A First Course In Design And Analysis Of Experiments

Transcription

A First Course inDesign and Analysisof Experiments

A First Course inDesign and Analysisof ExperimentsGary W. OehlertUniversity of Minnesota

Cover design by Victoria TomaselliCover illustration by Peter HamlinMinitab is a registered trademark of Minitab, Inc.SAS is a registered trademark of SAS Institute, Inc.S-Plus is a registered trademark of Mathsoft, Inc.Design-Expert is a registered trademark of Stat-Ease, Inc.Library of Congress Cataloging-in-Publication Data.Oehlert, Gary W.A first course in design and analysis of experiments / Gary W. Oehlert.p. cm.Includes bibligraphical references and index.ISBN 0-7167-3510-51. Experimental DesignI. TitleQA279.O34 2000519.5—dc2199-059934Copyright c 2010 Gary W. Oehlert. All rights reserved.This work is licensed under a “Creative Commons” license. Briefly, you are free tocopy, distribute, and transmit this work provided the following conditions are met:1. You must properly attribute the work.2. You may not use this work for commercial purposes.3. You may not alter, transform, or build upon this work.A complete description of the license may be found /.

For Beckywho helped me all the way throughand for Christie and Ericawho put up with a lot while it was getting done

ContentsPreface123Introduction1.1Why Experiment? . . . . . . . .1.2Components of an Experiment .1.3Terms and Concepts . . . . . . .1.4Outline . . . . . . . . . . . . .1.5More About Experimental Units1.6More About Responses . . . . .xvii.11457810.13141617192025262728Completely Randomized Designs3.1Structure of a CRD . . . . . . . . . . . . . . . . . . . . .3.2Preliminary Exploratory Analysis . . . . . . . . . . . . .3.3Models and Parameters . . . . . . . . . . . . . . . . . . .31313334.Randomization and Design2.1Randomization Against Confounding . . . . . . . . . .2.2Randomizing Other Things . . . . . . . . . . . . . . . .2.3Performing a Randomization . . . . . . . . . . . . . . .2.4Randomization for Inference . . . . . . . . . . . . . . .2.4.1 The paired t-test . . . . . . . . . . . . . . . . .2.4.2 Two-sample t-test . . . . . . . . . . . . . . . .2.4.3 Randomization inference and standard inference2.5Further Reading and Extensions . . . . . . . . . . . . .2.6Problems . . . . . . . . . . . . . . . . . . . . . . . . .

ting Parameters . . . . . . . . . . . .Comparing Models: The Analysis of VarianceMechanics of ANOVA . . . . . . . . . . . .Why ANOVA Works . . . . . . . . . . . . .Back to Model Comparison . . . . . . . . . .Side-by-Side Plots . . . . . . . . . . . . . .Dose-Response Modeling . . . . . . . . . . .Further Reading and Extensions . . . . . . .Problems . . . . . . . . . . . . . . . . . . .Looking for Specific Differences—Contrasts4.1Contrast Basics . . . . . . . . . . . . .4.2Inference for Contrasts . . . . . . . . .4.3Orthogonal Contrasts . . . . . . . . . .4.4Polynomial Contrasts . . . . . . . . . .4.5Further Reading and Extensions . . . .4.6Problems . . . . . . . . . . . . . . . .394445525254555860.65656871737575Multiple Comparisons775.1Error Rates . . . . . . . . . . . . . . . . . . . . . . . . . 785.2Bonferroni-Based Methods . . . . . . . . . . . . . . . . . 815.3The Scheffé Method for All Contrasts . . . . . . . . . . . 855.4Pairwise Comparisons . . . . . . . . . . . . . . . . . . . . 875.4.1 Displaying the results . . . . . . . . . . . . . . . 885.4.2 The Studentized range . . . . . . . . . . . . . . . 895.4.3 Simultaneous confidence intervals . . . . . . . . . 905.4.4 Strong familywise error rate . . . . . . . . . . . . 925.4.5 False discovery rate . . . . . . . . . . . . . . . . 965.4.6 Experimentwise error rate . . . . . . . . . . . . . 975.4.7 Comparisonwise error rate . . . . . . . . . . . . . 985.4.8 Pairwise testing reprise . . . . . . . . . . . . . . 985.4.9 Pairwise comparisons methods that do not controlcombined Type I error rates . . . . . . . . . . . . 985.4.10 Confident directions . . . . . . . . . . . . . . . . 100

