Two-sided Alternatives Research Articles

Modifying the exact test for a binomial proportion and comparisons with other approaches

In this note we provide a simple continuity and tail-corrected approach to the standard exact test for a single binomial proportion commonly used in practice. We redefine the p-value for the two-sided alternative by noting the skewed distribution of the sample proportion under the null hypothesis. We illustrate that for both one and two-sided alternatives the coverage probabilities of the new methodology approaches more closely the desired type I error α and thus recommend these modifications to the applied statistician for consideration.

Tests for a Simple Tree Order Restriction with Application to Dose–Response Studies

Summary We propose ‘Dunnett-type’ test procedures to test for simple tree order restrictions on the means of p independent normal populations. The new tests are based on the estimation procedures that were introduced by Hwang and Peddada and later by Dunbar, Conaway and Peddada. The procedures proposed are also extended to test for ‘two-sided’ simple tree order restrictions. For non-normal data, nonparametric versions based on ranked data are also suggested. Using computer simulations, we compare the proposed test procedures with some existing test procedures in terms of size and power. Our simulation study suggests that the procedures compete well with the existing procedures for both one-sided and two-sided simple tree alternatives. In some instances, especially in the case of two-sided alternatives or for non-normally distributed data, the gains in power due to the procedures proposed can be substantial.

More on the inadmissibility of step-up

Cohen and Sackrowitz [Characterization of Bayes procedures for multiple endpoint problems and inadmissibility of the step-up procedure, Ann. Statist. 33 (2005) 145–158] proved that the step-up multiple testing procedure is inadmissible for a multivariate normal model with unknown mean vector and known intraclass covariance matrix. The hypotheses tested are each mean is zero vs. each mean is positive. The risk function is a 2 × 1 vector where one component is average size and the other component is one minus average power. In this paper, we extend the inadmissibility result to several different models, to two-sided alternatives, and to other risk functions. The models include one-parameter exponential families, independent t-variables, independent χ 2 -variables, t-tests arising from the analysis of variance, and t-tests arising from testing treatments against a control. The additional risk functions are linear combinations where one component is the false discovery rate (FDR).

Optimality of the 2-CUSUM drift equalizer rules for detecting two-sided alternatives in the Brownian motion model

This work employs the Brownian motion model in which observations are taken sequentially. The objective is to detect a two-sided change in the constant drift by means of a stopping rule. As a performance measure, an extended Lorden criterion is used. The goal is to minimize the worst-case detection delay subject to a constraint in the frequency of false alarms. In a companion paper, attention is drawn to a first category of 2-CUSUM rules for which the harmonic mean rule holds. It is further seen that a special class of 2-CUSUM stopping rules within this category, called drift equalizer rules, perform strictly better than non-equalizer rules, according to this specific performance measure.

Optimality of the 2-CUSUM drift equalizer rules for detecting two-sided alternatives in the Brownian motion model

This work employs the Brownian motion model in which observations are taken sequentially. The objective is to detect a two-sided change in the constant drift by means of a stopping rule. As a performance measure, an extended Lorden criterion is used. The goal is to minimize the worst-case detection delay subject to a constraint in the frequency of false alarms. In a companion paper, attention is drawn to a first category of 2-CUSUM rules for which the harmonic mean rule holds. It is further seen that a special class of 2-CUSUM stopping rules within this category, called drift equalizer rules, perform strictly better than non-equalizer rules, according to this specific performance measure.

Testing hypotheses about the equality of several risk measure values with applications in insurance

This paper explores statistical tests about the equality of risk measure values obtained using a distortion-based risk measure. We consider both the case in which the risk measure value is specified in the null hypothesis and the case in which it is not. In the former case, one- and two-sided alternatives are considered, and in the latter case, ordered and unordered alternatives are considered. Asymptotically most powerful tests are obtained, and asymptotic distributions of the test statistics are found using results about the asymptotic distributions of the risk measure values. Finally, we consider a numerical example and conclude the paper with notes on when the results of the paper could, or could not, be safely used.

