One-way analysis of variance - Wikipedia

In statistics, one-way analysis of variance (abbreviated one-way ANOVA) is a technique that can be used to compare whether two sample's means are ... One-wayanalysisofvariance FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Instatistics,one-wayanalysisofvariance(abbreviatedone-wayANOVA)isatechniquethatcanbeusedtocomparewhethertwosample'smeansaresignificantlydifferentornot(usingtheFdistribution).Thistechniquecanbeusedonlyfornumericalresponsedata,the"Y",usuallyonevariable,andnumericalor(usually)categoricalinputdata,the"X",alwaysonevariable,hence"one-way".[1] TheANOVAteststhenullhypothesis,whichstatesthatsamplesinallgroupsaredrawnfrompopulationswiththesamemeanvalues.Todothis,twoestimatesaremadeofthepopulationvariance.Theseestimatesrelyonvariousassumptions(seebelow).TheANOVAproducesanF-statistic,theratioofthevariancecalculatedamongthemeanstothevariancewithinthesamples.Ifthegroupmeansaredrawnfrompopulationswiththesamemeanvalues,thevariancebetweenthegroupmeansshouldbelowerthanthevarianceofthesamples,followingthecentrallimittheorem.Ahigherratiothereforeimpliesthatthesamplesweredrawnfrompopulationswithdifferentmeanvalues.[1] Typically,however,theone-wayANOVAisusedtotestfordifferencesamongatleastthreegroups,sincethetwo-groupcasecanbecoveredbyat-test(Gosset,1908).Whenthereareonlytwomeanstocompare,thet-testandtheF-testareequivalent;therelationbetweenANOVAandtisgivenbyF = t2.Anextensionofone-wayANOVAistwo-wayanalysisofvariancethatexaminestheinfluenceoftwodifferentcategoricalindependentvariablesononedependentvariable. Contents 1Assumptions 1.1Departuresfrompopulationnormality 2Thecaseoffixedeffects,fullyrandomizedexperiment,unbalanceddata 2.1Themodel 2.2Thedataandstatisticalsummariesofthedata 2.3Thehypothesistest 2.4Analysissummary 3Example 4Seealso 5Notes 6Furtherreading Assumptions[edit] Theresultsofaone-wayANOVAcanbeconsideredreliableaslongasthefollowingassumptionsaremet: Responsevariableresidualsarenormallydistributed(orapproximatelynormallydistributed). Variancesofpopulationsareequal. Responsesforagivengroupareindependentandidenticallydistributednormalrandomvariables(notasimplerandomsample(SRS)). Ifdataareordinal,anon-parametricalternativetothistestshouldbeusedsuchasKruskal–Wallisone-wayanalysisofvariance.Ifthevariancesarenotknowntobeequal,ageneralizationof2-sampleWelch'st-testcanbeused.[2] Departuresfrompopulationnormality[edit] ANOVAisarelativelyrobustprocedurewithrespecttoviolationsofthenormalityassumption.[3] Theone-wayANOVAcanbegeneralizedtothefactorialandmultivariatelayouts,aswellastotheanalysisofcovariance.[clarificationneeded] ItisoftenstatedinpopularliteraturethatnoneoftheseF-testsarerobustwhentherearesevereviolationsoftheassumptionthateachpopulationfollowsthenormaldistribution,particularlyforsmallalphalevelsandunbalancedlayouts.[4]Furthermore,itisalsoclaimedthatiftheunderlyingassumptionofhomoscedasticityisviolated,theTypeIerrorpropertiesdegeneratemuchmoreseverely.