Dennis Wackerly

Exact conditional tests for cross-classifications: Approximation of attained significance levels

A procedure is proposed for approximating attained significance levels of exact conditional tests. The procedure utilizes a sampling from the null distribution of tables having the same marginal frequencies as the observed table. Application of the approximation through a computer subroutine yields precise approximations for practically any table dimensions and sample size.

Measuring nominal scale agreement between a judge and a known standard

Two designs for comparing a judges ratings with a known standard are presented and compared. Design A pertains to the situation where the judge is asked to categorize each ofN subjects into one ofr (known) classes with no knowledge of the actual number in each class. Design B is employed when the judge is given the actual number in each class and is asked to categorize the individuals subject to these constraints. The probability distribution of the total number of correct choices is developed in each case. A power comparison of the two procedures is undertaken.

Some exact conditional tests of independence forR ×C cross-classification tables

Exact conditional tests of independence in cross-classification tables are formulated based on theχ2 statistic and statistics with stronger operational interpretations, such as some nominal and ordinal measures of association. Guidelines for the table dimensions and sample sizes for which the tests are economically implemented on a computer are given. Some selected sample sizes and marginal distributions are used in a numerical comparison between the significance levels of the approximate and exact conditional tests based on theχ2 statistic.

A more powerful method for testing for agreement between a judge and a known standard

We assume that a judges task is to categorize each ofN subjects into one ofr known classes. The design of primary interest is employed if the judge is presented withs groups, each containingr subjects, such that each group of sizer consists of exactly one subject of each of ther types. The probability distribution for the total number of correct choices is developed and used to test the null hypothesis that the judge is “guessing” in favor of the alternative that he or she is operating at a better than chance level. The power of the procedure is shown to be superior to two other procedures which appear in the literature.

On Rank Transformation Techniques for Balanced Incomplete Repeated-Measures Designs

Abstract Asymptotic properties of statistics designed to detect general alternatives in compound symmetric balanced incomplete repeated-measures designs with fixed treatment effects are investigated. Included in this study are the analysis of variance (ANOVA) F statistic, its rank transform Durbins statistic. By making asymptotic relative efficiency comparisions among these statistics when they have been computed with and without mean alignment, valuable new insight into their large-sample performance characteristics is gained. Evidence is presented corroborating recent empirical studies that suggest that mean alignment can improve the performance of rank transformation statistics. Finally, it is noted that the rank transform of the ANOVA F statistic when it is computed using mean aligned data is generally the most efficient among the statistics studied here.

Nonparametric estimation in one-way random effects models

We present a rank based method for obtaining point and interval estimates of a scale version of the intraclass correlation coefficient in a one-way random effects model. When compared to the method of Arvesen and Schmitz (1970), the new method is not only applicable to a broader class of situations, but also much easier to implement. Results of a simulation study indicate that the new procedure compares favorably with the Arvesen-Schmitz procedure and the classical normal theory procedure especially If the random components have heavy tailed distributions.

Asymptotic Theory of Sequential Fixed-Width Confidence Interval Procedures

Abstract Consider a sequence of confidence intervals {I n }. Sequential procedures are based on stopping rules which, for specified constants w and p, terminate sampling at a value n for which I n has width approximately w and noncoverage probability approximately p. Asymptotic analyses of such procedures have previously concerned only w → 0 with p fixed. This article presents a novel and general approach for the case p → 0 with w fixed. Application is made to procedures based on the sample mean and median, and the corresponding asymptotic relative efficiencies are evaluated for selected values of w for the normal, Laplace, logistic, and uniform distributions.

Sequential and two-stage point estimation for the range in a power family distribution

Sequential and two-stage point estimation procedures for the range of a power family distribution are discussed using a loss function from a fairly general class. Some asymptotic characteristics of the sequential procedure are presented, including approximate expressions for the distribution of the sample size. For the two stage procedure, we present asymptotic properties and exact expressions for the risk of the procedure, and distribution and mean of the stopping rule. The results of numerical simulations are presented to demonstrate some practical merits of our procedures for moderate sample sizes.

SOME OBSERVATIONS ON THE MAKUCH/SIMON APPROACH TO SAMPLE SIZE DETERMINATION IN CLINICAL TRIALS WITH HISTORICAL CONTROLS

The widely implemented Makuch and Simon method for using historical control data to determine sample size for efficacy studies is carefully explained. An extensive Monte Carlo study of the method clearly delineates its performance characteristics. A second Monte Carlo study describes the poor overall performance achieved when the Makuch and Simon method is used with the natural two-sample statistic for detecting a difference in proportions and results in the unequivocal recommendation that the composition of the angular and square root transformations be used in the context of the historical control problem.

An Alternative Sequential Approach to the Problem of Selecting the Best of k Populations

Abstract Previous sequential solutions for selecting the population with the largest center of symmetry among k populations differing only in location have taken the following approach: Let cs denote the event of correct selection and derive a stopping rule which is such that, for α > 0 preassigned, lim inf P (cs) ≥ 1 − α as δ → 0 when δ = θ[k] − θ[k−1] and θ[i] is the ith smallest center of symmetry. In the present article the fixed δ case is of primary concern, and rules which perform well as α → 0 are sought. A general method of solution is presented and applied to yield a procedure based on the sample means. Comparison with a previously developed rule is made.