Table Of ContentSpringer Series in Statistics
Advisors:
P.Bickel,P.Diggle,S.Fienberg,U.Gather,
I.Olkin,S.Zeger
Springer Series in Statistics
Alho/Spencer:Statistical Demography and Forecasting.
Andersen/Borgan/Gill/Keiding:Statistical Models Based on Counting Processes.
Atkinson/Riani:Robust Diagnostic Regression Analysis.
Atkinson/Riani/Cerioli:Exploring Multivariate Data with the Forward Search.
Berger:Statistical Decision Theory and Bayesian Analysis,2nd edition.
Borg/Groenen:Modern Multidimensional Scaling:Theory and Applications,
2nd edition.
Brockwell/Davis:Time Series:Theory and Methods,2nd edition.
Bucklew:Introduction to Rare Event Simulation.
Cappé/Moulines/Rydén:Inference in Hidden Markov Models.
Chan/Tong:Chaos:A Statistical Perspective.
Chen/Shao/Ibrahim:Monte Carlo Methods in Bayesian Computation.
Coles:An Introduction to Statistical Modeling of Extreme Values.
David/Edwards:Annotated Readings in the History of Statistics.
Devroye/Lugosi:Combinatorial Methods in Density Estimation.
Efromovich:Nonparametric Curve Estimation:Methods,Theory,and Applications.
Eggermont/LaRiccia:Maximum Penalized Likelihood Estimation,Volume I:Density
Estimation.
Fahrmeir/Tutz:Multivariate Statistical Modelling Based on Generalized Linear
Models,2nd edition.
Fan/Yao:Nonlinear Time Series:Nonparametric and Parametric Methods.
Farebrother:Fitting Linear Relationships:A History of the Calculus of Observations
1750-1900.
Federer:Statistical Design and Analysis for Intercropping Experiments,Volume I:
Two Crops.
Federer:Statistical Design and Analysis for Intercropping Experiments,Volume II:
Three or More Crops.
Ferraty/Vieu:Nonparametric Functional Data Analysis:Models,Theory,Applications,
and Implementation
Ghosh/Ramamoorthi:Bayesian Nonparametrics.
Glaz/Naus/Wallenstein:Scan Statistics.
Good:Permutation Tests:Parametric and Bootstrap Tests of Hypotheses,3rd edition.
Gouriéroux:ARCH Models and Financial Applications.
Gu:Smoothing Spline ANOVA Models.
Györfi/Kohler/Krzyz•ak/Walk:A Distribution-Free Theory of Nonparametric
Regression.
Haberman:Advanced Statistics,Volume I:Description of Populations.
Hall:The Bootstrap and Edgeworth Expansion.
Härdle:Smoothing Techniques:With Implementation in S.
Harrell:Regression Modeling Strategies:With Applications to Linear Models,Logistic
Regression,and Survival Analysis.
Hart:Nonparametric Smoothing and Lack-of-Fit Tests.
Hastie/Tibshirani/Friedman:The Elements of Statistical Learning:Data Mining,
Inference,and Prediction.
Hedayat/Sloane/Stufken:Orthogonal Arrays:Theory and Applications.
Heyde:Quasi-Likelihood and its Application:A General Approach to Optimal
Parameter Estimation.
(continued after index)
Rahul Mukerjee
C.F. Jeff Wu
A Modern Theory of
Factorial Designs
Rahul Mukerjee C.F.Jeff Wu
Indian Institute ofManagement School ofIndustrial and Systems Engineering
Calcutta Georgia Institute of Technology
Joka,Diamond Harbour Road Atlanta,GA 30332-0205
Kolkata 700 104 USA
India [email protected]
[email protected]
Library ofCongress Control Number:2005939038
ISBN-10:0-387-31991-3
ISBN-13:978-0387-31991-9
Printed on acid-free paper.
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To my father and wife,
and to the memory of my mother, RM
To my wife, CFJW
Springer Series in Statistics
(continued from p.ii)
Huet/Bouvier/Poursat/Jolivet:Statistical Tools for Nonlinear Regression:A Practical
Guide with S-PLUS and R Examples,2nd edition.