[5] However,thisisamisconception,basedonworkdoneinthe1950sandearlier.ThefirstcomprehensiveinvestigationoftheissuebyMonteCarlosimulationwasDonaldson(1966).[6]Heshowedthatundertheusualdepartures(positiveskew,unequalvariances)"theF-testisconservative",andsoitislesslikelythanitshouldbetofindthatavariableissignificant.However,aseitherthesamplesizeorthenumberofcellsincreases,"thepowercurvesseemtoconvergetothatbasedonthenormaldistribution".Tiku(1971)foundthat"thenon-normaltheorypowerofFisfoundtodifferfromthenormaltheorypowerbyacorrectiontermwhichdecreasessharplywithincreasingsamplesize."[7]Theproblemofnon-normality,especiallyinlargesamples,isfarlessseriousthanpopulararticleswouldsuggest. Thecurrentviewisthat"Monte-Carlostudieswereusedextensivelywithnormaldistribution-basedteststodeterminehowsensitivetheyaretoviolationsoftheassumptionofnormaldistributionoftheanalyzedvariablesinthepopulation.Thegeneralconclusionfromthesestudiesisthattheconsequencesofsuchviolationsarelessseverethanpreviouslythought.Althoughtheseconclusionsshouldnotentirelydiscourageanyonefrombeingconcernedaboutthenormalityassumption,theyhaveincreasedtheoverallpopularityofthedistribution-dependentstatisticaltestsinallareasofresearch."[8] Fornonparametricalternativesinthefactoriallayout,seeSawilowsky.[9]FormorediscussionseeANOVAonranks. Thecaseoffixedeffects,fullyrandomizedexperiment,unbalanceddata[edit] Themodel[edit] Thenormallinearmodeldescribestreatmentgroupswithprobability distributionswhichareidenticallybell-shaped(normal)curveswith differentmeans.Thusfittingthemodelsrequiresonlythemeansof eachtreatmentgroupandavariancecalculation(anaveragevariance withinthetreatmentgroupsisused).Calculationsofthemeansand thevarianceareperformedaspartofthehypothesistest. Thecommonlyusednormallinearmodelsforacompletely randomizedexperimentare:[10] y i , j = μ j + ε i , j {\displaystyley_{i,j}=\mu_{j}+\varepsilon_{i,j}} (themeansmodel) or y i , j = μ + τ j + ε i , j {\displaystyley_{i,j}=\mu+\tau_{j}+\varepsilon_{i,j}} (theeffectsmodel) where i = 1 , … , I {\displaystylei=1,\dotsc,I} isanindexoverexperimentalunits j = 1 , … , J {\displaystylej=1,\dotsc,J} isanindexovertreatmentgroups I j {\displaystyleI_{j}} isthenumberofexperimentalunitsinthejthtreatmentgroup I = ∑ j I j {\displaystyleI=\sum_{j}I_{j}} isthetotalnumberofexperimentalunits y i , j {\displaystyley_{i,j}} areobservations μ j {\displaystyle\mu_{j}} isthemeanoftheobservationsforthejthtreatmentgroup μ {\displaystyle\mu} isthegrandmeanoftheobservations τ j {\displaystyle\tau_{j}} isthejthtreatmenteffect,adeviationfromthegrandmean ∑ τ j = 0 {\displaystyle\sum\tau_{j}=0} μ j = μ + τ j {\displaystyle\mu_{j}=\mu+\tau_{j}} ε ∼ N ( 0 , σ 2 ) {\displaystyle\varepsilon\thicksimN(0,\sigma^{2})} , ε i , j {\displaystyle\varepsilon_{i,j}} arenormallydistributedzero-meanrandomerrors. Theindex i {\displaystylei} overtheexperimentalunitscanbeinterpretedseveral ways.Insomeexperiments,thesameexperimentalunitissubjectto arangeoftreatments; i {\displaystylei} maypointtoaparticularunit.Inothers, eachtreatmentgrouphasadistinctsetofexperimentalunits; i {\displaystylei} may simplybeanindexintothe j {\displaystylej} -thlist. Thedataandstatisticalsummariesofthedata[edit] Oneformoforganizingexperimentalobservations y i j {\displaystyley_{ij}} iswithgroupsincolumns: ANOVAdataorganization,Unbalanced,Singlefactor ListsofGroupObservations I 1 {\displaystyleI_{1}} I 2 {\displaystyleI_{2}} I 3 {\displaystyleI_{3}} … {\displaystyle\dotso} I j {\displaystyleI_{j}} 1 y 11 {\displaystyley_{11}} y 12 {\displaystyley_{12}} y 13 {\displaystyley_{13}} y 1 j {\displaystyley_{1j}} 2 y 21 {\displaystyley_{21}} y 22 {\displaystyley_{22}} y 23 {\displaystyley_{23}} y 2 j {\displaystyley_{2j}} 3 y 31 {\displaystyley_{31}} y 32 {\displaystyley_{32}} y 33 {\displaystyley_{33}} y 3 j {\displaystyley_{3j}} ⋮ {\displaystyle\vdots} ⋮ {\displaystyle\vdots} i {\displaystylei} y i 1 {\displaystyley_{i1}} y i 2 {\displaystyley_{i2}} y i 3 {\displaystyley_{i3}} … {\displaystyle\dotso} y i j {\displaystyley_{ij}} GroupSummaryStatistics GrandSummaryStatistics #Observed I 1 {\displaystyleI_{1}} I 2 {\displaystyleI_{2}} … {\displaystyle\dotso} I j {\displaystyleI_{j}} … {\displaystyle\dotso} I J {\displaystyleI_{J}} #Observed I = ∑ I j {\displaystyleI=\sumI_{j}} Sum ∑ i y i j {\displaystyle\sum_{i}y_{ij}} Sum ∑ j ∑ i y i j {\displaystyle\sum_{j}\sum_{i}y_{ij}} SumSq ∑ i ( y i j ) 2 {\displaystyle\sum_{i}(y_{ij})^{2}} SumSq ∑ j ∑ i ( y i j ) 2 {\displaystyle\sum_{j}\sum_{i}(y_{ij})^{2}} Mean m 1 {\displaystylem_{1}} … {\displaystyle\dotso} m j {\displaystylem_{j}} … {\displaystyle\dotso} m J {\displaystylem_{J}} Mean m {\displaystylem} Variance s 1 2 {\displaystyles_{1}^{2}} … {\displaystyle\dotso} s j 2 {\displaystyles_{j}^{2}} … {\displaystyle\dotso} s J 2 {\displaystyles_{J}^{2}} Variance s 2 {\displaystyles^{2}} Comparingmodeltosummaries: μ = m {\displaystyle\mu=m} and μ j = m j {\displaystyle\mu_{j}=m_{j}} .Thegrandmeanandgrandvariancearecomputedfromthegrandsums, notfromgroupmeansandvariances. Thehypothesistest[edit] Giventhesummarystatistics,thecalculationsofthehypothesistest areshownintabularform.WhiletwocolumnsofSSareshownfortheir explanatoryvalue,onlyonecolumnisrequiredtodisplayresults. ANOVAtableforfixedmodel,singlefactor,fullyrandomizedexperiment Sourceofvariation Sumsofsquares Sumsofsquares Degreesoffreedom Meansquare F ExplanatorySS[11] ComputationalSS[12] DF MS Treatments ∑ T r e a t m e n t s I j ( m j − m ) 2 {\displaystyle\sum_{Treatments}I_{j}(m_{j}-m)^{2}} ∑ j ( ∑ i y i j ) 2 I j − ( ∑ j ∑ i y i j ) 2 I {\displaystyle\sum_{j}{\frac{(\sum_{i}y_{ij})^{2}}{I_{j}}}-{\frac{(\sum_{j}\sum_{i}y_{ij})^{2}}{I}}} J − 1 {\displaystyleJ-1} S S T r e a t m e n t D F T r e a t m e n t {\displaystyle{\frac{SS_{Treatment}}{DF_{Treatment}}}} M S T r e a t m e n t M S E r r o r {\displaystyle{\frac{MS_{Treatment}}{MS_{Error}}}} Error ∑ T r e a t m e n t s ( I j − 1 ) s j 2 {\displaystyle\sum_{Treatments}(I_{j}-1)s_{j}^{2}} ∑ j ∑ i y i j 2 − ∑ j ( ∑ i y i j ) 2 I j {\displaystyle\sum_{j}\sum_{i}y_{ij}^{2}-\sum_{j}{\frac{(\sum_{i}y_{ij})^{2}}{I_{j}}}} I − J {\displaystyleI-J} S S E r r o r D F E r r o r {\displaystyle{\frac{SS_{Error}}{DF_{Error}}}} Total ∑ O b s e r v a t i o n s ( y i j − m ) 2 {\displaystyle\sum_{Observations}(y_{ij}-m)^{2}} ∑ j ∑ i y i j 2 − ( ∑ j ∑ i y i j ) 2 I {\displaystyle\sum_{j}\sum_{i}y_{ij}^{2}-{\frac{(\sum_{j}\sum_{i}y_{ij})^{2}}{I}}} I − 1 {\displaystyleI-1} M S E r r o r {\displaystyleMS_{Error}} isthe estimateofvariancecorrespondingto σ 2 {\displaystyle\sigma^{2}} ofthe model. Analysissummary[edit] ThecoreANOVAanalysisconsistsofaseriesofcalculations.The dataiscollectedintabularform.Then Eachtreatmentgroupissummarizedbythenumberofexperimentalunits,twosums,ameanandavariance.Thetreatmentgroupsummariesarecombinedtoprovidetotalsforthenumberofunitsandthesums.Thegrandmeanandgrandvariancearecomputedfromthegrandsums.Thetreatmentandgrandmeansareusedinthemodel. ThethreeDFsandSSsarecalculatedfromthesummaries.ThentheMSsarecalculatedandaratiodeterminesF. Acomputertypicallydeterminesap-valuefromFwhichdetermineswhethertreatmentsproducesignificantlydifferentresults.Iftheresultissignificant,thenthemodelprovisionallyhasvalidity. Iftheexperimentisbalanced,allofthe I j {\displaystyleI_{j}} termsare equalsotheSSequationssimplify. Inamorecomplexexperiment,wheretheexperimentalunits(or environmentaleffects)arenothomogeneous,rowstatisticsarealso usedintheanalysis.Themodelincludestermsdependenton i {\displaystylei} .Determiningtheextratermsreducesthenumberof degreesoffreedomavailable. Example[edit] Consideranexperimenttostudytheeffectofthreedifferentlevelsofafactoronaresponse(e.g.threelevelsofafertilizeronplantgrowth).Ifwehad6observationsforeachlevel,wecouldwritetheoutcomeoftheexperimentinatablelikethis,wherea1,a2,anda3arethethreelevelsofthefactorbeingstudied. a1 a2 a3 6 8 13 8 12 9 4 9 11 5 11 8 3 6 7 4 8 12 Thenullhypothesis,denotedH0,fortheoverallF-testforthisexperimentwouldbethatallthreelevelsofthefactorproducethesameresponse,onaverage.TocalculatetheF-ratio: Step1:Calculatethemeanwithineachgroup: Y ¯ 1 = 1 6 ∑ Y 1 i = 6 + 8 + 4 + 5 + 3 + 4 6 = 5 Y ¯ 2 = 1 6 ∑ Y 2 i = 8 + 12 + 9 + 11 + 6 + 8 6 = 9 Y ¯ 3 = 1 6 ∑ Y 3 i = 13 + 9 + 11 + 8 + 7 + 12 6 = 10 {\displaystyle{\begin{aligned}{\overline{Y}}_{1}&={\frac{1}{6}}\sumY_{1i}={\frac{6+8+4+5+3+4}{6}}=5\\{\overline{Y}}_{2}&={\frac{1}{6}}\sumY_{2i}={\frac{8+12+9+11+6+8}{6}}=9\\{\overline{Y}}_{3}&={\frac{1}{6}}\sumY_{3i}={\frac{13+9+11+8+7+12}{6}}=10\end{aligned}}} Step2:Calculatetheoverallmean: Y ¯ = ∑ i Y ¯ i a = Y ¯ 1 + Y ¯ 2 + Y ¯ 3 a = 5 + 9 + 10 3 = 8 {\displaystyle{\overline{Y}}={\frac{\sum_{i}{\overline{Y}}_{i}}{a}}={\frac{{\overline{Y}}_{1}+{\overline{Y}}_{2}+{\overline{Y}}_{3}}{a}}={\frac{5+9+10}{3}}=8} whereaisthenumberofgroups. Step3:Calculatethe"between-group"sumofsquareddifferences: S B = n ( Y ¯ 1 − Y ¯ ) 2 + n ( Y ¯ 2 − Y ¯ ) 2 + n ( Y ¯ 3 − Y ¯ ) 2 = 6 ( 5 − 8 ) 2 + 6 ( 9 − 8 ) 2 + 6 ( 10 − 8 ) 2 = 84 {\displaystyle{\begin{aligned}S_{B}&=n({\overline{Y}}_{1}-{\overline{Y}})^{2}+n({\overline{Y}}_{2}-{\overline{Y}})^{2}+n({\overline{Y}}_{3}-{\overline{Y}})^{2}\\[8pt]&=6(5-8)^{2}+6(9-8)^{2}+6(10-8)^{2}=84\end{aligned}}} wherenisthenumberofdatavaluespergroup. Thebetween-groupdegreesoffreedomisonelessthanthenumberofgroups f b = 3 − 1 = 2 {\displaystylef_{b}=3-1=2} sothebetween-groupmeansquarevalueis M S B = 84 / 2 = 42 {\displaystyleMS_{B}=84/2=42} Step4:Calculatethe"within-group"sumofsquares.Beginbycenteringthedataineachgroup a1 a2 a3 6−5=1 8−9=−1 13−10=3 8−5=3 12−9=3 9−10=−1 4−5=−1 9−9=0 11−10=1 5−5=0 11−9=2 8−10=−2 3−5=−2 6−9=−3 7−10=−3 4−5=−1 8−9=−1 12−10=2 Thewithin-groupsumofsquaresisthesumofsquaresofall18valuesinthistable S W = ( 1 ) 2 + ( 3 ) 2 + ( − 1 ) 2 + ( 0 ) 2 + ( − 2 ) 2 + ( − 1 ) 2 + ( − 1 ) 2 + ( 3 ) 2 + ( 0 ) 2 + ( 2 ) 2 + ( − 3 ) 2 + ( − 1 ) 2 + ( 3 ) 2 + ( − 1 ) 2 + ( 1 ) 2 + ( − 2 ) 2 + ( − 3 ) 2 + ( 2 ) 2 =   1 + 9 + 1 + 0 + 4 + 1 + 1 + 9 + 0 + 4 + 9 + 1 + 9 + 1 + 1 + 4 + 9 + 4 =   68 {\displaystyle{\begin{aligned}S_{W}=&(1)^{2}+(3)^{2}+(-1)^{2}+(0)^{2}+(-2)^{2}+(-1)^{2}+\\&(-1)^{2}+(3)^{2}+(0)^{2}+(2)^{2}+(-3)^{2}+(-1)^{2}+\\&(3)^{2}+(-1)^{2}+(1)^{2}+(-2)^{2}+(-3)^{2}+(2)^{2}\\=&\1+9+1+0+4+1+1+9+0+4+9+1+9+1+1+4+9+4\\=&\68\\\end{aligned}}} Thewithin-groupdegreesoffreedomis f W = a ( n − 1 ) = 3 ( 6 − 1 ) = 15 {\displaystylef_{W}=a(n-1)=3(6-1)=15} Thusthewithin-groupmeansquarevalueis M S W = S W / f W = 68 / 15 ≈ 4.5 {\displaystyleMS_{W}=S_{W}/f_{W}=68/15\approx4.5} Step5:TheF-ratiois F = M S B M S W ≈ 42 / 4.5 ≈ 9.3 {\displaystyleF={\frac{MS_{B}}{MS_{W}}}\approx42/4.5\approx9.3} Thecriticalvalueisthenumberthattheteststatisticmustexceedtorejectthetest.Inthiscase,Fcrit(2,15)=3.68atα=0.05.SinceF=9.3 > 3.68,theresultsaresignificantatthe5%significancelevel.Onewouldnotacceptthenullhypothesis,concludingthatthereisstrongevidencethattheexpectedvaluesinthethreegroupsdiffer.Thep-valueforthistestis0.002. AfterperformingtheF-test,itiscommontocarryoutsome"post-hoc"analysisofthegroupmeans.Inthiscase,thefirsttwogroupmeansdifferby4units,thefirstandthirdgroupmeansdifferby5units,andthesecondandthirdgroupmeansdifferbyonly1unit.Thestandarderrorofeachofthesedifferencesis 4.5 / 6 + 4.5 / 6 = 1.2 {\displaystyle{\sqrt{4.5/6+4.5/6}}=1.2} .Thusthefirstgroupisstronglydifferentfromtheothergroups,asthemeandifferenceismorethan3timesthestandarderror,sowecanbehighlyconfidentthatthepopulationmeanofthefirstgroupdiffersfromthepopulationmeansoftheothergroups.However,thereisnoevidencethatthesecondandthirdgroupshavedifferentpopulationmeansfromeachother,astheirmeandifferenceofoneunitiscomparabletothestandarderror. NoteF(x, y)denotesanF-distributioncumulativedistributionfunctionwithxdegreesoffreedominthenumeratorandydegreesoffreedominthedenominator. Seealso[edit] Analysisofvariance