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A First Course in Design and Analysis of Experiments A First Course in Design and Analysis of Experiments Gary W. Oehlert University of Minnesota CoverdesignbyVictoriaTomaselli CoverillustrationbyPeterHamlin MinitabisaregisteredtrademarkofMinitab,Inc. SASisaregisteredtrademarkofSASInstitute,Inc. S-PlusisaregisteredtrademarkofMathsoft,Inc. Design-ExpertisaregisteredtrademarkofStat-Ease,Inc. LibraryofCongressCataloging-in-PublicationData. Oehlert,GaryW. Afirstcourseindesignandanalysisofexperiments/GaryW.Oehlert. p. cm. Includesbibligraphicalreferencesandindex. ISBN0-7167-3510-5 1. ExperimentalDesign I.Title QA279.O342000 519.5—dc21 99-059934 Copyright c 2010GaryW.Oehlert. Allrightsreserved. (cid:13) Thisworkislicensedundera“CreativeCommons”license. Briefly,youarefreeto copy,distribute,andtransmitthisworkprovidedthefollowingconditionsaremet: 1. Youmustproperlyattributethework. 2. Youmaynotusethisworkforcommercialpurposes. 3. Youmaynotalter,transform,orbuilduponthiswork. Acompletedescriptionofthelicensemaybefoundat http://creativecommons.org/licenses/by-nc-nd/3.0/. For Becky who helped me all the way through and for Christie and Erica who put up with a lot while it was getting done Contents Preface xvii 1 Introduction 1 1.1 WhyExperiment? . . . . . . . . . . . . . . . . . . . . . . 1 1.2 ComponentsofanExperiment . . . . . . . . . . . . . . . 4 1.3 TermsandConcepts . . . . . . . . . . . . . . . . . . . . . 5 1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 MoreAboutExperimentalUnits . . . . . . . . . . . . . . 8 1.6 MoreAboutResponses . . . . . . . . . . . . . . . . . . . 10 2 RandomizationandDesign 13 2.1 RandomizationAgainstConfounding . . . . . . . . . . . 14 2.2 RandomizingOtherThings . . . . . . . . . . . . . . . . . 16 2.3 PerformingaRandomization . . . . . . . . . . . . . . . . 17 2.4 RandomizationforInference . . . . . . . . . . . . . . . . 19 2.4.1 Thepairedt-test . . . . . . . . . . . . . . . . . . 20 2.4.2 Two-samplet-test . . . . . . . . . . . . . . . . . 25 2.4.3 Randomizationinferenceandstandardinference . 26 2.5 FurtherReadingandExtensions . . . . . . . . . . . . . . 27 2.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3 CompletelyRandomizedDesigns 31 3.1 StructureofaCRD . . . . . . . . . . . . . . . . . . . . . 31 3.2 PreliminaryExploratoryAnalysis . . . . . . . . . . . . . 33 3.3 ModelsandParameters . . . . . . . . . . . . . . . . . . . 34 viii CONTENTS 3.4 EstimatingParameters . . . . . . . . . . . . . . . . . . . 39 3.5 ComparingModels: TheAnalysisofVariance . . . . . . . 44 3.6 MechanicsofANOVA . . . . . . . . . . . . . . . . . . . 45 3.7 WhyANOVAWorks . . . . . . . . . . . . . . . . . . . . 52 3.8 BacktoModelComparison . . . . . . . . . . . . . . . . . 52 3.9 Side-by-SidePlots . . . . . . . . . . . . . . . . . . . . . 54 3.10 Dose-ResponseModeling . . . . . . . . . . . . . . . . . . 55 3.11 FurtherReadingandExtensions . . . . . . . . . . . . . . 58 3.12 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4 LookingforSpecificDifferences—Contrasts 65 4.1 ContrastBasics . . . . . . . . . . . . . . . . . . . . . . . 65 4.2 InferenceforContrasts . . . . . . . . . . . . . . . . . . . 68 4.3 OrthogonalContrasts . . . . . . . . . . . . . . . . . . . . 71 4.4 PolynomialContrasts . . . . . . . . . . . . . . . . . . . . 73 4.5 FurtherReadingandExtensions . . . . . . . . . . . . . . 75 4.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5 MultipleComparisons 77 5.1 ErrorRates . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.2 Bonferroni-BasedMethods . . . . . . . . . . . . . . . . . 81 5.3 TheScheffe´ MethodforAllContrasts . . . . . . . . . . . 85 5.4 PairwiseComparisons. . . . . . . . . . . . . . . . . . . . 87 5.4.1 Displayingtheresults . . . . . . . . . . . . . . . 88 5.4.2 TheStudentizedrange . . . . . . . . . . . . . . . 89 5.4.3 Simultaneousconfidenceintervals . . . . . . . . . 90 5.4.4 Strongfamilywiseerrorrate . . . . . . . . . . . . 92 5.4.5 Falsediscoveryrate . . . . . . . . . . . . . . . . 96 5.4.6 Experimentwiseerrorrate . . . . . . . . . . . . . 97 5.4.7 Comparisonwiseerrorrate . . . . . . . . . . . . . 