An utter refutation of the ‘Fundamental Theorem of the HapMap’ by Terwilliger and Hiekkalinna

Terwilliger and Hiekkalinna¹ (T&H) have provided a thought-provoking discussion of some fundamental issues underlying the HapMap Project and its use in genome-wide association (GWA) studies with the new generation of high-density SNP genotyping technologies. They make three important points: first, that estimates of r² will be upwardly biased in small samples and also in case–control studies; second, that (unlike recombination fraction estimates in linkage analysis) pairwise r²'s do not necessarily have a multiplicative relationship across sets of three or more loci in the presence of three-way interactions; and third, that a causal association of a particular SNP with disease does not necessarily imply the absence of a three-way interaction involving disease, the causal locus, and some other marker locus in LD with it. In particular, they provide an insightful example of the latter situation involving genetic heterogeneity, and discuss in general terms how population stratification or gene–environment interactions could produce a similar phenomenon. Taken together, these three observations lead to their conclusion that the expected sample size required to demonstrate an association between disease and some marker in LD with a causal variant can be underestimated, possibly considerably so – their extreme example illustrates how it would be theoretically possible for an association with a marker in strong LD with a causal locus to be undetectable with even an infinite sample size.

We agree with these general principles, but question their relevance to the HapMap and its application to GWA studies. Regarding the bias in r² estimates, our own simulations indicate that for sample sizes typically used by HapMap and for reasonably large r²'s that are of interest as potential markers, this upwards bias is generally modest. Of greater concern is the bias in the maximum r² over a set of markers in a region, which could be quite substantial. As it is this maximum (pairwise or multivariate) that is typically used in one-way or another by most tag SNP selection algorithms, it is quite possible that the ability of a tag SNP panel to predict an unobserved variant in a new sample, based on estimates from small samples, could be exaggerated. Although T&H are correct that estimation of r² from a pooled case–control sample would be biased by the over-representation of cases, most practitioners do not in fact use this approach. More typically, study of haplotype and LD structure and the selection of tag SNPs are carried out in controls only or completely independent samples from populations similar to those under study. We have analyzed the bias in haplotype relative risk estimates that can arise from case–control samples when the population haplotype frequencies are estimated from the combined data² and have provided a simple correction for this ascertainment bias using an appropriate ascertainment-corrected prospective likelihood. See also Epstein and Satten^{3, 4} for an alternative approach based on the retrospective likelihood.

With regard to T&H's second contention, we question its relevance in the case of the association of a disease (C) with a causal variant (B) and a marker (A) in LD with it. If B were truly the causal variable, then we would expect A and C to be conditionally independent, given B. The bounds on the possible r_AC² given r_AB² and r_BC² are then irrelevant, because r_AC²=r_AB²r_BC² owing to the absence of three-way interaction. T&H admit this argument at the beginning of their Discussion, but counter with their worked example illustrating what can go wrong in the case of genetic heterogeneity. In this case, B is not the sole cause of C, but there is another causal locus D, also in LD with A, which just happens to counteract the association in such a way as lead to no association between C and A. In this situation, we would respond that the correct analysis would be to identify additional markers that would effectively tag the real causal B–D haplotype. If we denote this expanded set of markers as A^*, then we contend that conditional on B–D, A^* and D would be conditionally independent, so again the conditions for multiplicativity would be met and the sample size inflation that would be required would be simply proportion to the inverse of the multivariate r² for the A^* by B–D haplotype association.

Racial heterogeneity is a well-known concern about disease association studies using unrelated individuals,⁵ but can be addressed in the usual ways by matching on self-reported ethnic origins, genomic control, or use of family studies. Gene–environment interactions are an issue only if the gene and environmental factor are correlated in the source population. Although there certainly are examples where gene–environment independence might be questionable in candidate gene association studies, this seems unlikely for most situations involving genome-wide scans with SNPs for which there is no prior hypothesis about environmental modifiers.

Going beyond the example of T&H, it is important to remember that balanced interactions between variants in their effects on risk can obscure main effects entirely, even without any added complications brought about by the ‘measurement error’ problem discussed by T&H (ie even if all possible variants were directly genotyped) and whether or not the markers are in LD. Does this mean that we should despair of ever finding any single-marker main effects? Of course not, but it does raise important design and analysis questions about what data-mining approaches to unearth effects involving interactions can be effectively applied at the whole genome scale, and how large sample sizes should be to have a hope of winnowing out the false from true-positive associations from such undertakings. Our contention is that it ultimately will be this question that will produce the greatest challenges in the future, as the r² problem is likely to be rapidly overcome by increasingly better knowledge of haplotype structure of the human genome and by increasingly more sophisticated SNP chips.