Journal of Applied Mathematics and Decision Sciences
Volume 2005 (2005), Issue 1, Pages 47-58

A two-stage procedure on comparing several experimental treatments and a control—the common and unknown variance case

John Zhang,1 Pinyuen Chen,2 and Yue Fang3

1Department of Mathematics, Indiana University of Pennsylvania, 15705, PA, USA
2Department of Mathematics, Syracuse University, 13244, NY, USA
3Lundquist College of Business, University of Oregon, Eugene 97403, OR, USA

Received 23 May 2001; Revised 6 December 2002

Copyright © 2005 John Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper introduces a two-stage selection rule to compare several experimental treatments with a control when the variances are common and unknown. The selection rule integrates the indifference zone approach and the subset selection approach in multiple-decision theory. Two mutually exclusive subsets of the parameter space are defined, one is called the preference zone (PZ) and the other, the indifference zone (IZ). The best experimental treatment is defined to be the experimental treatment with the largest population mean. The selection procedure opts to select only the experimental treatment which corresponds to the largest sample mean when the parameters are in the PZ, and selects a subset of the experimental treatments and the control when the parameters fall in the IZ. The concept of a correct decision is defined differently in these two zones. A correct decision in the preference zone (CD1) is defined to be the event that the best experimental treatment is selected. In the indifference zone, a selection is called correct (CD2) if the selected subset contains the best experimental treatment. Theoretical results on the lower bounds for P(CD1) in PZ and P(CD2) in IZ are developed. A table is computed for the implementation of the selection procedure.