Aprepitant has little inductive effect on CYP3A4 activity when coadministered with dexamethasone

Background Aprepitant (APR) is a neurokinin-1 receptor antagonist approved, in combination with a corticosteroid and a 5-hydroxytryptamine3 receptor antagonist, for the prevention of chemotherapy-induced nausea and vomiting. Corticosteroids and APR both induce CYP3A4 activity and thus this study was conducted to determine if coadministration of APR and a corticosteroid results in significant CYP3A4 induction in vivo. Methods This double-blind, 2-period, crossover study was conducted in 12 healthy subjects to assess the potential inductive effect of a triple-therapy antiemetic regimen versus a dual antiemetic regimen on CYP3A4 activity. Single doses of oral (2 mg) and IV (1 mg) stable isotope (13C515N1) labeled midazolam (MDZ) were simultaneously administered on Days -1 (baseline), 6, 8, 15, and 22 as probes to independently assess the effects on systemic vs systemic+ first pass CYP3A4 activity. (See Table) Results The fold-changes from baseline in oral and intravenous MDZ AUC0-∞ were used as the primary assessments of CYP3A4 activity and are summarized in the table below. The difference between the dual vs triple regimen on days 8, 15, and 22 was primarily due to slight CYP3A4 inhibition by the dual regimen. The mean fold-changes from baseline following the triple antiemetic regimen were all close to 1.0. Conclusion APR, when used in combination with DEX as part of a triple antiemetic regimen, is unlikely to have significant inductive effects on CYP3A4 activity. Clinical Pharmacology & Therapeutics (2005) 77, P73–P73; doi: 10.1016/j.clpt.2004.12.169 Regimen APR Ondansetron (OND) Dexamethasone (DEX) Triple1 D1: 125 mg PO D1: 32 mg IV D1: 12 mg PO D2–3: 80 mg PO D2–4: 8 mg PO Dual D1: 32 mg IV D1: 20 mg PO D2–4: 16 mg PO 1 Doses of DEX were reduced to achieve comparable exposure to dual regimen. MDZ AUC0–∞ Geometric Mean Fold Change† from Baseline†† Ratio of Geometric Mean Fold Change†† (Triple/Double) with 90% CI1 Day TripleRegimen DualRegimen P-value†† Oral MDZ 6 0.84 0.84 1.00 (0.80, 1.25) >0.250 8 0.91 1.45 0.63 (0.49, 0.78) <0.010 15 0.93 1.40 0.66 (0.53, 0.83) <0.010 22 0.83 1.18 0.70 (0.56, 0.88) <0.010 IV MDZ 6 0.90 1.02 0.88 (0.74, 1.03) 0.184 8 0.93 1.25 0.74 (0.63, 0.87) <0.010 15 0.90 1.14 0.79 (0.67, 0.93) 0.017 22 0.94 1.07 0.88 (0.75, 1.04) 0.198 CI= Confidence interval. † Geometric means, CIs, ratios, and fold changes back-transformed from least-squares means from ANOVA, performed on natural log-transformed values †† Tests null hypothesis of no between-treatment difference versus two-sided alternative

Tests for comparing mark-specific hazards and cumulative incidence functions.

It is of interest in some applications to determine whether there is a relationship between a hazard rate function (or a cumulative incidence function) and a mark variable which is only observed at uncensored failure times. We develop nonparametric tests for this problem when the mark variable is continuous. Tests are developed for the null hypothesis that the mark-specific hazard rate is independent of the mark versus ordered and two-sided alternatives expressed in terms of mark-specific hazard functions and mark-specific cumulative incidence functions. The test statistics are based on functionals of a bivariate test process equal to a weighted average of differences between a Nelson-Aalen-type estimator of the mark-specific cumulative hazard function and a nonparametric estimator of this function under the null hypothesis. The weight function in the test process can be chosen so that the test statistics are asymptotically distribution-free. Asymptotically correct critical values are obtained through a simple simulation procedure. The testing procedures are shown to perform well in numerical studies, and are illustrated with an AIDS clinical trial example. Specifically, the tests are used to assess if the instantaneous or absolute risk of treatment failure depends on the amount of accumulation of drug resistance mutations in a subject's HIV virus. This assessment helps guide development of anti-HIV therapies that surmount the problem of drug resistance.

Some notes about nonparametric tests for the equality of two populations

A new nonparametric test for the equality of two populations is proposed. The test is a generalization of the Girone-Cifarelli test (see Girone 1964, and Cifarelli 1974, 1975) which has been shown to be occasionally more powerful than other nonparametric tests, such as the Kolmogorov-Smirnov test. The test is based on a measure of dissimilarity between the sequences of ranks corresponding to the samples drawn from the two populations. The test can be adapted to verify the hypothesis of equality against one-sided and two-sided alternatives. Exact and asymptotic critical values of the test are provided. The asymptotic distribution of the test-statistic under H0 shows an interesting link with Brownian motion in [0,1].

