A test of homogeneity of Hardy-Weinberg disequilibrium across strata

Abstract

For genotype data being sampled from several strata with different allele frequencies, it is necessary to verify the assumption of homogeneity of Hardy–Weinberg disequilibrium across strata before testing Hardy–Weinberg law across strata. In practice, disequilibrium can be measured via fixation coefficients (ie, ratios of genotypic frequencies) or disequilibrium coefficients (ie, differences of genotypic frequencies). Test for homogeneity of Hardy–Weinberg disequilibrium using data from several populations has been derived according to fixation coefficients. In this article, using the likelihood score theory extended to nuisance parameters, we derive a homogeneity score test for comparing disequilibrium coefficients across several independent strata. Simulation results demonstrate that the homogeneity score test performs satisfactorily in the sense that its empirical size seldom exceeds the pre-chosen nominal level by more than 10% even for small sample sizes. Corresponding power and sample size formulae are provided as well. We illustrate our test with a real glyoxalase genotype data set.

Introduction

The law of Hardy–Weinberg equilibrium (HWE) states that in a large random mating population that is not affected by the evolutionary processes of mutation, migration, or selection, both the allele frequencies and the genotype frequencies are constant from generation to generation.^{1, 2} Furthermore, the genotype frequencies are related to the allele frequencies by the square expansion of those allele frequencies. In other words, the law of HWE states that under a restrictive set of assumptions, it is possible to calculate the expected frequencies of genotypes in a population if the frequency of the different alleles in a population is known. The original descriptions of HWE become an important landmark in the history of population genetics,³ and it is now a common practice to verify whether observed genotypes conform to Hardy–Weinberg expectations.^{4, 5}

In a diallelic locus with alleles A₁ and A₂ across K strata, let the genotypic array of the kth (k=1, …, K) stratum be

Let p_k be the allelic frequency of A₁ in the kth stratum and q_k=1−p_k (k=1, …, K). Populations with genotypic frequencies satisfying p_11k=p_k², p_12k=2p_kq_k, and p_22k=q_k² (k=1, …, K) are said to be in HWE at the locus under consideration. In studies of HWE, there are two widely used coefficients, namely the fixation and disequilibrium coefficients.⁶ For stratum k (k=1, …, K), the fixation and disequilibrium coefficients are defined by and D_k=p_kq_k−p_12k/2, respectively. Hence, the problem of testing HWE when individuals are sampled from several strata is equivalent to testing one of the following hypotheses:

where θ_k=f_k or D_k. For statistical tests based on disequilibrium coefficient, one can refer to the work of Haldane⁷ and Smith.⁸ For test procedures based on functions of fixation coefficients (eg, (1−f_k)²), one can consult the work of Emigh,⁹ Troendle and Yu,¹⁰ and Nam.¹¹

It is noteworthy that any statistical procedure for testing the null hypothesis in (1) assumes that the measure of disequilibrium (ie, θ_k) is constant across the strata. In this regard, it is important that one should consider testing the assumption of homogeneity of the measure of disequilibrium across strata before any testing of the null hypothesis in (1). For this purpose, we consider the following hypotheses:

Olson and Foley¹² proposed a large-sample test and an exact test for verifying the null hypothesis H₀ via a function of fixation coefficients, (1−f_k)². They also approximate the P-value of the exact test using a Markov chain Monte Carlo approach. Although the use of fixation coefficients to describe departures from HWE has some merit, it has the disadvantage that these parameters are estimated as ratios of genotypic frequencies. It is difficult to study sampling properties of ratio statistics.^{4, 6} Besides, functions of fixation coefficients such as (1−f_k)² may possess infinite upper bound. On the other hand, there are advantages in working with a composite kind of quantity such as the disequilibrium coefficient. This is simply the difference between a frequency and its values expected when there are no association between alleles. Moreover, it is easy to show that disequilibrium coefficient D_k satisfies max{−p_k², −q_k²}⩽D_k⩽p_kq_k. Unfortunately, test of homogeneity of disequilibrium coefficients across several strata has not been considered in the literature yet. Therefore, the objective of this study is to develop a new homogeneity score statistic for testing the null hypothesis in (2) based on disequilibrium coefficients. We first develop the theory and method and then demonstrate the advantage of our method over the method proposed by Olson and Foley¹² via Monte Carlo simulation studies. We also derive the approximate power and sample size formulae, which are necessary in design of studies. Finally, we illustrate our test with a real glyoxalase genotype data set.

