A Two-Stage Selection and Testing Design for Comparing Several Normal Means with a Standard: The Case of Common, Known Variance

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 known 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. Tables of procedure parameters for achieving desired size and power are provided. Sample size requirements are compared with the single-stage 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. Journal of the American Statistical Association 1978, 73, 385–392).

Keywords:

• Ranking and selection
• Hypothesis testing
• Least favorable configuration
• Two-stage design
• Comparison with specified standard