In genetic association studies, several test statistics are often used, including the score test Z(θ) and the maximin efficient robust test statistic Z_MERT. Although numerical behavior of the two tests are reported in various genetic association studies based on simulations, to our knowledge, a formal theoretical comparison of the two tests hasn’t been seen in the literature. It is of meaning to compare their asymptotic performances. Although for likelihood ratio based test statistic for testing hypothesis of simple null versus simple alternative, there is a uniformly most powerful test under some regularity conditions. However, most test statistics are not constructed directly from likelihood ratio, the hypothesis are composite, and there is generally no such optimal test. Therefore, the classical method to compare any two test statistics is to evaluate the Asymptotic Relative Efficiency (ARE) between them.

The ARE is a well studied area, with vast literatures and numerous different definitions. But often the computation of ARE is very difficult in the general case, some of the classical methods for ARE require that the test statistics have some standard forms, such as they have the same asymptotic distribution, or have the forms of i.i.d. summations. However, in practice, such as in genetic association studies, some test statistics do not have these forms. Sitlani and McKnight [1] studied AREs for the trend test under different models and stratifications. In this communication, wecompare the asymptotic behavior of two commonly used test statistics the score statistic Z(θ) and the maximin efficient robust test statistic Z_MERT, arise in case-control genetic association study, as given in Zheng, Li and Yuan [2], hereafter ZLY, by evaluate their AREs relative to each other. Four commonly used ARE measures, the Pitman ARE, Chernoff ARE, Hodges-Lehmann ARE and the Bahadur ARE are considered. Pitman’s ARE does not apply directly. We found the Chernoff, Hodges-Lehmann and the Bahadur AREs are suitable for our setting. Some modifications of these methods are made to simplify the computations.

Existing studies on ARE are mainly focused on two categories. One is to compare efficiencies of estimators of the same parameter; the other is to compare test statistics of the same hypothesis, in which the test statistics may not estimate the same parameter. The latter study can be under the assumption that the test statistics in comparison are asymptotic normality. In this case, the ARE’s can often be easily computed. There are also methods for compare ARE of different test statistics in general, in which different test statistics of the same hypothesis may have different asymptotic distributions. In this general case, Pitman, Bahadure and Hodges-Lehmann proposed different ways to compute the ARE, and it is often difficult. Although, when the test statistics have the same asymptotic distribution, the ARE can be computed easily. We also give a simple definition of ARE, so that it can be computed in the case of different asymptotic distributions, as long as the asymptotic distributions of the test statistics are known.

In Section 2, we describe the background of the genetic association study problem and a brief review of the classical definitions of ARE. In Section 3 we compare the ARE of the test statistics arose from our genetic association study. We found that he performances, or the efficiencies of the two test statistic varies for different criterion used, and for different parameter values under the same criterion, which described in the context. Section 4 gives brief numerical examples in simulation and application of the two tests in genetic association study, from our previous study, to illustration their usage.

Denote the log-likelihood function as, where Y_i is the outcome, R² are the parameters of interest, is a vector of parameters (m ≥ 0) for the covariate X_i = (x_1,....,x_im)^T, and n is the sample size. The goal is to test the null hypothesis against the alternative H₁ (λ₁, λ₂) \ {(1, 1)}, where has two edges with known slopes θ ₀ and θ₁, and the null point (1, 1) is on the boundary of . We assume - ∞ < θ ₀ < θ₁ <∞ and the endpoints θ ₀ and θ₁ satisfy some constraints as specified in ZLY. If θ₁ = ∞ which corresponds to a vertical edge, we can switch λ₁ and λ₂ and define new (θ₁, θ₂) so - ∞ < θ ₀ < θ₁ <∞ is satisfied by the new (θ₁, θ₂). For example, we can write λ₁ = 1 + (λ₂ - 1)/ λ₁^* (λ₂-1) and λ₁ = 1 + (λ₂ - 1)/ θ ₀ = 1 + θ ₀^* (λ₂ - 1) where - ∞ < θ ₀ < θ₁ <∞.

Assume θ ₀ and θ₁ are known from the problem of interest and/or scientific knowledge. Given λ₁ = λ ≥ 1, λ₂ can be written as . We treat η as a nuisance parameter not estimable under H ₀ λ = 1, but it is estimable under ₀. Then the log-likelihood becomes. l_n (λ, η, θ) The score test statistic H ₀ λ = 1 for is given by;

(1)

where is the MLE of η under H ₀. It would be difficult to deal with l_n (λ, η, θ) because θ in Z (θ) is implicitly expressed.

