Exploiting Population Samples to Enhance Genome-Wide Association Studies of Disease

Abstract

It is widely acknowledged that genome-wide association studies (GWAS) of complex human disease fail to explain a large portion of heritability, primarily due to lack of statistical power—a problem that is exacerbated when seeking detection of interactions of multiple genomic loci. An untapped source of information that is already widely available, and that is expected to grow in coming years, is population samples. Such samples contain genetic marker data for additional individuals, but not their relevant phenotypes. In this article we develop a highly efficient testing framework based on a constrained maximum-likelihood estimate in a case–control–population setting. We leverage the available population data and optional modeling assumptions, such as Hardy–Weinberg equilibrium (HWE) in the population and linkage equilibrium (LE) between distal loci, to substantially improve power of association and interaction tests. We demonstrate, via simulation and application to actual GWAS data sets, that our approach is substantially more powerful and robust than standard testing approaches that ignore or make naive use of the population sample. We report several novel and credible pairwise interactions, in bipolar disorder, coronary artery disease, Crohn’s disease, and rheumatoid arthritis.

GENOME-WIDE association studies (GWAS) have implicated thousands of single-nucleotide polymorphisms (SNPs) in the human genome as associated with hundreds of phenotypes (Johnson and O’Donnell 2009). However, as many researchers have pointed out (Manolio et al. 2009; Eichler et al. 2010), the results from GWAS fail to explain the observed heritability of many phenotypes, including complex human diseases, whose genetic architectures remain largely unknown. One often-cited reason for this problem is that the high multiple-testing burden requires an exceedingly stringent statistical significance level. Furthermore, while most studies have employed univariate (locus-by-locus) testing approaches, complex diseases are likely to be affected by interactions between loci (Eichler et al. 2010). Such interactions arise when there is a dependence of genotypic effects of one locus on genotypes at other loci (Cordell 2009).

In the case of interactions, due to the overwhelming number of locus subsets, the multiple-testing problem becomes a serious computational and statistical challenge. Even when limiting exploration to pairwise SNP–SNP interactions, a modest study including 300,000 usable loci requires testing ∼45 billion SNP pairs, and associations must have P-values < ∼10⁻¹² (the 0.05 Bonferroni-corrected significance level) to be declared statistically significant genome-wide. Recently, several authors have suggested sophisticated approximate and exhaustive methods for detecting pairwise interactions (Brinza et al. 2010; Liu et al. 2011; Prabhu and Pe’er 2012). These methods constitute a major step in dealing with the computational issue of carrying out the large number of tests, but their application to actual studies has led to surprisingly few replicable discoveries (for example, one pair in Liu et al. 2011 and another in Prabhu and Pe’er 2012). Many of the other reported pairwise discoveries fail after careful scrutiny (see, for example, the discussion in Liu et al. 2011 and the Discussion in the present article). Given findings in other organisms (Shao et al. 2008; Bloom et al. 2013), and the biological plausibility of the existence of interactions, the likely explanation for the limited GWAS results is that modest interaction effects comprising common SNPs do exist; however, due to the aforementioned statistical challenge, the tests employed are not powerful enough to detect them given available sample sizes.

Improving power of tests used in GWAS is therefore an extremely important research question today, especially when interactions are considered. One solution is to increase sample sizes by collecting more case–control data. Indeed, this is an ongoing trend, and ever-larger studies are in preparation by various worldwide consortia. Often, however, independent samples from the studied population but with unknown case–control status are already available. Sources include, for example, (i) population studies (Waye 2005; Siva 2008), (ii) quantitative trait studies (Yang et al. 2010), and (iii) binary trait studies that employ a case–population design [like the Wellcome Trust Case Control Consortium (WTCCC) study, discussed in more detail below]. The coming years are expected to make such population samples available in increasingly large numbers, given recently published government plans in multiple countries to sequence hundreds of thousands of genomes (http://news.sciencemag.org/scienceinsider/2012/12/uk-unveils-plan-to-sequence-whol.html).

A key challenge is to make optimal use of these population samples to increase power of GWAS in general and interaction detection efforts in particular. The common existing practice is to use only the case–control data and ignore the population data completely. Some studies employ population data as reference panels from which linkage disequilibrium structure is inferred. This can improve power by reducing the effective multiple-comparisons burden, as well as by the finer localization of detected associations through imputation. These methods are different from the one we develop and could be used as well. In another approach, which is closer to the one studied here, population samples are treated as additional controls with the intention of increasing sample size and thus power (Burton et al. 2007; Lippert et al. 2013). Arguably, for rare diseases the latter approach is satisfactory (because controls are very similar to the population). However, since this leads to mislabeling of any cases in the population sample as controls, for common disease this becomes a problem, possibly causing more harm than good. We suggest a more appropriate approach here that models the joint likelihood of the case–control and population samples, in the expectation that correctly using all available data would result in significantly improved power. This expectation is verified in our simulations and is further reflected in application to real GWAS data for seven different phenotypes studied by the WTCCC.

An important feature of our approach is its ability to incorporate assumptions about the studied population. The idea is that by exploiting additional properties of the data, one can reduce the parameter space describing associations of interest (Song and Nicolae 2009). This improves power to detect associations that comply with the assumptions, at the often acceptable cost of reduced power to detect implausible associations. Two popular and broadly applicable population assumptions that constrain the parameter space are Hardy–Weinberg equilibrium (HWE) (the independence of maternal and paternal allele values that make up each genotype locus) and linkage equilibrium (LE) (the independence of the two loci that make up a pair) (Yang et al. 1999; Chatterjee and Carroll 2005; Zhao et al. 2006). Imperfect as they may be (see Discussion), these assumptions are expected to hold for the vast majority of associations considered in a typical GWAS.

