Two-sided Alternatives Research Articles

On Tests Against One-Sided Hypotheses in Some Generalized Linear Models

One-sided hypotheses arise naturally in many situations. When testing against such hypotheses, it is desirable to take the available one-sided information into account, rather than simply applying a two-sided test. What we expect to gain by applying a one-sided test instead of a two-sided test is an increase in the power of the test. We consider various tests of one-sided hypotheses in a class of models that includes generalized linear and Cox regression models. The tests are likelihood ratio, Wald, score, generalized distance, and a Pearson chi-square. It is shown that these test statistics are asymptomatically equivalent in terms of local power; this is a generalization of the well-known corresponding result for two-sided alternatives. Two examples are also discussed. They are on (1) testing for interaction in binomial response models, and (2) comparison of treatments with ordinal categorical responses.

The Prediction Problem as the Dual Form of the Two-Sample Problem with Applications to the Poisson and the Binomial Distribution

Abstract Let X 1 and X 2 be independent random variables with the same distribution function F v, with the parameter v being unknown. A prediction interval for X 2, given an observation of X 1, is constructed on the basis of the acceptance region of the two-sample problem with the hypothesis H 0 that the parameters v1 and v2 of the distributions of X 1 and X 2 are the same against the one- or two-sided alternative. Although this simple idea can be found in the statistical literature, it deserves more attention in statistical textbooks. Applications are given to the Poisson and the binomial distribution. Besides the description of different procedures to compute exact prediction limits in the Poisson and binomial case, approximation formulas are given with an error term o(1) with corresponding reliable bounds for the approximation error, that seem to be new.

A note on the application of simple linear regression methods for trend detection at multiple sites and visits.

In comparing running median, tolerance, cusum, and regression methods for trend detection over a small number of visits, Yang et al. found that application of multiple Z-tests on the basis of a simple linear regression for each site separately was the most efficient for detection of trends at several sites simultaneously. Because the use of multiple Z-tests completely ignores the covariance among measurements taken from different sites, to improve the power we propose a global chi 2-test. Assuming the covariance matrix known, we have found that the proposed chi 2-test procedure is more powerful than multiple Z-tests for two-sided alternatives when both the correlation among measurements and the number of sites are small. We also have found that the former procedure can have power uniformly larger than the latter when ratios of slopes to standard deviations of measurements at different sites vary and the number of sites is large. In fact, in the latter situation, the proposed global chi 2-test procedure, usually used only for two-sided alternatives, can even have power larger than that of multiple Z-tests for one-sided alternatives. In the situation where the ratios of slopes to standard deviations of measurements are all equal, however, the proposed multivariate approach based on the chi 2-test distribution is the least efficient, especially when the number of sites and the correlation are moderate or large. Finally, to account for the effect of multiple tests over a series of visits on the overall alpha-level, on the basis of Monte Carlo simulations, we compute critical values for sequential use of the proposed multivariate test procedure.

Estimation of the slope in linear non-normal structural relationships

This paper studies the sensitivity to nonnormality of the normal-theory test for the null hypothesis that the slope is a specific value against a two-sided alternative. Edgeworth expansion and thus the asymptotic variance for the normal-theory maximum likelihood estimator of the slope are derived.

Optimal Designs for Comparing the Variances of Several Treatments with That of a Standard Treatment

Optimal designs and associated tables of probability points for comparing the variances of p experimental treatments with that of a standard treatment under the assumption of normality are presented. One-sided and two-sided alternatives are considered. The optimality criteria are those of maximizing the probability of declaring an experimental treatment variance to be smaller than, larger than, or different from that of the standard for a fixed total sample size and a fixed experimentwise error rate. Both small-sample and asymptotic results are given. It is shown that the proportion of observations assigned to the standard treatment for comparing variances in the limiting case is the same as that for comparing means. These optimal proportions are in some cases quite different for small samples.

Small sample properties of a sign test carried out under a Curtailed Inverse Sampling Scheme

The standard procedure of carrying out a sign test of location by attribute inspection under a Curtailed Inverse Sampling Scheme (CISS), for detecting both one-and two-sided alternatives, and also its error controlling properties, has been discussed. The nature of the Average Amount of Inspection (AAI) function of the procedure in small sample situations have been examined under four different symmetric models-normal, Laplace, Cauchy and logistic. The small sample efficiency of the sign test carried out under CISS has been compared with that of the UMP test based on the sample mean, under normal model.

