Multiple Comparisons With the Best Treatment

Abstract

Let π₁, π₂, …, π _k be k ≥ 2 sources of observations (treatments, populations) and suppose the “goodness” of treatment π _i is characterized by the size of an unknown real-valued parameter θ _i . Let θ_[k] = max_1≤i≤k θ _i . If π _i is preferred to π _j when θ _i > θ _j , the parameters δ _i = θ_[k] — θ _i , i = 1, 2, …, k reflect in an inverse sense the “goodness” of each treatment relative to the “best” treatment. A general technique for obtaining simultaneous confidence intervals on the δ _i is demonstrated with several examples. This technique can be applied in any setting where comparison-with-control intervals can be computed regarding any π _j as the control. These results have special importance in ranking and selection problems in that the process of generating upper bounds on the δ _i generates traditional confidence statements of both the indifference zone and the subset selection schools, simultaneously, as established by Hsu (1981).