A Two-Stage Selection and Testing Design for Comparing Several Normal Means with a Standard

Abstract

A two-stage design is proposed for the goal of selecting one among k experimental treatments, provided that it is better than a specified standard. In the first stage, ranking and selection techniques are used to screen out the one most promising treatment. In the second stage, hypothesis testing techniques are used to determine if the treatment chosen in the first stage is better than the standard. The design allows for early termination of the procedure at stage one if none of the treatments seem promising. All the treatments are assumed to follow normal distributions with a common unknown variance. A large population mean is assumed to be desirable, and hence “better than the standard” means that the population mean is sufficiently larger than the standard. Appropriate definitions of size and power are given. Sample size requirements are compared with an analogous two-stage selection procedure of Bechhofer and Turnbull [Bechhofer, R.E.; Turnbull, B.W. Two (k + 1)-decision selection procedures for comparing k normal means with a specified standard. J. Amer. Statist. Assoc. 1978, 73, 385–392].

Key Words:

• Known standard
• Least favorable configuration
• Ranking and selection
• Unknown control

Acknowledgments

We wish to express our gratitude to the Associate Editor and the two referees for their painstaking reviews and constructive comments which have made this manuscript a much better article. We are also grateful to be able to contribution in the Abraham Wald Centennial Numbers.

Notes

Recommended by T. K. S. Solanky