Hypothesis testing with a restricted parameter space

This paper considers hypothesis tests when the parameter space is restricted under the alternative hypothesis. Multivariate one-sided tests are a leading example. Optimal tests, called directed tests, are derived using a weighted average power criterion. Exact results are established first for Gaussian linear regression models with known variance. Asymptotic analogues are then established for dynamic nonlinear models. Simulation is used to compare the tests discussed in the paper. The D- W ∞ directed test is found to perform best in an overall sense for multivariate one-sided alternatives with the likelihood ratio test being a close second. The D- W ∞ and likelihood ratio tests are found to perform best for mixed one- and two-sided alternatives.

Hypothesis Estimates and Acceptability Profiles for 2 × 2 Contingency Tables

By estimating, rather than testing, hypotheses regarding the degree of dependence between the factors in 2 × 2 tables, the technical difficulties associated with small sample sizes are avoided. The estimators proposed here attempt to estimate 1 when the alternative hypothesis is true and 0 when the null hypothesis is true, subject to a bound on the squared error loss under the hypothesis. Such estimators provide guarded weights of evidence for the alternative hypothesis. Guarded weights of evidence based on the likelihood ratio are compared with those based on the p value or mid-p value, and they are shown to have lower risk functions except when the alternative is far from the hypothesis. For the case of two independent binomial distributions, it is shown that the conditional likelihood ratio estimator for the hypothesis of homogeneity against the two-sided alternative has a smaller unconditional risk than the unconditional likelihood ratio estimator, except when the binomial probabilities are far apart. Inversion of a family of guarded weights of evidence leads to acceptability profiles. These profiles provide more information than traditional confidence intervals regarding the unknown parameter. Two-sided profiles are found for the degree of dependence as measured by the odds ratio and log-odds ratio, and one-sided profiles are found for Yule's Q.

Weighing the evidence for hypotheses with small samples of right-censored exponential data.

The p-value evidence for an alternative to a null hypothesis regarding the mean lifetime can be unreliable if based on asymptotic approximations when there is only a small sample of right-censored exponential data. However, a guarded weight of evidence for the alternative can always be obtained without approximation, no matter how small the sample, and has some other advantages over p-values. Weights of evidence are defined as estimators of 0 when the null hypothesis is true and 1 when the alternative is true, and they are judged on the basis of the ensuing risks, where risk is mean squared error of estimation. The evidence is guarded in that a pre-assigned bound is placed on the risk under the hypothesis. Practical suggestions are given for choosing the bound and for interpreting the magnitude of the weight of evidence. Acceptability profiles are obtained by inversion of a family of guarded weights of evidence for two-sided alternatives to point hypotheses, just as confidence intervals are obtained from tests; these profiles are arguably more informative than confidence intervals, and are easily determined for any level and any sample size, however small. They can help understand the effects of different amounts of censoring. They are found for several small size data sets, including a sample of size 12 for post-operative cancer patients. Both singly Type I and Type II censored examples are included. An examination of the risk functions of these guarded weights of evidence suggests that if the censoring time is of the same magnitude as the mean lifetime, or larger, then the risks in using a guarded weight of evidence based on a likelihood ratio are not much larger than they would be if the parameter were known.

Optimal Tests for Parameter Instability in the Generalized Method of Moments Framework

This paper presents optimal tests for parameter instability in the GMM framework. The new tests include tests that are optimal for both one-sided and two-sided alternatives. One of the optimal tests for two-sided alternatives is the GMM generalization of the test presented in Andrews and Ploberger (1994) for the likelihood framework. The new tests include a class of optimal tests that direct the test's power to specific locations in the sample. One of these optimal tests has the attractive feature of a normal distribution under the null hypothesis.

Adaptive tests for the c-sample location problem - the case of two-sided alternatives

In the c-sample location problem the classical F-test is the appropriate test under the model of normality. For nonnormal data, however, there are rank tests which have higher power than the F-test, e.g, the Gastwirth test for symmetric distributions with short tails or the Hogg-Fisher-Randles test for asymmetric distributions. But usually the practicing statistician has no information about the underlying distribution. Therefore, an adaptive test should be applied which takes the given data set into account. Two versions of such an adaptive test are proposed including a new test for distributions with long tails. These adaptive tests are compared with each of the single rank tests in the adaptive scheme and also with the classical F-test. It is shown via Monte Carlo simulation that the adaptive tests behave well over a broad class of distributions.

