Robert E. Bechhofer

Incomplete Block Designs for Comparing Treatments Wth a Control: General Theory

In this paper we develop a theory of optimal incomplete block designs for comparing several treatments with a control. This class of designs is appropriate for comparing simultaneously p ≥ 2 test treatments with a control treatment (the so-called multiple comparisons with a control (MCC) problem) when the observations are taken in incomplete blocks of common size k < p + 1. For this problem we propose a new general class of incomplete block designs that are balanced with respect to (wrt) test treatments. We shall use the abbreviation BTIB to refer to such designs. We study their structure and give some methods of construction. A procedure for making exact joint confidence statements for this multiple comparisons problem is described. By using a new concept of admissibility of designs, it is shown how “inferior” designs can be eliminated from consideration, and attention limited to a small class of BTIB designs that can be constructed from so-called generator designs in the minimal complete class of such d...

A Two-stage minimax procedure with screening for selecting the largest normal mean

The problem of selecting the normal population with the largest population mean when the populations have a common known variance is considered. A two-stage procedure is proposed which guarantees the same probability requirement using the indifference-zone approach as does the single-stage procedure of Bechhofer (1954). The two-stage procedure has the highly desirable property that the expected total number of observations required by the procedure is always less than the total number of observations required by the corresponding single-stage procedure, regardless of the configuration of the population means. The saving in expected total number of observations can be substantial, particularly when the configuration of the population means is favorable to the experimenter. The saving is accomplished by screening out “non-contending” populations in the first stage, and concentrating sampling only on “contending” populations in the second stage. The two-stage procedure can be regarded as a composite one whic...

Two (k + 1)-Decision Selection Procedures for Comparing k Normal Means with a Specified Standard

Abstract The problem of comparing k normal means with a specified (absolute) standard is considered. A single-stage and a two-stage (k + 1)-decision procedure are proposed for the cases of common known variance and common unknown variance, respectively. The procedures guarantee that (1) with probability at least P 0* (specified), no population is selected when the largest population mean is sufficiently less than the standard, and (2) with probability at least P 1* (specified), the population with the largest population mean is selected when that mean is sufficiently greater than its closest competitor and the standard. Tables to implement the procedures are provided. Applications and generalizations are described.

Multiple Comparisons for Orthogonal Contrasts: Examples and Tables

In many experimental situations the pertinent inferences are made on the basis of orthogonal contrasts among the treatment means (as in 2 n factorial experiments). In this setting a particularly useful form of inference is one involving multiple comparisons. The present article describes situations in which such inferences are meaningful, gives examples of their use, and provides a table of constants needed to implement such multiple comparison procedures. The procedures can also be used for statistically legitimate “data snooping” (in the sense ofScheffe 1959, p. 80) to help decide which contrasts within a specified set warrant further study.

A Note on the Limiting Relative Efficiency of the Wald Sequential Probability Ratio Test

Abstract The efficiency (measured in terms of ratio of average sample size to fixed sample size) of the Wald sequential probability ratio test relative to the best competing fixed sample procedure for testing H 0:θ = θ0 versus H 1:θ = θ1 (θ0 0) this limiting relative efficiency is equal to (θ1 – θ0)/(4|θ1+θ0 − 2θ|). The practical implications of this result are discussed.

A two-stage minimax procedure with screening for selecting the largest normal mean (ii): an improved pcs lower bound and associated tables

This paper is a follow-up to an earlier article by the authors in which they proposed a two-stage procedure with screening to select the normal population with the largest population mean when the populations have a common known variance. The two-stage procedure has the highly desirable property that the expected total number of observations required by the procedure is always less than the total number of observations required by the corresponding single-stage procedure of Bechhofer (1954), regardless of the configuration of the population means. The present paper contains new results which make possible the more efficient implementation of the two-stage procedure. Tables for this purpose are given, and the improvements achieved (which are substantial) are assessed.

