Assuming the model used in Box 4.1, give the estimated heritabilities for BMI from the four studies in Table 4.3. Do the non-zero correlations observed between spouses support the assumptions of our model? Box 4.1 Estimation of h Using Parent-Child Trios Assuming that we have phenotypic data on both parents, we define the mid- parental value of the trait, Yp as the average phenotypic value of the two parents; we let Yo denote the phenotype of the offspring. We correspond- ingly let Xp denote the average of the two parental genotypes, and Xo be the genotype of the offspring. We assume that the genotype probabilities for each parent are independently distributed as B(2, p); the probabilities for each child's genotype, conditional on the parents, are given by Mendel's law. In the first step, we derive the covariance between the offspring phenotype and the mid-parental phenotype, assuming for simplicity M = 1 and d = 0. In the following derivations, we provide the results of each intermediate step. The algebraic derivation of each step is a homework assignment. It is straightfor- ward to see that for any member of the trio, E(YO) = E(Yp) = E(aXo) = + 2ap, and V = var(aXo) = 2ap(1 p). By properties of the covariance function, we have Cov(Yo, Yp) = acov(Xo, Xp) = a[E(XoXp) 4p]. Tedious algebra using Mendel's laws shows that cov(Xo, Xp) = p(1 p) and thus Cov(Yo, Yp) = ap(1 p) = V/2. This implies that the phenotypic covariance between the offspring and the mid-parental value is half the genetic variance V. Since this covariance and the phenotypic variance can be estimated based on the corresponding empiri- cal estimators, Cov(Yo, Yp) and Var(Yo), an estimator for the narrow-sense heritability is given by h = 2Cov(Yo, Yp)/Var(Y) = 2p, where p is the correlation between Yo and Yp. Table 4.3 Familial correlations for body mass index derived from four large family studies. Source Coon et al. (2007) Relationship Spouses Parent-Offspring Uncle/Aunt- Nephew/Niece Grandparent- Grandchild Dizygotic Twins Monozygotic Twins Framingham heart study 0.19 0.23 0.08 Canada fitness survey 0.12 0.20 -0.11 0.05 Quebec family survey 0.10 0.26 0.14 0.34 0.88 Norway study 0.12 0.24 0.00 0.07 0.20 0.58