Thus, the main contribution of our present work is the proposal of a method for use in GWAS that combines (a) the correct handling of case–population or case–control–population designs and (b) inclusion of population assumptions, to maximize efficiency of association testing. We show in simulation results that our direct constrained maximum-likelihood (CMLE) approach is substantially more powerful than ignoring extraneous population data in a case–control study or using them naively to extend controls. We then apply this method to the seven phenotypes from the WTCCC study, which employs a case–population design, with the cases of unrelated diseases potentially serving as a source for additional population samples. Our pairwise analysis of these data reveals multiple new associations in several diseases. Examination of all significant findings shows that a large number of them include SNPs in genomic regions not implicated by the original WTCCC study. Interestingly, these regions include several loci that were identified and replicated in later studies employing standard analysis methods but larger sample sizes—illustrating and validating the improved power of our method. We report several novel and credible associations, including pairwise interactions in bipolar disorder, coronary artery disease, Crohn’s disease, and rheumatoid arthritis.

An implementation of all tests described herein for pairwise and univariate GWAS is available in the R statistical computing software package CCpop. As our theory and results demonstrate, our new approach can lead to substantial gains in power (up to sevenfold in standard models, see Results). We believe that practically every case–control GWAS can identify relevant population samples to use as additional data and that many of these studies can further justify population assumptions when analyzing the data (any doubt about such assumptions can be eliminated by using more general tests during replication). Thus our recommendation is that previous studies be revisited with our approach and that new studies use our tests rather than the traditional ones.

Methods

Association testing with case–control–population data

Here we describe our new approach that correctly handles case–control–population designs within a maximum-likelihood framework. First, the method is presented in the context of a single SNP. Second, we extend it to the case of two or more SNPs. Finally, we show how to incorporate population assumptions into the model.

The univariate case:

Let x be a diploid SNP locus with genotypes in {0, 1, 2}, and let y be a binary disease phenotype coded 0 for controls (which are truly unaffected) and 1 for cases. Let $\overrightarrow{n}={\left\{n_{i,j}\right\}}_{i=0,j=0}^{1\hspace{1em}2}$ denote the observed number of samples in a standard retrospective case–control sample $\mathcal{C}$ with y = i and x = j. Let $n_{i,}={\displaystyle {\sum }_j{n}_{i,j}}$ denote the number of individuals with phenotype i and $n_{,j}={\displaystyle {\sum }_i{n}_{i,j}}$ denote the number of individuals with genotype j. Suppose there exists, in addition, an independent population sample $\mathcal{P}$ with the observed genotype counts at x denoted $\overrightarrow{m}={\left\{m_j\right\}}_{j=0}^2$. Let $\overrightarrow{p}={\left\{p_{i|j}\right\}}_{i=0,j=0}^{1 2}$ be the probability of disease status i given the genotype j. For i = 1, this is referred to as disease penetrance. Denote $\overrightarrow{\pi }={\left\{{\pi }_j\right\}}_{j=0}^2$ the marginal distribution of x in the population and K the prevalence of the disease, which is assumed known.

The log-likelihood of the pooled data ($\mathcal{C}$, $\mathcal{P}$) may be written as

$$\begin{array}{c}l\left(\overrightarrow{p},\overrightarrow{\pi };\overrightarrow{n},\overrightarrow{m}\right)={\displaystyle \sum _{i,j}{n}_{i,j}\log \left(p_{i|j}\frac{{\pi }_j}{K}\right)}+{\displaystyle \sum _j{m}_j\log {\pi }_j}\\ \propto {\displaystyle \sum _{i,j}{n}_{i,j}\log p_{i|j}}+{\displaystyle \sum _j\left(n_{,j}+m_j\right)\log {\pi }_j}.\end{array}$$ (1)

Without further assumptions on the natural parameter space $\left\{\overrightarrow{p},\overrightarrow{\pi }\right\}$, this space is of dimension 5, since $\overrightarrow{\pi }$ must sum to 1, and p_1❘j = 1 − p_0❘j. The known prevalence K imposes a constraint on this space:

$${\displaystyle \sum _j{p}_{1|j}{\pi }_j}=K.$$ (2)

Solving (1) for the maximum-likelihood estimator (MLE) under the alternative hypothesis (of association between y and x) may be approached directly as a nonlinear optimization problem with a nonlinear equality constraint (2) and “box” inequality constraints that keep all parameters in [0, 1]. Under the null hypothesis (of no association), an additional constraint is imposed that the penetrance be flat p_1❘j ≡ c. Trivially, due to (2),

$$c=K,\hspace{1em}{\pi }_j=\frac{n_{,j}+m_j}{n+m}.$$ (3)

This estimation approach works well, but convergence under the alternative may not be fast enough to manage the large number of tests that must be performed during GWAS. Rewriting the problem in standard convex form (Boyd and Vandenberghe 2004) and using modern solvers lead to faster model fitting. Define q_j = p_1❘jπ_j; solving (1) subject to (2) now takes the following convex optimization problem form:

$$\begin{array}{l}\underset{\overrightarrow{q},\overrightarrow{\pi }}{\text{minimize}}\hspace{0.17em}\hspace{0.5em}- {\displaystyle \sum _j\left(n_{0,j}\log \left({\pi }_j-q_j\right)+n_{1,j}\log q_j+m_j\log {\pi }_j\right)}\\ \text{subject}\hspace{0.17em}\text{to}\hspace{0.17em}\hspace{0.5em}{\displaystyle \sum _j}{q}_j=K,\\ \hspace{1em}{\displaystyle \sum _j{\pi }_j}=1.\end{array}$$ (4)

Notably, problem (4) is also subject to the implicit domain constraints q_j ≥ 0, π_j ≥ 0, and q_j ≤ π_j, but all constraints are linear. It is possible to solve on the order of tens of thousands of such small problems per second on a modern personal computer, using a variety of off-the-shelf open-source or commercial solvers. For example, one can substitute the equality constraint into the objective function, add a logarithmic barrier for the box constraints, and solve with Newton’s method or the BFGS algorithm (Ruszczynski 2011).

Problem (4) facilitates association testing within the template of the generalized likelihood-ratio test (GLRT), contrasting the maximum likelihood under the alternative hypothesis with that under the null. The GLRT statistic under the null is asymptotically distributed as a centered chi-square random variable with 3 d.f. As described below, this procedure readily incorporates standard population-level assumptions that can be used to decrease the degrees of freedom of the test.

The multivariate case:

Case–control–population analysis can be similarly applied for detecting associations that involve multiple SNPs working together. The methods we develop here are equally applicable to interactions of an arbitrary number of SNPs. For exposition purposes, however, we describe our methods and provide results in the context of pairwise interactions, where power is desperately needed but detection is still a feasible task.

