8 Incorporating Data from Singleton Twins

In most twin studies, there are many twin pairs in which only one twin agrees to cooperate. We call these pairs discordant-participant as opposed to concordant-participant pairs, in which data are collected from both members of the pair. Sadly, data from discordant-participant pairs are often just thrown away. This is unfortunate not only because of the wasted effort on the part of the twins, researchers, and data entry personnel, but also because they provide valuable information about the representativeness of the sample for the variable under study. If sampling is satisfactory, then we would expect to find the same mean and variance in concordant-participant pairs as in discordant-participant pairs. Thus, the presence of mean differences or variance differences between these groups is an indication that biased sampling may have occurred with respect to the variable under investigation. To take a concrete example, suppose that overweight twins are less likely to respond to a mailed questionnaire survey. Given the strong twin pair resemblance for BMI demonstrated in previous sections, we might expect to find that individuals from discordant-participant pairs are on average heavier than individuals from concordant-participant pairs. Such sampling biases will have differential effects on the covariances of MZ and DZ twin pairs, and thus may lead to biased estimates of genetic and environmental parameters (Lykken et al., 1987; Neale et al., 1989b). Table 6.7 reports means and variances for transformed BMI from

Table 6.7: Means and variances for BMI of twins whose cotwin did not cooperate in the 1981 Australian survey.

	Young Cohort ($<$=30)				Older Cohort ($>$30)
Group	N	$\bar{x}^\prime$	$\sigma^2$		N	$\bar{x}^\prime$	$\sigma^2$
MZ Female Twins	33	0.1795	1.0640		44	0.6852	1.1461
DZ Female Twins	55	0.5836	0.8983		62	1.0168	1.7357
MZ Male Twins	24	1.3266	1.2477		36	1.3585	1.1036
DZ Male Twins	47	1.2705	1.5309		48	1.0379	1.6716
Opp-Sex Pair Females	65	0.6551	1.4390		81	0.9756	1.2690
Opp-Sex Pair Males	28	0.8724	0.9754		27	1.7149	1.0019

individuals from discordant-participant pairs in the 1981 Australian survey. Zygosity assignment for MZ twins must be regarded as somewhat tentative, since most algorithms for zygosity diagnosis based on questionnaire data require reports from both members of a twin pair to confirm monozygosity (e.g., Eaves et al., 1989b). In most groups, comparing Table 6.7 to Table 6.3, we observe both higher means and higher variances in the discordant-participant pairs. It is clearly important to test whether these differences are statistically significant. To fit a model simultaneously to the means, variances, and covariances of concordant-participant pairs and the means and variances of discordant-participant pairs, requires that we analyze data where there are different numbers of observed variables per group, which is easily done in Mx. Appendix  presents a Mx script for testing for differences in mean or variance. We constrain the means of the responding twin in groups four (MZ discordant-participant) and five (DZ discordant-participant) to equal those of twins from the concordant-participant pairs. Our test for significant differences in means between the concordant-participant and discordant-participant groups is the improvement in goodness-of-fit obtained when we allow these latter, discordant-participant pairs, to take their own mean value. Table 6.8 summarizes the results of model-fitting. Model I is the no heterogeneity model of means and variances between concordant-participant versus discordant-participant twins. Model II allows for heterogeneity of variances, Model III for heterogeneity of means. Finally, Model IV tests both differences in means and variances. For these analyses, we considered only the best-fitting genetic model based on the results of the analyses ignoring means, and allowed for zygosity differences in means only if these were found to be significant in the analyses of the previous Section (i.e., in the younger twin pairs; young female pairs are the only group in which we find no difference between concordant-participant pairs and discordant-participant pairs). In the two older cohorts a model allowing for heterogeneity of means (Model 3) gives a substantially better fit than one that assumes no heterogeneity of means or variances (Model 1: older females: $\chi_{2}^{2}=12.86, p< 0.001$; older males: $\chi^{2}_2=30.87, p<0.001$). Specifying heterogeneity of variances in addition to heterogeneity of means does not produce a further improvement in fit (older females: $\chi_2^{2}=2.02, p=0.36$; older males: $\chi^2_{2}=1.99, p=0.37$). Such a result is not atypical because the power to detect differences in mean is much greater than that to detect a difference in variance.

Table 6.8: Results of fitting models to twin pair covariance matrices and twin means for Body Mass Index: Two like-sex twin groups, plus data from twins from incomplete pairs. Models test for heterogeneity of means or variances between twins from pairs concordant vs discordant for cooperation in 1981 survey.

		Female				Male
		Young		Older		Young		Older
	df	$\chi^2$	$p$	$\chi^2$	$p$	$\chi^2$	$p$	$\chi^2$	$p$
Model I	11	8.16	.70	20.62	.08	54.97	.001$^*$	48.55	.001$^*$
Model II	9	6.03	.74	17.84	.09	29.22	.001$^*$	44.58	.001$^*$
Model III	9	5.70	.77	7.76	.74	22.76	.01	7.68	.74
Model IV	7	3.93	.79	5.74	.77	7.72	.36	5.69	.77
Genetic Model		ADE		AE#		ADE		AE#
Means Model		$\overline{MZ}\neq\overline{DZ}$		$\overline{MZ} = \overline{DZ}$		$\overline{MZ}\neq\overline{DZ}$		$\overline{MZ} = \overline{DZ}$
$^*$ $p<.001$
# AE models have two more degrees of freedom than shown in the df column

When considering these results, we must bear in mind several possibilities. Numbers of twins from the discordant-participant groups are small, and estimates of mean and variance in these groups will be particularly vulnerable to outlier-effects; that is, to inflation by one or two individuals of very high BMI. Further outlier analyses (e.g., Bollen, 1989) would be needed to determine whether this is an explanation of the variance difference. In the young males, it is also possible that age differences between concordant-participant pairs and discordant-participant pairs could generate the observed mean differences.