Inter-class orthogonal main effect plans for asymmetrical experiments

TL;DR

This paper introduces inter-class orthogonal main effect plans (MEP) for asymmetrical experiments, including methods for construction and analysis, and explores concepts like partial orthogonality and orthogonality through another factor.

Contribution

It develops new construction methods for inter-class orthogonal MEPs, introduces the concept of partial orthogonality, and provides a computational approach for data analysis in factorial designs.

Findings

Constructed inter-class orthogonal MEPs for various experiments.

Achieved some plans that are fully orthogonal or almost orthogonal.

Provided a computational method for analyzing factorial design data.

Abstract

In this paper we construct `inter-class orthogonal' main effect plans (MEP) for asymmetrical experiments. In such a plan, a factor is orthogonal to all others except possibly the ones in its own class. 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. Interestingly some of these happen to be orthogonal MEP (OMEP), for example we have constructed an OMEP for a $3^{30}$ experiment on 64 runs. Further, many of the `inter-class orthogonal' MEPs are `almost orthogonal' in the sense that each factor is orthogonal to all others except possibly one. In many of the other MEPs factors are "orthogonal through another factor", thus leading to simplification in the analysis. Plans of small size ($\leq 15$ runs) are also constructed by ad-hoc methods. Finally, we present a user-friendly computational method for analysing data obtained from any general factorial design.