Graphical comparison of means with a standard in the one-way layout

Some simple test procedures are considered for comparing several group means with a standard value when the data are in a one-way layout. The underlying distributions are assumed to be normal with possibly unequal variances. The tests are based on a union-intersection formulation and can be applied in a form similar to a Shewhart control chart. Both two-sided and one-sided alternatives are considered. The power of the tests can be obtained from tables of a non-central t distribution. Implementation of the tests is illustrated with a numerical example. The tests help identify any group means different from the standard and might lead to a decision about rejecting the null hypothesis before all the group means are observed. The resulting savings in time and resources might be valuable in applications where the number of groups is large and the cost of acquiring data is high. For situations where the normality assumption is untenable, a non-parametric procedure, based on one-sample sign tests is considered.

The double triangular test: a sequential test for the two-sided alternative with early stopping under the null hypothesis

The double triangular test is a sequential procedure for testing the null hypothesis that a parameter is zero against the two-sided alternative that it is non-zero The test is devised to produce small samples if the parameter is large in magnitude, whether positive or negative, and also to produce small samples if the parameter is close to zero. In intermediate situations larger samples are likely to be necessary. In this paper some new results concerning the theory of the double triangular test are presented. These are then used to explore in detail its properties, and how data collected using the test may be analysed. In particular, significance levels and point and interval estimates are considered.

A sequential two-sample test developed by stochastic simulation

Only a few sequential tests have been developed for the two-sample design common in clinical trials. By stochastic simulation, we have developed a new sequential test based on the Wilcoxon–Mann–Whitney two-sample test for fixed sample size. The development is based on a shift model of normal distributions. The resulting test can be used against both one-sided and two-sided alternatives. Formulas for calculating the parameters in the equations for the boundaries make it possible to choose significance level and power against specified alternatives. Simulation investigations of this sequential test show that it requires substantially fewer patients to reach a conclusion than the corresponding test for fixed sample size. Monte Carlo investigations of its robustness properties show that it is robust and approximately distribution free.

Distribution-Free Tests for the Changepoint Problem

SYNOPTIC ABSTRACTLet X1, X2, …, Xr+1, …, Xn be independent, continuous random variables such that Xi, i = l, …, r, has an unknown distribution function F(x), and Xj, j = r + 1, …, n, has distribution function F(x - Θ), with Θ unknown (−∞ < Θ < ∞). The integer r, which is called the changepoint, is also taken to be unknown. The hypothesis to be tested is H0 : Θ = 0 (i.e., no change) vs. either one- or two-sided alternatives. We consider distribution-free tests for this problem based on Mann-Whitney-Wilcoxon statistics, and study their large sample properties. We also report on some Monte Carlo power comparisons involving these procedures and several parametric competitors.

Exact Tests for 2 × 2 Contingency Tables

Abstract Fisher's exact test, difference in proportions, log odds ratio, Pearson's chi-squared, and likelihood ratio are compared as test statistics for testing independence of two dichotomous factors when the associated p values are computed by using the conditional distribution given the marginals. The statistics listed above that can be used for a one-sided alternative give identical p values. For a two-sided alternative, many of the above statistics lead to different p values. The p values are shown to differ only by which tables in the opposite tail from the observed table are considered more extreme than the observed table.

The flume design — a methodology for evaluating material fluxes between a vegetated salt marsh and the adjacent tidal creek

An experimental flume is described which can be used as a tool to assess whether a vegetated marsh surface is a source or sink for nutrients via tidal inundation. An initial calibration study (two tidal cycles) was conducted to determine the optimum sampling design and aid in model development for flux calculations. A statistical analysis of the data showed a negligible concentration difference as a function of water depth for most of the constituents analyzed. This coupled with the low tidal velocities over the marsh surface (<1.5cm/s) suggested that a volumetric model was adequate for calculations of instantaneous discharge and nutrient flux through any station perpendicular to tidal flow. The resultant instantaneous mass flux calculations showed that water discharge was one of the dominant factors controlling the movement of material. A sine-cosine statistical model utilizing the main tidal periodicities was designed to: (1) model the instantaneous fluxes, (2) calculate the average net flux of suspended and dissolved materials, and (3) test the hypothesis that the average net flux equals zero versus a two-sided alternative using a standard regression t-test.

A nonparametric test for detecting changes in location

Procedure for the changepoint problem based on Mann-Whitney-Wilcoxon statistics is studied in Schechtman and Wolfe (1981). In this paper we give tables for the null distributions of the statistics for the one-sided and two-sided alternatives. We also report on some Monte Carlo power comparisons involving another nonparametric competitor, proposed by Pettitt (1979).