Truncation of the Bechhofer-Kiefer-Sobel Sequential Procedure for Selecting the Multinomial Event which has the Largest Probability (II): Extended Tables and an Improved Procedure

In an earlier article the authors studied the performance of a truncated version of a vector-at-a-time sequential sampling procedure proposed by Bechhofer, Kiefer and Sobel for selecting the multinomial event which has the largest probability. Among the performance characteristics studied for both the original (open) basic procedure (P*B:) and the truncated (closed) procedure (P*B:T:) were the achieved probability of a correct selection, and BT the expected number of vector-observations (n) to terminate sampling when the event probabilities are in the so-called least-favorable and equal-parameter configurations. These were compared with the same quantities for a competing (closed) procedure of Ramey and Alam (R-A). All three procedures guarantee the same requirement on the probability of a correct selection. The truncated procedure was shown to be greatly superior to the untruncated version both in terms of E{n} and Var{n}, and also to be superior to the R-A procedure.¶A limited set of truncation numbers ...

Truncation of the Bechhofer-Kiefer-Sobel Sequential Procedure for Selecting the Multinomial Event which has the Largest Probability

In this article we study the effect of truncation on the performance of an open vector-at-a-time sequential sampling procedure (P* B) proposed by Bechhofer, Kiefer and Sobel , for selecting the multinomial event which has the largest probability. The performance of the truncated version (P* B T) is compared to that of the original basic procedure (P* B). The performance characteristics studied include the probability of a correct selection, the expected number of vector-observations (n) to terminate sampling, and the variance of n. Both procedures guarantee the specified probability of a correct selection. Exact results and Monte Carlo sampling results are obtained. It is shown that P* B Tis far superior to P* B in terms of E{n} and Var{n}, particularly when the event probabilities are equal.The performance of P* B T is also compared to that of a closed vector-at-a-time sequential sampling procedure proposed for the same problem by Ramey and Alam; this procedure has here to fore been claimed to be the bes...

Closed Sequential Procedures for Selecting the Multinomial Events which Have the Largest Probabilities

Single-stage and closed sequential procedures for selecting the multinomial events which have the largest probabilities areconsidered. Two goals, Goal I (Selecting the s best categories without regard to order) and Goal II (Selecting the s best categories with regard to order) are studied in detail; here k ≥ 2 is the number of categories in the multjinomial distribution. Goal I includes as special cases, the goals of Bechhofer, Elmaghraby and Morse (1959) and Alam and Thompson (1972) which correspond here to the cases s = 1 and s = k-l, respectively; both foregoing articles gave single-stage procedures whi ch when used with an appropri ate si ng1e-stage slamp1e size n guarantee a probability requirement which employs the Isocalled indifference-zone approach. The sequenti a1 procedures tha t we propose achi eve the same probability of a correct selection as do the corresponding silngle stage procedures, uniformly in the unknown event probabilities . Moreover, this is accomplished with a smaller expected nu...

Optimal Allocation of Observations when Comparing Several Treatments with a Control, II: 2-Sided Comparisons

A setup involving p + 1 normal populations Π i with unknown population means μ p and known population variances σ2 i (0 ≤ i ≤ p) is considered; Π0 is the “control” population and Π i (1 ≤ i ≤ p) are the “test” populations. Based on independent observations zii (j = 1, 2, …, Ni ) from Π i (0 ≤ i ≤ p) it is desired to make an exact joint confidence statement of the form {x0 – x i – d −1, and d > 0 are specified prior to experimentation. The problem of choosing the Ni (0 ≤ i ≤ p) to maximize the confidence coefficient is considered. A procedure is described which is globally optimal if σ2 1 = σ2 2 = … = σ2 p and is optimal in a more restricted sense if not all of the σ2 i (1 ≤ i ≤ p) are equal. It is shown that the globally optimal proportion of observations i to be taken from II; (0 ≤ i ≤ p) depends only on the θ i = σ2 i /σ2 0 (1 ≤ i ≤ p) and λ = d √...