Multistage design procedures for identifying two-factor interactions, when higher effects are negligible

Abstract

Abstract Consider a 2m factorial experiment. Let L(ν×1) be the vector of factorial effects which are nonnegligible. Let μ denote the general mean. It will always be assumed that μ∈L, even though in some situations μ may be zero. All factorial effects which are not in L are thus assumed zero. We do not know what parameters are elements of L, and furthermore, we do not know the value of the various elements of L. The problem of model identification is to determine the elements of L, and also estimate the same. In this paper we shall investigate this problem in a certain limited context. We shall assume that all elements of L are included in the vector of parameters L ∗ (ν ∗ ×1) , except possibly for a set of p parameters. In this paper we shall take L ∗ to consist of the set of main effects, and the general mean μ. We assume p to be small and L to contain possibly a set of p 2-factor interactions. Three-factor and higher interactions are assumed negligible. We start with a saturated resolution III design of the 2m−q type, where m=2k−1. For a given value of (m−q) we shall also consider values of m smaller than the maximum. The purpose of the experiment is to estimate μ and the main effects, and furthermore identify all the p two-factor interactions which are nonnegligible, and also estimate the same. For this purpose, we consider using a multistage design, i.e., a design which allows the experiment to be done in several parts or stages. In the first stage, the resolution three design plus possibly a few extra treatments may be used. After looking at the resulting data, a further set of treatments is selected and observed. This may resolve the problem. If the problem is still not resolved we continue the same way, that is, we take a few more treatments and see if we can resolve the problem. The attempt is to have a design in which the average sample number is as small as possible. Examples are given for k=3,4.