The Annals of Statistics

Decomposition tables for experiments I. A chain of randomizations

C. J. Brien and R. A. Bailey

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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.

Article information

Ann. Statist. Volume 37, Number 6B (2009), 4184-4213.

First available in Project Euclid: 23 October 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62J10: Analysis of variance and covariance
Secondary: 62K99: None of the above, but in this section

Analysis of variance balance decomposition table design of experiments efficiency factor multiphase experiments multitiered experiments orthogonal decomposition pseudofactor structure tier


Brien, C. J.; Bailey, R. A. Decomposition tables for experiments I. A chain of randomizations. Ann. Statist. 37 (2009), no. 6B, 4184--4213. doi:10.1214/09-AOS717.

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