5.03	.28
ACE (3)	8.06	$<$.05	2.38	.50	10.88	.01	5.03	.17
ADE (3)	3.71	.29	1.97	.58	7.28	.06	5.03	.17

separate analyses of the data from younger or older, female or male like-sex twin pairs. Let us consider the results for young females first. The non-genetic model (CE) yields a chi-squared of 160.72 for 4 degrees of freedom, which is highly significant and implies a very poor fit to the data indeed. In stark contrast, the alternative model of additive genes and random environment (AE) is not rejected by the data, but fits moderately well ($p=.09$). Adding common environmental effects (the ACE model) does not improve the fit whatsoever, but the loss of a degree of freedom makes the $\chi^2$ significant at the .05 level. Finally, the ADE model which substitutes genetic dominance for common environmental effects, fits the best according to the probability level. We can test whether the dominance variation is significant by using the likelihood ratio test. The difference between the $\chi^2$ of a general model ($\chi^2_G$) and the that of a submodel ($\chi^2_S$ ) is itself a $\chi^2$ and has $\mbox{df}_S-\mbox{df}_G$ degrees of freedom (where subscripts $S$ and $G$ respectively refer to the submodel and general model, in other words, the difference in df between the general model and the submodel). In this case, comparing the AE and the ADE model gives a likelihood ratio $\chi^2$ of $8.06-3.71=4.35$ with $4-3=1$df. This is significant at the .05 level, so we say that there is significant deterioration in the fit of the model when the parameter $d$ is fixed to zero, or simply that the parameter $d$ is significant. Now we are in a position to compare the results of model-fitting in females and males, and in young and older twins. In each case, a non-genetic (CE) model yields a significant chi-squared, implying a very poor fit to the data: the deviations of the observed covariance matrices from the expected covariance matrices under the maximum-likelihood parameter estimates are highly significant. In all groups, a full model allowing for additive plus dominance genetic effects and random environmental effects (ADE) gives an acceptable fit to the data, although in the case of young males the fit is somewhat marginal. In the two older cohorts, however, a model which allows for only additive genetic plus random environmental effects (AE) does not give a significantly worse fit than the full (ADE) model, by likelihood-ratio $\chi^2$ test. In older females, for example, the likelihood-ratio chi-square is $2.38-1.97 = 0.41$, with degrees of freedom equal to $4-3=1$, i.e., $\chi^{2}_{1}=0.41$ with probability $p=0.52$; while in older males we have $\chi^{2}_{1}=0.00, p=1.00$. For the older cohorts, therefore, we find no significant evidence for genetic dominance. In young adults, however, significant dominance is observed in females (as noted above) and the dominance genetic effect is almost significant in males ( $\chi^{2}_{1}=3.6, p=0.06$). Table 6.5 summarizes

Table 6.5: Standardized parameter estimates under best-fitting model. Two-group analyses, complete pairs only.

	Estimate
	a$^2$	c$^2$	e$^2$	d$^2$
Young females	0.40	0	0.22	0.38
Older females	0.69	0	0.31	0
Young males	0.36	0	0.20	0.44
Older males	0.70	0	0.30	0

variance component estimates under the best-fitting models. Random environment accounts for a relatively modest proportion of the total variation in BMI, but appears to be having a larger effect in older than in younger individuals (30-31% versus 20-22%). Although the estimate of the narrow heritability (i.e., proportion of the total variance accounted for by additive genetic factors) is higher in the older cohort (69-70% vs 36-40%), the broad heritability (additive plus non-additive genetic variance) is higher in the young twins (78-80%).