Order-restricted Alternatives Research Articles

Order-restricted inference for means with missing values.

Missing values appear very often in many applications, but the problem of missing values has not received much attention in testing order-restricted alternatives. Under the missing at random (MAR) assumption, we impute the missing values nonparametrically using kernel regression. For data with imputation, the classical likelihood ratio test designed for testing the order-restricted means is no longer applicable since the likelihood does not exist. This article proposes a novel method for constructing test statistics for assessing means with an increasing order or a decreasing order based on jackknife empirical likelihood (JEL) ratio. It is shown that the JEL ratio statistic evaluated under the null hypothesis converges to a chi-bar-square distribution, whose weights depend on missing probabilities and nonparametric imputation. Simulation study shows that the proposed test performs well under various missing scenarios and is robust for normally and nonnormally distributed data. The proposed method is applied to an Alzheimer's disease neuroimaging initiative data set for finding a biomarker for the diagnosis of the Alzheimer's disease.

Tukey’s Contrast Test Versus Two-Sample Test in a Dose-Response Clinical Trial

Tukey, Ciminera, and Heyse (1985) proposed a linear trend test to establish a dose response; or if used sequentially, by eliminating the highest dose at each stage, to select the no-statistical-significance-of-trend (NOSTASOT) dose. The traditional alternatives have been to test the highest dose versus the control, ignoring contributions from the middle doses; or its sequential analog, to establish a minimum effective dose by testing first the highest dose versus the control, and if significant, testing the subsequent doses against the contrast until no significance is observed or doses are exhausted. In dose-response trials and/or in dose-finding trials, clinicians may believe a priori that the ordering of the responses is known, so the efficacy is assumed to increase monotonically with dose. Under order restricted alternatives, that is, under monotonic dose-response assumption, it is generally believed that Tukey’s contrast test, an analog of Tukey’s linear trend test in an analysis of variance set up, is superior to the test based on the highest versus control contrast in most practical or reasonable choices of dose response patterns. In this article we examine these two tests mathematically to establish areas of the parameter space where one is superior to the other in the order restricted parameter space when the number of doses is less than or equal to 7. We demonstrate that the highest versus control test is superior to Tukey’s contrast test only when all the middle doses have identical or near identical responses, which is a region of the alternative parameter space of minimal interest to researchers and dose-response modelers. Hence, we conclude that the traditional wisdom holds that for most of the popular and reasonable dose-response patterns Tukey’s contrast test is superior to the highest versus control test and should be employed by researchers in a typical dose-response trial.

Approximate Simultaneous Confidence Intervals for Multiple Contrasts of Binomial Proportions

Simultaneous confidence intervals for contrasts of means in a one-way layout with several independent samples are well established for Gaussian distributed data. Procedures addressing different hypotheses are available, such as all pairwise comparisons or comparisons to control, comparison with average, or different tests for order-restricted alternatives. However, if the distribution of the response is not Gaussian, corresponding methods are usually not available or not implemented in software. For the case of comparisons among several binomial proportions, we extended recently proposed confidence interval methods for the difference of two proportions or single contrasts to multiple contrasts by using quantiles of the multivariate normal distribution, taking the correlation into account. The small sample performance of the proposed methods was investigated in simulation studies. The simple adjustment of adding 2 pseudo-observations to each sample estimate leads to reasonable coverage probabilities. The methods are illustrated by the evaluation of real data examples of a clinical trial and a toxicological study. The proposed methods and examples are available in the R package MCPAN. ((c) 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim).

A defense of the likelihood ratio test for one-sided and order-restricted alternatives

It has been asserted recently that the likelihood ratio tests (LRT) for certain multi-parameter hypothesis-testing problems with one-sided and order-restricted alternatives exhibit anomalous behaviour (Cohen and Sackrowitz (Ann. Statist. 26 (1998) 2321); Cohen et al. (J. Mult. Analysis 72 (2002) 50). For sample points x and x′ such that x′ apparently lies “deeper” inside the alternative than does x, the LRT may accept the null hypothesis for x′ but reject it for x. Like Silvapulle (Amer. Statist. 51 (1997) 178), we argue that this conclusion is not anomalous but correct for the usual formulation of the null hypothesis accompanying one-sided and order-restricted alternatives. This alleged anomaly of the LRT occurs only for an alternative hypothesis given by an obtuse convex cone C. In this case, the reformulated null hypothesis “not C” both avoids the apparent anomaly and may be more appropriate scientifically.

