A predictive approach for the selection of a fixed number of good treatments

This paper offers a predictive approach for the selection of a fixed number (= t) of treatments from k treatments with the goal of controlling for predictive losses. For the ith treatment, independent observations X_ij (j = 1,2,…,n) can be observed where X_ij's are normally distributed N(θ_i; σ²). The ranked values of θ_i's and X_i's are θ₍₁₎ ≤… ≤θ X_(k) and X_[1] ≤… ≤X_(k) and the selected subset S = {[k], [k – 1],…, [k - t + 1]} will be considered. This paper distinguishes between two types of loss functions. A type I loss function associated with a selected subset S is the loss in utility from the selector's view point and is a function of θ_i with i εS. A type I1 loss function associated with S measures the unfairness in the selection from candidates' viewpoint and is a function of θ_i with i ε S. This paper shows that under mild assumptions on the loss functions S is optimal and provides the necessary formulae for choosing n so that the two types of loss can be controlled individually or sinlultaneously with a high probability. Predictive bounds for the losses are provided. Numerical examples support the usefulness of the predictive approach over the design of experiment approach. Copyright © 1994 Taylor & Francis Group, LLC. All rights reserved.