98 5.4.8 Pairwisetestingreprise . . . . . . . . . . . . . . 98 5.4.9 Pairwisecomparisonsmethodsthatdonotcontrol combinedTypeIerrorrates . . . . . . . . . . . . 98 5.4.10 Confidentdirections . . . . . . . . . . . . . . . . 100 CONTENTS ix 5.5 ComparisonwithControlortheBest . . . . . . . . . . . . 101 5.5.1 Comparisonwithacontrol . . . . . . . . . . . . . 101 5.5.2 Comparisonwiththebest . . . . . . . . . . . . . 104 5.6 RealityCheckonCoverageRates . . . . . . . . . . . . . 105 5.7 AWarningAboutConditioning. . . . . . . . . . . . . . . 106 5.8 SomeControversy. . . . . . . . . . . . . . . . . . . . . . 106 5.9 FurtherReadingandExtensions . . . . . . . . . . . . . . 107 5.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6 CheckingAssumptions 111 6.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . 111 6.2 Transformations . . . . . . . . . . . . . . . . . . . . . . . 113 6.3 AssessingViolationsofAssumptions. . . . . . . . . . . . 114 6.3.1 Assessingnonnormality . . . . . . . . . . . . . . 115 6.3.2 Assessingnonconstantvariance . . . . . . . . . . 118 6.3.3 Assessingdependence . . . . . . . . . . . . . . . 120 6.4 FixingProblems . . . . . . . . . . . . . . . . . . . . . . . 124 6.4.1 Accommodatingnonnormality . . . . . . . . . . 124 6.4.2 Accommodatingnonconstantvariance . . . . . . 126 6.4.3 Accommodatingdependence . . . . . . . . . . . 133 6.5 EffectsofIncorrectAssumptions . . . . . . . . . . . . . . 134 6.5.1 Effectsofnonnormality . . . . . . . . . . . . . . 134 6.5.2 Effectsofnonconstantvariance . . . . . . . . . . 136 6.5.3 Effectsofdependence . . . . . . . . . . . . . . . 138 6.6 ImplicationsforDesign . . . . . . . . . . . . . . . . . . . 140 6.7 FurtherReadingandExtensions . . . . . . . . . . . . . . 141 6.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 143 7 PowerandSampleSize 149 7.1 ApproachestoSampleSizeSelection . . . . . . . . . . . 149 7.2 SampleSizeforConfidenceIntervals. . . . . . . . . . . . 151 7.3 PowerandSampleSizeforANOVA . . . . . . . . . . . . 153 7.4 PowerandSampleSizeforaContrast . . . . . . . . . . . 158 7.5 MoreaboutUnitsandMeasurementUnits . . . . . . . . . 158 x CONTENTS 7.6 AllocationofUnitsforTwoSpecialCases . . . . . . . . . 160 7.7 FurtherReadingandExtensions . . . . . . . . . . . . . . 161 7.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 162 8 FactorialTreatmentStructure 165 8.1 FactorialStructure. . . . . . . . . . . . . . . . . . . . . . 165 8.2 FactorialAnalysis: MainEffectandInteraction . . . . . . 167 8.3 AdvantagesofFactorials . . . . . . . . . . . . . . . . . . 170 8.4 VisualizingInteraction . . . . . . . . . . . . . . . . . . . 171 8.5 ModelswithParameters. . . . . . . . . . . . . . . . . . . 175 8.6 TheAnalysisofVarianceforBalancedFactorials . . . . . 179 8.7 GeneralFactorialModels . . . . . . . . . . . . . . . . . . 182 8.8 AssumptionsandTransformations . . . . . . . . . . . . . 185 8.9 SingleReplicates . . . . . . . . . . . . . . . . . . . . . . 186 8.10 PoolingTermsintoError . . . . . . . . . . . . . . . . . . 191 8.11 Hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . 192 8.12 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 197 9 ACloserLookatFactorialData 203 9.1 ContrastsforFactorialData . . . . . . . . . . . . . . . . . 203 9.2 ModelingInteraction . . . . . . . . . . . . . . . . . . . . 209 9.2.1 Interactionplots . . . . . . . . . . . . . . . . . . 209 9.2.2 One-cellinteraction . . . . . . . . . . . . . . . . 210 9.2.3 Quantitativefactors . . . . . . . . . . . . . . . . 212 9.2.4 Tukeyone-degree-of-freedomfornonadditivity . . 217 9.3 FurtherReadingandExtensions . . . . . . . . . . . . . . 220 9.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 222 10 FurtherTopicsinFactorials 225 10.1 UnbalancedData . . . . . . . . . . . . . . . . . . . . . . 225 10.1.1 Sumsofsquaresinunbalanceddata . . . . . . . . 226 10.1.2 Buildingmodels . . . . . . . . . . . . . . . . . . 227 10.1.3 Testinghypotheses . . . . . . . . . . . . . . . . . 230 10.1.4 Emptycells . . . . . . . . . . . . . . . . . . . . . 233 10.2 MultipleComparisons . . . . . . . . . . . . . . . . . . . 234

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