Ibrahim/Chen/Sinha:Bayesian Survival Analysis.
Jolliffe:Principal Component Analysis,2nd edition.
Knottnerus:Sample Survey Theory:Some Pythagorean Perspectives.
Kolen/Brennan:Test Equating:Methods and Practices.
Kotz/Johnson (Eds.):Breakthroughs in Statistics Volume I.
Kotz/Johnson (Eds.):Breakthroughs in Statistics Volume II.
Kotz/Johnson (Eds.):Breakthroughs in Statistics Volume III.
Küchler/Sørensen:Exponential Families of Stochastic Processes.
Kutoyants:Statistical Influence for Ergodic Diffusion Processes.
Lahiri:Resampling Methods for Dependent Data.
Le Cam:Asymptotic Methods in Statistical Decision Theory.
Le Cam/Yang:Asymptotics in Statistics:Some Basic Concepts,2nd edition.
Liu:Monte Carlo Strategies in Scientific Computing.
Longford:Models for Uncertainty in Educational Testing.
Manski:Partial Identification of Probability Distributions.
Mielke/Berry:Permutation Methods:A Distance Function Approach.
Molenberghs/Verbeke:Models for Discrete Longitudinal Data.
Mukerjee/Wu:A Modern Theory of Factorial Designs.
Nelsen:An Introduction to Copulas.2nd edition
Pan/Fang:Growth Curve Models and Statistical Diagnostics.
Parzen/Tanabe/Kitagawa:Selected Papers of Hirotugu Akaike.
Politis/Romano/Wolf:Subsampling.
Ramsay/Silverman:Applied Functional Data Analysis:Methods and Case Studies.
Ramsay/Silverman:Functional Data Analysis,2nd edition.
Rao/Toutenburg:Linear Models:Least Squares and Alternatives.
Reinsel:Elements of Multivariate Time Series Analysis.2nd edition.
Rosenbaum:Observational Studies,2nd edition.
Rosenblatt:Gaussian and Non-Gaussian Linear Time Series and Random Fields.
Särndal/Swensson/Wretman:Model Assisted Survey Sampling.
Santner/Williams/Notz:The Design and Analysis of Computer Experiments.
Schervish:Theory of Statistics.
Shao/Tu:The Jackknife and Bootstrap.
Simonoff:Smoothing Methods in Statistics.
Singpurwalla and Wilson:Statistical Methods in Software Engineering:Reliability and
Risk.
Small:The Statistical Theory of Shape.
Sprott:Statistical Inference in Science.
Stein:Interpolation of Spatial Data:Some Theory for Kriging.
Taniguchi/Kakizawa:Asymptotic Theory of Statistical Inference for Time Series.
Tanner:Tools for Statistical Inference:Methods for the Exploration of Posterior
Distributions and Likelihood Functions,3rd edition.
Tillé:Sampling Algorithms.
van der Laan:Unified Methods for Censored Longitudinal Data and Causality.
van der Vaart/Wellner:Weak Convergence and Empirical Processes:With Applications
to Statistics.
Verbeke/Molenberghs:Linear Mixed Models for Longitudinal Data.
Weerahandi:Exact Statistical Methods for Data Analysis.
West/Harrison:Bayesian Forecasting and Dynamic Models,2nd edition.
To my father and wife, and to
the memory of my mother, RM
To my wife, CFJW
Preface
Factorial design has always played a prominent role in the theory and prac-
tice of experimental design. It allows efficient and economic experimentation
with multiple input variables and has been successfully used in a wide range
of applications. Much research has been done and texts have been written
on factorial design in the 70 years since its inception. For economic reasons,
fractionalfactorialshavebeenextremelypopular,especiallywhenthenumber
of factors is large and the runs are expensive. The first and perhaps the most
important issue faced by experimenters is the choice of a fractional factorial
design. Given the long history of factorial design, an “optimality” theory for
design selection should have emerged long ago. Surprisingly, the first serious
attemptinthisdirectionwasmadeonlyintheearlysixtieswiththenotionof
resolution. It became apparent later that this notion was not discriminating
asacriterionfordesignselection.Equallysurprisingly,ittookalmostanother
20 years to see the birth of the minimum aberration (MA) criterion, which
has since become the major criterion for selecting fractional factorial designs.
