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3 Planning, Construction, and Statistical Analysis of Comparative Experiments

4 WILEY SERIES IN PROBABILITY AND STATISTICS Established by WALTER A. SHEWHART and SAMUEL S. WILKS Editors: David J. Balding, Noel A. C. Cressie, Nicholas I. Fisher, Iain M. Johnstone, J. B. Kadane, Geert Molenberghs, Louise M. Ryan, David W. Scott, Adrian F. M. Smith, JozefL. Teugels Editors Emeriti: Vic Barnett, J. Stuart Hunter, David G. Kendall A complete list of the titles in this series appears at the end of this volume.

5 Planning, Construction, and Statistical Analysis of Comparative Experiments FRANCIS G. GIESBRECHT MARCIA L. GUMPERTZ,WILEY~ INTERSCIENCE A JOHN WILEY & SONS, INC., PUBLICATION

6 Copyright 2004 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, , fax , or on the web at Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., Ill River Street, Hoboken, NJ 07030, (201) , fax (201) Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services please contact our Customer Care Department within the U.S. at , outside the U.S. at or fax Wiley also publishes its books in a variety of electronic formats. Some content that appears in print, however, may not be available in electronic format. Library of Congress Cataloging-in-Publication Data: Giesbrecht, Francis G., Planning, construction, and statistical analysis of comparative experiments / Francis G. Giesbrecht, Marcia L. Gumpertz. p. cm. (Wiley series in probability and statistics) Includes bibliographical references and index. ISBN (acid-free paper) 1. Experimental design. I. Gumpertz, Marcia L., II. Title. III. Series. QA279.G dc

7 Contents Preface xüi 1. Introduction Role of Statistics in Experiment Design Organization of This Book Representativeness and Experimental Units Replication and Handling Unexplained Variability Randomization: Why and How Ethical Considerations 10 Review Exercises Completely Randomized Design Introduction Completely Randomized Design Assumption of Additivity Factorial Treatment Combinations Nested Factors 23 Review Exercises Linear Models for Designed Experiments Introduction Linear Model Principle of Least Squares Parameterizations for Row-Column Models 37 Review Exercises 41 Appendix 3A: Linear Combinations of Random Variables 43 Appendix 3B: Simulating Random Samples 44

8 CONTENTS Testing Hypotheses and Determining Sample Size Introduction Testing Hypotheses in Linear Models with Normally Distributed Errors Kruskal-Wallis Test Randomization Tests Power and Sample Size Sample Size for Binomial Proportions Confidence Interval Width and Sample Size Alternative Analysis: Selecting and Screening 74 Review Exercises 78 Methods of Reducing Unexplained Variation Randomized Complete Block Design Blocking Formal Statistical Analysis for the RCBD Models and Assumptions: Detailed Examination Statistical Analysis When Data Are Missing in an RCBD Analysis of Covariance 109 Review Exercises 112 Appendix 5A: Interaction of a Random Block Effect and a Fixed Treatment Effect 116 Latin Squares Introduction Formal Structure of Latin Squares Combining Latin Squares Graeco-Latin Squares and Orthogonal Latin Squares Some Special Latin Squares and Variations on Latin Squares Frequency Squares Youden Square 145 Review Exercises 150 Appendix 6A: Some Standard Latin Squares 153 Appendix 6B: Mutually Orthogonal Latin Squares 155 Appendix 6C: Possible Youden Squares 157 Split-Plot and Related Designs Introduction Background Material 158

9 CONTENTS vii 7.3 Examples of Situations That Lead to Split-Plots Statistical Analysis of Split-Plot Experiments Split-Split-Plot Experiments Strip-Plot Experiments Comments on Further Variations Analysis of Covariance in Split-Plots Repeated Measures 193 Review Exercises Incomplete Block Designs Introduction Efficiency of Incomplete Block Designs Distribution-Free Analysis for Incomplete Block Designs Balanced Incomplete Block Designs Lattice Designs Cyclic Designs «-Designs Other Incomplete Block Designs 231 Review Exercises 232 Appendix 8A: Catalog of Incomplete Block Designs Repeated Treatments Designs Introduction Repeated Treatments Design Model Construction of Repeated Treatments Designs Statistical Analysis of Repeated Treatments Design Data Carryover Design for Two Treatments Correlated Errors Designs for More Complex Models 267 Review Exercises Factorial Experiments: The 2 N System Introduction N Factorials General Notation for the 2 N System Analysis of Variance for 2 N Factorials Factorial Experiments: The 3 N System Introduction 288

10 viii CONTENTS x3 Factorial General System of Notation for the 3 N System Analysis of Experiments without Designed Error Terms Introduction Techniques That Look for Location Parameters Analysis for Dispersion Effects 304 Review Exercises Confounding Effects with Blocks Introduction Confounding 2 3 Factorials General Confounding Patterns Double Confounding N System Detailed Numerical Example: 3 3 Factorial in Blocks of Nine Confounding Schemes for the 3 4 System 337 Review Exercises Fractional Factorial Experiments Introduction Organization of This Chapter Fractional Replication in the 2 N System Resolution Constructing Fractional Replicates by Superimposing Factors Foldover Technique Franklin-Bailey Algorithm for Constructing Fractions Irregular Fractions of the 2 N System Fractional Factorials with Compromised Resolution A Caution About Minimum Aberration Direct Enumeration Approach to Constructing Plans Blocking in Small 2 N ' k Plans Fractional Replication in the 3^ System 383 Review Exercises 387 Appendix 14A: Minimum Aberration Fractions without Blocking 390 Appendix 14B: Minimum Aberration Fractions with Blocking 393

