A New Approach to Equivalence Assessment in Standard Comparative Bioavailability Trials by Means of the Mann-Whitney Statistic

By a suitable transformation of the pairs of observations obtained in the successive periods of the trial, bioequivalence assessment in a standard comparative bioavailability study reduces to testing for equivalence of two continuous distributions from which unrelated samples are available. Let the two distribution functions be given by F(x) = P[X ≤ x], G(y) = P[Y ≤ y] with (X, Y) denoting an independent pair of real-valued random variables. An intuitively appealing way of putting the notion of equivalence of F and G into nonparametric terms can be based on the distance of the functional P[X > Y] from the value it takes if F and G coincide. This leads to the problem of testing the null hypothesis H_o P[X > Y] ≤ 1/2 - ε₁ or P[X > Y] ≥ 1/2 + ε₂ versus H₁ : 1/2 ? ε₁ < P[X > Y] < 1/2 + ∈₂, with sufficiently small ε₁, ε₂ ∈ (0, 1/2). The testing procedure we derive for (₀, H₁) and propose to term Mann-Whitney test for equivalence, consists of carrying out in terms of the U-statistics estimator of P[X > Y] the uniformly most powerful level a test for an interval hypothesis about the mean of a Gaussian distribution with fixed variance. The test is shown to be asymptotically distribution-free with respect to the significance level. In addition, results of an extensive simulation study are presented which suggest that the new test controls the level even with sample sizes as small as 10. For normally distributed data, the loss in power as against the optimal parametric procedure is found to be almost as small as in comparisons between the Mann-Whitney and the t-statistic in the conventional one or two-sided setting, provided the power of the parametric test does not fall short of 80%.     

A Comparison of Algorithms for the Computation of the Mann-Whitney Test

An algorithm is proposed for the computation of the Mann-Whitney Test which only requires the separate ordering of the individual samples and is therefore more suitable than the usual one. It is also more suitable as to computation time than the algorithm described by KUMMER , 1981, if the two sample sizes are of the same magnitude.     

Linear Rank Statistics for the Simple Tree Alternatives

The problem of comparing several treatments with a control is considered. It is formulated as a test of homogeneity with one-sided alternatives that at least one treatment is better than the control, while no treatment is worse than the control. A class of linear rank tests for the simple tree alternatives is proposed, assuming a location model. The most efficient test of the proposed class for detecting a specified pattern is derived. Two optimal tests for use in the case of vaguely specified tree alternatives are derived. In a simulation study the power of these two tests are compared with the power of Kruskal-Wallis test. Recommendations about the use of the proposed tests are made.     

A Note on Optimality of Certain Rank Tests for Ordered Alternatives in Randomized Blocks

Nonparametric tests for ordered alternatives in randomised block designs based on within block rankings have been proposed by many authors. This note is concerned with the optimality of the choice of so-called regression constants usually considered in such rank tests. Some examples are discussed.     

A Study of Distribution-free Tests for Umbrella Alternatives

In this paper we are concerned with test procedures for umbrella alternatives in the k-sample location problem. Distribution-free tests are considered for both cases where the peak of the umbrella is known or unknown. Comparative results of a Monte Carlo power study are presented.     

Notes on the Mack-Wolfe and Chen-Wolfe Tests for Umbrella Alternatives

In this paper we find a class of umbrella alternatives for which the MACK -WOLFE (1981) peak known test is optimal in the sense of Pitman efficiency. The asymptotic null distribution of the CHEN -WOLFE (1989) statistic for the peak-unknown umbrella alternatives problem is obtained. Some percentiles of the asymptotic distribution computed by simulation are presented.     

A New Rank Test Against Ordered Alternatives in K-Sample Problems

The distribution-free test against ordered alternatives proposed by Jonckheere (1954) is based on the Kendall's rank correlation coefficient τ. A new rank test is proposed and illustrated with a numerical example. The proposed test is based on Spearman's σ and has similar functional structure as the Kruskal-Wallis test. A useful by-product is a test for departure from a trend.     

A Note on Binary Contingency Tables

This paper considers contingency tables in which the marginal frequencies for one variable are all 1. This could occur with two-category binary data or when a continuous variable is treated in categorical fashion. Some results concerning the expectation of goodness-of-fit statistics are reported. In particular it is noted that the expectation of the Pearson statistic is independent of the model being fitted.     

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.     

Use of the Score Statistic for Testing Goodness-of-Fit of Some Generalised Distributions

The use of the score statistic to test whether a generalised distribution gives an improved fit over a non-generalised distribution is recommended. The score statistic for a generalised exponential family is derived. Several specific examples are given.     

Tests of Occupancy Rates for Clustered Populations: Case Where the Cluster Sizes are Known

A modified chi-squared statistic Z is proposed for testing hypotheses about category occupancy rates for individuals distributed by clusters, when the cluster sizes are observed. This statistic is the Pearson chi-square statistic based on the individuals' counts divided by 1 + M* where M* is the mean number of other individuals per cluster per individual. The kind of alternative hypothesis for which the Z-based test compares favourably in power with the Pearson chi-square test based on the cluster frequencies is given. However, we prove that this latter test is more powerful than the former one as long as the equidistribution of the random choice vectors is assumed.     

An Algorithm for the Mann-Whitney U-Test

An algorithm and the corresponding FORTRAN-program for the computation of the MANN-WHITNEY U-Test test statistic are introduced which are simpler than the familiar procedure using ranks.