So we work with l_n (λ,1 - θ + θλ, η), where θ is explicitly expressed. It is convenient to view l_n (λ, η, θ) as a tri-variate function with variables x₁ = λ, x₂ = 1 - θ + θλ and x₃ = η. Denote l_n,u = ∂l_n/ ∂x_u for, u = 1,2,3, l_{n, uv} = ∂²l_n/∂x_u∂x_v for u = 1,2 and, v = 1.2.3, and l_n.33 = ∂²l_n/∂x₃∂x^T₃. Assume and v = 3. Denote L_vu (η) = E_Hnl_lvu (1.1, η).

Suppose we have a family of asymptotically normally distributed tests , where under H₁ λ = 1 for a given , which determines the data-generating model under H ₀: λ = 1. When is the true value Z(θ⁽⁾), is asymptotically most powerful (optimal). In this case, θ⁽¹⁾ ≠ θ⁽⁰⁾ when is used, the Pitman ARE of Z(θ⁽¹⁾) relative to Z(θ⁽¹⁾) is given by (Gastwirth [3, 4])

(2)

where is the asymptotic null correlation coefficient between and. Let be a set of all convex linear combinations of. A simple robust test derived under efficiency robust theory (Gastwirth [3, 4]; Birnbaum and Laska [5],) is the maximin efficient robust test (MERT), denoted as. When, is given by;

(3)

When T ₀ has more than two members, generally exists and is unique (Gastwirth [3]), but its computation needs quadratic programming methods (Rosen [6]). However, when there is an extreme pair (Z(θ_i), Z(θ _i)) in T ₀i.e. p_{θi, θi} = is MERT for if and only if (Gastwirth [7]).

and thus

(4)

That is, the MERT reaches the maximin ARE due to model uncertainty. The MERT was first derived for linear rank tests for the two-sample problem (Gastwirth [3]; Birnbaum and Laska [5],) and later extended to a family of asymptotically normally distributed tests (Gastwirth [4]).

The Z (θ) statistic has the following property (ZLY): Let. Then where and.

Let be the MLE of η under H ₀, and be that of (η, λ) under H₁. For given θ, the X² likelihood ratio test statistic is . For fixed θ, the number of parameters under H₁ is just 1 more than that under H ₀, so by Wilk’s theorem, under H ₀,

the chi-squared distribution with one degree of freedom. The likelihood ratio test is also widely used in genetic association studies, its properties, including its ARE is well studied in the literature, so we will not investigate it here.

Let the MLE here 0 presents a vector of 0’s. Let η ₀ be the true value (unknown) of η under either H ₀ or H₁, we define the score function as;

and the test statistic for H ₀ as;

(5)

where “~” means asymptotically equivalent, in the above is replaced by it is approximated by n^-1l_{n, vu} (1.1, η).

Denote . For a vector v (v₁, v₂, v₃)^T, denote . be the true density of the data y. The null model f (1, 1, η) is and the alternative model is . The following notation is also used under H₁. For fixed, (λ, θ) let;

(6)

Under H₁, the empirical version of η ₀ is just . We denote the Fisher information and its inverse in the blocked forms as;

Let

by is replaced by Note that with defined in the above,

Below we give a brief review of the notions of ARE for test statistics in the general case, more detailed account can be found in Serfling (1980) [8] and Nikitin (2011) [9].

The calculation of the existing of versions of ARE is generally not easy, as in the examples (Serfling, 1980 [8]; Nikitin, 1995 [10]; van der Varrt, 1998 [11]). We only point out that the Pitman ARE is based on the central limit theorem for test statistics, that the Bahadur ARE requires the large deviation asymptotics of test statistics under the null-hypothesis, while the Hodges-Lehmann ARE is connected with large deviation asymptotics under the alternative. Each type of ARE has its own advantage and dis-advantage, and the different notions of ARE are not always give consistent conclusion.

If the condition of asymptotic normality (or common asymptotic distribution) fails, considerable difficulties will arise in calculating the Pitman ARE as it may not at all exist or may depend on α and β. Usually one considers limiting Pitman ARE as α → 0 Wieand (1976) [12] established the correspondence between this kind of ARE and the limiting approximate Bahadur efficiency which is easy to compute.

