COMPARISONS AND SELECTION OF TWO-PARAMETER EXPONENTIAL POPULATIONS

For two-parameter exponential populations with the same scale parameter (known or unknown) comparisons are made between the location parameters. This is done by constructing confidence intervals, which can then be used for selection procedures. Comparisons are made with a control, and with the (unknown) “best” or “worst” population. Emphasis is laid on finding approximations to the confidence so that calculations are simple and tables are not necessary. (Since we consider unequal sample sizes, tables for exact values would need to be extensive.)     

Subset selection of the t best populations

Selection of the “best” t out of k populations has been considered in the indifferece zone formulation by Bachhofer (1954) and in the subset selection formulation by Carroll, Gupta and Huang (1975). The latter approach is used here to obtain conservative solutions for the goals of selecting (i) all the “good” or (ii) only “good” populations, where “good” means having a location parameter among the largest t. For the case of normal distributions, with common unknown variance, tables are produced for implementing these procedures. Also, for this case, simulation results suggest that the procedure may not be too conservative.     

△-correct decision for location and scale parameters

In selecting t out of k populations, a △-correct decision is said to be made if the smallest location parameter for the selected populations is not more than △ below the largest location parameter for the non-selected populations. (For seal? parameters there is a similar definition in terms of ratio3.) The minimum probability of △-correct decision over the entire parameter pace is shown to be equal to the minimum probability of correct selection over a preference zone determined by △.     

Partitioning with Respect to a Control: Unequal Sample Sizes Case

This article describes a method for partitioning with respect to a control for the situation in which the treatment sample sizes are unequal and also for the situation where the treatment sample sizes are equal except for a few missing values. Calculation of the critical values required for finding confidence limits is discussed and tables are presented for the “almost equal” sample size case. An application of this method to length of stay data for congestive heart failure patients is also provided.     

Least significant spacing for ‘one versus the rest’ normal populations

Consider sample means from k(≥2) normal populations where the variances and sample sizes are equal. The problem is to find the ‘least significant difference’ or ‘spacing’ (LSS) between the two largest means, so that if an observed spacing is larger we have confidence 1 - α that the population with largest sample mean also has the largest population mean.

When the variance is known it is shown that the maximum LSS occurs when k = 2, provided a < .2723. In other words, for any value of k we may use the usual (one-tailed) least significant difference to demonstrate that one population has a population mean greater than (or equal to) the rest.

When the variance is estimated bounds are obtained for the confidence which indicate that this last result is approximately correct.     

An upper bound on a ratio of variances

Several processes may be monitored in terms of their variances relative to each other by the maximum ratio of variances. Based on the observed value an upper confidence bound for the population value is derived and tables are supplied for its implementation. Such bounds may be used when the experimenter wishes to establish that the variances are "not too different". The usual testing of variances for equality is noticed to be inappropriate in such cases.     

Homogeneity of Two Straight Lines: Equivalence Approach Using Fitted Regressions

The paper examines the homogeneity of a pair of straight lines, regarded as the expected values of two different linear regressions, from an equivalence point of view. This seems more appropriate than the usual testing of the null hypothesis of homogeneity when the aim is to establish that the lines are close to homogeneous. Upper confidence bounds on the maximum difference between the lines are based on the usual least squares regression estimators, assuming normal distributions. These bounds are constructed for fixed points, or over a fixed interval, and it is concluded that the lines are 1-homogeneous if the bounds are not greater than 1: Also, intervals are constructed over which the lines are concluded to be 1-homogeneous.     

SELECTING THE t BEST POPULATIONS — SCALE PARAMETER CASE

Selection from k independent populations of the t (< k) populations with the smallest scale parameters has been considered under the Indifference Zone approach by Bechhofer k Sobel (1954). The same problem has been considered under the Subset Selection approach by Gupta & Sobel (1962a) for the normal variances case and by Carroll, Gupta & Huang (1975) for the more general case of stochastically increasing distributions. This paper uses the Subset Selection approach to place confidence bounds on the probability of selecting all “good” populations, or only “good” populations, for the Case of scale parameters, where a “good” population is defined to have one of the t smallest scale parameters. This is an extension of the location parameter results obtained by Bofinger & Mengersen (1986). Special results are obtained for the case of selecting normal populations based on variances and the necessary tables are presented.     

Selecting Closest to Control

Consider the usual one-way fixed effect analysis of variance model where the populations Π_i ( I = 0, 1, . . . , k ) have independent normal distributions with unknown means and common unknown variance. Let Π₀ be a control population with which the other (treatment) populations are to be compared. The basic problem is to select the treatment that is closest to the control mean. This situation occurs when one of the Π_i must be chosen, regardless of how many are equivalent to the control in the sense of having means sufficiently close. This paper follows the approach of Hsu (1996) and is based on a set of simultaneous confidence intervals. It provides a table of critical values which allows direct implementation of the new inference procedure. The applications given are of the balanced cross-over design type with negligible carry-over effects, for which the results of this paper may be used. One of the applications refers to the selection of a drug, which may not be bioequivalent to a reference formulation but is the closest of those drugs that are readily available to the group of patients considered.