Method

Homogeneity test

Let X_ijk (i⩽j=1, 2 and k=1, …, K) be the number of individuals with genotype A_iA_j in the kth population with n_k=X_11k+X_12k+X_22k. Let M(n_k, {p_ijk}) denote the trinomial distribution with parameter vector (p_11k, p_12k, p_22k). Hence, we have {X_ijk: i, j=1, 2; i⩽j}∼M(n_k, {p_ijk}) for k=1, …, K. In this article, we are interested to test the homogeneity hypothesis in (2) with θ_k=D_k. That is,

where D_k=p_kq_k−p_12k/2. All subsequent results are obtained under the assumptions that K is fixed and n_k is sufficiently large for k=1, 2, …, K.

Note that p_11k=p_k²+D_k, p_12k=2(p_kq_k−D_k) and p_22k=q_k²+D_k, the log-likelihood for the kth strata can be expressed in terms of D_k and p_k (k=1, …, K) as

Let D denote the common disequilibrium coefficient under H₀ and p=(p₁, …, p_K)′ the nuisance parameter vector. Under H₀, the total log-likelihood for all K strata is given by

Hence, the efficient scores for the kth stratum (ie, the first-order derivatives of l_k(D,p_k) with respect to D and p_k) are given by

Let D̂ and p̂ be the maximum-likelihood estimates (MLEs) of D and p under the null hypothesis H₀. In this case, D̂ and p̂ must satisfy the following K+1 equations:

Denote

In addition, denote , where

Hence, the likelihood score test for testing H₀: D₁=⋯=D_K is given by

which is asymptotically distributed as a χ² variate with K−1 degrees of freedom under H₀. Unfortunately, we note that D̂ and p̂ cannot be expressed in closed form and this makes the likelihood score test X² less appealing in real applications. To over this issue, using the theory of homogeneity score test extended to nuisance parameters,¹³ we consider the following modified score statistic:

where D^* and p^* are any consistent estimators of D and p, respectively. To this end, we choose D^* to be and p^*_k be the solution to the following equation:

or equivalently the following quintic polynomial equation,

where a₀=x_12kD*(1+D*)+2x_22k(D^*)², a₁=−2(n_kD* (1+D^*)+x_12kD^*), a₂=6n_kD^*+2x_11k+x_12k, a₃=−2(2n_kD*+ n_k+2x_11k+x_12k), a₄=4n_k+2x_11k+x_12k, and a₅=−2n_k. Here, D^* is analogous to the Mantel–Haenszel estimator¹⁴ and is a consistent estimator to D. However, it is not an efficient estimator to D in general. The proof of consistency and the condition to attain asymptotic efficiency for D* is given in Appendix A. We note that the calculation of in (3) could be tedious. Nonetheless, it is easy to show that is simply given by n_k/w_k(D, p_k) with w_k(D, p_k)=(p_k²+D) (q_k²+D)²+2(p_kq_k−D)³+(p_k²+D)²(q_k²+D)−4D² (see Appendix B for the proof). Similarly, X²* is asymptotically distributed as a χ² variate with K−1 degrees of freedom under H₀. Therefore, the homogeneity hypothesis H₀ is rejected at level α if X²*⩾χ_{K−1,(1−α)}², where χ_{K−1,(1−α)}², is the 100 × (1−α) percentile point of the χ² distribution with K−1 degrees of freedom. Finally, it is noteworthy that if the consistent estimators of D and p are the constrained maximum-likelihood estimators under H₀, then the second term of (3) vanishes, since ∑_k=1^K H_kD(D*, p_k^*)=0, and (3) reduces to the likelihood score statistic.

