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Introduction

Although the development of cheap high-throughput genotyping assays have made large-scale association studies a reality, most ongoing association studies genotype only a small proportion of SNPs in the region of study (be that the whole genome, or a set of candidate regions). Because of correlation (linkage disequilibrium, LD) among nearby markers, many untyped SNPs in a region will be highly correlated with one or more nearby typed SNPs. Thus, intuitively, testing typed SNPs for association with a phenotype will also have some power to pick up associations between the phenotype and untyped SNPs. In practice, typical analyses involve testing each typed SNP individually, and in some cases combinations of typed SNPs jointly (e.g., haplotypes), for association with phenotype, and hoping that these tests will indirectly pick up associations due to untyped SNPs. Here, we present a framework for more directly and effectively interrogating untyped variation.

In outline, our approach improves on standard analyses by exploiting available information on LD among untyped and typed SNPs. Partial information on this is generally available from the International HapMap project [1]; in some cases more detailed information (e.g., resequencing data) may also be available, either through public databases (e.g., SeattleSNPs [2]), or through data collected as a part of the association study design (e.g., [3]). Our approach combines this background knowledge of LD with genotypes collected at typed SNPs in the association study, to explicitly predict (“impute”) genotypes in the study sample at untyped SNPs, and then tests for association between imputed genotypes and phenotype. We use statistical models for multi-marker LD to perform the genotype imputation, with uncertainty, and a Bayesian regression approach to perform the test for association, allowing for potential errors in the imputed genotypes. Although we focus specifically on methods for analyzing quantitative phenotypes in candidate gene studies, the same general framework can also be applied to discrete traits, and/or genome-wide scans.

These imputation-based methods can be viewed as a natural analysis complement to the “tag SNP” design strategy for association studies, which attempts to choose SNPs that are highly correlated with, and hence good predictors of, untyped SNPs. We are simply directly exploiting this property, together with recently developed statistical models for multi-locus LD ([4,5]) to infer the untyped SNP genotypes. Our approach is also somewhat analogous to multipoint approaches to linkage mapping (e.g., [6]), in which observed genotypes at multiple markers predict patterns of identity by descent (IBD) at nearby positions without markers, and test for correlation between these patterns of IBD and observed phenotypes. In the association context, we are predicting identity by state rather than IBD, and the methods of predicting identity by state versus IBD differ greatly, but the approaches share the idea of using multipoint information to predict single-point information, and, at least in their simplest form, subsequently assessing correlation with phenotype at the single-point level. This strategy provides a clean and rigorous way to avoid the “curse of dimensionality” that can plague haplotype-based analyses, without making ad hoc decisions such as pooling rare haplotypes into a single class.

Although our methods are developed in a Bayesian framework, they can also be used to compute p-values assessing significance of observed genotype–phenotype associations. Our approach should therefore be of interest to practitioners whether or not they favor Bayesian procedures in general. It has two main advantages over more standard approaches. First, it provides greater power to detect associations. Part of this increased power comes from incorporating extra information (knowledge on patterns of LD among typed and untyped SNPs), but, unexpectedly, we also found an increased power of our Bayesian approach even when all SNPs were actually typed. Second, and perhaps more importantly, it provides more interpretable explanations for potential associations. Specifically, for each SNP (typed and untyped), it provides a probability that it is causal. This contrasts with standard single-SNP tests, which provide a p-value for each SNP, but no clear way to decide which SNPs with small p-values might be causal.

Methods

We focus on an association study design in which genotype data are available for a dense set of SNPs on a panel of individuals, and genotypes are available for a subset of these SNPs (which for convenience we refer to as “tag SNPs”) on a cohort of individuals who have been phenotyped for a univariate quantitative trait. We assume the cohort to be a random sample from the population, and consider application to other designs in the discussion.

Our strategy is to use patterns of LD in the panel, together with the tag SNP genotypes in the cohort, to explicitly predict the genotypes at all markers for members of the cohort, and then analyze the data as if the cohort had been genotyped at all markers. There are thus two components to our approach: (i) predicting (“imputing”) cohort genotypes, and (ii) analyzing association between cohort genotypes and phenotypes. For (i), we use existing models for population genetic variation across multiple markers [4,5], which perform well at estimating missing genotypes, and provide a quantitative assessment of the uncertainty in these estimates [5]. For (ii), we introduce a new approach based on Bayesian regression, and describe how this approach can yield not only standard Bayesian inference, but also p-values for testing the null hypothesis of no genotype–phenotype association. We chose to take a Bayesian approach partly because it provides a natural way to consider uncertainty in estimated genotypes. However, the Bayesian approach has other advantages; in particular, it provides a measure of the strength of the evidence for an association (the Bayes factor, BF) that is, in some respects, superior to conventional p-values. Furthermore, in our simulations, p-values from our Bayesian approach provide more powerful tests than standard tests, even if the cohort is actually genotyped at all markers (including all causal variants).

