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WORKINGPAPERAnalyzingANOVADesignsBiometricsInformationHandbookNo.507/1995ProvinceofBritishColumbiaMinistryofForestsResearchProgramAnalyzingANOVADesignsBiometricsInformationHandbookNo.5VeraSitProvinceofBritishColumbiaMinistryofForestsResearchProgramCitation:Sit.V.1995.AnalyzingANOVADesigns.Biom.Info.Hand.5.Res.Br.,B.C.Min.For.,Victoria,B.C.Work.Pap.07/1995.Preparedby:VeraSitfor:B.C.MinistryofForestsResearchBranch31BastionSquareVictoria,BCV8W3E7©1995ProvinceofBritishColumbiaCopiesofthisreportmaybeobtained,dependinguponsupply,from:B.C.MinistryofForestsResearchBranch31BastionSquareVictoria,BCV8W3E7iiiACKNOWLEDGEMENTSIwouldliketothanktheregionalstaffintheB.C.MinistryofForestsforrequestingadocumentonanalysisofvariancein1992.Theirneedspromptedthepreparationofthishandbookandtheirencouragementfuelledmydrivetocompletethehandbook.ThefirstdraftofthishandbookwasbasedontwoinformalhandoutspreparedbyWendyBergerud(1982,1991).IamgratefultoWendyforallowingtheuseofherexamples(Bergerud1991)inChapter6ofthishandbook.AspecialthanksgoestoJeffStoneforhelpingmefocusontheneedsofmyintendedaudience.IthankGordonNighforreadingthemanuscriptateachandeverystageofitsdevelopment.IamindebtedtoMichaelStoehrfortheinvaluablediscussionswehadaboutANOVA.IamgratefulfortheassistanceIreceivedfromDavidIzardinproducingthefiguresinthishandbook.Last,butnotleast,Iwanttoexpressmyappreciationforthevaluablecriticismandsuggestionsprovidedbymyreviewers:WendyBergerud,RobBrockley,PhilComeau,MikeCurran,NolaDaintith,GraemeHope,AmandaNemec,PasiPuttonen,JohnStevenson,DavidTanner,andChrisThompson.vCONTENTSAcknowledgements.........................................................................iii1Introduction............................................................................12TheDesignofanExperiment..................................................12.1Factor..............................................................................22.2QuantitativeandQualitativeFactors...................................22.3RandomandFixedFactors................................................22.4Variance...........................................................................32.5CrossedandNestedFactors................................................32.6ExperimentalUnit.............................................................52.7Element............................................................................62.8Replication.......................................................................62.9Randomization..................................................................72.10BalancedandUnbalancedDesigns......................................73AnalysisofVariance.................................................................93.1Example...........................................................................103.2HypothesisTesting.............................................................103.3Terminology.....................................................................113.3.1Model......................................................................113.3.2Sumofsquares.........................................................113.3.3Degreesoffreedom...................................................123.3.4Meansquares...........................................................133.4ANOVAAssumptions........................................................133.5TheIdeabehindANOVA...................................................143.6ANOVAF-test..................................................................173.7ConcludingRemarksaboutANOVA...................................184MultipleComparisons..............................................................194.1WhatisaComparison?.....................................................194.2DrawbacksofComparisons................................................204.3PlannedContrasts.............................................................214.4MultipleComparisons.......................................................224.5Recommendations.............................................................235CalculatingANOVAwithSAS:anExample...............................245.1Example...........................................................................245.2ExperimentalDesign.........................................................245.3ANOVATable...................................................................265.4SASProgram....................................................................286ExperimentalDesigns..............................................................336.1CompletelyRandomizedDesigns........................................346.1.1One-waycompletelyrandomizeddesign.....................346.1.2Subsampling.............................................................366.1.3Factorialcompletelyrandomizeddesign......................38vi6.2RandomizedBlockDesigns................................................406.2.1One-wayrandomizedblockdesign.............................416.2.2Factorialrandomizedblockdesign............