7 Testing the Equality of Means

Applications of structural equation modeling to twin and other family data typically tend to ignore means. That is, observed measures are treated as deviations from the phenotypic mean (and are thus termed deviation phenotypes), and likewise genetic and environmental latent variables are expressed as deviations from their means, which usually are fixed at 0. Most simple genetic models predict the same mean for different groups of relatives, so, for example, MZ twins, DZ twins, males from opposite-sex twin pairs, and males from like-sex twin pairs should have (within sampling error) equal means. Where significant mean differences are found, they may indicate sampling problems with respect to the variable under study or other violations of the assumptions of the basic genetic model. Testing for mean differences also may be important in follow-up studies, where we are concerned about the bias introduced by sample attrition, but can compare mean scores at baseline for those relatives who remain in a study with those who drop out. Fortunately, Mx facilitates tests for mean differences between groups. For Mx to fit a model to means and covariances, both observed means and a model for them must be supplied. Appendix  contains a Mx script for fitting a univariate genetic model which also estimates the means of first and second twins from MZ and DZ pairs. The first change we make is to feed Mx the observed means in our sample, which we do with the Means command:

Means 0.9087 0.8685

Second, we declare a matrix for the means, e.g. M Full 1 2 in the matrices declaration section. Third, we can equate parameters for the first and second twins by using a Specify statement such as

Specify M 101 101

where 101 is a parameter number that has not been used elsewhere in the script. By using the same number for the two means, they are constrained to be equal. Fourth, we include a model for the means:

Means M;

In the DZ group we also supply the observed means, and adjust the model for the means. We can then either (i) equate the mean for MZ twins to that for DZ twins by using the same matrix M, 'copied' from the MZ group or equated to that of the MZ group as follows:

M Full 1 2 = M2

where M2 refers to matrix M in group 2; to fit a no heterogeneity model (Model I); or (ii) equate DZ twin 1 and DZ twin 2 means but allow them to differ from the MZ means by declaring a new matrix (possibly called M too; matrices are specific to the group in which they are defined, unless they are equated to a matrix or copied from a previous group) to fit a zygosity dependent means model ( $\overline{MZ}\neq\overline{DZ}$, Model II); or (iii) estimate four means, i.e., first and second twins in each of the MZ and DZ groups; to fit the heterogeneity model (Model III). This third option gives a perfect fit to the data with regard to mean structure, so that the only contribution to the fit function comes from the covariance structure. Hence the four means model gives the same goodness-of-fit $\chi^2$ as in the analyses ignoring means. Table 6.6 reports the results of fitting models incorporating means

Table 6.6: Results of fitting models to twin pair covariance matrices and twin means for Body Mass Index: Two-group analyses, complete pairs only.

		Female				Male
		Young		Older		Young		Older
	df	$\chi^2$	$p$	$\chi^2$	$p$	$\chi^2$	$p$	$\chi^2$	$p$
Model I	6	7.84	.25	5.74	.57	12.81	.05	5.69	.58
Model II	5	3.93	.56	4.75	.58	7.72	.17	5.36	.50
Model III	3	3.71	.29	2.38	.67	7.28	.06	5.03	.17
Genetic Model		ADE		AE$^*$		ADE		AE$^*$
$^*$ AE models have one more degree of freedom than shown in the df column

to the like-sex twin pair data on BMI. In each analysis, we have considered only the best-fitting genetic model identified in the analyses ignoring means. Again we subtract the $\chi^2$ of a more general model from the $\chi^2$ of a more restricted model to get a likelihood ratio test of the difference in fit between the two. For the two older cohorts we find no evidence for mean differences either between zygosity groups or between first and second twins. That is, the model that assumes no heterogeneity of means (model 1) does not give a significantly worse fit than either (i) estimating separate MZ and DZ means (model 2), or (ii) estimating 4 means. For older females, likelihood-ratio chi-squares are $\chi^{2}_{1}=0.99, p=0.32$ and $\chi^{2}_{3}=3.36, p=0.34$; and for older males, $\chi^{2}_{1}=0.36, p=0.55$ and $\chi^{2}_{3}=0.43, p=0.33$. Maximum-likelihood estimates of mean log BMI in the older cohort are, respectively, 21.87 and 22.26 for females and males; estimates of genetic and environmental parameters are unchanged from those obtained in the analyses ignoring means. In the younger cohorts, however, we do find significant mean differences between zygosity groups, both in females ( $\chi^{2}_{1}=3.91, p< 0.05$) and in males ( $\chi^{2}_{1}=5.09, p< 0.02$). In both sexes, mean log BMI values are lower in MZ pairs (21.35 for females, 21.63 for males) than for DZ pairs (21.45 for females, 21.79 for males). As these data are not age-corrected, it is possible that BMI values are still changing in this age-group, and that the zygosity difference reflects a slight mean difference in age. We shall return to this question in Section 6.2.9.