Optimal Rank Tests Research Articles

Assessment of statistical significance and clinical relevance

In drug development, it is well accepted that a successful study will demonstrate not only a statistically significant result but also a clinically relevant effect size. Whereas standard hypothesis tests are used to demonstrate the former, it is less clear how the latter should be established. In the first part of this paper, we consider the responder analysis approach and study the performance of locally optimal rank tests when the outcome distribution is a mixture of responder and non-responder distributions. We find that these tests are quite sensitive to their planning assumptions and have therefore not really any advantage over standard tests such as the t-test and the Wilcoxon-Mann-Whitney test, which perform overall well and can be recommended for applications. In the second part, we present a new approach to the assessment of clinical relevance based on the so-called relative effect (or probabilistic index) and derive appropriate sample size formulae for the design of studies aiming at demonstrating both a statistically significant and clinically relevant effect. Referring to recent studies in multiple sclerosis, we discuss potential issues in the application of this approach.

R-Estimator of Location of the Generalized Secant Hyperbolic Distribution

ABSTRACT The generalized secant hyperbolic distribution (GSHD) proposed in Vaughan (2002) includes a wide range of unimodal symmetric distributions, with the Cauchy and uniform distributions being the limiting cases, and the logistic and hyperbolic secant distributions being special cases. The current article derives an asymptotically efficient rank estimator of the location parameter of the GSHD and suggests the corresponding one- and two-sample optimal rank tests. The rank estimator derived is compared to the modified MLE of location proposed in Vaughan (2002). By combining these two estimators, a computationally attractive method for constructing an exact confidence interval of the location parameter is developed. The statistical procedures introduced in the current article are illustrated by examples.

A method of constructing rank tests in survival analysis

It is typical in survival analysis that the quantities of interest are only partially observable. One encounters data being censored, truncated or interval censored. When testing a control versus a treatment group in such a situation one gets an additional complication, namely infinite-dimensional nuisance parameters representing the censoring and truncation distributions. In this paper, we review a method for constructing in some sense asymptotically optimal rank tests when data are only partially observed.

Statistical evaluation of the attainment of interim cleanup standards at hazardous waste sites

Cleanup standards at hazardous waste sites include (i) numeric standards (often risk-based), (ii) background standards in which the remediated site is compared with data from a supposedly clean region, and (iii) interim standards in which the remediated site is compared with preremediation data from the same site. The latter are especially appropriate for verifying progress when an innovative, but unproven, technology is used for remediation. Standards of type (i) require one-sample statistical tests, while those of type (ii) and type (iii) call for two-sample tests. This paper considers two-sample tests with an emphasis upon the type (iii) scenario. Both parametric (likelihood ratio) and nonparametric (linear rank) protocols are examined. The methods are illustrated with preremediation data from a site on the National Priorities List. The results indicate that nonparametric procedures can be quite competitive (in terms of power) with distributional modelling provided a near optimal rank test is selected. Suggestions are given for identifying such rank tests. The results also confirm the importance of sound baseline sampling; no amount of post-remediation sampling can overcome baseline deficiencies.

A SIMPLE SEQUENTIAL WILCOXON TEST

SummaryA simple sequential non‐parametric test for the two‐sample problem is proposed. A method of deriving its O.C. and A.S.N. funtions is given, and their adequacy confirmed by simulation. The test is found to require about 10 percent more observations than an optimal rank test; however it is much easier to apply. The test is found to be relatively robust.

Rank test statistics for umbrella alternatives

This paper proposes an optimal rank test procedure for testing an umbrella alternative when the peak of the umbrella is known. It is referred to as maximin efficient linear rank test. Also when the peak of the umbrella is unknown, a test procedure is proposed and its performance is discussed.

Asymptotically optimal rank tests for the two-sample problem with randomly censored data

Using the methods of asymptotic decision theory asymptotically optimal rank tests are constructed in the two-sample testing problem for translation families and positive scale families under the assumption of equal censoring in both samples. The resulting tests have a simple form extending the known tests for un-censored data in a natural way. Relations to a recent proposal by Albers and Akritas are discussed.

On a condition for the log-rank test to be a locally optimal rank test

In the Cox (1972) regression model for survival data combined with random censoring, Sen (1984) obtains a condition for the log-rank test of a null hypothesis to be a locally optimal rank test and he gives two situations where the condition holds. In this paper we show that these two situations exhaust all the cases where the condition is satisfied.

One-sided and multi-sided rank tests against trend in the case of concordant observers

Optimal rank tests are derived for testing homogeneity ofk populations observed bym independent andconcordant observers against oriented respectively non-oriented contiguity alternatives with respect to a location parameter. By means of the theorems obtained appropriate rank tests are proposed for testing against a one-sided respectively a multi-sided trend.

The Cox regression model, random censoring and locally optimal rank tests

Conditions on the hazard functions under the usual log-rank test remains locally optimal for the Cox regression model under random censoring (withdrawal) are examined. In the light of these, the asymptotic efficiency results pertaining to the Cox partial likelihood statistic and the log-rank statistic are studied.

Locally optimal rank tests for independence

The problem of testing the hypothesis of independence against multiparametrical set of alternatives is considered. Rank tests, having some locally maximin property are studied and a certain characterization of these tests is given. Finite sample and asymptotic test statistics in a restricted class of tests are derived.

Locally most powerful rank tests for multiple-censored data

A locally most powerful rank test is derived for multiple censored data in the context of a two-sample problem where the alternatives involve a single parameter. With independent random samples , from F and G respectively, let the combined sample order statistics be denoted by where N = m+n, We present results for the triple censoring scheme where two blocks of order statistics are observed in addition to recording the number of X's and Y’s among the censored values. These extend the derivation of optimal rank tests for single censored data to the multiple censored situation.

Distribution-Free Tests of Independence

The object of this article is to characterize the family of all distribution-free (DF) tests of independence, and those subfamilies which are optimal for specified alternative classes. It is found (Theorem 3.2) that each DF statistic is a function of a Pitman (or permutation) statistic; and (Theorem 3.4) that the rank statistics are those whose distributions depend appropriately on the maximal invariant [Lehmann (1959), p. 227]. For parametric alternatives the MP (most powerful) [Lehmann and Stein (1949)] and the locally MP tests are found to be Pitman tests based on the likelihood function (Lemma 4.1 and Theorem 4.1); while the corresponding optimal rank tests are analogous (Lemmas 4.2, 4.3) to those in the 2-sample case [e.g. Capon (1961)], and are closely related to those of Bhuchongkul (1964). In Section 5 randomized statistics similar to those of [4] are shown (Corollary 5.2) to be asymptotically equivalent to the optimal nonrandomized statistic for specified parametric alternatives. For one reasonable nonparametric class of alternatives one proves (Theorem 6.1) that the normal scores test is minimax; while for the other class an unexpected statistic (Theorem 6.2) is minimax. Finally, in Section 7 the ideas of monotone tests developed by Chapman (1958) and others are extended, and analogous results (Theorem 7.1) are obtained for minimum power.