A Shrinkage Method for Testing the Hardy–Weinberg Equilibrium in Case-Control Studies

Introduction

The Hardy–Weinberg equilibrium (HWE) is one of the most important properties in population genetics. More than a century ago, G. H. Hardy and W. Weinberg individually noted that for a large, self-contained, and randomly mating population, assuming a bi-allelic locus with alleles A and a and corresponding allele frequencies p and q, the genotype frequencies are p², and q² for genotypes , and , respectively [Hardy, 1908; Weinberg, 1908]. As genotyping errors distort the genotype distribution and thus break the HWE, testing for this equilibrium has been routinely conducted as a way of checking the quality of genotyping [Gomes et al., 1999; Hosking et al., 2004; Xu et al., 2002]. Deviation from the HWE often indicates a poor quality of genotyping.

A straightforward way to test the HWE is the χ² test, which is based on pooling samples from the cases and controls [Weir, 1996]. We refer to this test as the pooled χ² test in order to distinguish it from other versions of χ² tests described later. The pooled χ² test has high statistical power but requires the subjects under investigation to be a random sample from the target population. For case-control data, subjects are retrospectively ascertained based on their disease status. Therefore, if a candidate marker is associated with the disease, the corresponding genotypes in the case-control sample are no longer representative of the target population. Consequently, the pooled χ² test may yield misleading results and mistakenly conclude a violation of the HWE when the target population is actually within the HWE [Wittke et al., 2005]. Nevertheless, it is worth noting that if the candidate marker is independent of the disease, the genotypes of the case-control samples are a random sample from the population, and thus the pooled χ² test is a valid and efficient test for assessing the HWE.

Many methods have been proposed to test the HWE for case-control samples without making the strong assumption that the candidate marker is independent of the disease. A widely used approach is to conduct the χ² test using only the controls, while discarding the cases. Unfortunately, this control-only χ² test is (approximately) valid only when the disease prevalence is low, and therefore the controls provide a good approximation of the general population. When the disease prevalence is moderate or high, the control-only χ² test leads to inflated type I errors [Li and Li, 2008]. In addition, because of discarding the information obtained from cases, the control-only χ² test is not efficient. To address these issues, Li and Li [2008] developed a likelihood ratio test (LRT) to assess the HWE using data from both cases and controls, while taking into account the potential association between the marker and the disease. Compared to the control-only χ² test, the LRT is more powerful for detecting departures from the HWE for common diseases and has comparable power for use in analyzing rare diseases. Yu et al. [2009] proposed a similar test based on the likelihood ratio framework. Wang and Shete [2010] developed a bootstrapping test that also accounts for the underlying genetic association. Because the performances of these tests are quite comparable, herein we focus on the LRT.

Although the LRT is valid and more powerful than the control-only χ² test, it has two limitations. First, because the LRT requires the estimation of extra nuisance parameters (e.g. penetrances), it can be substantially less powerful than the pooled χ² test if the marker actually is independent of the disease. This is a concern for some genetic studies, such as a genome-wide association study (GWAS), in which most of the markers are expected to be weakly or not associated with the disease. In addition, the LRT typically requires numerical optimization to calculate the test statistic (i.e. the maximum likelihood estimates); therefore it can be time consuming to apply the LRT to large-scale case-control studies, e.g. a GWAS, in which hundreds of thousands of single nucleotide polymorphisms (SNPs) need to be tested.

In this article, we propose a novel shrinkage test to circumvent the limitations of the LRT. Toward this goal, we first propose an extension of the pooled χ² test, called the generalized χ² test, for assessing the HWE. The generalized χ² test is valid regardless of whether or not the marker is associated with the disease. Based on that characteristic, we then propose a shrinkage test statistic, which takes a form of the weighted average of the pooled χ² test statistic and the generalized χ² test statistic. When the marker is independent of the disease, the proposed shrinkage test converges to the pooled χ² test, and therefore achieves high statistical power. When the marker is associated with the disease, the proposed shrinkage test converges to the generalized χ² test, and therefore remains statistically valid. A simulation study shows that, compared to the LRT, the shrinkage test is more powerful to detect departures from the HWE when the marker is weakly or not associated with the disease, and has comparable power when the marker is strongly associated with the disease. In addition, as the proposed shrinkage test has a closed form, it is easy to calculate and is particularly suitable for testing the HWE for large case-control datasets.

The remainder of this article is organized as follows. We first briefly review the pooled χ² test, and then propose the generalized χ² test and the shrinkage test. We compare the performances of the proposed test to existing methods using simulation studies and a GWAS dataset. We conclude the article with a brief discussion.