Similarly to the univariate case, let $\overrightarrow{x}=\left(x_1,x_2\right)$, with x₁ ∈ {0, 1, 2} and x₂ ∈ {0, 1, 2}, be a pair of diploid SNPs. Let n_i,j,k denote the observed counts in $\mathcal{C}$ with y = i, x₁ = j, x₂ = k, and let $n_i={\displaystyle {\sum }_{j,k}{n}_{i,j,k}}$ and $n_{j,k}={\displaystyle {\sum }_i{n}_{i,j,k}}$ be the disease status and pairwise genotypic counts, respectively. The pairwise genotype counts in $\mathcal{P}$ are denoted m_j,k. Let p_i❘j,k be the probability of disease status i given the pair of genotypes and π_j,k be the bivariate marginal distribution of the pair.

Testing for pairwise genetic association with disease is typically formulated using a logistic parameterization of the penetrance,

$$p_{1|j,k}=\frac{1}{1+\text{exp}\left(-{\xi }_{j,k}\right)},$$

where ξ_j,k may take different forms, depending on the specific formulation (see Discussion). The approach we focus on here tests for association while allowing for interaction (Cordell 2009). In this approach one is interested in the overall significance of the model for $\overrightarrow{p}$ that includes effects at both loci, compared to a null model that includes neither,

$$\begin{array}{cc}{\xi }_{j,k}^{\text{flat}}=& \mu \hfill \\ {\xi }_{j,k}^{\text{full}}=& \mu +{\alpha }_1{\mathbb{I}}_{j=1}+{\alpha }_2{\mathbb{I}}_{j=2}+{\beta }_1{\mathbb{I}}_{k=1}+{\beta }_2{\mathbb{I}}_{k=2}\hfill \\ & +{\gamma }_{1,1}{\mathbb{I}}_{j=1,k=1}+{\gamma }_{1,2}{\mathbb{I}}_{j=1,k=2}+{\gamma }_{2,1}{\mathbb{I}}_{j=2,k=1}\hfill \\ & +{\gamma }_{2,2}{\mathbb{I}}_{j=2,k=2},\hfill \end{array}$$ (5)

with $\mathbb{I}$ being the indicator function.

By writing the log-likelihood of the pooled data and repeating the steps taken in the univariate case, we obtain the MLE as the standard convex form,

$$\begin{array}{l}\underset{\overrightarrow{q},\overrightarrow{\pi }}{\text{minimize}}\hspace{0.17em}\hspace{0.5em}- {\displaystyle \sum _{j,k}\left(n_{0,j,k}\log \left({\pi }_{j,k}-q_{j,k}\right)+n_{1,j,k}\log q_{j,k}+m_{j,k}\log {\pi }_{j,k}\right)}\\ \text{subject\hspace{0.5em}to\hspace{0.17em}\hspace{0.5em}}{\displaystyle \sum _{j,k}{q}_{j,k}}=K,\\ {\displaystyle \sum _{j,k}{\pi }_{j,k}}=1,\end{array}$$ (6)

with q_j,k = p_1|j,kπ_j,k. Now the full parameter space is of dimension 17 (again, population assumptions will reduce this dimension). The same optimizers used for solving problem (4) can be used for problem (6) to fit thousands of pairwise models per second. While on a single personal computer this may still be too computationally demanding for a genome-wide exhaustive pairwise search with its billions of candidates, we show in Results that this approach can be combined with a filter such as the fast methods mentioned in the Introduction. An appropriate filter rapidly generates a ranking of all pairs according to a score that is roughly related to the likelihood-ratio statistic of interest. Subsequently, millions of top-ranking candidate pairs can be processed with our approach, and the whole analysis takes <1 hr in the case of the WTCCC data.

The flat model, ${\xi }_{j,k}^{\text{flat}}$, is again trivially solved by

$$\mu =K,\hspace{1em}{\pi }_{j,k}=\frac{n_{j,k}+m_{j,k}}{n+m}.$$

Incorporating distributional assumptions

The nonparametric tests described above (where the specifications of $\overrightarrow{p}$ and $\overrightarrow{\pi }$ are saturated with respect to the included SNPs) are consistent against all forms of dependence and genotypic distributions. However, when additional assumptions can be made about the nature of plausible effects and distributions, the space of underlying estimated parameters is reduced, and testing can become more efficient (Song and Nicolae 2009; Zheng et al. 2012). A considerable amount of attention has been given in the literature to developing such testing approaches. Some of the most widely used assumptions for simplifying $\overrightarrow{p}$ are allelic, dominant, recessive, and additive SNP effects (Sasieni 1997; Freidlin et al. 2009), which can be tested with the aforementioned nonparametric approaches similarly to standard ones, i.e., by using transformed genotypes or treating them as continuous (Zheng et al. 2012).

As for population assumptions that constrain the parameter space for π, two popular and broadly applicable ones are HWE (the independence of maternal and paternal allele values that make up each SNP) and LE (the independence of the two loci that make up a pair) (Yang et al. 1999; Chatterjee and Carroll 2005; Zhao et al. 2006). Although such assumptions have their limitations (Albert et al. 2001; Mukherjee and Chatterjee 2008), HWE can be expected to hold for homogeneous populations, and LE is plausible in the case of distal locus pairs, i.e., almost all pairs considered in a typical GWAS (Chen and Chatterjee 2007; Song and Nicolae 2009). Consider, for example, the pairwise case. Under LE, ${\pi }_{j,k}={\pi }_j^{\left(1\right)}{\pi }_k^{\left(2\right)}$, where ${\pi }^{\left(1\right)}$ and ${\pi }^{\left(2\right)}$ are the marginal distributions of x₁ and x₂ genotypes in the population, respectively. This assumption reduces the dimension of the parameter space (and the degrees of freedom of the GLRT) by 4. Further, under HWE at x₁, suppose the minor allele frequency is f₁; then we have that ${\pi }_0^{\left(1\right)}={\left(1-f_1\right)}^2,\hspace{0.5em}{\pi }_1^{\left(1\right)}=\left(1-f_1\right)f_1,\hspace{0.5em}{\pi }_2^{\left(1\right)}=f_1^2$—eliminating an additional parameter. Assume HWE at x₂ has the same effect (with the minor allele frequency f₂ parameterizing π⁽²⁾). Thus problem (6), for example, simplifies to

