## Mathematics - Dissertations

#### Title

Selection procedures for binomial populations

8-2006

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

Mathematics

Pinyuen Chen

#### Keywords

Binomial, Ranking, Curtailment, Least-favorable configuration

Mathematics

#### Abstract

In this thesis we consider the problem of selecting the best among several experimental treatments in comparison to a control treatment, under the binomial setting. The goal is to select the experimental treatment that produces the largest value of the probability of getting a success in a single trial, when this probability is better than the one corresponding to the control population. Otherwise, we select the control treatment.

We propose selection procedures based on the single-stage procedure defined by Dunnett (1984) and the hybrid selection and testing procedure defined by Thall et al.(1988). Our procedures are exact. That is, they are based on the binomial distribution only, as opposed to most of the original procedures that involve the normal approximation to the binomial. We evaluate the probability of a correct selection P(CS) for our procedures exactly and then derive the least favorable configurations (LFC). With our results on the LFC, the redefined procedures can be applied to any sample size.

Based on the fixed sample size procedure proposed by Sobel and Hyuett (1957), with the desire of minimizing the number of observations taken from the poorer populations, Bechoffer and Kulkarni (1982) proposed a sequential procedure which achieves the same probability of a correct selection as does the Sobel-Hyuett procedure, but requires fewer observations. Along the same lines, but based on Dunnett's fixed-sample-size procedure (1984) where a control is involved, we have used strong curtailment to define a procedure which requires fewer observations from the experimental treatments and from the control, but achieves the same probability of a correct selection as does Dunnett's procedure. Then, based on Thall et al.'s two-stage design (1988), we use strong curtailment again to define a hybrid selecting and testing design which reaches the same probability of a correct selection as the original design, but requires fewer observations.

#### Access

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