Bayesian Regression Approach

We now provide further details of our Bayesian regression approach. The literature on Bayesian regression methods is too large to review here, but papers particularly relevant to our work include [7–9].

For simplicity, we focus on the situation where cohort genotypes are known at all SNPs (tag and non-tag). Extension to the situation, where the cohort is genotyped only at tag SNPs and other genotypes are imputed using sampling-based algorithms such as PHASE [10,11] or fastPHASE [5], is relatively straightforward (see below).

Let G denote the cohort genotypes for all n individuals in the cohort, and y = (y₁, …, y_n) denote the corresponding (univariate, quantitative) phenotypes. We model the phenotypes by a standard linear regression:

where y_i is the phenotype measurement for individual i, μ is the phenotype mean of individuals carrying the “reference” genotype, the x_ijs are the elements of a design matrix X (which depends on the genotype data; see below), the β_js are the corresponding regression coefficients, and ɛ_i is a residual. We assume ɛ_is are independent and identically distributed ~N(0,1/τ), where τ denotes the inverse of the variance, usually referred to as the precision (we choose this parameterization to simplify notation in later derivations). Thus and:

We assume a genetic model where the genetic effect is additive across SNPs (i.e., no interactions) and where the three possible genotypes at each SNP (major allele homozygote, heterozygote, and minor allele homozygote) have effects 0, a + ak and 2a, respectively [12]. We achieve this by including two columns in the design matrix for each SNP, one column being the genotypes (coded as 0, 1, or 2 copies of the minor allele), and the other being indicators (0 or 1) for whether the genotype is heterozygous. The effect of SNP j is then determined by a pair of regression coefficients (β_j1, β_j2), which are, respectively, the SNP additive effect a_j and dominance effect d_j = a_jk_j. While there are other ways to code the correspondence between genotypes and the design matrix, we chose this coding to aid specifying sensible priors (see below).

Priors for (β, μ, τ).

Prior specification is intrinsically subjective, and specifying priors that satisfy everyone is probably a hopeless goal. Our aim is to specify “useful” priors, which avoid some potential pitfalls (discussed below), facilitate computation, and have some appealing properties, while leaving some room for context-specific subjective input. In particular, we describe two priors below, which we refer to as prior D₁ and D₂, that were developed based on the following considerations: (i) inference should not depend on the units in which the phenotype is measured; (ii) even if the phenotype is affected by SNPs in this region, the majority of SNPs will likely not be causal; (iii) for each causal variant there should be some allowance for deviations from additive effects (i.e., dominant/recessive effects) without entirely discarding additivity as a helpful parsimonious assumption; and (iv) computations should be sufficiently rapid to make application to genome-wide studies practical (this last consideration refers to prior D₂).

Priors on the phenotype mean and variance.

The parameters μ and τ relate to the mean and variance of the phenotype, which depend on units of measurement. It seems desirable that estimates (and, more generally, posterior distributions) of these parameters scale appropriately with the units of measurement, so, for example, multiplying all phenotypes by 1,000 should also multiply estimates of μ by 1,000. Motivated by this, for prior D₁ we used Jeffreys' prior for these parameters:

This prior is well known to have the desired scaling properties in the simpler context where observed data are assumed to be N(μ, 1/τ) [13], and we conjecture that our prior D₁ also possesses these desired scaling properties in the more complex context considered here, although we have not proven this.

For prior D₂ we used a slightly different prior, based on assuming a prior for (μ, τ) of the form

Specifically, our prior D₂ assumes the limiting form of this prior as κ,λ → 0 and . In Protocol S1 we show that the posterior distributions obtained using this limiting prior scale appropriately.

Both prior distributions above are “improper” (meaning that the densities do not integrate to a finite value). Great care is necessary before using improper priors, particularly where one intends to compute BFs to compare models, as we do here. However, we believe results obtained using these priors are sensible. For prior D₂, as we show in Protocol S1 the posteriors are proper, and the BF tends to a sensible limit. For prior D₁ we believe this to be true, although we have not proven it.