Asymptotic power and sample size formulae

In this section, we aim to derive the asymptotic power and sample size formulae¹⁵ based on X²*. For these purposes, we assume n_k=nb_k for some n and b_k>0. Let D̄_k and p̄_k be the true parameter values for D_k and p_k under the alternative H₁, where k=1, 2, …, K and D̄_k≠D̄_j for some k≠j. Hence, the asymptotic power of the homogeneity score test X^2* at α level is given by

where χ_K−1² (δ) denotes the non-central χ² distribution with K–1 degrees of freedom and the non-centrality parameter δ is equal to

with q̄_k=1−p̄_k,

and p_k is the solution to the following equation:

where ā₀=2(p̄_kq̄_k−D̄_k)D(1+D)+2(q̄_k²+D̄_k)D², ā₁=−2D(1+D)+4(p̄_kq̄_k−D̄_k)D, ā₂=6D+2p̄_k, ā₃=−2(2D+1+p̄_k), ā₄=4+p̄_k, and ā₅=−2.

As a result, the desirable sample size n required to attain the power at 1−β with D̄_k and p̄_k being the true parameter values for D_k and p_k under the alternative H₁ at nominal level α can be determined from the following equality:

where χ_K−1,β² (δ) is the 100 × β percentage point of the non-central χ² distribution with K−1 degrees of freedom with non-centrality parameter being δ. The value of n can be readily obtained by solving the equation given in (4).

Simulation

We evaluate the performance of our proposed homogeneity score test in terms of type I error rate and power. For type I error rate, we include the homogeneity test proposed by Olson and Foley¹² in our comparison study. In their case, they adopted a function of fixation coefficients as the measure for Hardy–Weinberg disequilibrium. Specifically, they were interested to test the homogeneity hypothesis in (2) with θ_k=(1−f_k)². That is,

H₀^*: (1−f₁)²=⋯=(1−f_K)² versus H₁^*: Not all (1−f_k)² are equal, and their proposed statistic for testing the above hypotheses is given by

where θ̂=(∑_k=1^K((x_12k²−x_12k)/(2(2n_k−1)))/(∑_k=1^K((2x_11k²− x_22k)/(2n_k−1))), h_k(θ)=x_12k²−x_12k−4θx_11kx_22k, and Vâr [h_k(θ)]= 4(x_11k³−3x_11k²+2x_11k)(1−θ)+2(x_11k²−x_11k) (2n_kθ−3θ +2) for k=1, …, K. We would like to point out here that our proposed homogeneity score test (ie, X^2*) and Olson and Foley's test (ie, T_homog²) can be fairly comparable only when θ_k=0 for k=1, …, K in the null hypotheses H₀ and H₀^* (ie, H′₀ in (1)). In the present comparisons, we consider both the asymptotic (denoted as T_homog,a²) and exact (denoted as T_homog,e) versions of T_homog². For the implementation of T_homog,e², one can refer to Olson and Foley (1996, p 975). Here, we investigate type I error rates of X^2* and T_homog,a for small (eg, n_k=20 and 30) to large sample sizes (eg, n_k=50–200) when θ_k=0 for k=1, …, K. As T_homog,e² is computationally intensive for large sample sizes, we consider its small-sample behavior only. Results of Monte Carlo experiments with 5 000 repetitions for different designed allele probabilities p′_ks with k=1, …, K and K=3 and 5 at 0.05 nominal level are summarized in Tables 1 (for small sample sizes) and 2 (for moderate to large sample sizes).

Table 1 Empirical type I error rates for X^2*, T_homog,a² and T_homog,e² under H′₀ when K=3 and K=5

Table 2 Empirical type I error rates for X^2* and T_homog,,a² under H′₀ when K=3 and K=5

As expected, the exact test T²_homog,e is always conservative (ie, its type I error rates are always less than the pre-assigned nominal level). The empirical type I error rates of our asymptotic homogeneity score test X^2* are satisfactorily close to the nominal 0.05 level for allelic probabilities being bounded away from 0 and 1, whereas those of the T_homog,a² are generally liberal (eg, more than 11 times of the given nominal level) even for large sample sizes. It is noteworthy that X^2* appears to be conservative than T_homog,e², for small allele probabilities (eg, p_k's being 0.1). However, the conservativeness of X^2* vanishes with an increase in sample sizes and the computation of X^2* is much more simpler than T_homog,e².

In view of the above observations, we prefer the proposed homogeneity score test X^2* (based on disequilibrium coefficients) to the existing homogeneity tests based on function of fixation coefficients (ie, T_homog,a²_, and T_homog,e²). Hence, we exclude T_homog,a²_, and T_homog,e² in all subsequent evaluation and discussion. Table 3 further summarizes the type I error rate of X^2* for some non-zero (common) disequilibrium coefficients (ie, D≠0) under different settings. Again, the propose homogeneity score test performs satisfactorily in the sense that its empirical type I error rates are close to the pre-chosen nominal level and seldom exceed the nominal level by more than 10%.

