Tight orthogonal main effect plans

Abstract

In this thesis, we study orthogonal main effect plans (OMEPs), also known as orthogonal resolution III fractional designs. OMEPs are are a generalization of orthogonal arrays, and play a role in experimental design, in particular in screening experiments. We show that for any OMEP parameter set P, there is a special parameter set P', called a tight parameter set, so that if an OMEP with parameters P' exists then it can be used to obtain an OMEP with parameters P easily. Tight OMEPs are more structured than general OMEPs and therefore are easier to analyze. We find all tight OMEPs on three, four, and five rows, and use this to answer the existence question for four and five-factor OMEPs. The same procedure can be used to help answer the existence question for OMEPs on any number of rows. We also show that, asymptotically, for any tight parameter set there is a corresponding OMEP (with one small class of exceptions). We use this information to gain insight into Jacroux's lower bound on the number of runs in an OMEP. We demonstrate that any OMEP (not just every OMEP parameter set) having three rows can be uncollapsed to a tight OMEP, so in the case of OMEPs having three rows all the structural information about OMEPs is contained in the subset of three-row tight OMEPs. We also develop recursive constructions for equally replicated OMEPs. Often these constructions produce OMEPs having more rows than a direct product construction could achieve. Sometimes the OMEPs produced are tight. One of these constructions produces resolvable orthogonal arrays. Other miscellaneous results concerning OMEPs are also proven.