CONTENTS5.55.65.75.85.95.1067Comparison with Control or the Best5.5.1 Comparison with a control .5.5.2 Comparison with the best .Reality Check on Coverage Rates .A Warning About Conditioning . . .Some Controversy . . . . . . . . . .Further Reading and Extensions . .Problems . . . . . . . . . . . . . .ix.Checking Assumptions6.1Assumptions . . . . . . . . . . . . . . . . . .6.2Transformations . . . . . . . . . . . . . . . . .6.3Assessing Violations of Assumptions . . . . . .6.3.1 Assessing nonnormality . . . . . . . .6.3.2 Assessing nonconstant variance . . . .6.3.3 Assessing dependence . . . . . . . . .6.4Fixing Problems . . . . . . . . . . . . . . . . .6.4.1 Accommodating nonnormality . . . .6.4.2 Accommodating nonconstant variance6.4.3 Accommodating dependence . . . . .6.5Effects of Incorrect Assumptions . . . . . . . .6.5.1 Effects of nonnormality . . . . . . . .6.5.2 Effects of nonconstant variance . . . .6.5.3 Effects of dependence . . . . . . . . .6.6Implications for Design . . . . . . . . . . . . .6.7Further Reading and Extensions . . . . . . . .6.8Problems . . . . . . . . . . . . . . . . . . . .Power and Sample Size7.1Approaches to Sample Size Selection . .7.2Sample Size for Confidence Intervals . . .7.3Power and Sample Size for ANOVA . . .7.4Power and Sample Size for a Contrast . .7.5More about Units and Measurement 8158

xCONTENTS7.67.77.889Allocation of Units for Two Special Cases . . . . . . . . . 160Further Reading and Extensions . . . . . . . . . . . . . . 161Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 162Factorial Treatment Structure8.1Factorial Structure . . . . . . . . . . . . . . . . .8.2Factorial Analysis: Main Effect and Interaction .8.3Advantages of Factorials . . . . . . . . . . . . .8.4Visualizing Interaction . . . . . . . . . . . . . .8.5Models with Parameters . . . . . . . . . . . . . .8.6The Analysis of Variance for Balanced Factorials8.7General Factorial Models . . . . . . . . . . . . .8.8Assumptions and Transformations . . . . . . . .8.9Single Replicates . . . . . . . . . . . . . . . . .8.10 Pooling Terms into Error . . . . . . . . . . . . .8.11 Hierarchy . . . . . . . . . . . . . . . . . . . . .8.12 Problems . . . . . . . . . . . . . . . . . . . . .165165167170171175179182185186191192197A Closer Look at Factorial Data9.1Contrasts for Factorial Data . . . . . . . . . . . . . . . .9.2Modeling Interaction . . . . . . . . . . . . . . . . . . .9.2.1 Interaction plots . . . . . . . . . . . . . . . . .9.2.2 One-cell interaction . . . . . . . . . . . . . . .9.2.3 Quantitative factors . . . . . . . . . . . . . . .9.2.4 Tukey one-degree-of-freedom for nonadditivity .9.3Further Reading and Extensions . . . . . . . . . . . . .9.4Problems . . . . . . . . . . . . . . . . . . . . . . . . 10 Further Topics in Factorials10.1 Unbalanced Data . . . . . . . . . . . . . .10.1.1 Sums of squares in unbalanced data10.1.2 Building models . . . . . . . . . .10.1.3 Testing hypotheses . . . . . . . . .10.1.4 Empty cells . . . . . . . . . . . . .10.2 Multiple Comparisons . . . . . . . . . . .

CONTENTS10.310.410.510.6Power and Sample Size . . . . .Two-Series Factorials . . . . . .10.4.1 Contrasts . . . . . . . .10.4.2 Single replicates . . . .Further Reading and ExtensionsProblems . . . . . . . . . . . .xi.11 Random Effects11.1 Models for Random Effects . . . . . . . . . . .11.2 Why Use Random Effects? . . . . . . . . . . .11.3 ANOVA for Random Effects . . . . . . . . . .11.4 Approximate Tests . . . . . . . . . . . . . . .11.5 Point Estimates of Variance Components . . . .11.6 Confidence Intervals for Variance Components11.7 Assumptions . . . . . . . . . . . . . . . . . .11.8 Power . . . . . . . . . . . . . . . . . . . . . .11.9 Further Reading and Extensions . . . . . . . .11.10 Problems . . . . . . . . . . . . . . . . . . . .12 Nesting, Mixed Effects, and Expected Mean Squares12.1 Nesting Versus Crossing . . . . . . . . . . . .12.2 Why Nesting? . . . . . . . . . . . . . . . . . .12.3 Crossed and Nested Factors . . . . . . . . . . .12.4 Mixed Effects . . . . . . . . . . . . . . . . . .12.5 Choosing a Model . . . . . . . . . . . . . . . .12.6 Hasse Diagrams and Expected Mean Squares .12.6.1 Test denominators . . . . . . . . . . .12.6.2 Expected mean squares . . . . . . . .12.6.3 Constructing a Hasse diagram . . . . .12.7 Variances of Means and Contrasts . . . . . . .12.8 Unbalanced Data and Random Effects . . . . .12.9 Staggered Nested Designs . . . . . . . . . . .12.10 Problems . . . . . . . . . . . . . . . . . . . 275.279279283283285288289290293296298304306307