Table 3 Empirical type I error rates for X^2* under H₀

For power performance, the parameters and sample size are quite similar to those adopted in Table 3, except that {D_k} are now specifically designed under H₁. For this purpose, we set D_k=D₀+Δ(k−1). For K=3, we consider: (i) D₀=−0.03, Δ=0.03 and (ii) D₀=−0.05, Δ=0.05. For K=5, we consider: (i) D₀=−0.06, Δ=0.03 and (ii) D₀=−0.1, Δ=0.05. The results are reported in Table 4. From the simulation results, the power of X^2* increases with the sample size n or Δ. For those settings with the same {D_k}, the one with varied allele probabilities across strata usually have power greater than that with equal allele probabilities across strata.

Table 4 Empirical power for X^2*

Real example

Ghosh reported genotype frequencies of red cell glyoxalase 1 (GLO) polymorphism from several populations.¹⁶ We consider the data, reproduced in Table 5, from four populations in the Western Pacific Area. The gene frequencies of four populations highly vary from 0.0455 in the Eastern Carolines to 0.3611 in the Tokelau Islands, Samoa and Fuji in between. The estimated disequilibrium coefficients (ie, D̂_k) in the four populations are ranging from 0.0145–0.019, which are close to zero. This seems to suggest that the homogeneity of HWE across the four populations, although the gene frequencies vary appreciably. Our proposed homogeneity score test yields X^2*=2.33 with P-value being 0.51. Hence, it is now safe to assume that the HWE is simultaneously valid across the four populations. We apply the Olson and Foley's test to the same glyoxalase genotype data set in the Western Pacific Area. Function of fixation coefficients (ie, (1−f_K)²) was adopted and the corresponding homogeneity test yields T_homog,a²=2.78 with P-value being 0.43. In this case, both tests reach the same conclusion.

Table 5 Glyoxalase genotype data in Western Pacific Area

Discussion

In practice, one is tempted to test the Hardy–Weinberg law across several independent populations without verifying the underlying assumption of homogeneity of Hardy–Weinberg disequilibrium across populations. Verification of the latter assumption is critical in genotype data analysis. Olson and Foley proposed a homogeneity test for this purpose. Unfortunately, our simulations show that their asymptotic version test is not reliable (ie, inflated type I error rates) even in large sample size. Although an exact version test was also proposed to overcome the liberty issue, such a test is however always conservativeness and computationally intensive for large sample sizes.

In this paper, we consider a homogeneity score test based on disequilibrium coefficients. Empirical results from our simulation studies support that our homogeneity score test is a reliable asymptotic testing procedure even for small sample sizes. However, our test may suffer the drawback that it may be quite conservative for rare allelic probabilities (eg, ⩽0.1). In this case, one may require larger sample sizes to overcome the conservativeness issue. In this regard, we also provide a sample size formula for design purpose. We have implemented the test procedures described in this manuscript in a Matlab program, which can be downloaded from the web site: http://math.nenu.edu.cn/jhguo/program.htm.

We also applied the Kolmogorov–Smirnov test to study the asymptotic behaviors of our test (ie, X^2*). Briefly, for allele frequency greater than or equal to 0.1, we find that the asymptotic χ² sampling distribution property follows for moderate sample sizes (eg, n_k⩾50). For rare allele frequency (ie, <0.1), larger sample sizes are required. In fact, after some straightforward algebra, we observe that H_kD (D*, p_k*) has larger variance for rare p_k. This may explain the severe conservativeness of X²* for rare p_k. We are now undertaking an investigation of possible modification of X²* for conservative correction.

We note that exact (conditional) method works in Olson and Foley¹² as they considered fixation coefficient f's which in turn are odds ratio. In their case, sufficient statistics for those nuisance parameters exist and can be eliminated by conditioning on their sufficient statistics. On the contrary, we consider the disequilibrium coefficient D's, which are actually rate differences. In our case, sufficient statistics do not exist for the corresponding nuisance parameters and the exact conditional method hence is not applicable.¹⁷

Finally, the theories developed in this paper can be readily extended to genotype data with multiple alleles.