$$\begin{array}{l}\underset{\overrightarrow{q},f_1,f_2}{\text{minimize}}\hspace{0.5em}\hspace{0.17em}- {\displaystyle \sum _{j,k}(n_{0,j,k}\log \left({\pi }_j\left(f_1\right){\pi }_k\left(f_2\right)-q_{j,k}\right)}\end{array}\begin{array}{l}+\hspace{0.5em}{n}_{1,j,k}\log q_{j,k}+m_{j,k}\log \left({\pi }_j\left(f_1\right){\pi }_k\left(f_2\right)\right))\end{array}\begin{array}{l}\text{subject\hspace{0.5em}to\hspace{1em}}{\displaystyle \sum _{j,k}{q}_{j,k}}=K.\end{array}$$ (7)

Other methods and design of simulation study

Assuming known population parameters:

Since population data contain only information about population parameters, it is interesting to compare in our simulation study the performance of the GLRT based on problems (4), (6), and (7) to an unrealistic test, where the genotype frequencies are known (or, under HWE, and under LE for the pairwise case, where the minor allele frequencies are known). This represents a theoretical upper bound on performance when infinitely many population samples are available. In the univariate case, for example, one has to maximize with respect to $\overrightarrow{p}$ the log-likelihood:

$$l\left(\overrightarrow{n};\overrightarrow{p}\right)={\displaystyle \sum _{i,j}{n}_{i,j}\log \left(p_{i|j}\frac{{\pi }_j}{K}\right)}\propto {\displaystyle \sum _{i,j}{n}_{i,j}\log \left(p_{i|j}\right)}.$$ (8)

This is already a concave optimization problem with the affine equality constraint

$${\displaystyle \sum _j{p}_{1|j}{\pi }_j}=K,$$

and box constraints, and as such can be handled efficiently. Treatment of the multivariate case is similar.

The existing naive approach:

A simple way of testing for association, while exploiting an extraneous population sample, is to include the samples $\mathcal{P}$ as additional controls and perform standard association testing on the resulting extended case–control sample, denoted $\mathcal{C}$⁺. This approach is taken, for example, in the univariate “expanded reference group analysis” of the WTCCC study (Burton et al. 2007) (although there are some additional issues to consider in that context, as we discuss in Results). While this approach mislabels as controls any cases that exist in $\mathcal{P}$, leading to a potential loss of power, this loss will be small for a rare disease. Also, the mislabeling of true cases in $\mathcal{P}$ acts as another (independent) source of noise, meaning that tests that are valid on $\mathcal{C}$ are also valid when applied to $\mathcal{C}$⁺ and do not suffer increased type I error. Finally, the simplest approach ignores the extraneous population data altogether, keeping to $\mathcal{C}$ alone.

To represent existing testing methods, we consider four popular approaches. The first approach is standard case–control analysis, which is represented by the logistic GLRT contrasting the saturated and flat models of Equation 5. This test is also known as a “G test” of independence in the two-dimensional contingency table formed by $y\times \overrightarrow{x}$. For univariate testing this table is 2 × 3, while for pairwise testing it is 2 × 9. This table may be compiled either from $\mathcal{C}$ or from $\mathcal{C}$⁺. This test does not exploit the HWE and LE assumptions, nor does it make use of the known prevalence of disease. It is worth noting that the commonly used Pearson chi-square test is an asymptotic approximation to the G test. We refrain from using the chi-square test because it generally gives less accurate P-values at the far tails that are of interest in pairwise testing. The second approach is that of Chen and Chatterjee (2007). It applies to the univariate case only and assumes HWE among controls (and therefore, for rare disease, approximately HWE in the population).

The two remaining approaches are relevant only for the pairwise testing scenario. In the case-only 3 × 3 GLRT, one uses the contingency table for x₁ and x₂ in cases only. Thus using $\mathcal{C}$⁺ (which has the same cases, but more controls) for compiling the table provides no benefit over using $\mathcal{C}$. This test assumes LE among the controls population (or in the general population, under a rare disease assumption) and ignores any marginal association signal (Piegorsch et al. 1994; Song and Nicolae 2009). While this test is a popular approach to harnessing the LE assumption, by definition, it cannot benefit from including extraneous population samples as additional controls. We therefore consider the related “three-way independence” 2 × 3 × 3 G test, which uses the full three-dimensional case–control contingency table y × x₁ × x₂. This test assumes LE in the population under the null of no association, but not under the alternative, thus wasting degrees of freedom on modeling a dependence between the SNPs. If the disease is rare, then the degrees of freedom invested in modeling the dependence between the SNPs among controls go to waste as well, as this can capture only spurious dependence signals. Contrary to the case-only test, the three-way independence test is sensitive to marginal effects.

Design of the simulation study:

The association testing methods described above are summarized in Table 1. The methods differ in the assumptions they make and are further classified as (i) ignoring $\mathcal{P}$ and using $\mathcal{C}$ only as done in most GWAS to date, (ii) extending $\mathcal{C}$ by including $\mathcal{P}$ as if they were additional controls (denoted $\mathcal{C}$⁺), or (iii) correctly analyzing all samples [referred to as ($\mathcal{C}$, $\mathcal{P}$) analysis]. The “known P_x” and case-only G tests are applied only to $\mathcal{C}$ (since the former cannot benefit from population samples, and the latter ignores any control data), and the case–control and three-way G tests and the method of Chen and Chatterjee (2007) are applicable either to $\mathcal{C}$ or to $\mathcal{C}$⁺. The CMLE test can be applied to $\mathcal{C}$ and $\mathcal{C}$⁺ and is the only method that can be applied to ($\mathcal{C}$, $\mathcal{P}$).

Table 1. Summary of testing approaches.

