AnalysisofVariance(ANOVA) Analysis of Variance (ANOVA) Janette Walde [email protected] DepartmentofStatistics UniversityofInnsbruck AnalysisofVariance(ANOVA) Outline I Introduction Problems Whatis Analysis ofVariance? One-WayANOVA(SingleFactorANOVA) AreaofApplication ModelDescription ModelAssumptions Descriptive Analysis ModelEstimation andHypothesisTesting Post-HocAnalysis Application PowerAnalysis PostHocPowerAnalysis A-PrioriPowerAnalysis Movingpast One-wayANOVA AnalysisofVariance(ANOVA) Outline II Two-WayANOVA(Factorial Design) ModelandTerminology ModelandAssumptions ModelResults InteractionofFactors andInteraction Plot Application FurtherModelExtensions Literature AnalysisofVariance(ANOVA) Introduction Problems Problems/Questions I Dofertilizer have differenteffects ondifferentkind of wheat? I Reactfemalesdifferentlyon anti-cancerdrugsas males? I Doeswaterevaporationofsoil dependon thekind of vegetationgrowing,controlling forclimate conditions? I A newtreatmentmeantto helpthose with chronicarthritis painwasdevelopedandtested for its long-term effectiveness. Participantsin the experimentrated their levelofpainon a 0(no pain)to 9 (extreme pain)scale at three-monthintervals.Was the treatmenteffective? I Doesthe exposure ofplantsto variousamountsofCO2 affect characteristics oftheplant? AnalysisofVariance(ANOVA) Introduction WhatisAnalysisofVariance? ANOVA - Preliminaries I ANalysis OfVAriance. I Use variancesandvariance like quantitiesto study the equalityornon-equalityofpopulationmeans. I So,althoughit is analysisofvariance we are actually analyzingmeans,notvariances. I Thereareothermethodswhichanalyze thevariances betweengroups. I Partitionsthe observedvariance basedon explanatory (independent)variables. I Comparespartitionsto test significance on explanatory variables. AnalysisofVariance(ANOVA) One-WayANOVA(SingleFactorANOVA) AreaofApplication Basics I One-wayANOVAis usedwhen I Onlytestingtheeffectofoneexplanatoryvariable. I Eachsubjecthasonlyonetreatmentorcondition.Thus,a between-subjectdesign. I Usedto testfor differencesamongtwo ormore independentgroups(in orderto ’avoid’themultiple testing problem). I Givesthesameresultsastwosamplet-tests ifexplanatory variable hasto levels. AnalysisofVariance(ANOVA) One-WayANOVA(SingleFactorANOVA) ModelDescription Model Description - Notation I If wehave K groupsdenotethe meansofthegroupsas µ ,µ ,...,µ . 1 2 K I Subjecti in groupj hasobservationy : ij I yij =µj +(cid:15)ij I where(cid:15)ij areindependently distributedN(0,σ2). I Cancombinethisandsaythatsubjectsfromgroupj have distributionN(µ,σ2). j I With randomassignmentthesample meanfor any treatmentgroupis representativeofthe populationmean for thatgroup. AnalysisofVariance(ANOVA) One-WayANOVA(SingleFactorANOVA) ModelAssumptions Model Assumptions 1. Theerrors (cid:15) are normallydistributed. ij 2. Across the conditionsthe errors haveequalspread, referred to asequalvariances. I Ruleofthumb:theassumptionismetifthelargestvariance islessthantwicethesmallestvariance. I Ifunequalvariancesneedtomakeacorrection.If deviationsaresmallthisisusuallyα/2. 3. Theerrors areindependentfrom eachother. AnalysisofVariance(ANOVA) One-WayANOVA(SingleFactorANOVA) DescriptiveAnalysis Typical exploratory analysis include I Tabulationofthenumberofsubjectsinexperimentalgroup. I Side-by-side boxplots. I Statistics abouteachgroup. AnalysisofVariance(ANOVA) One-WayANOVA(SingleFactorANOVA) DescriptiveAnalysis Application: Medley & Clements (1998) [MC1998] Medley& Clements(1998)samplesa numberofstations (betweenfour andseven)onsix streams knownto bepolluted by heavymetalsin theRocky MountainregionofColorado, USA.Theyrecordedzinc concentration,andspecies richness andspecies diversity ofthe diatomcommunityandproportion ofdiatomcells thatwere theearly-successional species, Achanthesminutissima.