Methods

Consider a bi-allelic candidate marker with two alleles, A and a, having frequencies p and , respectively, where p is the minor allele frequency (MAF). Denote three genotypes by , and with genotype frequencies for When the HWE holds, , and . Denote the penetrance by , and the disease prevalence by , then the genotype frequencies in the cases and controls are given by and , respectively, for . Define the genetic relative risks (GRRs) as , , to characterize the underlying genetic association. When the candidate marker is not associated with the disease, ; otherwise, with at least one inequality holding. When the genetic association is present, the genetic model can be used to describe the relationship between 's. A genetic model is called recessive (REC) if and ; additive (ADD) if and , and dominant (DOM) if and [Sasieni, 1997].

Consider case-control data consisting of r cases and s controls. Let and denote the genotype counts of G₀, G₁, and G₂ in cases and controls, respectively. Let , , and . The genotype counts for the cases and controls are displayed in Table 1.

Table 1. Genotype data for the case-control study of a bi-allelic candidate marker

				Total
Case	r0	r1	r2	r
Control	s0	s1	s2	s
Total	n0	n1	n2	n

The Pooled χ² Test

Weir [1996] proposed a (pooled) χ² test to assess the HWE through the Hardy–Weinberg disequilibrium coefficient, defined as

Obviously, when the HWE holds. For case-control data, when the candidate marker is not associated with the disease, genotypes in the case-control samples are a random sample from the general population. Therefore, we can pool the cases and controls together and estimate , , and

The pooled χ² test statistic is defined as

which asymptotically follows a χ² distribution with 1 degree of freedom under the null hypothesis that the HWE holds.

The pooled χ² test is efficient and possesses high statistical power because of the use of the pooled data from cases and controls. However, when the candidate marker is associated with the disease, the χ² test is invalid because the genotypes in the case-control sample are no longer a random sample from the general population and therefore , , and are biased. Under this circumstance, the control-only χ² test, which uses only control data, is often used to examine the HWE.

The Generalized χ² Test

Before we introduce the shrinkage test, we propose a new χ² test for assessing the HWE, upon which the shrinkage test will be built. For convenience, we call this new test the generalized χ² test because it makes the χ² test [Weir, 1996] generally applicable to case-control data regardless of the association between the marker and the disease. The generalized χ² test is developed based on the basis of the following key observations: for the case-control data, the minor allele frequency p can be expressed as

and the genotype frequency g₂ can be expressed as

Similar to the LRT [Li and Li, 2008], we assume that the disease prevalence k is known. This assumption is plausible because disease prevalence can often be assessed from data external to that of the case-control study. Given the case-control data shown in Table 1, , , and can be consistently estimated by , , and , respectively. Therefore, we estimate p and g₂ by

Then, the generalized estimate of the Hardy–Weinberg disequilibrium (HWD) coefficient is given by

and correspondingly, the generalized χ² test statistic for the HWE is defined as

(see the Appendix for derivation of a closed form of ) Under the null hypothesis that the HWE holds, T_gen asymptotically follows a χ² distribution with 1 degree of freedom.

Because the development of the generalized χ² test does not require any assumptions of the association between the marker and the disease, this test is generally applicable to case-control samples. Actually, the generalized χ² test is a Wald test that is asymptotically equivalent to the LRT. Compared to the LRT, the main advantage of the generalized χ² test is its computational simplicity. The generalized χ² test statistic has a closed form and thus is more suitable to test the HWE for modern large-scale case-control studies involving millions of markers. However, like the LRT, if the marker is not associated with the disease, the generalized χ² test is less powerful than the pooled χ² test. To address this issue, we propose a shrinkage test as follows.

The Shrinkage Test

The shrinkage test is based upon the following shrinkage estimate of the HWD coefficient

which takes a form of the weighted average of the generalized estimate of the HWD coefficient and pooled estimate of the HWD coefficient . The shrinkage factor or weight w () controls how much shrinks toward or . As described later, we select w in a data-adaptive way so that inherits the merits from both the pooled χ² test and the generalized χ² test. Specifically, we require that when the marker is associated with the disease, w goes to 1 and thus converges to to ensure the validity of the test; and when the marker is not associated with the disease, w goes to 0 and thus converges to to achieve the high statistical power of the pooled χ² test.

Correspondingly, we define the shrinkage test statistic as

A closed form of can be obtained through the Delta method (see the Appendix for details). Under the null hypothesis that the HWE holds, T_shrink follows a χ² distribution with 1 degree of freedom.