Prior on SNP effects.

For brevity, we refer to SNPs that affect phenotype as QTNs, for quantitative trait nucleotides. Our prior on the SNP effects has two components: a prior on which SNPs are QTNs and a prior on the QTN effect sizes.

Prior on which SNPs are QTNs.

We assume that with some probability, p₀, none of the SNPs is a QTN; that is, the “null model” of no genotype–phenotype association holds. Otherwise, with probability (1 − p₀), we assume there are l QTNs, where l has some distribution p(l) on {1, 2, …, n_s} where n_s denotes the number of SNPs in the region. Given l, we assume all subsets of l SNPs are equally likely. Both p₀ and p(l) can be context-dependent, and choice of appropriate values is discussed below.

Prior on effect sizes.

If SNP j is a QTN, then its effect is modeled by two parameters, a_j and d_j = a_jk_j. The parameter a_j measures a deviation from the mean μ and will depend on the unit of measurement of the phenotype. To reflect this, we scale the prior on a_j by the phenotypic standard deviation within each genotype class, . Specifically, our prior on a_j is , where σ_a reflects the typical size of a QTN effect compared with the phenotype standard deviation within each genotypic class. Choice of σ_a may be context-dependent, and is discussed below.

The parameter d_j = a_jk_j measures the dominance effect of a QTN. If k_j = 0, then the QTN is additive: the heterozygote mean is exactly between the means of the two homozygotes. If k_j = 1 (respectively, −1), allele 1 (respectively, 0) is dominant. The case corresponds to overdominance of allele 1 or allele 0. We investigate two different priors for the dominance effect:

Prior D₁: We assume that k_j is a priori independent of a_j, with . We chose σ_k = 0.5, which gives , reflecting a belief that overdominance is relatively rare.

Prior D₂: We assume that d_j is a priori independent of a_j, with , where we took σ_d = 0.5σ_a. This prior on d_j induces a prior on k_j in which k_j is not independent of a_j.

Prior D₁ has the attractive property that the prior probability of overdominance is independent of the QTN additive effect a_j. However, the posterior distributions of a_j and k_j must be estimated via a computationally intensive Markov Chain Monte Carlo (MCMC) scheme (see Protocol S2). (An alternative, which we have not yet pursued, would be to approximate BFs under prior D₁ by numerical methods, such as Laplace Approximation; e.g., [14]). Prior D₂ is more convenient, as, when combined with the priors on μ and τ in Equation 4, posterior probabilities of interest can be computed analytically (Protocol S1).

For both priors D₁ and D₂ we assume effect parameters for different SNPs are, a priori, independent (given the other parameters).

Choice of p₀, p(l), and σ_a.

The above priors include “hyperparameters,” p₀ and σ_a, and a distribution p(l) that must be specified. The hyperparameter p₀ gives the prior probability that the region contains no QTNs. While choice of appropriate value is both subjective and context-specific, for candidate regions we suggest p₀ will typically fall in the range 10⁻² to 0.5. If data on multiple regions are available, then it might be possible to estimate p₀ from the data, although we do not pursue this here. Instead, we mostly sidestep the issue of specifying p₀ by focusing on the BF (described below), which allows readers of an analysis to use their own value for p₀ when interpreting results.

In specifying the prior, p(l), for the number of QTNs, we suggest concentrating most of the mass on models with relatively few QTNs. Indeed, here we focus mainly on the extreme case in which p(l) is entirely concentrated on l = 1: that is, the “alternative” model is that the region contains a single QTN. Although rather restrictive, this seems a good starting point in practice, particularly since our results show that it can perform well even if multiple QTNs are present. Nonetheless, there are advantages to considering models with multiple QTNs, and so we also consider a prior where p(l) puts equal mass on l = 1, 2, 3, or 4. This prior suffices to illustrate the potential of our approach, although in practice it would probably be preferable to place decreasing probabilities on larger numbers of QTNs (e.g., p(l = 2) < p(l = 1)). An alternative would be to sidestep specifying p(l) by computing BFs comparing, say, 4-QTN, 3-QTN, 2-QTN, and 1-QTN models versus the “null” model. However, interpreting and acting on these BFs will inevitably correspond to implicit assumptions about the relative prior plausibility of these multi-QTN models.

