The score statistic Z(θ) and the maximin efficient robust test statistic Z_{MERT} are commonly used in genetic association study, but according to our knowledge there is no formal comparison of them.
In this report, we compare the asymptotic behavior of Z(θ) and Z_{MERT}, by computing their Asymptotic Relative Efficiencies (AREs) relative to each other. Four commonly used ARE measures, the Pitman ARE, Chernoff ARE, Hodges-Lehmann ARE and the Bahadur ARE are considered. Some modifications of these methods are made to simplify the computations. We found that the Chernoff, Hodges-Lehmann and Bahadur AREs are suitable for our setting.
Based on our study, the efficiencies of the two test statistic varies for different criterion used, and for different parameter values under the same criterion, so each test has its advantages and dis-advantages according to the criterion used and the parameters involved, which are described in the context. Numerical examples are given to illustrate the use of the two statistics in genetic association study.
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 [1C.M. Sitlani, and B. McKnight, "Relative efficiency of trend tests with misspecified genetic models in stratified analyses of case-control or cohort data", Hum. Hered., vol. 71, no. 4, pp. 246-255.[http://dx.doi.org/10.1159/000328858] [PMID: 21811075] ] 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 [2G. Zheng, "Some statistical properties of efficiency robust tests with applications to genetic association studies", Scand. J. Stat., pp. 762-774.[http://dx.doi.org/10.1111/sjos.12060] ], 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, R^{2} 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 H_{1} (λ_{1}, λ_{2}) \ {(1, 1)}, where has two edges with known slopes θ_{ 0} and θ_{1}, and the null point (1, 1) is on the boundary of . We assume - ∞ < θ_{ 0} < θ_{1} <∞ and the endpoints θ_{ 0} and θ_{1} satisfy some constraints as specified in ZLY. If θ_{1} = ∞ which corresponds to a vertical edge, we can switch λ_{1} and λ_{2} and define new (θ_{1}, θ_{2}) so - ∞ < θ_{ 0} < θ_{1} <∞ is satisfied by the new (θ_{1}, θ_{2}). For example, we can write λ_{1} = 1 + (λ_{2} - 1)/ λ_{1}^{*} (λ_{2}-1) and λ_{1} = 1 + (λ_{2} - 1)/ θ_{ 0} = 1 + θ_{ 0}^{*} (λ_{2} - 1) where - ∞ < θ_{ 0} < θ_{1} <∞.
Assume θ_{ 0} and θ_{1} are known from the problem of interest and/or scientific knowledge. Given λ_{1} = λ ≥ 1, λ_{2} 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 x_{1} = λ, x_{2} = 1 - θ + θλ and x_{3} = η. Denote l_{n,u} = ∂l_{n}/ ∂x_{u} for, u = 1,2,3, l_{n, uv} = ∂^{2}l_{n}/∂x_{u}∂x_{v} for u = 1,2 and, v = 1.2.3, and l_{n.33} = ∂^{2}l_{n}/∂x_{3}∂x^{T}_{3}. Assume and v = 3. Denote L_{vu} (η) = E_{Hn}l_{lvu} (1.1, η).
