4 Model for Age-Correction of Twin Data

We now turn to a slightly more elaborate example of univariate analysis, using data from the Australian twin sample that were used in the BMI example earlier, but in this case data on social attitudes. Factor analysis of the item responses revealed a major dimension with low scores indicating radical attitudes and high scores indicating attitudes commonly labelled as ``conservative.'' Our a priori expectation is that variation in this dimension will be largely shaped by social environment and that genetic factors will be of little or no importance. This expectation is based on the differences between the MZ and DZ correlations; $r_{MZ} = 0.68$ and $r_{DZ} = 0.59$, indicating little, if any, genetic influence on social attitudes. We also might expect that conservatism scores are affected by age. We can use the Mx script in Appendix  to examine the age effects, reading in the age of each twin pair and the conservatism scores for twin 1 (Cons_t1) and twin 2 (Cons_t2). Since in this specification we have 3 indicator variables, we adjust NInput_vars=3. If we initially ignore age, as an exploratory analysis, we can select only the conservatism scores for analysis, using the Select command (note that the list of variables selected must end with a semicolon `;'). The script fits the ACE model. The results of this model are presented in the fourth line of the standardized results of Table 6.11, which shows that the squares of parameters estimated from the model sum to one, because these correspond to the proportions of variance associated with each source (A, C, and E).

Table 6.11: Conservatism in Australian females: standardized parameter estimates for additive genotype (A), common environment (C), random environment (E) and dominance genotype (D).

	Parameter Estimates				Fit statistics
Model	$a$	$c$	$e$	$d$	$\chi^2$	df	$p$
$E$	--	--	1.000	--	823.76	5	.000
$CE$	--	0.804	0.595	--	19.41	4	.001
$AE$	0.836	--	0.549	--	56.87	4	.000
$ACE$	0.464	0.687	0.559	--	3.07	3	.380
$ADE$	0.836	--	0.549	0.000	56.87	3	.000

The significance of common environmental contributions to variance in conservatism may be tested by dropping $c$ (AE model) but this leads to a worsening of $\chi^2$ by 53.8 for 1 d.f., confirming its importance. Similarly, the poor fit of the CE model confirms that genetic factors also contribute to individual differences (significance of $a$ is $19.41-3.07=16.34$ for 1 df, which is highly significant). The $e$ model, which hypothesizes that there is no family resemblance for conservatism, is overwhelmingly rejected, illustrating of the great power of this data set to discriminate between competing hypotheses. For interest, we also present the results of the ADE model. Since we have already noted that $r_{DZ}$ is appreciably greater than half the MZ correlation, it is clear that this model is inappropriate. Symmetric with the results of fitting an ACE model to the BMI data (where $2r_{DZ}$ was still less than $r_{MZ}$, indicating dominance), we now find that the estimate of $d$ gets ``stuck" on its lower bound of zero. The BMI and conservatism examples illustrate in a practical way the perfect reciprocal dependence of $c$ and $d$ in the classical twin design of which only one may be estimated. The issue of the reciprocal confounding of shared environment and genetic non-additivity (dominance or epistasis) in the classical twin design has been discussed in detail in papers by Martin et al., (1978), Grayson (1989), and Hewitt (1989). It is clear from the results above that there are major influences of the shared environment on conservatism. One aspect of the environment that is shared with perfect correlation by cotwins is their age. If a variable is strongly related to age and if a twin sample is drawn from a broad age range, as opposed to a cohort sample covering a narrow range of birth years, then differences between twin pairs in age will contribute to estimated common environmental variance. This is the case for the twins in the Australian sample, who range from 18 to 88 years old. It is clearly of interest to try to separate this variance due to age differences from genuine cultural differences contributing to the estimate of $c$. Fortunately, structural equation modeling, which is based on linear regression, provides a very easy way of allowing for the effects of age regression while simultaneously estimating the genetic and environmental effects (Neale and Martin, 1989). Figure 6.2 illustrates the method with a path diagram, in which the regression of $Cons_{t1}$ and $Cons_{t2}$ on $Age$ is $s$ (for senescence), and this is specified in the script excerpt below.

Figure 6.2: Path model for additive genetic ($A$), shared environment ($C$) and specific environment ($E$) effects on phenotypes ($P$) of pairs of twins ($T1$ and $T2$). $\alpha$ is fixed at 1 for MZ twins and at .5 for DZ twins. The effects of age are modelled as a standardized latent variable, $Age_L$, which is the sole cause of variance in observed $Age$.

We now work with the full $3 \times 3$ covariance matrices (so the Select statement is dropped from the previous job). We estimate simultaneously the contributions of additive genetic, shared and unique environmental factors on conservatism, the variance of age V*V, and the contribution of age to conservatism S*V.

