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Background: The maximum likelihood estimator of D' – a standard measure of linkage disequilibrium – is biased toward disequilibrium, and the bias is particularly evident in small samples and rare haplotypes. Results: This paper proposes a Bayesian estimation of D' to address this problem. The reduction of the bias is achieved by using a prior distribution on the pair-wise associations between single nucleotide polymorphisms (SNP)s that increases the likelihood of equilibrium with increasing physical distances between pairs of SNPs. We show how to compute the Bayesian estimate using a stochastic estimation based on MCMC methods, and also propose a numerical approximation to the Bayesian estimates that can be used to estimate patterns of LD in large datasets of SNPs. Conclusion: Our Bayesian estimator of D' corrects the bias toward disequilibrium that affects the maximum likelihood estimator. A consequence of this feature is a more objective view about the extent of linkage disequilibrium in the human genome, and a more realistic number of tagging SNPs to fully exploit the power of genome wide association studies. Background Single nucleotide polymorphisms (SNPs) are an invaluable resource to identify regions of the human genome that may be associated with disease. A key to this process is linkage disequilibrium (LD) that is defined as the non-random association between the alleles of SNPs [1]. Although LD may occur between SNPs that are not in linkage but are associated, we will focus on the LD due to the spatial structure of the genome. In this situation, the nonrandom association implies that pairs of alleles in the same haplotype occur differently from what we would expect in a random pairing and several measures of LD have been proposed to capture the departure from independent pairing of the alleles of SNPs [2]. In this paper we will limit attention to D, its normalized version D', and the well known bias of the Maximum Likelihood Estimate (MLE) of D' toward disequilibrium [2,3]. This bias is particularly large in small samples and SNPs with rare alleles to the point that SNPs whose alleles occur independently may be inferred to be in strong LD [4]. However, relying on small samples to identify patterns of LD is not unusual: for example, the International HapMap Project (IHMP) aims to establish genome-wide patterns of LD using genotype data of at most 30 trios or 45 unrelated individuals [5]. The genotype data typed in this small number of samples are used to describe the extent of LD in the human genome, and derive a map of the haplotypes and the SNPs that are sufficient to tag the human genome. These results will have a deep impact on Published: 25 June 2007 BMC Genetics 2007, 8:36 doi:10.1186/1471-2156-8-36 Received: 4 February 2007 Accepted: 25 June 2007 This article is available from: http://www.biomedcentral. com/1471-2156/8/36 © 2007 Sebastiani and Abad-Grau; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Page 1 of 13 (page number not for citation purposes) BMC Genetics 2007, 8:36 http://www.biomedcentral. com/1471-2156/8/36 genome wide association studies and, in particular, inferring larger blocks of LD than the real ones may lead to the selection of an insufficient number of SNPs and hence decrease the power of genome wide association studies. In this scenario, biasing the estimate of LD toward equilibrium appears to be a safer alternative. Several solutions have been proposed to reduce the bias of the MLE of D' toward disequilibrium [4]. A pragmatic solution is to impose some "ad hoc" threshold on the minimum allele frequency (MAF) of those SNPs that can be used to infer the pattern of LD [6]. Imposing this threshold leads to a non-random selection of SNPs and may introduce ascertainment bias [7,8]. The thought behind our approach is that the bias of the MLEs of D and D' is due to the lack of information in the data to discriminate between equilibrium and different magnitude of disequilibrium, and any attempt to correct this bias is due to fail as it was acknowledged in [3]. However, it is known that, on average, the strength of LD due to linkage decreases as the physical distance between SNPs increases [9,10]. Therefore, we propose a Bayesian estimator of D' that allows us to integrate data with prior information about the pattern of LD decay. To this end, we use a prior distribution on the pairwise dependencies between different SNPs that is a decreasing function of their physical distance. We show how to compute the posterior estimate of D' using Markov Chain Monte Carlo methods, and provide a numerical approximation that can be used for fast estimation of LD in large regions of the human genome. As we show in simulated and real data from the IHMP, the effect of the prior distribution is to drastically reduce the bias toward disequilibrium even in small samples, and to remove the need of arbitrary thresholds on the MAF. We also show that, compared to the MLE, our estimators lead to infer patterns of LD decay that are much closer to published results [10], and confirms the existence of haplotype blocks as regions of low recombination. The method is implemented in a computer program called Bayesian Linkage (BLink) [11]. Results and discussion The traditional D and D' Given two SNPs L1 and L2, with alleles A/a and B/b, A and B the major alleles, we define the probability of the haplotype ij by pij = p(L1 = i, L2 = j), i = A, a, j = B, b. As in [12], we assume the relation pA ≥ pB on the probabilities pA = p(L1 = A), pB = p(L2 = B) from which the inequality pAb ≥ paB follows. The two SNPs are in linkage equilibrium when the co-occurrence of two alleles on the same haplotype is random, e.g. pij = pipj for all i = A, a, j = B, b. On the other hand, LD implies some form of dependency in the alleles on the same haplotype and hence departure from independence of the probabilities pij. Although there are many ways to measure departure from independence in a 2 × 2 table [13], a widely used measure of LD is the parameter D defined by D = pAB pApB papb ≤ D ≤ papB. Because the domain of D is a function of the allele frequencies, different normalization methods have been proposed to facilitate the interpretation [2]. The most common one is the measure that was suggested by Lewontin [14] and is defined as D/max D: is defined in the interval [-1, 1], with = ± 1 describing perfect disequilibrium and = 0 describing equilibrium. It is also common to take the absolute value of , say D' = | | to have a measure in the interval [0,1]. This is for example the default measure of LD in the popular program HaploView [6]. Maximum likelihood estimation Suppose now that we have a data set of N individuals and n = 2N known haplotypes for the two SNPs (we assume here known phase for all haplotypes and discuss the phasing issue at the end of this section). We denote by nij (i = A, a, j = B, b) the frequencies of the four haplotypes, and by ni and nj the allele frequencies with nA ≥ nB. Assuming that the four haplotypes follow a multinomial distribution with probabilities pij, the likelihood function can be written as: and the MLE of pij, pA and pB are from which we derive the MLE of D, and D': ′ Ds