Tukey's range test - Wikipedia

Tukey's range test, also known as Tukey's test, Tukey method, Tukey's honest significance test, or Tukey's HSD (honestly significant difference) test, ... Tukey'srangetest FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Statisticaltestformultiplecomparisons NottobeconfusedwithTukeymean-differencetest. Tukey'srangetest,alsoknownasTukey'stest,Tukeymethod,Tukey'shonestsignificancetest,orTukey'sHSD(honestlysignificantdifference)test,[1]isasingle-stepmultiplecomparisonprocedureandstatisticaltest.Itcanbeusedtofindmeansthataresignificantlydifferentfromeachother. NamedafterJohnTukey,[2]itcomparesallpossiblepairsofmeans,andisbasedonastudentizedrangedistribution(q)(thisdistributionissimilartothedistributionoftfromthet-test.Seebelow).[3] Tukey'stestcomparesthemeansofeverytreatmenttothemeansofeveryothertreatment;thatis,itappliessimultaneouslytothesetofallpairwisecomparisons μ i − μ j {\displaystyle\mu_{i}-\mu_{j}\,} andidentifiesanydifferencebetweentwomeansthatisgreaterthantheexpectedstandarderror.Theconfidencecoefficientfortheset,whenallsamplesizesareequal,isexactly 1 − α {\displaystyle1-\alpha} forany 0 ≤ α ≤ 1 {\displaystyle0\leq\alpha\leq1} .Forunequalsamplesizes,theconfidencecoefficientisgreaterthan1 − α.Inotherwords,theTukeymethodisconservativewhenthereareunequalsamplesizes. AcommonmistakenbeliefisthattheTukeyhsdshouldonlybeusedfollowingasignificantANOVA.TheANOVAisnotnecessarybecausetheTukeytestcontrolstheTypeIerrorrateonitsown.[citationneeded] ThistestisoftenfollowedbytheCompactLetterDisplay(CLD)statisticalproceduretorendertheoutputofthistestmoretransparenttonon-statisticianaudiences. Contents 1Assumptions 2Theteststatistic 3Thestudentizedrange(q)distribution 4Confidencelimits 5ComparingANOVAandTukey-Kramertests 6Seealso 7Notes 8Furtherreading 9Externallinks Assumptions[edit] Theobservationsbeingtestedareindependentwithinandamongthegroups.[citationneeded] Thegroupsassociatedwitheachmeaninthetestarenormallydistributed.[citationneeded] Thereisequalwithin-groupvarianceacrossthegroupsassociatedwitheachmeaninthetest(homogeneityofvariance).[citationneeded] Theteststatistic[edit] Tukey'stestisbasedonaformulaverysimilartothatofthet-test.Infact,Tukey'stestisessentiallyat-test,exceptthatitcorrectsforfamily-wiseerrorrate. TheformulaforTukey'stestis q s = Y A − Y B S E , {\displaystyleq_{s}={\frac{Y_{A}-Y_{B}}{SE}},} whereYAisthelargerofthetwomeansbeingcompared,YBisthesmallerofthetwomeansbeingcompared,andSEisthestandarderrorofthesumofthemeans. Thisqsvaluecanthenbecomparedtoaqvaluefromthestudentizedrangedistribution.Iftheqsvalueislargerthanthecriticalvalueqαobtainedfromthedistribution,thetwomeansaresaidtobesignificantlydifferentatlevel α : 0 ≤ α ≤ 1 {\displaystyle\alpha:0\leq\alpha\leq1} .[3] SincethenullhypothesisforTukey'steststatesthatallmeansbeingcomparedarefromthesamepopulation(i.e.μ1=μ2=μ3=...=μk),themeansshouldbenormallydistributed(accordingtothecentrallimittheorem).ThisgivesrisetothenormalityassumptionofTukey'stest. Thestudentizedrange(q)distribution[edit] TheTukeymethodusesthestudentizedrangedistribution.SupposethatwetakeasampleofsizenfromeachofkpopulationswiththesamenormaldistributionN(μ,σ2)andsupposethat y ¯ {\displaystyle{\bar{y}}} ministhesmallestofthesesamplemeansand y ¯ {\displaystyle{\bar{y}}} maxisthelargestofthesesamplemeans,andsupposeS2isthepooledsamplevariancefromthesesamples.ThenthefollowingrandomvariablehasaStudentizedrangedistribution. q = y ¯ max − y ¯ min S 2 / n {\displaystyleq={\frac{{\overline{y}}_{\max}-{\overline{y}}_{\min}}{S{\sqrt{2/n}}}}} Thisvalueofqisthebasisofthecriticalvalueofq,basedonthreefactors: α(theTypeIerrorrate,ortheprobabilityofrejectingatruenullhypothesis) k(thenumberofpopulations) df(thenumberofdegreesoffreedom(N – k)whereNisthetotalnumberofobservations) Thedistributionofqhasbeentabulatedandappearsinmanytextbooksonstatistics.Insometablesthedistributionofqhasbeentabulatedwithoutthe 2 {\displaystyle{\sqrt{2}}} factor.Tounderstandwhichtableitis,wecancomputetheresultfork = 2andcompareittotheresultoftheStudent'st-distributionwiththesamedegreesoffreedomandthesame α. Inaddition,Roffersacumulativedistributionfunction(ptukey)andaquantilefunction(qtukey)for q. Confidencelimits[edit] TheTukeyconfidencelimitsforallpairwisecomparisonswithconfidencecoefficientofatleast1 − αare y ¯ i ∙ − y ¯ j ∙ ± q α ; k ; N − k 2 σ ^ ε 2 n i , j = 1 , … , k i ≠ j . {\displaystyle{\bar{y}}_{i\bullet}-{\bar{y}}_{j\bullet}\pm{\frac{q_{\alpha;k;N-k}}{\sqrt{2}}}{\widehat{\sigma}}_{\varepsilon}{\sqrt{\frac{2}{n}}}\qquadi,j=1,\ldots,k\quadi\neqj.