Group 2: female MZ twin pairs
Data NInput_vars=3 NOberservations=941
Labels age cons_t1 cons_t2
CMatrix Symmetric File=ozconmzf.cov
Matrices= Group 1
Covariances V*V' | V*S'    | V*S'   _
            S*V' | A+C+E+G | A+C+G  _
            S*V' | A+C+G   | A+C+E+G;

The matrix algebra here is more complex than usual, and for univariate analysis it would be easier to draw the diagram with the GUI. However, the algebraic approach has the advantage that it is much easier to generalize to the multivariate case. Results of fitting the ACE model with age correction are in the first row of Table 6.12. Standardized results are presented, from which we see that the standardized regression of conservatism on age (constrained equal in twins 1 and 2) is 0.422. In the unstandardized solution, the first loading on the age factor is the standard deviation of the sample for age, in this case 13.2 years. The latter is an estimated parameter, making five free parameters in total. In each group we have $\mbox{k(k+1)/2}$ statistics, where k is the number of observed variables, so there are $2\times$ $(\mbox{k(k+1)/2-5 =7}$ degrees of freedom. Dropping either $c$ or $a$ still causes significant worsening of the fit, and it also is very clear that one cannot omit the age regression itself (final ACE model; $\chi^{2}_{8} = 370.17, p = .000$).

Table 6.12: Age correction of Conservatism in Australian females: standardized parameter estimates for models of additive genetic (A), common environment (C), random environment (E), and senescence or age (S).

	Parameter Estimates				Fit statistics
Model	$a$	$c$	$e$	$s$	$\chi^2$	df	$p$
$ACES$	0.474	0.534	0.558	0.422	7.41	7	.388
$AES$	0.720	--	0.547	0.426	31.56	8	.000
$CES$	--	0.685	0.595	0.421	25.49	8	.001
$ACE$	0.464	0.687	0.559	--	370.17	8	.000

It is interesting to compare the results of the ACE model in Table 6.11 with those of the ACES model in Table 6.12. We see that the estimates of $e$ and $a$ are identical in the two tables, accounting for $0.559^2=31$% and $0.464^2=22$% of the total variance, respectively. However, in the first table the estimate of $c=0.687$, accounting for 47% of the variance. In the analysis with age however, $c=0.534$ and accounts for 29% of variance, and age accounts for $0.422^2=18\%$. Thus, we have partitioned our original estimate of 47% due to shared environment into 18% due to age regression and the remaining 29% due to `genuine' cultural differences. If we choose, we may recalculate the proportions of variance due to $a,c$, and $e$, as if we were estimating them from a sample of uniform age -- assuming of course that the causes of variation do not vary with age (see Chapter 9). Thus, genetic variance now accounts for $22/(100-18)=27$% and shared environment variance is estimated to be $29/82=35$%. Our analysis suggests that cultural differences are indeed important in determining individual differences in social attitudes. However, before accepting this result too readily, we should reflect that estimates of shared environment may not only be inflated by age regression, but also by the effects of assortative mating -- the tendency of like to marry like. Since there is known to be considerable assortative mating for conservatism (spouse correlations are typically greater than 0.6), it is possible that a substantial part of our estimate of $c^2$ may arise from this source (Martin et al., 1986). This issue will be discussed in greater detail in Chapter . Age is a somewhat unusual variable since it is perfectly correlated in both MZ and DZ twins (so long as we measure the members of a pair at the same time). There are relatively few variables that can be handled in the same way, partly because we have assumed a strong model that age causes variability in the observed phenotype. Thus, for example, it would be inappropriate to model length of time spent living together as a cause of cancer, even though cohabitation may lead to greater similarity between twins. In this case a more suitable model would be one in which the shared environment components are more highly correlated the longer the twins have been living together. Such a model would predict greater twin similarity, but would not predict correlation between cohabitation and cancer. Some further discussion of this type of model is given in Section  in the context of data-specific models. One group of variables that may be treated in a similar way to the present treatment of age consists of maternal gestation factors. Vlietinck et al. (1989) fitted a model in which both gestational age and maternal age predicted birthweight in twins. Finally we note that at a technical level, age and similar putative causal agents might most appropriately be treated as $x$-variables in a multiple regression model. Thus the observed covariance of the $x$-variables is incorporated directly into the expected matrix, so that the analysis of the remaining $y$-variables is conditional on the covariance of the $x$-variables. This type of approach is free of distributional assumptions for the $x$-variables, and is analogous to the analysis of covariance. However, when we fit a model that estimates a single parameter for the variance of age in each group, the estimated and observed variances are generally equal, so the same results are obtained.