gms | German Medical Science

To unravel the genetic basis of complex genetic disorders, intermediate quantitative traits are often analyzed in practice instead of the clinically relevant outcome [Ref. 1]. In family studies, these are often investigated with a sample of nuclear families, consisting in at least two offspring and their parents. Risch and Zhang [Ref. 2] proposed a new approach for sample size and power calculations with the Haseman-Elston method [Ref. 3] as tool for analysis. They started off with the simple Falconer model [Ref. 4]

x_it = μ + g_it + e_it,

where x_it is the trait value of offspring t, t = 1, 2, in family i, i = 1, …, n. μ is the general mean, g_it is the genetic effect attributed to the diallelic major locus. Polygenic and environmental effects are absorbed in the error term e_it. The genetic effect g takes on values a, d, and –a, when the offspring is homozygous for the high allele, heterozygous, and homozygous for the low alleles, respectively. The Haseman-Elston method then regresses the sib-pairs squared trait difference on the proportion τ_i of alleles shared identical by descent (IBD) and the probability z_1i for sib-pair i sharing one allele IBD:

y_i = (x_i1 – x_i2)² = α + β τ_i + γ z_1i + ε_i

Haseman and Elston [Ref. 3] have shown that the slope coefficient β is given by [Ref. 3] β = -2(1-2θ)² σ_g ², where θ is the recombination fraction between the marker and the trait locus, and σ_g ² is the genetic variance attributable to the trait locus. Formulae for σ_g ² can be found elsewhere [see, e.g., [Ref. 1]].

For power and sample size calculations, Risch and Zhang [Ref. 2] considered an additive genetic model, i.e., γ = 0, and a completely informative genetic marker for the null hypothesis H₀: σ_g ² = 0 against the one-sided alternative H₁: σ_g ² > 0.

In this presentation, we firstly demonstrate in Monte-Carlo simulation studies that the power calculated by Risch and Zhang do not match their theoretical levels. We argue that the hypothesis σ_g ² = 0 used by Risch and Zhang is inadequate. We next develop new formulae for power and sample size calculations for the Haseman-Elston method based on H₀: θ = ½ versus H₀: θ < ½. Finally, we illustrate the validity of our new formulae in Monte-Carlo simulation studies.

In conclusion, power and sample size calculations for the Haseman-Elston method should not be performed with the formulae developed by Risch and Zhang [Ref. 2]. For this purpose, one should either utilize the approach proposed by Amos et al. [Ref. 5] or our new method. For this, it is important to stress the limitations of these methods. Both are restricted to a single diallelic major trait locus. However, while Amos et al. assume absence of polygenic components and/or shared environmental effects, our approach is based on the existence of a completely informative genetic marker. We are convinced that the latter limitation is less crucial because today’s genome-wide linkage studies utilizing 10,000 or more single nucleotide polymorphisms lead to almost complete marker information at any chromosomal position.