Name	Description
Case–control G	The 2 × d GLRT of the phenotype y and a genotype (or a pair of genotypes) treated as a d = 3 (or d = 9) category variable.
Chen and Chatterjee G	The univariate case–control Wald test assuming HWE among controls (Chen and Chatterjee 2007).
Case-only G	The pairwise 3 × 3 GLRT examining the dependence between genotypes x1 and x2 among cases (y = 1).
Three-way G	The pairwise 2 × 3 × 3 GLRT, testing for three-way independence of genotype x1, genotype x2, and the phenotype.
Known Px	The (unrealistic) GLRT based on Equation 8, where marginal/pairwise SNP distribution and the prevalence are known.
CMLE	The GLRT based on Equation 4 (for a single SNP) or Equation 6 (in the pairwise case), assuming known prevalence.
CMLE HWE LE	Same as CMLE above, but also assuming HWE and LE.

In boldface type are the novel tests that are introduced in this article. “Known P_x” is unrealistic and merely represents a theoretical upper bound on the benefit of exploiting population samples. The remaining tests represent existing approaches.

Several problem parameters affect testing performance, notably samples sizes,

$$n_0={\displaystyle \sum _{i=0,j}{n}_{i,j}},\hspace{1em}{n}_1={\displaystyle \sum _{i=1,j}{n}_{i,j}},\hspace{1em}m={\displaystyle \sum _j{m}_j}$$

(here for the univariate testing scenario), and the true values of K and π, the latter possibly including deviation from LE and HWE.

As the basis for simulated alternatives (i.e., nonnull setups) in the univariate case we use the six partitions of the three genotypes in two risk levels: an “at risk” set of genotype values G₁ vs. a “protected” set G₀. These models include the recessive, dominant, and heterozygous partitions (e.g., recessive minor-at-risk: G₀ = {0, 1}, G₁ = {2}). An “effect size” parameter is defined, which governs $p_{1|j\in G_1}-p_{1|j\in G_0}$, the difference in risk levels between the two groups. Risk levels themselves for G₁ and G₀ are determined from the remaining parameters.

The minor allele frequency (MAF) is taken uniformly from [0.05, 0.5]. This mimics the prior distribution typical to data genotyped using SNP array technology (Waye 2005). We repeat the simulations with prevalence of disease K ∈ {0.05, 0.1, 0.2, 0.4} spanning the common disease spectrum, e.g., bipolar disorder to hypertension to obesity. Sample sizes are fixed at a balanced n₀ = n₁ = 1000, and we assume a modest m = 5000 independent population samples are also available. Nominal test levels are set according to the 0.05 Bonferroni correction for the number of loci in the study, M, assuming throughout M = 300,000.

In the pairwise GWAS scenario, we focus on studying association while allowing for interaction, as in Equation 5. Because in this case we are not testing directly for a “pure” interaction, marginal associations (“main effects”) alone can lead to a successful discovery. A pairwise GWAS is typically preceded, however, by a univariate study (see, for example, the classification in Liu et al. 2011 to marginal, conditional, and pairwise testing), and this makes pairs with marginally detectable SNPs less interesting. We thus perform a marginal association test at both loci and remove simulated data sets from consideration if either univariate test leads to a successful detection. In other words, what is measured is the power of the tests, conditional on not having a marginally detectable signal.

As in the univariate case, we use as alternative hypothesis setups all the possible partitions of the 3 × 3 pairwise SNP–SNP covariate space into two risk levels. These models are the same as the fully penetrant models of Li and Reich (2000), which have been used in multiple GWAS simulations (Evans et al. 2006; Song and Nicolae 2009; Wan et al. 2010), except that we admit partially penetrant models by controlling the risk levels. The MAF changes independently for each SNP, and we use the same sample sizes as in the univariate simulation described above. The nominal test level in the pairwise case is adjusted for all M(M − 1)/2 locus pairs.

Results

Simulation study

An extensive simulation was performed to compare the tests from Table 1 on the basis of statistical power to detect univariate associations and pairwise interactions under various disease models. Additional simulations compared performance under deviation from the modeling assumptions.

Power analysis:

We first examined the performance of the methods in the single-SNP scenario. Power is averaged over all models described in Methods and presented in Figure 1. The horizontal axis is chosen to objectively reflect problem difficulty—the true area under the curve (AUC) is the probability of the penetrance of a randomly chosen case being higher than that of a randomly chosen control (Jostins and Barrett 2011). It is immediately evident that, as expected, the unrealistic approach that uses an oracle for the marginal SNP distribution is uniformly most powerful among the methods compared. The least powerful approach in all scenarios except K = 0.4 is the one most often used in actual studies—the case–control analysis that ignores any population samples. Even with a modest population sample, naive use of these data can greatly improve detection power when K is small.

Figure 1.

Univariate power simulation. Power is averaged across univariate models with two risk levels. AUC is area under ROC curve, a measure of detection difficulty (see main text). A–D show results for prevalences (K) 0.05, 0.1, 0.2, and 0.4, respectively.

For higher prevalence of disease, however, the effect of mislabeling cases in the population sample as if they were controls becomes more evident, until for K = 0.4 this makes all methods using this approach substantially inferior to ignoring the population sample altogether. Exploiting HWE in controls pays off for low K. This is also expected, since when the disease is relatively rare, the HWE that holds in the population also holds approximately among controls. For K = 0.4 this is no longer true, and it is better to perform the standard case–control test.

The inclusion of four variants of our proposed CMLE test, combined with the selection of existing tests, allows us to examine the individual benefits obtained by the two ingredients of our approach, namely, correct handling of population samples and use of the HWE assumption. For low prevalence of disease, there is little to gain from the correct ($\mathcal{C}$, $\mathcal{P}$) analysis, but HWE (and use of the known K) provides a consistent modest edge. Comparing in this case CMLE HWE to the approach of Chen and Chatterjee (2007), again we see how there is little difference between assuming HWE in controls and in the population. These differences, however, grow with K, and the effect of each ingredient becomes apparent. Finally, we note that the power of the most appropriate test, CMLE HWE ($\mathcal{C}$, $\mathcal{P}$), is the highest of all the realistic tests in all cases and quickly approaches the theoretical upper bound of known marginals, which means the potential benefits from the exogenous data are fully realized by our approach.