Once the importance of the MA criterion was recognized in the late eighties,
research on the theory and algorithms for finding MA and related designs
has grown rapidly in the last 15 years. Besides building and improving upon
existing techniques in projective geometry and coding theory, such research
has led to the development of novel techniques like complementary designs
and efficient search algorithms. Factorial designs with increasing complexity
in the underlying structure (e.g., mixed-levels, blocking, split-plots, and ro-
bust parameter design) have also received considerable attention. A detailed
description of this history and an account of the nature and contents of the
book appear in Chapter 1.
In 2000, the present authors felt that the time was ripe to start planning
andwritingamodernbookonfactorialdesignswiththeMAperspective.Such
a book should contain the major theoretical tools and results, and tables of
optimalorefficientdesignsavailable intheliterature.Ittookseveralvisitsby
RM to CFJW at the Department of Statistics, University of Michigan, and
later at the School of Industrial and Systems Engineering, Georgia Institute
viii Preface
of Technology, to complete the project. The hospitality and support of both
institutions are gratefully acknowledged.
All the mathematical and theoretical prerequisites are given in Chapter 2
ofthebook.Formathematicallyorientedreaders,littleadditionalbackground
isrequiredtoreadandfollowthelogicandderivationsinthebook,sinceeach
new concept is accompanied with a brief statistical explanation, justification,
or reference. For statistically oriented readers, background in basic design
and analysis of experiments will help in understanding and appreciating the
significance and impact of the concepts and results in the book. The book
can be used as a text for a graduate course in design theory in statistics or
mathematics programs and also as a reference text for a graduate course in
combinatorial mathematics. We hope that it will be a useful reference for
design researchers in general. The extensive collection of design tables should
endear it to practitioners and others interested in the use of factorial designs.
During the course of writing this book, we have received comments and
assistance from colleagues, and former and current students, including Aloke
Dey,TirthankarDasgupta,XingweiDeng,GregDyson,YingHung,Abhyuday
Mandal,ZhiguangQian,XiangguiQu,andHongquanXu.Wesincerelythank
all of them. The writing of the book was supported in part by NSF grant
DMS-0072489. RM also received support for this project from the Centre
for Management and Development Studies, Indian Institute of Management
Calcutta.
Rahul Mukerjee C.F. Jeff Wu
Calcutta Atlanta
Contents
1 Introduction and Overview ................................ 1
1.1 Prologue ............................................... 1
1.2 Why This Book? ........................................ 3
1.3 What Is in the Book? .................................... 4
1.4 Beyond the Book........................................ 6
2 Fundamentals of Factorial Designs ......................... 9
2.1 Factorial Effects......................................... 9
2.2 Kronecker Product Formulation for Factorial Effects ......... 12
2.3 ARepresentationforFactorialEffectsinSymmetricalFactorials 18
2.4 Regular Fractions ....................................... 24
2.5 Optimality Criteria: Resolution and Minimum Aberration .... 32
2.6 Connection with Orthogonal Arrays ....................... 35
2.7 Connection with Finite Projective Geometry................ 39
2.8 Algebraic Coding Theory................................. 45
3 Two-Level Fractional Factorial Designs .................... 49
3.1 Basic Definitions Revisited ............................... 49
3.2 Minimum Aberration 2n−k Designs with k ≤4 .............. 52
3.3 Minimum Aberration Designs via Complementary Designs.... 57
3.4 Clear Effects and the MaxC2 Criterion ..................... 63
3.5 Description and Use of the Two-Level Design Tables ......... 67
4 Fractional Factorial Designs: General Case................. 85
4.1 Three-Level Designs ..................................... 85
4.2 Minimum Aberration sn−k Designs with Small k ............ 87
4.3 A General Result on Complementary Designs ............... 91
4.4 MinimumAberrationsn−k DesignsviaComplementary Designs 98
4.5 Description and Use of the Three-Level Design Tables........102