Weighted Logrank Testing Procedures for the Simple Tree Alternatives

Weighted logrank testing procedures for comparing r treatments with a control when some of the data are randomly censored are discussed. Four kinds of test statistics for the simple tree alternatives are considered. The weighted logrank statistics based on pairwise ranking scheme is proposed and the covariances of the test statistics are explicitly obtained. This class of test statistics can be viewed as the general statistics of constructing the test procedures for various order restricted alternatives by modifying weights. Four kinds of weighted logrank tests are illustrated with an example. Simulation studies are performed to compare the sizes and the powers of the considered tests with the other.

Approximating the powers of order-restricted log rank tests at local alternatives

Abstract Under a proportional hazards model with continuous, right-censored data, tests of homogeneity with order-restricted alternatives are considered for survival curves. For such situations, analogues to the log rank test have been derived by applying order-restricted inference procedures to the score statistics. For local alternatives, approximations to the power functions of these tests are obtained by relating them to the likelihood ratio tests of homogeneity of normal means with order-restricted alternatives. The accuracies of the approximations are studied using Monte Carlo techniques for the simple order and the simple tree order restrictions.

A Multivariate MultiSample Quantile Test for Ordered Alternatives

We propose and study a multivariate c-sample rank test for order-restricted alternatives when the underlying distributions have a contaminated component. An example concerning lens opacity in adults is presented. The asymptotic distributions are obtained under local mixture alternatives, and Pitman efficiencies are calculated. An empirical power study is also included.

An Extension of a Convolution Inequality forG-Monotone Functions and an Approach to Bartholomew's Conjectures

A variety of convolution inequalities have been obtained since Anderson's theorem. ?In this paper, we extend a convolution theorem for G -monotone functions by weakening the symmetry condition of G -monotone functions. Our inequalities are described in terms of several orderings obtained from a cone. It is noteworthy that the orderings detect differences in directions. A special case of the orderings induces a majorization-like relation on spheres. Applying our inequality, Bartholomew's conjectures, which concern directions yielding the maximum power and the minimum power of likelihood ratio tests for order-restricted alternatives, are partly settled.

Order-Restricted Tests for Stratified Comparisons of Binomial Proportions

The data set presented relates a binomial response to ordered levels of an explanatory variable, representing doses of a drug, with data collected at several centers. A study goal is to test independence of the response and the ordinal factor, assuming under the alternative only that the binomial parameter is a monotonically increasing function of the ordinal predictor. We present two likelihood-ratio tests that are sensitive to order-restricted alternatives. Simulating the exact distributions of the test statistics yields nearly exact P-values. We also discuss related analyses for comparing two groups on an ordinal response, and we propose a test that is sensitive to a stochastic ordering alternative.

A note on the symmetry of normal mean hypotheses and its implications

This paper considers the likelihood ratio tests for homogeneity of normal means versus order-restricted alternatives. Symmetry properties of the test statistics with respect to the parameter vector about a linear space are studied.

Inadmissibility of some tests for order-restricted alternatives

Lee (1987) proposes four test statistics for testing homogeneity of multinomial parameters against an order-restricted alternative. We show that two of the four tests are often not monotone in some direction dictated by the ordered alternative. In such cases, oftentimes the tests based on these statistics are inadmissible. In fact, for the simple order alternative two of the four tests are inadmissible for most sample sizes and most test sizes.

Projected Tests for Order Restricted Alternatives

Consider the model where X ij , i = 1, 2,..., k, j = 1, 2,..., n, are independent random variables distributed according to a one-parameter exponential family, with natural parameter θ i . We test H 0 : θ 1 =... = θ k versus H 1 : θ ∈ C − {θ: θ ∈ H 0 }, where θ = (θ 1 ,..., θ k )' and C is a cone determined by Aθ ≥ 0, where the rows of A are contrasts with two nonzero elements. We offer a method of generating «good» tests for H 0 versus H 1 . The method is to take a «good» test for H 0 versus H 2 : not H 0 , and apply the test to projected sample points, where the projection is onto C. «Good» tests for H 0 versus H 1 are tests that are Schur convex. «Good» tests for H 0 versus H 1 are tests which are monotone with respect to a cone order. We demonstrate that if the test function for H 0 versus H 2 is a constant size Schur convex test, then the resulting projected test is monotone

Rank tests for ordered alternatives in analysis of variance

A class of asymptotically X̄ 2 rank tests is proposed for testing homogeneity against order restricted alternatives in two-way layouts with unequal number of observations per cell. The asymptotic relative efficiency comparisons (in Pitman sense) with respect to the restricted least squares test is made. An extension to the case of linear models is also considered.

A Test for a Multiple Isotonic Regression Problem

A new procedure for testing the homogeneity of normal mean vectors versus order restricted alternatives is proposed and its performance discussed.

A test for a multiple isotonic regression problem

SUMMARY A new procedure for testing the homogeneity of normal mean vectors versus order restricted alternatives is proposed and its performance discussed.