Ftest(Includesaone-wayANOVAexample) Mixedmodel Multivariateanalysisofvariance(MANOVA) RepeatedmeasuresANOVA Two-wayANOVA Welch'st-test Notes[edit] ^abHowell,David(2002).StatisticalMethodsforPsychology.Duxbury.pp. 324–325.ISBN 0-534-37770-X. ^Welch,B.L.(1951)."OntheComparisonofSeveralMeanValues:AnAlternativeApproach".Biometrika.38(3/4):330–336.doi:10.2307/2332579.JSTOR 2332579. ^Kirk,RE(1995).ExperimentalDesign:ProceduresForTheBehavioralSciences(3 ed.).PacificGrove,CA,USA:Brooks/Cole. ^Blair,R.C.(1981)."Areactionto'Consequencesoffailuretomeetassumptionsunderlyingthefixedeffectsanalysisofvarianceandcovariance.'".ReviewofEducationalResearch.51(4):499–507.doi:10.3102/00346543051004499. ^Randolf,E.A.;Barcikowski,R.S.(1989)."TypeIerrorratewhenrealstudyvaluesareusedaspopulationparametersinaMonteCarlostudy".PaperPresentedatthe11thAnnualMeetingoftheMid-WesternEducationalResearchAssociation,Chicago. ^Donaldson,TheodoreS.(1966)."PoweroftheF-TestforNonnormalDistributionsandUnequalErrorVariances".PaperPreparedforUnitedStatesAirForceProjectRAND. ^Tiku,M.L.(1971)."PowerFunctionoftheF-TestUnderNon-NormalSituations".JournaloftheAmericanStatisticalAssociation.66(336):913–916.doi:10.1080/01621459.1971.10482371. ^"GettingStartedwithStatisticsConcepts".Archivedfromtheoriginalon2018-12-04.Retrieved2016-09-22. ^Sawilowsky,S.(1990)."Nonparametrictestsofinteractioninexperimentaldesign".ReviewofEducationalResearch.60(1):91–126.doi:10.3102/00346543060001091. ^Montgomery,DouglasC.(2001).DesignandAnalysisofExperiments(5th ed.).NewYork:Wiley.p. Section3–2.ISBN 9780471316497. ^ Moore,DavidS.;McCabe,GeorgeP.(2003).IntroductiontothePracticeofStatistics(4th ed.).WHFreeman&Co.p. 764.ISBN 0716796570. ^ Winkler,RobertL.;Hays,WilliamL.(1975).Statistics:Probability,Inference,andDecision(2nd ed.).NewYork:Holt,RinehartandWinston.p. 761. Furtherreading[edit] GeorgeCasella(18April2008).Statisticaldesign.Springer.ISBN 978-0-387-75965-4. Retrievedfrom"https://en.wikipedia.org/w/index.php?title=One-way_analysis_of_variance&oldid=1105611565" Categories:AnalysisofvarianceStatisticaltestsHiddencategories:WikipediaarticlesneedingclarificationfromSeptember2014 Navigationmenu Personaltools NotloggedinTalkContributionsCreateaccountLogin Namespaces ArticleTalk English Views ReadEditViewhistory More Search Navigation MainpageContentsCurrenteventsRandomarticleAboutWikipediaContactusDonate Contribute HelpLearntoeditCommunityportalRecentchangesUploadfile Tools WhatlinkshereRelatedchangesUploadfileSpecialpagesPermanentlinkPageinformationCitethispageWikidataitem Print/export DownloadasPDFPrintableversion Languages العربيةMagyar日本語PolskiSuomiУкраїнська粵語 Editlinks