11 CONTENTS x 15 Response Surface Designs Introduction Basic Formulation Some Basic Designs Rotatability Statistical Analyses of Data from Response Surface Designs Blocking in Response Surface Designs Mixture Designs Optimality Criteria and Parametric Modeling Response Surfaces in Irregular Regions Searching the Operability Region for an Optimum Examination of an Experimental Problem 426 Review Exercise Plackett-Burman Hadamard Plans Introduction Hadamard Matrix Plackett-Burman Plans Hadamard Plans and Interactions Statistical Analyses Weighing Designs Projection Properties of Hadamard Plans Very Large Factorial Experiments General p N and Nonstandard Factorials Introduction Organization of This Chapter p N System with p Prime ^ System ^ System Using Pseudofactors at Two Levels ^ Factorial System Asymmetrical Factorials N x 3 M Plans Plans for Which Run Order Is Important Introduction Preliminary Concepts Trend-Resistant Plans 487

12 X CONTENTS 18.4 Generating Run Orders for Fractions Extreme Number of Level Changes Trend-Free Plans from Hadamard Matrices Extensions to More Than Two Levels Small One-at-a-Time Plans Comments Sequences of Fractions of Factorials Introduction Motivating Example Foldover Techniques Examined Augmenting a 2 4_1 Fraction Sequence Starting from a Seven-Factor Main Effect Plan Augmenting a One-Eighth Fraction of Adding Levels of a Factor Double Semifold Planned Sequences Sequential Fractions Factorial Experiments with Quantitative Factors: Blocking and Fractions Introduction Factors at Three Levels Factors at Four Levels Based on the 2 N System Pseudofactors and Hadamard Plans Box-Behnken Plans 553 Review Exercises 555 Appendix 20A: Box-Behnken Plans Supersaturated Plans Introduction Plans for Small Experiments Supersaturated Plans That Include an Orthogonal Base Model-Robust Plans Multistage Experiments Introduction Factorial Structures in Split-Plot Experiments Splitting on Interactions 573

13 CONTENTS xi 22.4 Factorials in Strip-Plot or Strip-Unit Designs General Comments on Strip-Unit Experiments Split-Lot Designs 590 Appendix 22A: Fractional Factorial Plans for Split-Plot Designs Orthogonal Arrays and Related Structures Introduction Orthogonal Arrays and Fractional Factorials Other Construction Methods Nearly Orthogonal Arrays Large Orthogonal Arrays Summary Factorial Plans Derived via Orthogonal Arrays Introduction Preliminaries Product Array Designs Block Crossed Arrays Compound Arrays Experiments on the Computer Introduction Stratified and Latin Hypercube Sampling Using Orthogonal Arrays for Computer Simulation Studies Demonstration Simulations 657 References 661 Index 677

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15 Preface This is a book based on years of teaching statistics at the graduate level and many more years of consulting with individuals conducting experiments in many disciplines. The object of the book is twofold. It is designed to be used as a textbook for students who have already had a solid course in statistical methods. It is also designed to be a handbook and reference for active researchers and statisticians who serve as consultants in the design of experiments. We encountered very early in our work that the subject is not linear in the sense that there is a unique order in which topics must be presented that topics A, B, and C must be presented in that order. All too often, we found that in one sense, A should be before B, and in another, B should be before A. Our aim is to give sufficient detail to show interrelationships, that is, how to reach destinations, yet not hide important points with too much clutter. We assume that the reader of this book is comfortable with simple sampling, multiple regression, r-tests, confidence intervals, and the analysis of variance at the level found in standard statistical methods textbooks. The demands for mathematics are limited. Simple algebra and matrix manipulation are sufficient. In our own teaching, we have found that we pick and choose material from various chapters. Chapters 1 to 5 and Chapter 7 provide a review of methods of statistical analysis and establish notation for the remainder of the book. Material selected from the remaining chapters then provides the backbone of an applied course in design of experiments. There is more material than can be covered in one semester. For a graduate level course in design of experiments targeted at a mix of statistics and non-statistics majors, one possible course outline covers Chapter 1, Sections 3.3 and 3.4, Chapter 4, Section 5.5, and Chapters 6 and 8 to 15. If the students have a very strong preparation in statistical methods, we suggest covering selections from Chapters 6 to 16 and 19 to 24, with just the briefest review of Chapters 1 to 5. We have also used material from Chapters 10 to 14, 16, and 19 to 24 for a more specialized course on factorial experiments for students interested in industrial quality control. In a sense, it would have been much easier to write a book that contained a series of chapters that would serve as a straightforward text. However, our aim is also to provide a handbook for the investigator planning a research program. xiii