Decomposition tables for experiments I. A chain of randomizations

TL;DR

This paper develops a mathematical framework for analyzing complex experimental designs involving chains of randomizations, enabling the systematic decomposition of data spaces and evaluation of design properties.

Contribution

It introduces methods to combine orthogonal decompositions for single and multiple randomizations, providing a structured approach to evaluate complex experimental designs.

Findings

Decomposition tables effectively summarize sources of variation.

Properties of complex designs derive from simpler component designs.

Framework facilitates comparison of competing experimental designs.

Abstract

One aspect of evaluating the design for an experiment is the discovery of the relationships between subspaces of the data space. Initially we establish the notation and methods for evaluating an experiment with a single randomization. Starting with two structures, or orthogonal decompositions of the data space, we describe how to combine them to form the overall decomposition for a single-randomization experiment that is ``structure balanced.'' The relationships between the two structures are characterized using efficiency factors. The decomposition is encapsulated in a decomposition table. Then, for experiments that involve multiple randomizations forming a chain, we take several structures that pairwise are structure balanced and combine them to establish the form of the orthogonal decomposition for the experiment. In particular, it is proven that the properties of the design for such an experiment are derived in a straightforward manner from those of the individual designs. We show how to formulate an extended decomposition table giving the sources of variation, their relationships and their degrees of freedom, so that competing designs can be evaluated.