4 Application to Body Mass Index

In this section, we apply sex-limitation models to data on body mass index collected from twins in the Virginia Twin Registry and twins ascertained through the American Association of Retired Persons (AARP). Details of the membership of these two twin cohorts are provided in Eaves et al. (1991), in their analysis of BMI in extended twin-family pedigrees. In brief, the Virginia twins are members of a population based registry comprised of 7,458 individuals (Corey et al., 1986), while the AARP twins are members of a volunteer registry of 12,118 individuals responding to advertisements in publications of the AARP. The Virginia twins' mean age is 39.7 years (SD = 14.3), compared to 54.5 years (SD = 16.8) for the AARP twins. Between 1985 and 1987, Health and Lifestyle questionnaires were mailed to twins from both of these cohorts. Among the items on the questionnaire were those pertaining to physical similarity and confusion in recognition by others (used to diagnose zygosity) and those asking about current height and weight (used to compute body mass index). Questionnaires with no missing values for any of these items were returned by 5,465 Virginia and AARP twin pairs. From height and weight data, body mass index (BMI) was calculated for the twins, using the formula:

$$\mbox{BMI} = wt (kg) / ht (m)^{2}$$

The natural logarithm of BMI was then taken to normalize the data. Before calculating covariance matrices of log BMI, the data from the two cohorts were combined, and the effects of age, age squared, sample (AARP vs. Virginia), sex, and their interactions were removed. The resulting covariance matrices are provided in the Mx scripts in Appendices  and , while the correlations and sample sizes appear in Table 9.1 below.

Table 9.1: Sample sizes and correlations for BMI data in Virginia and AARP twins.

Zygosity Group	N	r
MZF	1802	0.744
DZF	1142	0.352
MZM	750	0.700
DZM	553	0.309
DZO	1341	0.251

We note that both like-sex MZ correlations are greater than twice the respective DZ correlations; thus, models with dominant genetic effects, rather than common environmental effects, were fit to the data. In Table 9.2, we provide selected results from fitting the following models: general sex-limitation (I); common effects sex-limitation (II-IV); and scalar sex-limitation (V). We first note that the general sex-limitation model provides a good fit to the data, with $p=0.32$. The estimate of $a'_{m}$ under this model is fairly small, and when set to zero in model II, found to be non-significant ($\chi^{2}_{1}$ = 2.54, $p > 0.05$). Thus, there is no evidence for sex-specific additive genetic effects, and the common effects sex-limitation model (model II) is favored over the general model. As an exercise, the reader may wish to verify that the same conclusion is reached if the general sex-limitation model with sex-specific dominant genetic effects is compared to the common effects model with $d'_{m}$ removed. Note that under model II the dominant genetic parameter for females is quite small; thus, when this parameter is fixed to zero in model III, there is not a significant worsening of fit, and model III becomes the most favored model. In model IV, we consider whether the dominant genetic effect for males can also be fixed to zero. The goodness-of-fit statistics indicate that this model fits the data poorly ($p < 0.01$) and provides a significantly worse fit than model III ($\chi^{2}_{1}$ = 26.73, $p < 0.01$). Model IV is therefore rejected and model III remains the favored one. Finally, we consider the scalar sex-limitation model. Since there is evidence for dominant genetic effects in males and not in females, it seems unlikely that this model, which constrains the variance components of females to be scalar multiples of the male variance components, will provide a good fit to the data, unless the additive genetic variance in females is also much smaller than the male additive genetic variance. The model-fitting results support this contention: the model provides a marginal fit to the data ($p$ = 0.05), and is significantly worse than model II ($\chi^2_{2}$ = 7.82, $p < 0.05$ ). We thus conclude from Table 9.2 that III is the best fitting model. This conclusion would also be reached if AIC was used to assess goodness-of-fit.

Table 9.2: Parameter estimates from fitting genotype $\times$ sex interaction models to BMI.

	MODEL
Parameter	I	II	III	IV	V
$a_{f}$	0.449	0.454	0.454	0.454	0.346
$d_{f}$	0.172	0.000	-	-	0.288
$e_{f}$	0.264	0.265	0.265	0.267	0.267
$a_{m}$	0.210	0.240	0.240	0.342	-
$d_{m}$	0.184	0.245	0.245	-	-
$e_{m}$	0.213	0.213	0.213	0.220	-
$a'_{m}$	0.198	-	-	-	-
$k$	-	-	-	-	0.778
$\chi^{2}$	9.26	11.80	11.80	38.53	19.62
$d.f.$	8	9	10	11	11
$p$	0.32	0.23	0.30	0.00	0.05
$AIC$	-6.74	-6.20	-8.20	16.53	-2.38

Using the parameter estimates under model III, the expected variance of log BMI (residuals) in males and females can be calculated. A little arithmetic reveals that the phenotypic variance of males is markedly lower than that of females (0.17 vs. 0.28). Inspection of the parameter estimates indicates that the sex difference in phenotypic variance is due to increased genetic and environmental variance in females. However, the increase in genetic variance in females is proportionately greater than the increase in environmental variance, and this difference results in a somewhat larger broad sense (i.e., $a^2+d^2$) heritability estimate for females (75%) than for males (69%). The detection of sex-differences in environmental and genetic effects on BMI leads to questions regarding the nature of these differences. Speculation might suggest that the somewhat lower male heritability estimate may be due to the fact that males are less accurate in their self-report of height and weight than are females. With additional information, such as test-retest data, this hypothesis could be rigorously tested. The sex-dependency of genetic dominance is similarly curious. It may be that the common environment in females exerts a greater influence on BMI than in males, and, consequently, masks a genetic dominance effect. Alternatively, the genetic architecture may indeed be different across the sexes, resulting from sex differences in selective pressures during human evolution. Again, additional data, such as that from reared together adopted siblings, could be used to explore these alternative hypotheses. One sex-limitation model that we have not considered, but which is biologically reasonable, is that the across-sex correlation between additive genetic effects is the same as the across-sex correlation between the dominance genetic effects. Fitting a model of this type involves a non-linear constraint which can easily be specified in Mx.