Inter-Class Orthogonal Main Effect Plans for Asymmetrical Experiments

Abstract

In this paper we construct ‘inter-class orthogonal’ main effect plans (MEPs) for asymmetrical experiments. In such a plan, the factors are partitioned into classes so that any two factors from different classes are orthogonal. We have also defined the concept of “partial orthogonality” between a pair of factors. In many of our plans, partial orthogonality has been achieved when (total) orthogonality is not possible due to divisibility or any other restriction. We present a method of obtaining inter-class orthogonal MEPs. Using this method and also a method of ‘cut and paste’ we have obtained several series of inter-class orthogonal MEPs. One of them happens to be a series of orthogonal MEP (OMEPs) [see Theorem 3.6], which includes an OMEP for a 330 experiment on 64 runs. We have also obtained a series of MEPs which are almost orthogonal in the sense that every contrast is non-orthogonal to at most one more. A member of this series is an MEP for a 310210 experiment on 32 runs in which the only non-orthogonality is between the linear contrasts of pairs of three-level factors. Plans of small size (≤ 15 runs) are also constructed by ad-hoc methods. Among these plans there are MEPs for a 42.32.2 and a 35.2 experiment on 12 runs and a 52.32 experiment on 15 runs.