Finally, specification of the standard deviation of the effect size, σ_a, involves subtle issues. Although it may seem tempting to use “large” σ_a to reflect relative “ignorance” about effect sizes [15], we believe this is inadvisable. Although large σ_a yields a flat prior on effect sizes, this prior is far from uninformative, in that it places almost all its mass on large effect sizes. The result would essentially allow only zero effects (i.e., the “null” model), or large effects (the “alternative” model). If in truth the causal SNPs have relatively small effect, which is probably generally realistic, then (for realistic sample sizes) the null model would be strongly favored over the alternative, because the data would be more consistent with zero effects than with large effects. Choice of σ_a can thus strongly affect inference, particularly the BF, which we use to summarize evidence for the region containing any QTNs. Partly because of this, in practice we suggest averaging results over several values for σ_a (equivalent to placing a prior on σ_a). It may also be helpful to examine sensitivity of results to σ_a. For example, if the BF is small for all values of σ_a, then there is no evidence for any QTN in the region; if the BF is large for some values and small for others, then the evidence depends on the extent to which you believe in large versus small effects. However, for simplicity, all results in this paper were obtained using a fixed value of σ_a = 0.5.

Inference

We focus on two key inferential problems: (i) detecting association between genotypes and phenotype, and (ii) explaining observed associations. In the model of Equation 1, these translate to answering (i) are any β_js non-zero? and (ii) which β_js are non-zero and how big are they? We view the ability to address both questions within a single framework to be an advantage of our approach.

Detecting association.

To measure the evidence for any association between genotypes and phenotypes, we use the BF, [16] given by

where H₀ denotes the null hypothesis that none of the SNPs is a QTN (a_j = d_j = 0 for all j), and H₁ denotes the complementary event (i.e., at least one SNP is a QTN). Computing the BF involves integrating out unknown parameters, as described in Protocols S1 and S2. In interpreting a BF, it is helpful to bear in mind the formula “posterior odds = prior odds × BF,” so, for example, if the prior odds are 1:1 (i.e., p₀ = 0.5, so association with genetic variation in the region is considered equally plausible, a priori, as no association) then a BF of 10 gives posterior odds of 10:1, or ~91% probability of an association.

In the special case where we allow at most one QTN, Equation 5 reduces to

where H_j denotes the event that SNP j is the QTN. The jth term in this sum corresponds to the BF for H_j versus the null model, and involves the genotype data at SNP j only. We refer to these terms as the “single-SNP” BFs, so in this special case the overall BF is the mean of the single-SNP BFs. This natural way for combining information across (potentially correlated) SNPs is an attractive property of BFs compared with single-SNP p-values. Furthermore, in terms of detecting a genotype–phenotype association it can work well even if multiple QTNs are present (see Results).

The Bayes/non-Bayes compromise.

From a Bayesian viewpoint, the BF provides the measure of the strength of evidence for genotype–phenotype association. That is, if one accepts our prior distributions and modeling assumptions, then the BF is all that is necessary to decide whether a genuine association is present. However, given the potential for debate over prior distributions, and for deviations from modeling assumptions, it is helpful to note that a p-value for testing H₀ can be obtained from a BF through permutation. Specifically, one can compute the BF for the observed data, and for artificial data sets created by permuting observed phenotypes among cohort individuals, and obtain a p-value as the proportion of permuted data sets for which the BF exceeds the BF for the observed data. Being based on permutation, the resulting p-value is valid irrespective of whether the model or priors are appropriate. This p-value also provides a helpful way to compare our approach with standard tests of association, and, as we show below, tests based on BF appear to perform well in a wide variety of situations. Using BFs as test statistics to obtain p-value is referred to as the “Bayes/non-Bayes compromise” by Good [17].

Explaining and interpreting associations.

To “explain” observed associations we compute posterior distributions for SNP effects (a_j + d_j and 2a_j for the heterozygote and minor-allele homozygote, respectively), with particular focus on the posterior probability that each SNP is a QTN P(a_j ≠ 0). Here, our Bayesian regression approach has an important qualitative advantage over standard multiple regression. Specifically, if a genetic region contains multiple highly correlated SNPs, each highly correlated with the phenotype, then the correct conclusion would be any of these SNPs could be causal, without identifying which one. This will be reflected in the posterior distribution of the effects: the overall probability that at least one SNP is a QTN will be high, but (at least in the simplest case where we assume at most one QTN) this probability will be spread out over the multiple correlated SNPs. In contrast, if multiple highly correlated SNPs are included in a standard multiple regression it is possible that no one of them will produce a significant p-value.