Testing Homogeneity of Uniform Scale Distributions against Two-Sided and One-Sided Alternatives

Abstract Consider the model where Xij, i = 1, 2, …, k; j = 1, …, n i are independent uniform random variables with scale parameter θ i > 0. Test H 0: θl = ··· = θ k versus H 2: not H 0. Also consider the alternative H 1: θl ≤ ··· ≤ θ k . For H 0 versus H 2 and n i = n, we obtain a complete class of constant-size permutation invariant tests and show that each test in the class is unbiased. The likelihood ratio test and others are in this class. For H 0 versus H 1 we obtain a complete class of constant-size tests based on a set of variables that is a transformation of the sufficient statistics. Again we show that each test in the class is unbiased. The likelihood ratio test is in this class. We derive the exact and asymptotic distribution of the likelihood ratio test. All results also hold for testing homogeneity of location parameters of exponential distributions. This latter case is of considerable practice importance in reliability.

Testing Homogeneity of Uniform Scale Distributions Against Two-Sided and One-Sided Alternatives

Testing Homogeneity of Uniform Scale Distributions Against Two-Sided and One-Sided Alternatives

A Score Test against One-Sided Alternatives

Abstract A score-type statistic, T s , is introduced for testing H: ψ = 0 against K: ψ ≥ 0 and more general one-sided hypotheses when nuisance parameters may be present; ψ is a vector parameter. The main advantages of T s , are that it requires estimation of the model only under the null hypothesis and that, it is asymptotically equivalent to the likelihood ratio statistic; these are precisely the reasons for the popularity of the score tests for testing against two-sided alternatives. In this sense, T s preserves the main attractive features of the classical two-sided score test. The theoretical results are presented in a general framework where the likelihood-based score function is replaced by an estimating function so that the test is applicable even if the exact population distribution is unknown. Computation of T s , is simplified by the fact that it can be computed easily once the corresponding two-sided statistic has been computed. The relevance and simplicity of T s are illustrated by discussing ...

An example of a two-sided Wilcoxon signed rank test which is not unbiased

Although the theory of rank tests is rather complete in the one-sided case, it was not even known in 1959, whether the Wilcoxon two-sample test and other similar tests are unbiased against the two-sided alternatives (Lehmann (1959,Testing Statistical Hypotheses, p. 240, Wiley, New York)). A partial answer to this question was given by Sugiura in 1965, who found, that the test named above may be biased (Sugiura (1965,Ann. Inst. Statist. Math.,17, 261–263)). According to Lehmann (1986,Testing Statistical Hypotheses, 2nd ed., pp. 322–324, Wiley, New York) it seems to be still open, whether the same is true for the WILCOXON one-sample test, which is also known as WILCOXON signed rank test. This will be shown in the present paper.

The Large Sample Correspondence between Classical Hypothesis Tests and Bayesian Posterior Odds Tests

This paper establishes a correspondence in large samples between classical hypothesis tests and Bayesian posterior odds tests for models without trends. More specifically, tests of point null hypotheses and one- or two-sided alternatives are considered (where nuisance parameters may be present under both hypotheses). It is shown that, for certain priors, the Bayesian posterior odds test is equivalent in large samples to classical Wald, Lagrange multiplier, and likelihood ratio tests for some significance level and vice versa. The priors considered under the alternative hypothesis are taken to shrink to the null hypothesis at rate n[superscript -1/2] as the sample size n increases. Copyright 1994 by The Econometric Society.

Two-Sided Tests and One-Sided Confidence Bounds

Based on the duality between tests and confidence sets we introduce a new method to derive one-sided confidence bounds following the rejection of a null hypothesis with two-sided alternatives. This method imputes that the experimenter is only interested in confidence bounds if the null hypothesis is rejected. Furthermore, we suppose that he is only interested in the direction and a lower confidence bound concerning the distance of the true parameter value to the parameter values in the null hypothesis. If the null hypothesis is rejected, the new one-sided confidence bounds are always not worse than the corresponding bounds of the two-sided confidence interval approach. If the true parameter is far away from the null hypothesis, the new bounds tend to be nearly equal to the corresponding one-sided confidence bounds with full confidence level $1 - \alpha$. The new method will be studied and illustrated in more detail in one-parameter exponential families and location families with unimodal Lebesgue densities, and, as an example where conditional tests are available, we consider the comparison of two Poission distributions. In case of the normal distribution with unknown variance we propose among others a modification of a procedure of Hodges and Lehmann. Here it may be surprising that there exist situations where the new method yields confidence bounds exactly matching the classical one-sided confidence bounds.