We now discuss how to construct the shrinkage factor w. In order to ensure that the value of w adaptively changes with the strength of the marker-disease association, we first define a measure of the marker-disease association and then use that as a basis for constructing the shrinkage factor.

We measure the strength of the marker-disease association using the Bayes factor. The Bayes factor is the cornerstone of Bayesian hypothesis testing [Jeffreys, 1961; Kass and Raftery, 1995] and provides an evidence-based measure of the likelihood of a hypothesis being true. As the standard Bayes factor involves high-dimensional integration and is sensitive to the prior of the unknown parameters, we herein adopt a variation of the Bayes factor, called the approximate Bayes factor (ABF; Wakefield [2007]; Xu et al. [2012]). The main difference between the ABF and the standard Bayes factor is that the ABF is based on the likelihood of a test statistic, whereas the standard Bayes factor is based on the likelihood of the observed data [Johnson, 2005; Wakefield, 2007].

Letting T denote a test statistic for testing the null hypothesis vs. the alternative , and assuming that the distribution of T depends only on θ, the ABF is defined as

Because the ABF does not depend on the nuisance parameters in the model, it is easy to calculate and is also immune to the influence of the prior of the nuisance parameters. There is a close relationship between the ABF and the posterior probabilities of the hypotheses. Letting denote the prior odds of H₁ vs. H₀ being true, the posterior probability of H₁ being true is

(1)

We gauge the strength of the marker-disease association using the ABF based on the trend test statistic. Given the case-control data listed in Table 1, the trend test statistic under the additive model is defined as [Sasieni, 1997]

Let β denote the logarithm of the odds ratio of the disease penetrance and define (no marker-disease association) and (having marker-disease association). Under H₀, and under H₁, . Following Xu et al. [2012], we assign β a normal prior distribution with . The ABF based on Z_tt is given by

The value of the ABF is between 0 and ∞. It converges to 0 in the absence of a marker-disease association and to ∞ in the presence of such an association [Johnson, 2005].

With the ABF at hand, a nature choice for the shrinkage factor w is the posterior probability of the marker being associated with the disease, i.e. as given in (1), because of its appealing feature that in the presence of a marker-disease association (as goes to 1) and in the absence of such an association (as goes to 0). To use this shrinkage factor, we found that the conventional noninformative prior (i.e. there is an equal prior probability that the marker is associated or not associated with the disease) did not work well in finite samples and led to inflated type I errors (see Table 1 in the Supplementary Materials). This is because in finite samples, the value of can be substantially smaller than 1 under H₁ although it asymptotically converge to 1. As a result, even there is a marker-disease association, T_pool still contributes to T_shrink, which causes inflated type I errors. One way to address this issue is to assign a higher prior probability to H₁, thus a lower prior weight to T_pool, to control the influence of T_pool. Based on the simulation study, we recommend with an addition set-off of 0.1 for w because it consistently yields good operating characteristics in terms of both the type I error and power. That is,

Obviously, with this finite-sample adjustment, the value of w sometimes can be larger than 1 if is very close to 1. In this case, we simply round w to 1 so that .

Simulation Studies

We carried out comprehensive simulation studies to investigate the performance of the proposed shrinkage test and compare our proposed approach to the control-only χ² test and the LRT. We first investigated the type I error rate of the different methods under the null hypothesis that the HWE holds. We assumed the MAF , disease prevalence or 0.1, GRR , 1.5 or 2.0, and sample size . We considered three genetic models, including REC, ADD, and DOM, under which the values of λ₁ were determined.

The genotype distributions among the case and control groups were determined according to p, λ₁, λ₂, and k under the constraint of the HWE. Under each simulation condition, 10,000 replicates were used to evaluate the empirical type I error rate for the tests.

To evaluate the power of the methods to detect different departures from the HWE, following Li and Li [2008], we simulated case-control data from two genotyping error models (denoted as S1 and S2) introduced by Leal [2005]. Model S1 assumes that heterozygosity may be incorrectly genotyped as homozygosity. In this case, given the MAF p and error rate δ, the observed genotype probabilities are , and . In S2, homozygosity may be incorrectly genotyped as heterozygosity with , and . The parameters setting for evaluating the power were the same as those for evaluating the type I error rate except that we considered , 1.25 or 1.5. We specified an error rate of or 0.075 and evaluated the empirical power using 10,000 replicates under a significance level 0.05.