Next we considered the pairwise GWAS scenario, where we were first interested in the power gains under the HWE and LE assumptions. Figure 2, A and B, summarizes the results across all considered models. Supporting Information, File S1, Figure S1, Figure S2, Figure S3, Figure S4 includes detailed results for each Li and Reich (2000)-like model class. Given that few interactions have been detected in all-pairs studies (exhaustive or otherwise) conducted so far, interest lies mainly in those setups where current methods have low power. Incorporating available population samples or known parameters clearly offers substantial power gains for interaction detection under many of the possible setups that fall in the region of interest. Not surprisingly, the best approach is again the unrealistic one, where the pairwise distribution is known (or in this case of HWE and LE, the MAFs at both loci are known). Here too, the performance of this approach defines an upper bound for the realistic approaches, and the important conclusion stemming from these results is that the suggested CMLE test approaches this bound. Notably, the gains from performing $\mathcal{C}$⁺ or ($\mathcal{C}$, $\mathcal{P}$) analysis beyond standard case–control $\mathcal{C}$ or case-only analysis are great, reaching sevenfold power gain, averaged over all models (when power increases from ∼0.1 of G ($\mathcal{C}$) to ∼0.7 of CMLE HWE LE ($\mathcal{C}$, $\mathcal{P}$) in Figure 2A). The differences between ($\mathcal{C}$, $\mathcal{P}$) and $\mathcal{C}$⁺ are not as great, but become more substantial for higher prevalence values.

Figure 2.

Pairwise power simulation. Power is averaged across all materially different models as described in Li and Reich (2000) with our extensions (see main text). The horizontal axis is defined as in Figure 1. (A and B) Under LE. (C and D) Under LD with ρ² = 0.01. (A and C) Disease prevalence 0.05. (B and D) Disease prevalence 0.2.

We repeated the simulation under a modest amount of linkage disequilibrium (LD) (ρ² = 0.01), applying only methods that do not rely on the LE assumption. The results were similar, but naturally some of the power gains in the CMLE approach were attenuated, to the point it is clearly superior to standard $\mathcal{C}$⁺ analysis only for prevalence >0.1 (see Figure 2, C and D).

Sensitivity analysis:

We compared the type I error of the tests to quantify the effects of deviation from HWE and LE assumptions. Figure 3 shows type I error rates for the pairwise scenario with n₀ = n₁ = 1000, m = 5000, and the MAF at both loci held at 0.3. In Figure 3, left, there is HWE, and the degree of LD between the loci, measured by Pearson’s product-moment correlation coefficient squared, is varied. In Figure 3, right, LE holds but there is a varying degree of Hardy–Weinberg disequilibrium (HWD) (measured again by the squared correlation, here between the two alleles at the same locus). In all tests, the nominal type I error rate is set to 0.05.

Figure 3.

Type I error under deviation from assumptions. Left, LE; right, HWE. ρ is the allelic Pearson correlation coefficient (for LD, between any allele in x₁ and any allele in x₂; for HWD, between the two alleles of the same SNP, with both SNPs simulated to have the same degree of HWD for simplicity).

As long as modeling assumptions are met, all tests maintain the nominal level as expected. In accordance with similar past observations (Albert et al. 2001; Mukherjee and Chatterjee 2008), methods that rely on LE, namely, the case-only, three-way independence, and CMLE LE tests, are sensitive to this assumption and quickly break down under LD. Of these methods, CMLE is the most robust to departure from LE. The sensitivity to deviation from HWE is high as well. This calls for discretion when applying such tests and for use of the general CMLE in the case of doubt regarding the validity of assumptions. As mentioned in the Introduction and as elaborated in the Discussion, the HWE and/or LE assumptions will often be applicable for random sampling in homogeneous populations and for distal pairs; however, it is good practice to perform replication analysis with tests that do not make these assumptions to minimize the chance of a false discovery.

Application to the WTCCC study

The proposed CMLE tests were used to analyze the data sets from the WTCCC study (Burton et al. 2007). The WTCCC data contain independent case cohorts for seven common diseases: bipolar disorder (BD), coronary artery disease (CAD), hypertension (HT), Crohn’s disease (CD), rheumatoid arthritis (RA), type I diabetes (T1D), and type II diabetes (T2D). Each case cohort is on the order of 2000 individuals, and an additional shared cohort is available with ∼3000 population controls (i.e., controls were not screened to exclude cases of the seven diseases). Each individual was genotyped at ∼500,000 SNP loci, which after standard quality control (QC) reduces to ∼350,000 loci. For pairwise testing we follow the recommendations from Liu et al. (2011) and further filter out SNPs with MAF < 0.1, leading to ∼300,000 loci.

In the main analysis undertaken by the WTCCC (Burton et al. 2007), each case cohort is analyzed separately against the shared controls, when the latter are treated as pure controls rather than population controls. This simplifies the work and is a reasonable approximation since the prevalences of the (sometimes nonstandard) phenotype definitions considered by WTCCC are not high [the prevalences for the seven phenotypes often quoted for the relevant UK population vary between 0.001 and 0.15, but the most common phenotypes, such as HT, are actually defined in the WTCCC study as extreme cases of HT, leading to lower prevalence (Burton et al. 2007)]. This approach is essentially a $\mathcal{C}$⁺ analysis with zero pure controls. In the expanded reference group analysis (ERGA), to increase power, Burton et al. (2007) extend the controls set for any one disease, using the cases from remaining diseases that are thought to be unrelated (autoimmune diseases CD, RA, and T1D are related and so are the cardiovascular phenotypes CAD and HT). Because there is no guarantee that a case in an unrelated data set is not also a case with respect to the current phenotype, this too is a $\mathcal{C}$⁺ analysis, using additional population samples. Both analyses in Burton et al. (2007) are univariate, but several other authors have since performed interaction testing on these data (e.g., Emily et al. 2009; Liu et al. 2011; Prabhu and Pe’er 2012; Lippert et al. 2013).

The pairwise problem is less extensively studied, more challenging, and according to the simulation results above holds the greatest potential for benefiting from our methodology. We therefore performed a pairwise analysis where the shared controls and extending case cohorts were taken as population samples. This is more appropriate almost everywhere in the genome, except for loci associated with the unrelated phenotypes—which we excluded. In addition, we imposed the prevalence numbers obtained from UK health organizations (Burton et al. 2007; Allender et al. 2012).