We also argue that the imputation-based approach brings us closer to being able to interpret estimated effects for each SNP as actual causal effects, rather than simply associations. Indeed, the key to making the leap from association to causality is controlling for all potential confounding factors, and by imputing genotypes at nearby SNPs, the imputation-based approach controls for one important set of confounding factors (the nearby SNPs), which would otherwise be ignored. Thus, while functional studies provide the ultimate route to convincingly demonstrating causal effects, our approach may help target such studies on the most plausible candidate SNPs.

Imputing genotypes

In the tagSNP design, observed genotypes G_obs consist of panel genotypes at all SNPs and cohort genotypes at tagSNPs only. To apply our methods in this situation, we use sampling-based algorithms (PHASE [10,11], or fastPHASE [5]) to generate multiple imputations for the complete genotype data (all individuals at all SNPs) by sampling from . We then incorporate these imputations into our inference: for prior D₁, this involves adding a step in the MCMC scheme to sample the imputed genotypes from their posterior distribution given all the data; for prior D₂ it involves simply averaging relevant calculations over imputations. Details are given in Protocols S1 and S2.

Availability of Software

Methods described here are implemented in a software package, Bim-Bam (Bayesian IMputation-Based Association Mapping), available from the Stephens Lab website http://stephenslab.uchicago.edu/software.html.

Results

“Power” and Comparisons with Other Approaches

We compared the power of our approach to other common approaches via simulation. We simulated genotype and phenotype data (with μ = 0 and τ = 1) for genetic regions of length 20 kb containing a single QTN, and genetic regions of length 80 kb containing four QTNs, as follows:

(1) Using a coalescent-based simulation program, msHOT [18], simulate 600 haplotypes from a constant-sized random mating population, under an “infinite sites” mutation model, with (population-scaled) mutation rate θ = 0.4/kb and “background” recombination rate ρ = 0.8/kb, and a recombination hotspot (width 1 kb; recombination rate 50ρ per kb) in the center of the region.

(2) Form genotypes for a “panel” of 100 individuals by randomly pairing 200 haplotypes, and a “cohort” of 200 individuals by randomly pairing the other 400 haplotypes.

(3) Select tag SNPs from the panel data using the approach of Carlson et al. [19] with an r² cutoff of 0.8. As in Carlson et al. [19], SNPs with panel minor allele frequency (MAF) <0.1 were not tagged.

(4) Select which SNPs are QTNs, and their effect sizes, and simulate phenotype data for each cohort individual according to Equation 1. We considered four scenarios: (A) a “common” (MAF>0.1) QTN, with a range of effect sizes a = 0.2, 0.3, 0.4, 0.5 and “mild” dominance for the minor allele (d = 0.4a); (B) a common QTN, with a = 0.3 and “strong” dominance for the major allele (d = −a); (C) a “rare” (MAF 0.01 − 0.05) QTN, with a = 1 and no dominance (d = 0); (D) four common, relatively uncorrelated, QTNs, each with a = 0.3 and d = 0.4a. In each situation, we randomly chose a QTN satisfying the relevant MAF requirements (in the 600 sampled haplotypes), except under scenario (D) we first chose four tag SNP “bins” at random and then randomly chose a QTN satisfying the MAF requirement in each bin, thereby ensuring the four QTNs were relatively uncorrelated. (While real data may contain multiple highly correlated QTNs, we did not explicitly consider this case, since their effect would be similar to a single QTN.)

We compared power of tests based on the BF (under prior D₂, allowing at most one QTN, using Equation 6 with four other significance tests:

(1) Two tests based on p_min, the minimum p-value obtained from testing each SNP individually (via standard ANOVA-based methods) for association with the phenotype. These two tests differed in whether the single SNP p-values were obtained using the 1 degree-of-freedom (df) “allelic” test, which assumes an additive model where the mean phenotype of heterozygotes lies midway between the two homozygotes (equivalent to linear regression of phenotype on genotype), or the 2 df “genotype” test, which treats the mean of the heterozygotes as a free parameter.

(2) A test based on p_reg, the global p-value obtained from linear regression of phenotype on all SNP genotypes (using the standard F statistic, coding the genotypes as 0,1 and 2 at each SNP, and assuming additivity across SNPs). See Chapman et al. [20] for example.

(3) A test based on BF_max, the maximum single-SNP BF. We included this test for comparison with the mean single-SNP BF (Equation 6), to examine whether averaging information across SNPs in Equation 6 improved power.