Figure 1 shows the type I error rates of the control-only χ² test, LRT and shrinkage test under different simulation conditions. We can see that the LRT and the shrinkage test consistently controlled the type I error rates at the nominal value (5%) across all conditions. In contrast, although the control-only χ² test performed reasonably well when the disease prevalence was low , it led to inflated type I error rates under REC and DOM with modest disease prevalence and strong genetic association (e.g. , ). For example, the type I error rate of the control-only χ² test was inflated up to 10.84% under REC with and . Therefore, if we were to use the control-only χ² test to evaluate the genotyping quality for large-sample studies (e.g. GWAS), we might falsely exclude the important candidate markers from further study.

In terms of power to detect departures from the HWE, the proposed shrinkage test outperformed the LRT and control-only χ² test, especially when the marker was weakly or not associated with the disease, as shown in Table 2. For example, when the marker is not associated with the disease, under genotyping error model S2 with disease prevalence , the power of the shrinkage test was about 12% and 14% higher than that of the LRT when the error rates are and 0.075, respectively. Such improvement stems from the fact that the shrinkage test automatically converged toward the pooled χ² test in the absence of a marker-disease association.

Table 2. Empirical power (%) of the control-only χ² test, LRT and the shrinkage test based on 10,000 replicates. The significance level is 0.05

Genetic		Error
model	λ2	model	δ	Control	LRT	Shrinkage	Control	LRT	Shrinkage
No	1.0	S1	0.05	34.2	34.2	43.2	35.4	41.1	48.6
association			0.075	65.7	66.0	78.4	65.1	73.2	82.4
		S2	0.05	35.0	35.8	47.8	35.6	41.8	51.4
			0.075	65.3	66.5	80.9	65.5	74.2	84.4
REC	1.25	S1	0.05	35.0	35.2	52.2	28.5	41.6	54.5
			0.075	63.8	65.7	79.3	57.6	73.5	83.7
		S2	0.05	35.7	36.0	37.0	41.2	41.5	41.6
			0.075	66.1	67.1	71.3	71.9	74.8	77.4
	1.5	S1	0.05	33.2	34.5	43.1	22.3	40.1	46.5
			0.075	63.9	65.9	71.8	50.9	74.9	78.1
		S2	0.05	36.6	36.5	34.5	49.2	43.0	41.4
			0.075	67.6	66.4	67.2	77.3	74.8	77.5
ADD	1.25	S1	0.05	34.3	34.3	38.4	34.6	41.3	43.8
			0.075	65.0	65.2	70.6	64.7	73.6	76.6
		S2	0.05	35.1	36.2	43.7	35.3	42.6	48.2
			0.075	64.6	65.9	74.6	65.1	75.2	80.3
	1.5	S1	0.05	33.9	33.9	33.8	33.7	40.6	39.9
			0.075	65.0	65.6	65.5	62.7	72.7	72.2
		S2	0.05	35.3	35.9	38.8	34.7	42.4	44.1
			0.075	66.3	66.8	70.0	65.7	75.7	77.5
DOM	1.25	S1	0.05	35.4	34.8	33.8	40.6	40.7	39.4
			0.075	66.1	65.8	66.2	69.9	72.9	72.8
		S2	0.05	34.3	36.0	43.4	29.5	41.8	47.3
			0.075	66.4	67.7	74.8	59.4	75.5	79.9
	1.5	S1	0.05	35.8	34.9	34.5	45.7	40.7	39.6
			0.075	66.1	65.3	64.6	73.6	73.2	72.8
		S2	0.05	33.5	35.2	36.9	25.5	42.3	44.0
			0.075	65.1	67.5	69.1	54.1	75.7	76.7

As the association between the marker and the disease becomes stronger (e.g. or 1.5), the power gain using the shrinkage test becomes smaller because the shrinkage test converges toward the generalized χ² test, which is asymptotically equivalent to the LRT. Even so, in general, we observed that the shrinkage test was slightly more powerful than the LRT in many cases, especially under the S2 genotyping error model. For instance, under the ADD and the moderate association with and disease prevalence , the shrinkage test was 7.5% and 8.7% more powerful than the LRT when the error rates were and 0.075 under genotyping error model S2.

The proposed shrinkage test also has a substantial edge over the LRT in terms of computing time. It took about 4 min to conduct 10,000 simulations using the LRT; whereas it took only 15 seconds using the proposed shrinkage test on a personal computer with a 3.20 GHZ Intel Core i5 CPU and 4.00 GB memory.

To further understand the behavior of the proposed shrinkage test, we investigated the relationship between the shrinkage factor w and strength of the marker-disease association (i.e. λ₂). As shown in Figure 2, across different genetic models, the value of the shrinkage factor w automatically adjusted according to the strength of the marker-disease association. When the association was weak (i.e. the value of λ₂ was close to 1), the value of the shrinkage factor w was small, thereby strongly converging the shrinkage test toward the pooled χ² test to achieve high statistical power. When λ₂ increased, the shrinkage factor w approached 1, thereby converging the shrinkage test toward the generalized χ² test to maintain the validity of the test. These results explain the underlying reason the shrinkage test yields higher power than the LRT, while also controlling the type I error rate at the nominal value, as described previously.

