Basic ideas and concepts for multiple comparison procedures



Around 1970, the author proposed a general theoretical approach to multiple decision problems (MDPs) of which multiple comparison problems (MCPs) are special cases. Suppose that a sample space equation image is given together with a set of probability distributions equation image defined over equation image. Let a finite partition of the parameter space equation image be given. Based on the observation equation image, an MDP is to decide, which ωa the true parameter θ belongs to. An MD confidence procedure is a mapping ψ from equation image to the class of subsets of A, such that the probability that equation image includes the true parameter θ is not smaller than 1−αθ. Here, 1−αθ is called the level of the confidence procedures and may vary depending on θ∈ωa. The MP confidence procedures are derived from the following proposition. When the ωa's are mutually disjoint, there is a one-to-one correspondence between an MD confidence procedure ψ and a collection of (non-randomized) tests for the hypotheses Ha : θ∈ωa with level αa by rejecting the hypothesis Ha if equation image. In this paper we discuss in detail the problems of determining the signs or the orderings of normal means. The resulting confidence procedures from the LR tests are seen to be too complicated and difficult to understand. We therefore propose simplified, less powerful methods. We define an overlapping partition of Ω into simple sets, such that the original ωa's can be expressed as an intersection of such simple sets. For each such set we define rejection regions corresponding to the levels α, α/2,...,α/k. Then we obtain the acceptance regions for Ha : θ∈ωa given as the intersection of all acceptance regions for the simple sets containing ωa at the level α/k, if there are k such simple sets. This method can be extended to obtain sequential confidence procedures.