The Bahadur (1960) [13] ARE is to fix the power of tests and compare the exponential rate of decrease of their sizes for the increasing number of observations and fixed alternative. Its computation is always non-trivial, and heavily depends on advancements in large deviation theory, as in Dembo and Zeitouni (1998) [14] and Deuschel and Strook (1989) [15].

It is proved that under some regularity conditions the likelihood ratio statistic is asymptotically optimal in Bahadur sense (Bahadur, 1967 [16]; Arcones, 2005 [17]). Often the Bahadur ARE is difficult to compute for any alternative but it is possible to calculate the limit of Bahadur ARE as θ approaches the null-hypothesis, to obtain the local Bahadur efficiency.

The Hodges-Lehmann ARE is, in contrast to Bahadur efficiency, it fixes the level of tests and compares the exponential rate of decrease of their type-II errors for the increasing number of observations and fixed alternative. The computation of Hodges-Lehmann ARE is also difficult as it requires large deviation asymptotics of test statistics under the alternative.

The drawback of Hodges-Lehmann efficiency is that most two-sided tests like Kolmogorov and Cramer-von Mises tests are all asymptotically optimal, and hence one cannot discriminate among them. On the other hand, under some regularity conditions the one-sided tests, such as linear rank tests can be compared, and their Hodges-Lehmann efficiency coincides locally with Bahadur efficiency (Nikitin, 1995 [10]).

The Chernoff ARE is to minimize, asymptotically, a linear combination of type I and type II errors, it does not depend on the nominal level nor the power. But it basically only applies to test statistics of the form of i.i.d. summation.

The local ARE is much easier to compute than the previous ones, but it only applies to test statistics which are asymptotical normal with rate . We will see that some test statistics used in genetic association studies do not satisfy this condition.

Besides the four commonly used AREs for hypothesis tests described above, there are some other interesting methods. Hoeffding’s (1965) ARE [18], based on the work of Sanov (1957) [19], is theoretically appealing, but ony applies to multinomial data; Rubin and Sethurman ARE (1965) [20] is based on Bayes risk; others including Kallenberg ARE (1983) [21], and the Borovkov-Mogulskii ARE (1993) [22], etc.

In this section, we investigate the uses of Pitman ARE, Chernoff ARE, Hodges-Lehmman ARE, and Bahadur ARE to the commonly used statistics in genetic association analysis. We focus on the statistics used in ZLY, Z(θ) and, Z_MERTand refer the notations there. Although some other commonly used test statistics in genetic association studies, such as the likelihood ratio statistic (chi-squared statistic), we will not discuss them here, as most of them are well studied in the literatures.

Pitman ARE. Consider testing Let S_n be a test statistic based on data of size n, with mean µ_n (λ) and standard deviation µ_n (λ). To use this method the following conditions are needed.

(P1). For some continuous strictly increasing distribution function F independent of λ, and some, δ > 0 as n → ∞,

(P2). For , is k times differentiable, with µ_n⁽¹⁾ (λ ₀) = ... =

(P3). For d(n) → ∞ some and some constant

(P4). For

Pitman appears as the first to introduce the notion of ARE for tests in his unpublished lectures, and the following result was stated in Noether’s works.

(Pitman, 1949 [23]; Noether, 1950 [24]). Assume (P1)-(P4), that α_n = P_{λ 0} (S_n > then , if and only if

(7)

(ii) Let S_1,n and S_2,n each satisfy (P1)-(P4) with the common F, K, n₁ and n₂ be the sample size required for S_1,n and S_2,n to have the same asymptotic power 1 - β, then

Thus, if d(n) = n^q (q > 0), then the Pitman ARE is given by; .

and Pitman ARE is then;

(8)

Let l (λ ₀) be the Fisher information at λ ₀. Under some additional conditions, Rao (1963) [25] proved that

Any test statistic S_n achieves the equality in the above is called Pitman efficient.

Under suitable conditions, Pitman ARE can be expressed in terms of correlation coefficient between the two test statistics in their standardized form, as given below.

(P5) are asymptotic joint normal uniformly in a neighborhood of λ₀.

Denote p(λ)the asymptotic correlation coefficient between them under, and and be the distribution and density function of. The following result is true.