To speed up the processing of the seven data sets, pairs were first filtered using the software package PIAM (Liu et al. 2011), such that SNPs with significant association signals at the univariate Bonferroni level were removed. Then pairwise testing was performed over all remaining SNPs that show some marginal signal (P < 0.1), using an optimized implementation of the unconstrained G test applied to $\mathcal{C}$⁺ (see Methods). The 100,000 pairs with the smallest P-value from this analysis that are also well separated (at least 5 Mbp away) were further analyzed by our CMLE HWE LE approach. The entire process was completed in ∼30 min per data set on a modern personal computer (using an exhaustive pairwise search filter is feasible as well and adds several hours of running time per data set).

Overall, across six of the seven diseases, 736 pairs were detected that are significant at the pairwise Bonferroni level (≈10⁻¹², varying slightly between data sets) and contain distal SNPs. After filtering pairs with SNPs that are marginally significant at the univariate Bonferroni level according to the univariate CMLE HWE test (which as we have seen above has more power than the $\mathcal{C}$⁺ G test used in the PIAM filter), 9 pairs in four diseases survive. Seven of these pairs, in three diseases, were not implicated by the original WTCCC study (defined as both SNPs being at least 1 Mb away from any WTCCC significant association). These results are summarized in Table 2. Table 3 briefly highlights some of the most promising pairwise findings overall, and Figure 4 and Figure 5 give the underlying contingency tables and estimated odds ratios for a few of these top pairs. The detailed listing of all results is provided in File S1, and a short discussion for each disease is given next.

Table 2. Significant pairwise findings in WTCCC.

Disease	BD	CAD	CD	HT	RA	T1D	T2D
Significant pairs	0	134	3	576	12	8	3
With weak marginals	0	2	3	0	2	2	0
Implicating novel loci	0	2	3	0	2	0	0

Pairs with weak marginals are defined as pairs where both SNPs have univariate CMLE P-values larger than the relevant univariate Bonferroni significance level. Novel loci are defined as pairs where both SNPs are >1 Mb away from any association reported in the original WTCCC study.

Table 3. Promising pairwise associations in WTCCC.

Phenotype	SNP1		SNP2		P1	P2	PCO	PCMLE
BD	rs9865654	3p25.2	rs17600642	10q22.1	7.64 × 10−4	1.04 × 10−6	2.66 × 10−6	1.13 × 10−12
CAD	rs16905928	10p12.31	rs3781575	11p13	6.15 × 10−3	1.30 × 10−5	5.12 × 10−9	1.83 × 10−13
CD	rs962087	5p14.1	rs7028357	9p24.1	3.83 × 10−4	3.24 × 10−5	2.11 × 10−8	1.28 × 10−13
CD	rs7554511	1q32.1	rs11945978	4p12	1.10 × 10−6	4.93 × 10−6	1.01 × 10−4	4.17 × 10−13
RA	rs1605705	3p26.1	rs6831911	4q34.3	3.58 × 10−5	5.28 × 10−7	1.73 × 10−10	2.78 × 10−18
RA	rs894848	1p13.2	rs6427122	1q24.2	1.18 × 10−4	2.04 × 10−5	2.57 × 10−7	3.18 × 10−13

We selected pairs that achieved or came close to achieving genome-wide significance at the 0.05 pairwise Bonferroni level, that have weak marginal signals, and that appear to be novel associations when compared to the discoveries of the original WTCCC study (Burton et al. 2007). P₁ and P₂ are P-values for the univariate CMLE HWE test for SNP₁ and SNP₂, respectively; P_CO is the P-value for the (pairwise) case-only test; and P_CMLE is the P-value for the pairwise CMLE HWE LE test. Strictly significant P-values are in boldface type.

Figure 4.

WTCCC promising pairs (part 1). Shown are contingency tables and odds ratios estimated using CMLE HWE LE for promising pairwise associations in CAD (top, OR truncated at 10 for presentation) and CD (bottom) data; see text for discussion.

Figure 5.

WTCCC promising pairs (part 2). Shown are contingency tables and odds ratios estimated using CMLE HWE LE for promising pairwise associations in RA data; see text for discussion.

BD:

Although no significant pairs were detected at the (somewhat arbitrary and probably quite conservative) Bonferroni level (1.09 × 10⁻¹²), it is interesting to note that the pair ranked first (rs9865654, rs17600642), with a pairwise P-value of 1.13 × 10⁻¹², has a leg on chromosome 10 (rs17600642), which was not detected in the ERGA, but has been noted more recently (Jiang and Zhang 2011). Because of our more stringent filtering, our data do not include rs10925490 (MAF ≈ 0.06) and thus the pair identified by Prabhu and Pe’er (2012).

CAD:

The immediate regions of the SNPs composing the topmost pair (rs5007171, rs2329902) have not been reported in past GWAS results [according to the GaP and dbGaP catalogs (Mailman et al. 2007)], but because rs5007171 has a strong marginal signal (P-value of 7 × 10⁻⁶), it is possible that this result is due to a marginal association here. Many of the following ranked pairs also include rs5007171 and are thus suspect as well. On the other hand, a noteworthy significant pair with weak marginals is (rs16905928, rs3781575) which repeats with small SNP location shifts. This pair shows a mostly recessive–recessive effect; see Figure 4.

CD:

The highest ranked pair (rs962087, rs7028357) appears to be a dominant–dominant association (see Figure 4), where neither SNP has a known association with any disease. The pair ranked second (rs7554511, rs11945978) involves SNPs not implicated in the WTCCC study, but a proximal locus of rs7554511 (rs11584383) was found later to be strongly associated with CD in a larger study (P-value of 10⁻¹¹) (Barrett et al. 2008), in which the WTCCC data were combined with additional sources. This pair is notably also detected as a significant pairwise association in Liu et al. (2011), where it is the only pair that is found to be a strictly significant association. Liu et al. (2011) thoroughly analyzed this interaction and have in fact validated it in an independent data set (IBDGC-non-Jewish, which is also a part of the data used in Barrett et al. 2008). Both rs7554511 and rs11945978 have already been suggested as marginal associations in Barrett et al. (2009) (which studied ulcerative colitis, a closely related condition) and Duerr et al. (2006) (studying CD), respectively. Pairs composed of respectively neighboring SNPs have small P-values as well and corroborate the association signal (although none are strictly significant).