For each test, we analyzed each dataset in two ways: as if data had been collected using (i) a “resequencing design” (i.e., all individuals were completely resequenced, so genotype data are available at all SNPs in all individuals); and (ii) a “tag SNP design” (i.e., in panel individuals genotype data are available at all SNPs, but in cohort individuals genotype data are available at tag SNPs only). For the tag SNP design, we assumed haplotypic phase is known in the panel (as it is, mostly, for the HapMap data for example), but not in the cohort; however our approach can also deal with unknown phase in the panel. For p_reg and p_min, tests were performed on all SNPs for the resequencing design, and on tag SNPs only for the tag SNP design. For BF and BF_max, single-SNP BFs were computed for all SNPs in both designs (averaging over imputed genotypes for non-tag SNPs in the tag SNP design). For p_reg, we computed a p-value assessing significance using the standard asymptotic distribution for the F statistic; for the other tests we found p-values by permutation, using 200–500 random permutations of phenotypes assigned to cohort individuals. (The relatively small number of permutations limits the size of the smallest possible p-value, causing discontinuities near the origin in Figure 1).

Figure 1. Power Comparisons

(A) single common variant, modest dominance; (B) single common variant, strong dominance for minor allele; (C) single rare variant, no dominance; (D) multiple common variants.

Each colored line shows power of test varying with significance threshold (type I error). Black: BF from our method (prior D₂); Green: p_min (allelic test); Red: p_min (genotype test); Blue: p_reg, multiple regression; Grey: BF_max. Each column of figures shows results for data analyzed under the “resequencing design” (left) and the “tag SNP design” (right). Each row shows results for the four different simulation scenarios.

doi:10.1371/journal.pgen.0030114.g001

Figure 1 shows power of each test versus type I error under both resequencing and tag SNP designs. For Scenario (A) (a single common QTN), the relative performances of methods were similar for all four effect sizes examined (unpublished data), and so we pooled these results in the figure.

Comparing p_min and p_reg, the single-SNP tests (p_min) were more powerful when all variants (including the causal variant) were typed, or when the QTN was a common SNP and therefore “tagged” by a tag SNP, while the regression-based approach (p_reg) was more powerful when the QTN was a rare SNP not “tagged” by any tag SNP. Among the two single-SNP tests, the 1 df allelic test performed as well as, or better than, the 2 df genotypic test, except in Scenario (B), where the major allele exhibits strong dominance. In particular, for Scenario (A), where the causal variant exhibits dominance, the allelic test (which assumes no dominance), performed better than the genotypic test. This is presumably because, with the effect and sample sizes considered, the extra parameter estimated in the genotypic test does not sufficiently improve model fit. Although relative performance of p_min and p_reg in the tag SNP design could depend on tag SNP selection scheme (and the one we used, based on pairwise LD, would seem to favor p_min), it seems reasonable to expect single-SNP tests to be effective at detecting “direct” associations between the phenotype and a causal variant, or “near-direct” association between a SNP that tags a causal variant, and the regression-based approach to be better at detecting indirect associations between a phenotype and a variant not “tagged” by a single SNP (the intuition, from Chapman et al. [20], is that such variants can be highly correlated with linear combinations of tag SNPs, and thus be detected by linear regression). In principle, p_reg could also effectively capture “direct” associations, but our empirical results suggest that it is less effective at this than the single SNP tests. (However, poor performance of p_reg under the resequencing design may be due in part to inadequacy of the asymptotic theory when large numbers of correlated covariates are used. This might be alleviated by assessing significance of p_reg by permutation.)

Turning now to our approach, except for Scenario (B) in the tag SNP design, the test based on the BF is as powerful or more powerful than the other tests. Thus, unlike p_reg and p_min, the BF performs well in detecting both “direct” and “indirect” associations: if the QTN is typed, the BF detects it using observed genotype data at that SNP; otherwise, it detects it using the imputed genotype data at the QTN. In Scenario (B), where the major allele exhibits strong dominance, our approach suffered slightly in power compared with the genotypic test, presumably because our prior places relatively low weight on strong dominance. However, the power loss was small compared with that of the allelic test. Thus our prior “allows” for dominance without suffering the full penalty incurred by the extra parameter in the genotypic test when dominance is less strong (Scenario [A]).