Suppose we have a family of asymptotically normally distributed tests , where under H_{1} 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 [3J. Gastwirth, "On robust procedures", J. Am. Stat. Assoc., pp. 929-948.[http://dx.doi.org/10.1080/01621459.1966.10482185] , 4J. Gastwirth, "The use of maximin efficiency robust tests in combining contingency tables and survival analysis", J. Am. Stat. Assoc., pp. 380-384.[http://dx.doi.org/10.1080/01621459.1985.10478127] ])
(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 [3J. Gastwirth, "On robust procedures", J. Am. Stat. Assoc., pp. 929-948.[http://dx.doi.org/10.1080/01621459.1966.10482185] , 4J. Gastwirth, "The use of maximin efficiency robust tests in combining contingency tables and survival analysis", J. Am. Stat. Assoc., pp. 380-384.[http://dx.doi.org/10.1080/01621459.1985.10478127] ]; Birnbaum and Laska [5A. Birnbaum, "Efficiency robust 2-sample rank tests", J. Am. Stat. Assoc., pp. 1241-1251.[http://dx.doi.org/10.1080/01621459.1967.10500929] ],) 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 [3J. Gastwirth, "On robust procedures", J. Am. Stat. Assoc., pp. 929-948.[http://dx.doi.org/10.1080/01621459.1966.10482185] ]), but its computation needs quadratic programming methods (Rosen [6J. Rosen, The gradient projection method for non-linear programming Part I: Linear constraints., SIAM J., pp. 181-217.]). However, when there is an extreme pair (Z(θ_{i}), Z(θ _{i})) in T_{ 0}i.e. p_{θi, θi} = is MERT for if and only if (Gastwirth [7J. Gastwirth, "On robust rank test", Nonparametric Techniques in Statistical Inference, Cambridge University Press: London, .]).
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 [3J. Gastwirth, "On robust procedures", J. Am. Stat. Assoc., pp. 929-948.[http://dx.doi.org/10.1080/01621459.1966.10482185] ]; Birnbaum and Laska [5A. Birnbaum, "Efficiency robust 2-sample rank tests", J. Am. Stat. Assoc., pp. 1241-1251.[http://dx.doi.org/10.1080/01621459.1967.10500929] ],) and later extended to a family of asymptotically normally distributed tests (Gastwirth [4J. Gastwirth, "The use of maximin efficiency robust tests in combining contingency tables and survival analysis", J. Am. Stat. Assoc., pp. 380-384.[http://dx.doi.org/10.1080/01621459.1985.10478127] ]).
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 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;
(5) |
where “~” means asymptotically equivalent, in the above is replaced by it is approximated by n^{-1}l_{n, vu} (1.1, η).
Denote . For a vector v (v_{1}, v_{2}, v_{3})^{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 H_{1}. For fixed, (λ, θ) let;
(6) |
Under H_{1}, 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) [8R. Serfling, Approximation Theorems of mathematical Statistics Wiley., Wiley, .[http://dx.doi.org/10.1002/9780470316481] ] and Nikitin (2011) [9Y. Nikitin, "Asymptotic relative efficiency in testing (version 5)", Stat Prob: The Encyclopedia Sponsored by Statistics and Probability Societies, .].
The calculation of the existing of versions of ARE is generally not easy, as in the examples (Serfling, 1980 [8R. Serfling, Approximation Theorems of mathematical Statistics Wiley., Wiley, .[http://dx.doi.org/10.1002/9780470316481] ]; Nikitin, 1995 [10Y. Nikitin, Asymptotic Efficiency of Nonparametric Tests., Cambridge University Press, .[http://dx.doi.org/10.1017/CBO9780511530081] ]; van der Varrt, 1998 [11A. van der Varrt, Asymptotic Statistics., Cambridge University Press, .[http://dx.doi.org/10.1017/CBO9780511802256] ]). 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) [12H. Wieand, "A condition under which the Pitman and Bahadur", Ann. Stat., pp. 1003-1011.[http://dx.doi.org/10.1214/aos/1176343600] ] established the correspondence between this kind of ARE and the limiting approximate Bahadur efficiency which is easy to compute.
The Bahadur (1960) [13R. Bahadur, "Stochastic comparison of tests", Ann. Math. Stat., pp. 276-295.[http://dx.doi.org/10.1214/aoms/1177705894] ] 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) [14A. Dembo, Large deviation techniques and applications 2^{nd}., Springer: New York, .[http://dx.doi.org/10.1007/978-1-4612-5320-4] ] and Deuschel and Strook (1989) [15J. Deuschel, Large deviations, Academic Press: Boston, .].