HT:

While the original (univariate) WTCCC study did not produce any significant findings, hundreds of alleged associations are detected in our pairwise analysis. Examining these associations more closely, it seems that all of these are in fact due to a few marginal associations that are just below the univariate Bonferroni threshold. One such association is represented by rs17509005, which repeats in many pairs. It is probably more accurately captured by rs11047543, which has been implicated more recently in Pfeufer et al. (2010) with a P-value of 10^−12.5, or by rs17287293, described in Eijgelsheim et al. (2010) with a P-value of 10^−9.7. This marginal association comes up only as a moderate signal (10⁻⁵), using standard univariate testing of the WTCCC ERGA, but comes up stronger in our results (10⁻⁶).

RA:

The first pair here is (rs1605705, rs6831911), showing a dominant–dominant deleterious effect (Figure 5, top); both SNPs are not mentioned in the WTCCC study results. rs1605705 is noted as a moderate association in dbGaP. The second pair (rs894848, rs6427122) has the dominant–dominant pattern again (see Figure 5, bottom). These SNPs were not found by the WTCCC analysis as well, but more recently rs864537, which is <1 Mb from rs6427122, has been implicated in RA (P-value of 10^−10.7) (Stahl et al. 2010). Another strong marginal proximal association is rs840016 (Zhernakova et al. 2011).

T1D:

Significant pairs in this data set can be attributed to marginal signals, mostly in rs1377748. The pair (rs2077749, rs10849946) is proximal to a region found on chromosome 12 in the original study; however, careful examination indicates that this is a separate association discovered more recently in two studies applying different methods to the WTCCC data (rs1265564 in Huang et al. 2012 with a P-value of 10⁻¹⁶ and rs3184504 in Plagnol et al. 2011 with a P-value of 10⁻³⁷).

T2D:

Of the three significant pairs, both SNPs making up the first pair are also implicated in the WTCCC ERGA. The remaining two pairs are due to a marginal effect in rs962087.

Discussion

Existing tests used in GWAS are often not powerful enough to detect modest associations given the available sample sizes. Specifically, the search for effects that are statistically significant only when considering multiple loci jointly is an important ongoing effort that has so far produced very few replicable results. We introduce a powerful new approach for univariate and multivariate testing that is based on constrained maximum-likelihood estimation. Our method makes efficient use of extraneous population samples in the context of case–control, case–population, and case–control–population studies of complex human disease. The underlying modeling approach has the important capability to exploit population assumptions, such as HWE and LE of constituent SNPs.

The resulting association testing approach is quite flexible and shown through extensive simulation to be valid and powerful for a wide range of problem settings. The successful application of our approach to the WTCCC study data demonstrates this as well and shows in particular that an efficient pairwise analysis can uncover novel associations that, when approached by marginal testing, are visible only in larger samples (if at all). These results have an immediate implication for disease association studies in general—most studied populations have samples that are easily accessible to researchers and that can now be utilized to substantially improve power of standard testing approaches. The increasing number and richness of genetic data sources such as those mentioned in the Introduction will make the suggested approach only more relevant.

A critical examination of past results, especially from pairwise analyses, raises many suspicions regarding reported discoveries: there is strong reliance on sparse data due to low MAFs (Lippert et al. 2013) or LD (Brinza et al. 2010; Wan et al. 2010; Lippert et al. 2013) or on SNPs showing traces of a range of genotyping errors (Cordell 2009; Brinza et al. 2010; Liu et al. 2011), as well as a host of other problems. Although there are certainly other explanations (Marchini et al. 2005), this could help account for the low replicability of such association results. We reiterate the conclusions of Liu et al. (2011), who have identified similar problems in other studies. They suggested that quality control for pairwise analysis be stricter than for univariate GWAS, and we have implemented such control in this study. Another issue that contributes to the difficulty of replicating pairwise associations is due to the unconstrained search space employed by most studies. The present article is part of a larger body of work that attempts to improve the situation by restricting the search for multilocus associations to more biologically plausible ones (Chatterjee and Carroll 2005; Wang and Sheffield 2005; Chen and Chatterjee 2007; Song and Nicolae 2009; Han et al. 2012; Luss et al. 2012). Various combinations of the constraints considered by others and those described here are reasonable, and while it is straightforward to apply them to data, asymptotic theory to support fast testing is not always available.

Many of the limitations discussed in Liu et al. (2011) are relevant to our approach as well. It is worth noting a few limitations specific to the additional assumptions taken here. One issue is that joining samples from different sources often requires stratification adjustment, which we expect to address in future work. Another issue noted by various authors is the high sensitivity of tests based on HWE and LE to departure from these assumptions (as is also evident in our simulations), and ways of mitigating this sensitivity have been suggested (Albert et al. 2001; Mukherjee and Chatterjee 2008). Indeed, a prudent step following discoveries arising from our analysis is to confirm them on independent data, using an unconstrained test. On the other hand, additional simulations (results not shown) imply that the sensitivity of the suggested approach to misspecification of disease prevalence is low, which is important when accurate estimates are not available, for example in populations of developing countries.

While we have focused on univariate associations and on pairwise associations allowing for interaction, let us stress two points. First, the methods described are equally applicable to SNP subsets of arbitrary order. Second, the literature is rich with other formulations for the genetic interaction problem, and many of them plug naturally into our approach as well. For example, our framework also allows conditional association and pure interaction tests that (similarly to the case-only test) attempt to identify interactions that cannot be due to combinations of marginal effects alone (Cordell 2009; Liu et al. 2011). This task is significantly more challenging (that is, the optimization can be more difficult, and tests may have less power), but we expect our methods when combined with an appropriate filtering method such as that of Prabhu and Pe’er (2012) to be more powerful than existing alternatives in this context as well.

Finally, it would be interesting to examine the benefit of using the methods suggested here in case–control studies of late-onset diseases such as Alzheimer’s, because taking controls in a study like this as known to be unaffected is susceptible to the same issues as using population controls. Within our framework the “controls” of such studies can be assumed to be a random population sample and handled properly.