In Scenario (D), which involved multiple QTNs, tests based on the BF clearly outperformed other tests considered, even though the BF was computed allowing at most one QTN. Our explanation is that the BF, being the average of single-SNP BFs, has greater opportunity to capture the presence of multiple QTNs than does the minimum p-value. This explanation is supported by the fact that the maximum BF, BF_max, performs less well than BF. To examine whether power might be further increased by explicitly allowing for multiple QTNs, we compared power for BFs computed using 1-QTN and 2-QTN models (in the 2-QTN model p(l = 1) = p(l = 2) = 0.5). We found little difference in power, although BFs for the 2-QTN model tended to be larger than BFs for the 1-QTN model, so allowing for multiple QTNs may help if the BF itself, rather than a p-value based on the BF, is used to measure the strength of evidence for association. In addition, considering multiple-QTN models should have advantages when attempting to explain an association (see below).

A second, and perhaps more surprising, situation where the BF outperforms other methods is when all SNPs are typed and tested (i.e., Scenario (A), resequencing design). Here, in contrast to Scenario (D), BF_max performs similarly to the standard BF, suggesting that the power gain is due not to averaging, but to an intrinsic property of single-SNP BFs that makes them better measures of evidence than single-SNP p-values. Our explanation is that the BF tends to be less influenced by less informative SNPs (e.g., those with very small MAF, of which there are many in the resequencing design), whereas p-values tend to give equal weight to all SNPs, regardless of information content. Specifically, BFs for relatively uninformative SNPs will always lie close to 1, and should not greatly influence either the maximum or the average of the single-SNP BFs (or, more precisely, will not greatly influence differences in these test statistics among permutations of phenotypes). In contrast, p-values for each SNP are forced, by definition, to have a uniform distribution under H₀, and so p-values from a large number of uninformative SNPs unassociated with the phenotype could swamp any signal generated by a single informative SNP associated with the phenotype. Although the resequencing design is currently uncommon, this observation suggests that it may generally be preferable to rank SNPs according to their BFs, rather than by p-values (e.g., in genome scans). It also highlights a general (rarely considered, and perhaps underappreciated) drawback of p-values as a measure of evidence: the strength of evidence of a given p-value depends on the informativeness of the test being performed, or, more specifically, on the distribution on the p-values under the alternative hypothesis, which is generally not known. Thus, for example, a p-value of 10⁻⁵ in a study involving few individuals may be less impressive than the same p-value in a larger study. In contrast, the interpretation of a BF does not depend on study size or similar factors.

Resequencing versus Tag SNP Designs

An important feature of Figure 1 is that, for Scenarios (A), (B), and (D), where the causal SNPs are common, power is similar for the resequencing and tag SNP designs. Indeed, in these cases most other aspects of inference are also similar. For example, Figure 2 shows that, under Scenarios (A) and (B), estimated effect sizes, BFs, and posterior probability that the actual causal variant is a QTN, are typically similar for both designs. Thus under these scenarios, our imputation-based approach effectively recreates results that would have been obtained by resequencing all individuals.

Figure 2. Comparison of Results for Resequencing Design (x-axis) and Tag SNP Design (y-axis)

Panels show: (a) errors in the estimates (posterior means) of the heterozygote effect (a + d); (b) errors in the estimates (posterior means) of the main effect (a); and (c) posterior probability of being a QTN (P((a, d) ≠ (0, 0))) assigned to the causal variant.

doi:10.1371/journal.pgen.0030114.g002

In contrast, when the causal variant is rare, there is a noticeable drop in power for the tag SNP design versus the resequencing design, and the BFs, posterior probabilities, and effect size estimates under the two designs often differ substantially (unpublished data). This may seem slightly disappointing: one might have hoped that, even with tag SNPs chosen to capture common variants, they might also capture some rare variants. Indeed, this can happen: in some simulated data sets the rare causal variant was clearly identified by our approach, presumably because it was highly correlated with a particular haplotype background, and could thus be accurately predicted by tag SNPs. However, this occurred relatively rarely (just a few simulations out of 100).

We wondered whether a different tagging strategy, aimed at capturing rare variants, might improve performance when the causal variant is rare. The development of such strategies lies outside the scope of this paper, but, to assess potential gains that might be achieved, we analyzed rare-variant simulations assuming that all SNPs except the causal variant were typed in the cohort. Power from this approach (Figure 3) gives a conservative upper bound on what could be achieved using a more effective tagging design, without actually typing the causal variant. Although power was higher than with the r²-based tag SNP selection, it remained substantially lower than in the resequencing design, where the causal variant is typed.

