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%.     

Optimal Testing for Serial Correlation in a Large Number of Small Samples

In recent years, there has been an increased awareness of the potential one-sided nature of many testing problems in applied sciences. Usually, these testing problems can be reduced, either by conditioning on sufficient statistics or by invariant techniques. COX and SOLOMON (1988) considered testing the serial correlation coefficient of a stationary first order autoregressive process and concentrated on four independent samples, with each of size three. We outline a general method for testing the serial correlation coefficient, using locally best invariant, point optimal invariant and locally most mean powerful invariant test procedures. The first procedure optimizes power near the null hypothesis, the second optimizes it at a pre-determined point away from the null while the third optimizes the average curvature of the power hypersurface in the neighbourhood of the null hypothesis.     

Simultaneous Statistical Inference Concerning the Standardized Mortality Ratios (SMB) of Several Strata in an Epidemiologic Study

In the present paper, simultaneous prediction intervals are constructed for some mortality measures involving real data in a recent retrospective epidemiologic study of a cohort of man-made mineral fiber workers. The object is to compare a test population with a standard population in which the workers are exposed to several levels (say K) of a suspected carcinogen and the number of deaths in the K exposure groups are recorded. The SMR is chosen as a mortality measure for this comparison. If the total number of deaths d. in the entire study is known, the number of deaths in the various strata become dependent random variables. Using univariate statistical procedures to test the significance of the individual SMR's is inappropriate and may give misleading conclusions. The present paper shows how to test the simultaneous statistical significance of the SMR's of the K exposure groups, i.e., to test the simultaneous null hypothesis H_o : E(SMR₁) = E(SMR₂) =…E(SMR_k) = 100. If the null hypothesis H_o is rejected, it is shown how to identify the SMR (or the SMR's) which contributed to the rejection of H_o. The power function of such a test is also constructed for some simple and useful alternative hypotheses.