In this simulation studies, we focus on the single-SNP case-control studies. Actually, the proposed shrinkage test can also be used in the GWAS study where millions of SNPs are being tested. The only modification is that we should select a much stricter significance level (i.e. 0.0001) for the GWAS study due to the multiple comparison issue. In the following section, we applied the shrinkage test to a real GWAS dataset to investigate its performance in the GWAS study.

Application

We applied the proposed method to a GWAS dataset from the Genome Medicine Database of Japan (GeMDBJ) [Yoshida et al., 2003]. This dataset contains information from 763 patients diagnosed with Alzheimer's disease and 1,422 healthy volunteers from Japan. A total of 577,728 SNPs were genotyped to identify patterns of genomic variation associated with Alzheimer's disease. According to Matsui et al. [2009], the incidence rate of Alzheimer's disease was 14.6/1,000 persons in the Japanese population. To assess the genotyping quality of the data, we applied the proposed method to the SNPs. We first screened out the SNPs with estimated MAF < 0.05, and then applied the LRT and the shrinkage test to assess the HWE for the remaining 469,225 SNPs.

As shown in Table 3, across different cutoffs (of the p-value) for significance, compared to the LRT, the proposed shrinkage test identified more SNPs that had significant departure from the HWE, which suggests that the shrinkage test has higher power than the LRT. For example, with 0.05 as the significance cutoff, the LRT detected 33,006 significant SNPs, whereas the shrinkage test detected 34,952 significant SNPs. To obtain more insight into the variations in performance between the shrinkage test and the LRT, we calculated the p-values of the SNP-disease association for each of the SNPs based on the trend test (e.g. PV_assoc), and then stratified the SNPs into four groups according to PV_assoc; see Table 4. Consistent with our simulation results, compared to the statistical power of the LRT, the power gained by using the shrinkage test depended on the strength of the SNP-disease association. When the SNP-disease association was weak (i.e. PV_assoc ⩾ 0.05), the shrinkage test identified 1,896 more significant SNPs than the LRT (31,268 vs. 33,164). As the strength of the association became stronger, the difference between these two methods diminished. The shrinkage test identified 41, 8, and 1 more significant SNPs than the LRT when and , and , respectively. Given the fact that most candidate SNPs in the GWAS are not associated with the disease, the shrinkage test is preferred to the LRT for checking the quality of genotyping in the GWAS. In addition, the proposed shrinkage test required a substantially shorter computing time. To analyze this dataset, the LRT took more than 2.5 hr, whereas the shrinkage test took only about 10 min.

Table 3. Number of SNPs showing significant departure from HWE under different p-value cutoffs

	p-value less than
Method
LRT	396	437	673	2,636	33,006
Shrinkage	1,411	1,669	2,149	4,605	34,952

Table 4. Number of SNPs showing significant departure from HWE, stratified by the p-value of the marker-disease association, based on the LRT and shrinkage test. The significance level for the LRT and shrinkage test is 0.05

	p-value for marker-disease association
Method	[0, 10−4)				Total
LRT	3	374	1,361	31,268	33,006
Shrinkage	4	382	1,402	33,164	34,952

Conclusion

In this article, we propose a shrinkage test for assessing the HWE in case-control data. The proposed shrinkage test is more powerful than the LRT when the marker is independent of or weakly associated with the disease, and remains valid when the marker is strongly associated with the disease. Specifically, we propose a generalized χ² test that is asymptotically equivalent to the LRT but easier to calculate. Then, we develop a shrinkage test that takes the form of the weighted average of the pooled χ² test and the generalized χ² test. We construct the weight (or shrinkage factor) based on the ABF so that the weight adaptively shrinks the test toward the pooled χ² test or the generalized χ² test according to the strength of the marker-disease association. When the marker is independent of the disease, the shrinkage test converges to the pooled χ² test to achieve high statistical power; and when the marker is associated with the disease, the shrinkage test converges to the generalized χ² test to guarantee the validity of the test. In addition, the shrinkage test has a closed form and is easy to use. A simulation study and real data application show that the shrinkage test outperforms the existing methods with higher statistical power to detect departures from the HWE. The associated R code to implement the proposed shrinkage test can be downloaded from http://odin.mdacc.tmc.edu/yyuan/Software_release/HWE/simu.R