The Annals of Mathematical Statistics

Sequential Procedures for Selecting the Best One of Several Binomial Populations

Edward Paulson

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Abstract

A problem that seems to be of some practical importance is how to select the best one of $k$ experimental categories or populations when there is a fixed probability for each population that any measurement will be classified as a `success' and the best population is defined as the one with the greatest probability of a success. For example, we might be interested in determining which of $k$ new drugs offers the greatest probability of survival against a specified disease, or which of $k$ new production techniques has the greatest probability of producing a `good' item. A treatment of this problem using a fixed sample size approach was given by Sobel and Huyett [6]. To describe their formulation of the problem, denote the populations by $\prod_1, \prod_2, \cdots, \prod_k$, the corresponding probabilities by $p_1, p_2, \cdots, p_k$, and the ordered probabilities by $p_{\lbrack 1\rbrack} \geqq p_{\lbrack 2\rbrack} \geqq \cdots \geqq p_{\lbrack k\rbrack}$, and let $\prod_{\lbrack j\rbrack}$ be the population associated with $p_{\lbrack j\rbrack}$. Then [6] described a statistical procedure and gave tables for determining the common sample size required with each population so that population $\prod_{\lbrack 1\rbrack}$ will be selected with probability $\geqq P^\ast$ whenever $p_{\lbrack 1\rbrack} \geqq p_{\lbrack 2\rbrack} + d$, where $d$ and $P^\ast$ are constants selected in advance of the experiment. This formulation of the problem, which we will call the main formulation, seems satisfactory when nothing is known about the magnitude of $(p_1, p_2, \cdots, p_k)$ or if there is some a priori information available which indicates that $p_{\lbrack 1\rbrack}$ and $p_{\lbrack 2\rbrack}$ do not differ too much from .5, say .25 $\leqq p_{\lbrack 2\rbrack} \leqq p_{\lbrack 1\rbrack} \leqq .75$. An alternative formulation of the problem when the a priori information indicates that $p_{\lbrack 2\rbrack}$ and $p_{\lbrack 1\rbrack}$ differ substantially from .5 was given in [6] as follows: the sample size is determined so that population $\prod_{\lbrack 1\rbrack}$ is selected with probability $\geqq P^\ast$ whenever $p_{\lbrack 2\rbrack} \leqq p^\ast_{\lbrack 2\rbrack}$ and $p_{\lbrack 1\rbrack} \geqq p^\ast_{\lbrack 2\rbrack} + d$, where $p^\ast_{\lbrack 2\rbrack}$ is an additional constant determined in advance of the experiment on the basis of the a priori information about the probable value of $p_{\lbrack 2\rbrack}$. The present paper is based on a somewhat novel use of the Poisson distribution to obtain a random number of measurements from each population at every stage of experiment combined with the application of one-sided sequential confidence limits developed in [4]. Using these techniques we derive sequential procedures for selecting the best population both for the main formulation and for a generalization of the alternative formulation of the problem. Some Monte Carlo calculations which are summarized in Section 5 indicate that a substantial saving is possible with the sequential procedures.

Article information

Source
Ann. Math. Statist., Volume 38, Number 1 (1967), 117-123.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177699062

Digital Object Identifier
doi:10.1214/aoms/1177699062

Mathematical Reviews number (MathSciNet)
MR203890

Zentralblatt MATH identifier
0155.25302

JSTOR
links.jstor.org

Citation

Paulson, Edward. Sequential Procedures for Selecting the Best One of Several Binomial Populations. Ann. Math. Statist. 38 (1967), no. 1, 117--123. doi:10.1214/aoms/1177699062. https://projecteuclid.org/euclid.aoms/1177699062


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