} Noticethatthepointestimatorandtheestimatedvariancearethesameasthoseforasinglepairwisecomparison.Theonlydifferencebetweentheconfidencelimitsforsimultaneouscomparisonsandthoseforasinglecomparisonisthemultipleoftheestimatedstandarddeviation. Alsonotethatthesamplesizesmustbeequalwhenusingthestudentizedrangeapproach. σ ^ ε {\displaystyle{\widehat{\sigma}}_{\varepsilon}} isthestandarddeviationoftheentiredesign,notjustthatofthetwogroupsbeingcompared.Itispossibletoworkwithunequalsamplesizes.Inthiscase,onehastocalculatetheestimatedstandarddeviationforeachpairwisecomparisonasformalizedbyClydeKramerin1956,sotheprocedureforunequalsamplesizesissometimesreferredtoastheTukey–Kramermethodwhichisasfollows: y ¯ i ∙ − y ¯ j ∙ ± q α ; k ; N − k 2 σ ^ ε 1 n i + 1 n j {\displaystyle{\bar{y}}_{i\bullet}-{\bar{y}}_{j\bullet}\pm{\frac{q_{\alpha;k;N-k}}{\sqrt{2}}}{\widehat{\sigma}}_{\varepsilon}{\sqrt{{\frac{1}{n}}_{i}+{\frac{1}{n}}_{j}}}\qquad} wheren iandn jarethesizesofgroupsiandjrespectively.Thedegreesoffreedomforthewholedesignisalsoapplied. ComparingANOVAandTukey-Kramertests[edit] BothANOVAandTukey-Kramertestsarebasedonthesameassumptions.However,thesetwotestsforkgroups(i.e.μ1=μ2=...=μk)mayresultinlogicalcontradictionswhenk>2,eveniftheassumptionshold.Itispossibletogenerateasetofpseudorandomsamplesofstrictlypositivemeasuresuchthathypothesisμ1=μ2isrejectedatsignificancelevel 1 − α > 0.95 {\displaystyle1-\alpha>0.95} whileμ1=μ2=μ3isnotrejectedevenat 1 − α = 0.975 {\displaystyle1-\alpha=0.975} .[4] Seealso[edit] Familywiseerrorrate Newman–Keulsmethod Notes[edit] ^Lowry,Richard."OneWayANOVA–IndependentSamples".Vassar.edu.ArchivedfromtheoriginalonOctober17,2008.RetrievedDecember4,2008.Alsooccasionallyas"honestly,"seee.g.Morrison,S.;Sosnoff,J.J.;Heffernan,K.S.;Jae,S.Y.;Fernhall,B.(2013)."Aging,hypertensionandphysiologicaltremor:Thecontributionofthecardioballisticimpulsetotremorgenesisinolderadults".JournaloftheNeurologicalSciences.326(1–2):68–74.doi:10.1016/j.jns.2013.01.016. ^Tukey,John(1949)."ComparingIndividualMeansintheAnalysisofVariance".Biometrics.5(2):99–114.JSTOR 3001913. ^abLinton,L.R.,Harder,L.D.(2007)Biology315–QuantitativeBiologyLectureNotes.UniversityofCalgary,Calgary,AB ^Gurvich,V.;Naumova,M.(2021)."LogicalContradictionsintheOne-WayANOVAandTukey–KramerMultipleComparisonsTestswithMoreThanTwoGroupsofObservations".Symmetry.13(8:1387).doi:10.3390/sym13081387. Furtherreading[edit] Montgomery,DouglasC.(2013).DesignandAnalysisofExperiments(Eighth ed.).Wiley.Section3.5.7. Externallinks[edit] NIST/SEMATECHe-HandbookofStatisticalMethods:Tukey'smethod Retrievedfrom"https://en.wikipedia.org/w/index.php?title=Tukey%27s_range_test&oldid=1108648758" Categories:AnalysisofvarianceStatisticaltestsMultiplecomparisonsHiddencategories:ArticleswithshortdescriptionShortdescriptionmatchesWikidataAllarticleswithunsourcedstatementsArticleswithunsourcedstatementsfromAugust2022ArticleswithunsourcedstatementsfromJanuary2022 Navigationmenu Personaltools NotloggedinTalkContributionsCreateaccountLogin Namespaces ArticleTalk English Views ReadEditViewhistory More Search Navigation MainpageContentsCurrenteventsRandomarticleAboutWikipediaContactusDonate Contribute HelpLearntoeditCommunityportalRecentchangesUploadfile Tools WhatlinkshereRelatedchangesUploadfileSpecialpagesPermanentlinkPageinformationCitethispageWikidataitem Print/export DownloadasPDFPrintableversion Languages Français한국어Македонски日本語PolskiРусский Editlinks