(van Eden, 1963 [26]). Assume that S_1,n and S_2,n satisfy (P1)-(P5) in their standardized form with , and that p(λ_n) → p(λ λ_n): = p as λ_n → λ ₀ Then;

(i) For 0 ≤ λ ≤ 1, tests of the form satisfy (P1)-(P5), and the “best” S_yn which maximizes is the one with;

and

(9)

(ii) If S_1n is the best test satisfying (P1)-(P5), then;

(10)

In the typical case, S_n is an i.i.d. summation (upto scale), then µ_n(λ) = nµ(λ)

Note does not (α, β) depend on , thus if or, C₁ > C₂ then {S_1n} is better than {S_2n} for all (α, β).

Pitman ARE given by (3) or (4) are easy to use. However, they require the two comparing test statistics have the same asymptotic distribution (after standardization), (4) require further that they are jointly asymptotic normal. In practice, these conditions some times cannot be satisfied. For example the chi-squared test Z (θ ₀) and have different asymptotic distributions. Below we give a generalized version of (3) to the case the two comparing test statistics not necessarily have the same asymptotic distribution (after standardization). Similar generalizations may have already exist in the literature, we still state our version to see what form it has in this case. Let F_i be the asymptotic distribution of We have;

Assume (P1)-(P4) for S_in with µ_in, σ_in and F_i separately, but with the same K and nominal level α, n₁ and n₂ be the sample sizes required for S_1n and N_2n to have the same asymptotic power 1 - β(0 < β < 1 - α), then

Thus for d(n) = n^q (q > 0), we define the generalized Pitman ARE as;

(11)

In the typical case or 1/q = 2, and;

Note, unlike the case of F₁ = F₂, in this case, Pitman’s ARE depends on the values of level α and power β , and comparison of two tests may not have consistent result.

Can we have the corresponding form of (10) in the case S_1n and S_2n have different asymptotic distribution? For this we checked the proof for (4), and find in this case, although in principle there is a relationship among the asymptotic correlation coefficient p between S_1n and S_2n , the asymptotic distributions’s, F_i's, and the level α and power β , but its mathematically intractable. Below we give its actual value.

Proposition 1.

Remark: When some of the conditions (P1)-(P5) are not satisfied, ARE may not be characterized by correlation coefficient. For example, T₁ = Z is an estimate of θ = 0 under H ₀, and Z is symmetrically distributed around 0, so E_Ho (Z) = 0 and suppose VAR_Ho (Z) = 1 . Let, is an estimate of can also be used to test H ₀. However , but we cannot say that T₂ is a ‘bad’ test statistic, and .

Chernoff ARE. This notion only considers test statistic of the form with the s i.i.d. with be the moment generating function of Y, and;

Let and (assume µ ₀ ≤ µ₁), (i = 0,1), and is called the Chernoff index . be a linear combination of type I and type II errors evaluated at the critical value t, and Q_n = inf_{µ0 ≤ t ≤ µ,}Q_n (t) be the minimum of these errors for test statistic S_n. Chernoff (1952) [27] showed that Q_n tends to 0 at exponential rate, (so the faster the rate, or the larger absolute value of logQ_n, the better the test statistic), and established.

the result is independent of γ.

Let {S_1,n} and {S_2,n} both of the form of i.i.d. summation and have Chernoff indices p₁ and p₂ respectively, n₁ and n₂ be the corresponding sample sizes for which Q_1,n, ~ Q_2,n, the Chernoff ARE of {S_1,n} relative to {S_2,n} is defined and given by;

(12)

For test statistic not in the form of i.i.d summation, its Chernoff index is difficult to compute. The following result sometimes is very helpful in this case, and give an upper bound of Chernoff index.

(Kallenberg, 1982 [28]) Let for some

Then

In the case of simple null vs simple alternative, Kallenberg (1982) [28] also gives an upper bound of the Chernoff index, and any test statistic achieves this bound is said to be Chernoff efficient. As this bound itself is not easy to compute, we won’t pursue it here, interested readers can check the mentioned paper or the book by Nikitin (1995) [10].

As another way to simplify the computation, we consider a modified version of this Chernoff index. Let S be the weak limit of S_n, be the distribution function of S, and H_n: λ_n + λ_n = n^-1/2be a sequence of local alternatives. As the sample size increases, the test statistic S_n is expected to be able to distinguish the local alternatives from the null. Let (assume µ₁ ≥µ ₀), and be the asymptotic linear combination of type I and local type II errors evaluated at t, and . The smaller is , the better S_n as a test statistic for H ₀vs.H₁ For two test statistics S_1n and S_2n with we define the modified Chernoff ARE as;

(13)

Let, ;