Figure 3. Examination of Potential Effect of Different Tag SNP Strategies on Power, When the Causal Variant is Rare (0.01 < MAF < 0.05)

Solid line: Resequencing design; dashed line: tag SNP design, with tags selected using method from [19]; and dotted line: tag SNP design, with all SNPs except the causal SNP as tags.

doi:10.1371/journal.pgen.0030114.g003

We also wondered whether a different approach to impute missing genotypes (in the cohort at non-tag SNPs) might improve performance. For results above, we used the software fastPHASE [5] to impute the genotypes, so we re-ran the analysis using a different imputation algorithm [10,11]. Results for these two approaches (Figure 4) show little difference in terms of power, consistent with previous results [5] suggesting the two approaches have similar accuracy in imputing missing genotypes.

Figure 4. Power of the Multipoint Approach in the Rare Variant Scenario for Two Different Imputation Algorithms

doi:10.1371/journal.pgen.0030114.g004

In summary, imputation-based methods appear to increase power of the tag SNP design to detect rare variants, but nevertheless remain notably less powerful than BFs based on the complete resequencing data.

Comparison of Prior D₁ and D₂

Priors D₁ and D₂ differ in their assumed correlation between the dominance effect (d = ak) and main effect a: in D₁ the prior probability of overdominance is independent of a, whereas under D₂ overdominance is more likely for small a than for large a (Figure 5). In this respect, D₁ is perhaps more sensible than D₂; however, D₂ is computationally much simpler. To examine the effects of these priors on inference, we compared (i) the BF and (ii) the posterior probability assigned to the actual causal variant under each prior for the datasets from Scenarios (A) and (B). Results agreed quite closely (Figure 6), suggesting prior D₂ provides a reasonable approximation to prior D₁ in the scenarios considered. This is important, since prior D₂ is computationally practical for computing BFs for very large datasets (e.g., genome-wide association studies with hundreds of thousands of SNPs), for which sampling posterior distributions of parameters using an MCMC scheme would be computationally daunting.

Figure 5. Scatter Plot of Samples from Prior Distribution of a (x-axis) and a + d (y-axis), for Priors D₁ (Black) and D₂ (Blue)

The solid yellow line corresponds to d = 0 (additivity). The dashed red lines are the limits above and below which a SNP exhibits over-dominance.

doi:10.1371/journal.pgen.0030114.g005

Figure 6. Comparison of Inferences using Prior D₁ and D₂ for the BF (Left) and the Posterior Probability Assigned to the Causal Locus Being a QTN (Right)

Results shown are for all datasets for the common variant Scenario (A) and (B) and for both the resequencing design and the tag SNP design. The discrepancy between the larger estimated BFs is caused by the fact that we used insufficient MCMC iterations to accurately estimate very large BFs (>10⁶) under prior D₁.

doi:10.1371/journal.pgen.0030114.g006

Allowing for Multiple Causal Variants

When analyzing a candidate region, one would ideally like not only to detect any association, but also to identify the causal variants (QTNs). Since a candidate region could contain multiple QTNs, we implemented an MCMC scheme (using prior D₁) to fit multi-QTN models where the number of QTNs is estimated from the data; here, we consider a multi-QTN model with equal prior probabilities on 1, 2, 3, or 4 QTNs. (A similar MCMC scheme could also be implemented for prior D₂, and could exploit the analytical advantages of this prior to reduce computation. Indeed, for regions containing a modest number of SNPs it would be possible to examine all subsets of SNPs, and entirely avoid MCMC.)

We compare this multi-QTN model with a one-QTN model on a dataset simulated with four QTNs (scenario [D]). The estimated BF for a one-QTN model was ~6,000, while for the multi-QTN model it was >10⁵ (we did not perform sufficient iterations to estimate how much bigger than 10⁵). Thus, if a region contains multiple causal variants, then allowing for this possibility may provide substantially higher BFs. Figure 7 shows the marginal posterior probabilities for each SNP being a QTN, under the one-QTN and multi-QTN models, conditional on at least one SNP in the region being a QTN. (Summarising the more complex information on posterior probabilities for combinations of SNPs is an important future challenge.) Under the one-QTN model, only one of the four causal SNPs has a large marginal posterior probability, whereas under the multi-QTN model all four are moderately large. Of course, other SNPs correlated with the four QTNs were also associated with the phenotype, and so have elevated posterior probabilities. This example illustrates the potential for the multi-QTN model to provide fuller explanations for associations.

Figure 7. Illustration of How a Multi-QTN Model Can Provide Fuller Explanations Than a One-QTN Model for Observed Associations