The Optimal Design of Blocked and Split-Plot Experiments

by Peter Goos





Contents

Summary

Book reviews

Fortran 77 programs and sample input files

Erratum

Order the book

The author




Summary

Chapter 1: Introduction

Chapter 1 is an introductory chapter in which a broad overview of the experimental design literature is given. Special attention is given to the basics of the optimal design approach, as well as to the standard response surface designs and to categorical designs.

Chapter 2: Advanced topics in optimal design

In Chapter 2, a concise overview is given of the optimal design literature, as far as it is concerned with the design of experiments with nonhomogeneous error variance and correlated observations. The topic of blocking response surface designs receives elaborate attention as well. It is shown that orthogonal blocking is an optimal blocking strategy when the block effects are assumed to be fixed.

Chapter 3: Compound symmetric error structure

In Chapter 3, the desing problems considered in this thesis are described in detail and an appropriate statistical model is introduced. It is shown that the variance-covariance structure of the statistical model is compound symmetric. The chapter also contains sections on the analysis of blocked and split-plot designs and on the equivalence of ordinary least squares (OLS) and generalized least squares (GLS). Finally, it is shown that the asymptotic variance-covariance matrix is a good approximation to the finite sample one.

Chapter 4: Optimal designs in the presence of random block effects

In Chapter 4, D-optimal designs for blocked experiments are computed. The first part of the chapter is devoted to optimal designs that do not depend on the degree of correlation. The general case, in which the optimal designs depend on the degree of correlation, receives attention in the second part of the chapter. A design construction algorithm and an adjustment algorithm to improve the resulting designs are developed. It is also shown that orthogonal blocking is an optimal blocking strategy when the block effects are assumed to be random and the block sizes are homogeneous.

Chapter 5: Optimal designs for quadratic regression on one variable and blocks of size two

In this chapter, the design of an optometry experiment is examined in detail. This chapter provides useful insights in the relationship between discrete designs and continuous designs and in the relationship between random and fixed blocks.

Chapter 6: Constrained split-plot designs

In Chapter 6, D-optimal split-plot designs are computed for experiments with a prespecified number of whole plots and prespecified whole plot sizes. In this chapter, it is shown that an optimal design strategy is to arrange the observations such that the resulting design is crossed. It is also shown that two-level factorial and fractional factorial designs are D-optimal in the class of crossed split-plot designs.

Chapter 7: Optimal split-plot designs in the presence of hard-to-change factors

In this chapter, D-optimal designs are derived under the assumption that the levels of the hard-to-change or whole plot factors are changed as little as possible. As a result, the designs computed in this chapter possess only one whole plot per whole plot factor level. It is shown that split-plot designs sometimes provide better estimates than completely randomized designs.

Chapter 8: Optimal split-plot designs

In Chapter 8, it is shown that split-plot experiments become more efficient when their number of whole plots is increased. The resulting split-plot designs are far more efficient than completely randomized designs. As a result, split-plot designs are not only easier to carry out, but they are also statistically more efficient than a completely randomized design.

Chapter 9: Two-level factorial and fractional factorial designs

In this chapter, it is shown how two-level factorial and fractional factorial designs can be used in blocked and split-plot experiments. A review is given of the recent literature on this topic. For that purpose, the concept of minimum aberration is introduced.

Chapter 10: Summary and future research




Book reviews

Review in Journal of the American Statistical Association (JASA)
(by Timothy J. Robinson, March 2004)

Review in Biometrics
(by Alexander N. Donev, June 2004)

Review in Journal of Applied Statistics
(by Philip Prescott)




Fortran 77 programs


Blocked experiments

BLOCK.FOR / BLOCK.EXE

This program computes the best possible designs for a blocked experiment. The search is over a finite grid of candidate points, which must be specified in the input. The default set of candidate points assumes a hypercubic design region and uses the points of a full factorial design as the set of candidate points. Appendix A of Chapter 4 contains a concise overview of the program.

Input and Output Files


* * *


BLOCKAA.FOR / BLOCKAA.EXE

This program combines the program BLOCK.FOR and the adjustment algorithm (AA) described in Section 4.7.3. The adjustment algorithm is outlined in Appendix B of Chapter 4. The format of the input files is identical to that of the program BLOCK.FOR. Note however that the adjustment algorithm is not useful when the experimental factors are qualitative.

Input and Output Files


* * *


Split-plot experiments

SPD1.FOR / SPD1.EXE

This program computes the best possible design for a split-plot experiment with a prespecified number of whole plots and prespecified whole plot sizes. This design problem is tackled in Chapter 6. The search for the optimal design is over a finite grid of candidate points, which must be specified in the input. The default set of candidate points assumes a hypercubic design region and uses the points of a full factorial design as the set of candidate points. Appendix A of Chapter 6 contains a brief description of the program.

Input and Output Files


* * *


SPD2.FOR / SPD2.EXE

This program computes the best possible design for a split-plot experiment under the assumption that the levels of the whole plot factors are changed as little as possible. The search is over a finite grid of candidate points, which must be specified in the input. The default set of candidate points assumes a hypercubic design region and uses the points of a full factorial design as the set of candidate points. Appendix A of Chapter 7 contains a concise overview of the program.

Input and Output Files


* * *


SPD3.FOR / SPD3.EXE

The program SPD3.FOR computes the best possible split-plot designs when no restrictions are imposed on the number of whole plots or on the whole plot sizes. Nevertheless, the program has an option to specify an upper bound for the number of whole plots. The search for the optimal design is over a finite grid of candidate points, which must be specified in the input. The default set of candidate points assumes a hypercubic design region and uses the points of a full factorial design as the set of candidate points. The appendix to Chapter 8 contains an outline of the design construction algorithm.

Input and Output Files




Order the book

You can order the book online via the Springer website, www.springer-ny.com for North America and www.springer.de for Europe and the rest of the world. Alternatively, you can order the book via Amazon, or via Barnes & Noble. In order to find the cheapest alternative, AllBookstores.com or BookFinder might be able to help you.

If you would like to order the book in your local bookshop, it is helpful to mention the ISBN number 0-387-95515-1.


The author




Peter Goos is a Professor at the University of Antwerp, where he teaches basic statistics to economics students. He also teaches Total Quality Management at the International School of Management in St.-Petersburg (Russia) and is involved in executive training programs. He obtained a Ph.D. in Applied Economics from the Katholieke Universiteit Leuven (2001). Dr. Goos wote his dissertation, as well as a number of methodological articles, on various aspects of designed experiments.

A more detailed outline of his career can be found at his website http://users.chello.be/peter.goos/. Dutch speaking people might be interested in visiting his page at the University of Antwerp: http://www.ua.ac.be/peter.goos/