Asymptotic Relative Efficiencies of the Score and Robust Tests in Genetic Association Studies

Department of Biostatistics, Bioinformatics and Biomathematics, Georgetown University, Washington DC, 20057, USA Department of Statistics and Actuarial Science, University of Hong Kong, Hong Kong School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China LSC, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China


INTRODUCTION
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.

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]).
Then where and .

Let
be the MLE of η under H 0 , and be that of (η, λ) under H 1 .For given θ, the X 2 likelihood ratio test statistic is .For fixed θ, the number of parameters under H 1 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 H 1 , we define the score function as; and the test statistic for H 0 as; (( (1) ), ( (0) )) =   (0) , ( Under H 1 , the empirical version of η 0 is just .We denote the Fisher information and its inverse in the blocked forms as; 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 nullhypothesis, 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.

ARE OF TWO TESTS IN GENETIC ASSOCIATION STUDIES
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 MERT and 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.

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

𝑑(𝑛
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 λ.
Denote p(λ)the asymptotic correlation coefficient between them under, and and be the distribution and density function of.The following result is true.
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.
We have; ( , −  , ())/ , () In the typical case or 1/q = 2, and; Remark: When some of the conditions (P1)-(P5) are not satisfied, ARE may not be characterized by correlation coefficient.For example, T 1 = 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 T 2 is a 'bad' test statistic, and .

𝑌 ∼ 𝐹. Let 𝑀(𝑧) = 𝐸 𝐹 (𝑒 𝑧𝑌 )
Let and (assume µ 0 ≤ µ 1 ), (i = 0,1) 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 called the Chernoff index 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 p 1 and p 2 respectively, n 1 and n 2 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; 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 -1/2 be 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 µ 1 ≥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 0 vs.H 1 For two test statistics S 1n and S 2n with we define the modified Chernoff ARE as;

(13)
Below we give values p z(θ(0)) and p Z MERT 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 T 1 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 g 1 is the observed genotype of the i-th individual, x 1 is the corresponding covariates, and let;

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;  Let  (1) (, ) = (, )/, and  (1) = ( 1 (1) , . . .,   (1) )  = (  (1) ( 0 ,  1 ) ( 1 ) , . . .,  (1)   [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 0 vs,H 1 with d 1 (λ) and d 2 (λ), the Hodges-Lehmann ARE of {S 1n } relative to {S 2n } at is defined as; For probability density functions f and g, let be the Kullback-Leibler divergence between f) and g).For any test statistic S n (X 1 ,.....,X n ) based on (X 1 ,.....,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 H 1 its asymptotic distribution is a non-central chi-squared distribution, with noclosed form.So its Hodges-Lehmann ARE is not directly available.
0 :  ∈ Λ 0 vs  1 :  ∈ Λ 1 , (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 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 , 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 The limit C can be computed under the following conditions. (B1).
(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 (X 1 ,....,X 2 ) based on X 1 ,....,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 comparing test statistic have the same asymptotic distribution, nor they have the form of i.i.d.summations, so it has wide application scope. .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 Let P be the Minor Allele Frequency (MAF) of a marker of interest.We consider case-control data with r = 500 cases and s = 500 controls, and the disease prevalence K = 0.05.We generate 1000 datasets, and compute the means and standard deviations of For Z MERT , we choose θ i = 0 and θ j = 1.
Table T1 shows the result, the means of AREs and the standard deviations of AREs are in brackets.First we can see the mean of all three AREs are less than 1, which show that Z θ o is consistent better than Z MERT .Corresponding tothis fact when θ = θ(o) is the true value Z θ (o), is asymptotically most powerful.Then the three AREs are increased with the P or λ increased.Third, the e p has the lowest variance among the three AREs, next is

Application
We use 6 reported SNPs associated with breast cancer 2 (Hunter et al. 2007 [31]; Li et al., 2008 [32]) to illustrate the ARE of Z MERT .These 6 SNPs are rs10510126, rs12505080, rs17157903, rs1219648, rs7696175, and rs2420946.The counts of subjects with three types of genotypes in cases and controls are shown in Table 2, where (r, r 1 , r 2 ) is the number of three genotypes in cases and (s, s 1 , s 2 ) is the number of genotypes in controls.From the table, we find three AREs of E p , E c and E b are higher than 75%, sometimes it can reach 97%.For example, for SNP rs17157903, the AREs of, and are 0.8255, 0.8453 and 0.7642, respectively.It shows that Z MERT is a robust test.  (1(1,  0 )/( 0 )), ̃  = 1 − Φ(  (1) (1)).