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## A General Approach for Power Calculations for the Haseman-Elston Method

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Published: | September 1, 2006 |
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### Outline

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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 H_{0}: σ_{g}
^{2} = 0 against the one-sided alternative H_{1}: 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 H_{0}: θ = ½ versus H_{0}: 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.

### References

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- Ziegler A, König IR. A Statistical Approach to Genetic Epidemiology. Heidelberg: Wiley-VCH; 2006.
- 2.
- Risch N, Zhang H. Extreme Discordant Sib Pairs for mapping Quantitative Trait Loci in Humans. Science. 1995; 268:1584-1589.
- 3.
- Haseman JK, Elston RC. The investigation of linkage between a quantitative trait and a marker locus. Behav Genet. 1972; 2: 3-19.
- 4.
- Falconer D. Einführung in die quantitative Genetik. Stuttgart: UTB; 1989.
- 5.
- Amos CI, Elston RC. Robust methods fort he detection of genetic linkage for quantitative data from pedigrees. Genet Epidemiol. 1989; 6:349-360.