It is proved that under some regularity conditions the likelihood ratio statistic is asymptotically optimal in Bahadur sense (Bahadur, 1967 [16R. Bahadur, "Rates of convergence of estimates and test statistics", Ann. Math. Stat., pp. 303-324.[http://dx.doi.org/10.1214/aoms/1177698949] ]; Arcones, 2005 [17M. Arcones, "Bahadur efficiency of the likelihood ratio test", Math. Methods Stat., pp. 163-179.]). 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 [10Y. Nikitin, Asymptotic Efficiency of Nonparametric Tests., Cambridge University Press, .[http://dx.doi.org/10.1017/CBO9780511530081] ]).
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 [18W. Hoeffding, "Asymptotically optimal tests for multinomial distributions (with discussion)", Ann. Math. Stat., pp. 369-408.[http://dx.doi.org/10.1214/aoms/1177700150] ], based on the work of Sanov (1957) [19I. Sanov, On the probability of large deviations of random variables.Sel., Transl. Math. Statist. Prob, pp. 213-244.], is theoretically appealing, but ony applies to multinomial data; Rubin and Sethurman ARE (1965) [20H. Rubin, "Bayes risk efficiency", Sankhya, A., pp. 325-346.] is based on Bayes risk; others including Kallenberg ARE (1983) [21W. Kallenberg, "Intermediate efficiency, theory and examples", Ann. Stat., pp. 170-182.[http://dx.doi.org/10.1214/aos/1176346067] ], and the Borovkov-Mogulskii ARE (1993) [22A. Borovkov, "Large deviations and testing of statistical hypotheses", Sib. Adv. Math., .], 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_{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.
(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 [23E. Pitman, Lecture Notes on Nonparametric Statistical Inference, Columbia University: Mimeographed., 1949.]; Noether, 1950 [24G. Noether, "Asmptotic properties of the wald-wolfowitz test of randomness", Ann. Math. Stat., pp. 231-246.[http://dx.doi.org/10.1214/aoms/1177729841] ]). 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, n_{1} and n_{2} 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) [25C. Rao, "Criteria of estimation in large samples", Sankhya Ser. A, pp. 189-206.] 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_{0}.
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 [26C. van Eden, "The relationship between Pitman’s asymptotic relative efficiency of two tests and correlation coefficient between their test statistics", Ann. Math. Stat., pp. 1442-1451.[http://dx.doi.org/10.1214/aoms/1177703876] ]). 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, C_{1} > C_{2} 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, n_{1} and n_{2} 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 F_{1} = F_{2}, 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.
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 .
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} ≤ µ_{1}), (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) [27H. Chernoff, "A measure of asymptotic efficiency for tests of a hypothesis based on sums of observations", Ann. Math. Stat., pp. 493-507.[http://dx.doi.org/10.1214/aoms/1177729330] ] 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;
(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 [28W. Kallenberg, "Chernoff efficiency and deficiency", Ann. Stat., pp. 583-594.[http://dx.doi.org/10.1214/aos/1176345799] ]) Let for some
Then
In the case of simple null vs simple alternative, Kallenberg (1982) [28W. Kallenberg, "Chernoff efficiency and deficiency", Ann. Stat., pp. 583-594.[http://dx.doi.org/10.1214/aos/1176345799] ] 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) [10Y. Nikitin, Asymptotic Efficiency of Nonparametric Tests., Cambridge University Press, .[http://dx.doi.org/10.1017/CBO9780511530081] ].