Below we give values p_z(θ(0)) and p_ZMERT and so that their Chernoff ARE can be obtained. We also give and, so their modified Chernoff ARE can be obtained. For the chi-squared test T, under T₁ its asymptotic distribution is a non-central chi-squared distribution, with a non-closed form, its modified Chernoff index is not directly computable. Let , where g₁ is the observed genotype of the i-th individual, x₁ is the corresponding covariates, and let;

Let, and

Proposition 2. (i) Assume is normal with mean and variance . Then, for E to denote expectation with respect to (x_i, g_i), we have;

Hodges-Lehmann ARE. Consider testing the null hypothesis be given a level α test statistic S_n with critical value the type II error at λ is β_n (λ) = Typically, β_n (λ) tends to zero at exponential rate, the faster the better S_n is. Hodges and Lehmann (1956) [29] proposed;

as a measure of the performance of S_n and it called the Hodges-Lehmann index of the statistic S_n. For two test statistics S_1n and S_2n for the same H ₀vs,H₁ with d₁ (λ) and d₂ (λ), the Hodges-Lehmann ARE of {S_1n} relative to {S_2n} at is defined as;

(14)

For probability density functions f and g, let g(x]dx) be the Kullback-Leibler divergence between f) and g). For any test statistic S_n (X₁,.....,X_n) based on (X₁,.....,X_n) i.i.d. density , the Hodges-Lehmann index has the following property;

and any test statistic achieve the equality in the above is said to be Hodges-Lehmann efficient.

Compared to the Pitman and Chernoff ARE, the Hodges-Lehmman ARE does not require the comparing test statistic have the same asymptotic distribution, nor they have the form of i.i.d. summations, so it has wilder application scope.

Proposition 3. Under conditions of Theorem 4 in Zheng et al. (2010) [30], with , given in (2), for λ > 1, we have;

For the chi-squared test T, under H₁ its asymptotic distribution is a non-central chi-squared distribution, with no-closed form. So its Hodges-Lehmann ARE is not directly available.

Bahadur ARE. Consider testing the null hypothesis be Let F_n,λ(.) be the distribution function of a test statistic S_n under p_λ, and for , let;

the p-value of the observed S_n under the distribution p_λ, and;

if the limit exists. Typically, L_n tends to one and L_n tends to zero exponentially fast, and the faster, or the bigger c(.), the better S_n is. For two test statistics S_i,n (l = 1,2) for the same hypothesis with L_n, C_i (λ), and sample size n_i, to perform “equivalently” in the sense lim n₁^-1 log L_2,n2 = lim n₁^-1 Log L_1,n1, the Bahadur ARE of S_1,n log L_1n, the Bahadur ARE of relative to S_2,n, at , is defined as, and has the property

(15)

The limit C can be computed under the following conditions.

(B1). For

(B2). For the interval , there is a function g on l, such that;

(Bahadur, 1960 [13]). If S_n satisfies (B1)-(B2), then for ,

For any test statistic S_n (X₁,....,X₂) based on X₁,....,X_n i.i.d. density Bahadur (1967) [16] obtained the following;

Note although the above relationship is regarded as a dual to that of the Hodges-Lehmann index, the two are not equivalent as A test statistic is said to be Bahadur efficient if for each lim_n, log

Bahadur efficiency of likelihood ratio test has been studied by a number of researchers for some special distribution families. Arcones (2005 [17], Theorem 3.3) proved that, under some regularity conditions, the likelihood ratio statistic is Bahadur efficient. Let be the density function of the data, under his conditions of Theorem 3.3, for each fixed λ > 1 and θ, we have;

Like the Hodges-Lehmman ARE, Bahadur ARE does not require the comparing test statistic have the same asymptotic distribution, nor they have the form of i.i.d. summations, so it has wide application scope.

For computation easiness, we consider a local version of Bahadur ARE. Consider testing H ₀: λ = λ ₀vs the local alternative H ₀: λ = λ ₀ + n^-1/2. Let F ₀ be the asymptotic distribution function of S_n under H ₀, we define;

Typically, 0 < <1. The smaller , the better S_n is. For two test statistics S_i,n(i = 1,2) for the same hypothesis with G_i,n and , we define the local Bahadur ARE of S_1,n relative to S_2,nas;

(16)

Proposition 4. (i) with µ_MERT (λ) given in Proposition 3, we have;

(ii) Under conditions of Theorem 4 in ZLY, µ_MERT (λ) with be the derivative of µ_MERT (λ), θ ₀ be the value of θ H ₀ under, we have;