Decomposition tables for experiments. II. Two--one randomizations

TL;DR

This paper explores the structure of pairs of randomizations in experimental design, providing methods to construct orthogonal decompositions of data spaces for evaluating and comparing complex experimental designs.

Contribution

It introduces a framework for analyzing pairs of randomizations that are not in a chain, and offers guidelines for selecting appropriate multiple randomization strategies.

Findings

Provides a method to establish decomposition tables for complex randomizations

Offers criteria for choosing between different multiple randomization types

Enhances understanding of the relationships and degrees of freedom in experimental structures

Abstract

We investigate structure for pairs of randomizations that do not follow each other in a chain. These are unrandomized-inclusive, independent, coincident or double randomizations. This involves taking several structures that satisfy particular relations and combining them to form the appropriate orthogonal decomposition of the data space for the experiment. We show how to establish the decomposition table giving the sources of variation, their relationships and their degrees of freedom, so that competing designs can be evaluated. This leads to recommendations for when the different types of multiple randomization should be used.