Sequential factorial probing designs for identifying and estimating non-negligible factorial effects for the 2 m experiment under the tree structure

Abstract

Consider a 2 m factorial experiment. Let A (2 m × 1) be the vector of ‘factorial effects’. Let θ be a v × 1 subvector of A . Then θ is said to have a ‘tree structure’ if it is such that if A( i 1,…, i k ) (which, for all k, denotes the k-factor interaction between factors i 1,…, i k , where 1≤ i 1< i 2< i k ≤ m) belongs to θ , then at least one ( k−1)-factor interaction A( j 1,hellip;, j k−1 ) (where the set ( j 1,hellip;, j k−1 ) is a subset of ( i 1,…, i k )), also belongs to θ . Now, suppose that an element of A is non-negligible if and only if it belongs to θ . Suppose further that θ is unknown (except that it is known to have a tree structure, and also that v is known). In this paper, for m≤4, we obtain sequential designs for identifying and estimating θ (denoted later as L ∗). We employ the intersection sieve and other tentative decision rules introduced in Srivastava (1987). Note that the earlier work deals only with single stage search designs, or with optimal balanced designs or designs of the parallel flats types, the latter two being meant for estimating a given subset of A (under the assumption that the elements of A not in the subset are negligible). Thus, the present work is at a level of generality higher than ever attempted before. Also, it will be useful in signal processing.