xiiCONTENTS13 Complete Block Designs31513.1Blocking . . . . . . . . . . . . . . . . . . . . . . . . . . . 31513.2The Randomized Complete Block Design . . . . . . . . . 31613.2.1 Why and when to use the RCB . . . . . . . . . . 31813.2.2 Analysis for the RCB . . . . . . . . . . . . . . . 31913.2.3 How well did the blocking work? . . . . . . . . . 32213.2.4 Balance and missing data . . . . . . . . . . . . . 32413.3Latin Squares and Related Row/Column Designs . . . . . 32413.3.1 The crossover design . . . . . . . . . . . . . . . . 32613.3.2 Randomizing the LS design . . . . . . . . . . . . 32713.3.3 Analysis for the LS design . . . . . . . . . . . . . 32713.3.4 Replicating Latin Squares . . . . . . . . . . . . . 33013.3.5 Efficiency of Latin Squares . . . . . . . . . . . . 33513.3.6 Designs balanced for residual effects . . . . . . . 33813.4Graeco-Latin Squares . . . . . . . . . . . . . . . . . . . . 34313.5Further Reading and Extensions . . . . . . . . . . . . . . 34413.6Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 34514 Incomplete Block Designs14.1357Balanced Incomplete Block Designs . . . . . . . . . . . . 35814.1.1 Intrablock analysis of the BIBD . . . . . . . . . . 36014.1.2 Interblock information . . . . . . . . . . . . . . . 36414.2Row and Column Incomplete Blocks . . . . . . . . . . . . 36814.3Partially Balanced Incomplete Blocks . . . . . . . . . . . 37014.4Cyclic Designs . . . . . . . . . . . . . . . . . . . . . . . 37214.5Square, Cubic, and Rectangular Lattices . . . . . . . . . . 37414.6Alpha Designs . . . . . . . . . . . . . . . . . . . . . . . . 37614.7Further Reading and Extensions . . . . . . . . . . . . . . 37814.8Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 379

CONTENTS15 Factorials in Incomplete Blocks—Confounding15.1xiii387Confounding the Two-Series Factorial . . . . . . . . . . . 38815.1.1 Two blocks . . . . . . . . . . . . . . . . . . . . . 38915.1.2 Four or more blocks . . . . . . . . . . . . . . . . 39215.1.3 Analysis of an unreplicated confounded two-series 39715.1.4 Replicating a confounded two-series . . . . . . . 39915.1.5 Double confounding . . . . . . . . . . . . . . . . 40215.2Confounding the Three-Series Factorial . . . . . . . . . . 40315.2.1 Building the design . . . . . . . . . . . . . . . . 40415.2.2 Confounded effects . . . . . . . . . . . . . . . . 40715.2.3 Analysis of confounded three-series . . . . . . . . 40815.3Further Reading and Extensions . . . . . . . . . . . . . . 40915.4Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 41016 Split-Plot Designs41716.1What Is a Split Plot? . . . . . . . . . . . . . . . . . . . . 41716.2Fancier Split Plots . . . . . . . . . . . . . . . . . . . . . . 41916.3Analysis of a Split Plot . . . . . . . . . . . . . . . . . . . 42016.4Split-Split Plots . . . . . . . . . . . . . . . . . . . . . . . 42816.5Other Generalizations of Split Plots . . . . . . . . . . . . 43416.6Repeated Measures . . . . . . . . . . . . . . . . . . . . . 43816.7Crossover Designs . . . . . . . . . . . . . . . . . . . . . 44116.8Further Reading and Extensions . . . . . . . . . . . . . . 44116.9Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 44217 Designs with Covariates45317.1The Basic Covariate Model . . . . . . . . . . . . . . . . . 45417.2When Treatments Change Covariates . . . . . . . . . . . . 46017.3Other Covariate Models . . . . . . . . . . . . . . . . . . . 46217.4Further Reading and Extensions . . . . . . . . . . . . . . 46617.5Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 466

xivCONTENTS18 Fractional Factorials47118.1Why Fraction? . . . . . . . . . . . . . . . . . . . . . . . . 47118.2Fractioning the Two-Series . . . . . . . . . . . . . . . . . 47218.3Analyzing a 2k q . . . . . . . . . . . . . . . . . . . . . . 47918.4Resolution and Projection . . . . . . . . . . . . . . . . . . 48218.5Confounding a Fractional Factorial . . . . . . . . . . . . . 48518.6De-aliasing . . . . . . . . . . . . . . . . . . . . . . . . . 48518.7Fold-Over . . . . . . . . . . . . . . . . . . . . . . . . . . 48718.8Sequences of Fractions . . . . . . . . . . . . . . . . . . . 48918.9Fractioning the Three-Series . . . . . . . . . . . . . . . . 48918.10 Problems with Fractional Factorials . . . . . . . . . . . . 49218.11 Using Fractional Factorials in Off-Line Quality Control . . 49318.11.1 Designing an off-li

A First Course in Design and Analysis of Experiments Gary W. Oehlert University of Minnesota