Two-Stage Procedures for Comparing Treatments with a Control: Elimination at the First Stage and Estimation at the Second Stage

We consider the problem of comparing a set of p₁ test treatments with a control treatment. This is to be accomplished in two stages as follows: In the first stage, N₁ observations are allocated among the p₁ treatments and the control, and the subset selection procedure of Gupta and Sobel (1958) is employed to eliminate “inferior” treatments. In the second stage, N₂ observations are allocated among the (randomly) selected subset of p₂(≤p₁) treatments and the control, and joint confidence interval estimates of the treatment versus control differences are calculated using Dunnett's (1955) procedure. Here both N₁ and N₂ are assumed to be fixed in advance, and the so-called square root rule is used to allocate observations among the treatments and the control in each stage. Dunnett's procedure is applied using two different types of estimates of the treatment versus control mean differences: The unpooled estimates are based on only the data obtained in the second stage, while the pooled estimates are based on the data obtained in both stages. The procedure based on unpooled estimates uses the critical point from a p₂-variate Student t-distribution, while that based on pooled estimates uses the critical point from a p₁-variate Student t-distribution. The two procedures and a composite of the two are compared via Monte Carlo simulation. It is shown that the expected value of p₂ determines which procedure yields shorter confidence intervals on the average. Extensions of the procedures to the case of unequal sample sizes are given. Applicability of the proposed two-stage procedures to a drug screening problem is discussed.