Tests for the Independence Between Two Seemingly Unrelated Regression Equations

When the error terms in two different regression equations are correlated, Zellner proposed an alternative estimator for the coefficients of each equation based on an estimated covariance matrix between the two error terms. However, since an estimated covariance matrix is used, the OLSE seems better than Zellner's estimator when the correlation of the two equations is close enough to zero. This paper considers the problem of testing the independence between two regression equations and derives a locally best invariant test for a one-sided alternative hypothesis and a locally best unbiased and invariant test for a two-sided alternative.

Unbiasedness of two-sided nonparametric tests in the two-sample problem

For two independent random samples from absolutly continuous distributions Fand G, the problem of testing H0:F=Gagainst H0:F=G and other two-sided alternatives is considered. Then, the rank tests, that are invariant with respect to the remuneration of the samples of size two? are investigated, A description of the problem in geo metric terms' enables us to find the minimal essentially complete class as well as the necessary and sufficient condition of unbiasedness. As a result, we note that some well known tests are not unbiased.

A Test for Detecting Outlying Cells in the Multinomial Distribution and Two-Way Contingency Tables

Abstract A test based on the maximum adjusted residual from multinomial models, namely, the M test, is proposed. Sharp bounds on its critical values are provided for the multinomial case, while for the two-way table we present conservative critical values. The new test can be used to test null hypotheses against one- or two-sided alternatives and to detect outliers simultaneously with the rejection of the null hypothesis. Under certain alternatives, the M test is asymptotically more powerful than the chi-squared test.

Group sequential methods for clinical trials with a one-sided hypothesis

Pocock (1977) has recently proposed group sequential methods for sequential analysis of clinical trials. However, these methods are intended to test a null hypothesis against a two-sided alternative, and many trials of a new therapy against a standard have one-sided alternatives. Continuation of the trial to determine whether the new therapy is actually inferior to the standard therapy may be inappropriate or even unethical. In the present paper, three possible modifications of the group sequential method for sequential monitoring of clinical trials testing a one-sided hypothesis are considered.

Some complex variable transformations and exact power comparisons of two-sided tests of equality of two Hermitian covariance matrices

Abstract Power studies of tests of equality of covariance matrices of two p -variate complex normal populations σ 1 = σ 2 against two-sided alternatives have been made based on the following five criteria: (1) Roy's largest root, (2) Hotelling's trace, (4) Wilks' criterion and (5) Roy's largest and smallest roots. Some theorems on transformations and Jacobians in the two-sample complex Gaussian case have been proved in order to obtain a general theorem for establishing the local unbiasedness conditions connecting the two critical values for tests (1)–(5). Extensive unbiased power tabulations have been made for p =2, for various values of n 1 , n 2 , λ 1 and λ 2 where n 1 is the df of the SP matrix from the i th sample and λ 1 is the i th latent root of σ 1 σ -1 2 ( i =1, 2). Equal tail areas approach has also been used further to compute powers of tests (1)–(4) for p =2 for studying the bias and facilitating comparisons with powers in the unbiased case. The inferences have been found similar to those in the real case. (Chu and Pillai, Ann. Inst. Statist. Math. 31.

Power comparisons of two-sided tests of equality of two covariance matrices based on six criteria

Power studies of tests of equality of covariance matrices of twop-variate normal populations Σ1=Σ2 against two-sided alternatives have been made based on the following six criteria: 1) Roy's largest root, 2) Hotelling's trace, 3) Pillai's trace, 4) Wilks' criterion, 5) Roy's largest-smallest roots and 6) modified likelihood ratio. A general theorem has been proved establishing the local unbiasedness conditions connecting the two critical values for tests 1) to 5). Extensive unbiased power tabulations have been made forp=2, for various values ofn 1,n 2, λ1 and λ2 wheren i is the df of the SP matrix from theith sample and λ i is theith latent root of Σ1Σ 2 −1 (i=1,2). Further, comparisons of powers of tests 1) to 5) have been made with those of the modified likelihood ratio after obtaining the exact distribution of the latter forn 2=2n 1 andp=2. Equal tail areas approach has also been used further to compute powers of tests 1) to 4) forp=2 for studying the bias. Again, a separate study has been made to compare the powers of the largest-smallest roots test with its three biased approximate approaches as well as the largest root. Since the largest root test was observed to have some advantage over the others, critical values were also obtained for this test in the unbiased as well as equal tail areas case forp=3.

Comparison of Two Sequential Tests for Two-Sided Alternatives

Summary Wald's method based on weight functions provides sequential tests for two-sided alternatives. In this paper, this method is compared with a test due to Sobel and Wald for three decision problems used for two-sided alternatives. The boundaries of acceptance and rejection of the tests are compared, and it is shown that for the usual choice of stopping bounds, the region of continued observation for Wald's test is included in that of Sobel and Wald's test. The difference between the region of rejection and acceptance of the two tests is also discussed.