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) |
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 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;
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) [29J. Hodges, "The efficiency of some nonparametric competitors of the t-test", Ann. Math. Stat., pp. 324-335.[http://dx.doi.org/10.1214/aoms/1177728261] ] 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;
(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} (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) [30Y. Zang, "Simple algorithms to calculate asymptotic Simple algorithms to calculate asymptotic", J. Stat. Softw., pp. 1-24.], 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 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 n_{1}^{-1} log L_{2,n2} = lim n_{1}^{-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 [13R. Bahadur, "Stochastic comparison of tests", Ann. Math. Stat., pp. 276-295.[http://dx.doi.org/10.1214/aoms/1177705894] ]). 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) [16R. Bahadur, "Rates of convergence of estimates and test statistics", Ann. Math. Stat., pp. 303-324.[http://dx.doi.org/10.1214/aoms/1177698949] ] 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 [17M. Arcones, "Bahadur efficiency of the likelihood ratio test", Math. Methods Stat., pp. 163-179.], 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_{ 0}vs 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,n}as;
(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;
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_{θo} 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 , last is
We use 6 reported SNPs associated with breast cancer 2 (Hunter et al. 2007 [31D.J. Hunter, P. Kraft, K.B. Jacobs, D.G. Cox, M. Yeager, S.E. Hankinson, S. Wacholder, Z. Wang, R. Welch, A. Hutchinson, J. Wang, K. Yu, N. Chatterjee, N. Orr, W.C. Willett, G.A. Colditz, R.G. Ziegler, C.D. Berg, S.S. Buys, C.A. McCarty, H.S. Feigelson, E.E. Calle, M.J. Thun, R.B. Hayes, M. Tucker, D.S. Gerhard, J.F. Fraumeni Jr, R.N. Hoover, G. Thomas, and S.J. Chanock, "A genome-wide association study identifies alleles in FGFR2 associated with risk of sporadic postmenopausal breast cancer", Nat. Genet., vol. 39, no. 7, pp. 870-874.[http://dx.doi.org/10.1038/ng2075] [PMID: 17529973] ]; Li et al., 2008 [32Q. Li, G. Zheng, Z. Li, and K. Yu, "Efficient approximation of P-value of the maximum of correlated tests, with applications to genome-wide association studies", Ann. Hum. Genet., vol. 72, no. Pt 3, pp. 397-406.[http://dx.doi.org/10.1111/j.1469-1809.2008.00437.x] [PMID: 18318785] ]) 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_{0}, r_{1}, r_{2}) is the number of three genotypes in cases and (s_{0}, 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.
Derivation of : From (P3), we have . Also, as in the proof in Serfling (1980 [8R. Serfling, Approximation Theorems of mathematical Statistics Wiley., Wiley, .[http://dx.doi.org/10.1002/9780470316481] ], p. 317-318), if and only if
Thus, for β_{i,n} (θ _{n}) → β, we must have;
Proof of Proposition 1: We use (4) to compute e_{p} (Z_{MERT,}Z(θ^{(0)})). By definition of Z(θ^{(0)})) and CLT we have , and by Theorem 3 in ZLY, Also Z(θ^{(0)})), and Z_{MERT})are jointly asymptotic normal with correlation . Thus the condition of (4) are satisfied, and it gives;
Proof of Proposition 2. (i) By assumption As in the proof of Theorem 4 in ZLY, we have that where the V_{i} = V_{i} (θ) ’s are i.i.d. with;
Under with and Under with and So we have
By example A in Serfling (1980 [8R. Serfling, Approximation Theorems of mathematical Statistics Wiley., Wiley, .[http://dx.doi.org/10.1002/9780470316481] ], p. 330), we have;
similar to that for (Z(θ^{(0)})).
(ii). We first compute . In this case, let be the weak limit of (Z(θ^{(0)})). Then,
Proof of Proposition 3. Since under , we have t_{n}(α)→ Ф^{-1} (1- α); and under is continuous on (- ∞, ∞), the distribution function of converges to uniformly Ф (.). Note µ(λ, θ) > 0, so for λ > 1 we have;
Let , using L’hopital’s rule twice, we get;
Similarly, under where The same way we get;
Proof of Proposition 4. i). In our case and when uniformly in S_{n}. From proof of Theorem 4 in ZLY, we have that for (a.s.). Now we compute, for
Let , and use L’Hopital’s rule,
Since , and by L’hopital’s rule, , so use L’Hopital’s rule on the above again,
Thus by Bahadur’s (1960) [13R. Bahadur, "Stochastic comparison of tests", Ann. Math. Stat., pp. 276-295.[http://dx.doi.org/10.1214/aoms/1177705894] ] Theorem,
is similarly computed;
Similarly, under , (a.s.), so
Not applicable.
The authors declare no conflict of interest, financial or otherwise.
Declared none.
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