Duncan's new multiple range test - Wikipedia

In statistics, Duncan's new multiple range test (MRT) is a multiple comparison procedure developed by David B. Duncan in 1955. Duncan's MRT belongs to the ... Duncan'snewmultiplerangetest FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Instatistics,Duncan'snewmultiplerangetest(MRT)isamultiplecomparisonproceduredevelopedbyDavidB.Duncanin1955.Duncan'sMRTbelongstothegeneralclassofmultiplecomparisonproceduresthatusethestudentizedrangestatisticqrtocomparesetsofmeans. DavidB.DuncandevelopedthistestasamodificationoftheStudent–Newman–Keulsmethodthatwouldhavegreaterpower.Duncan'sMRTisespeciallyprotectiveagainstfalsenegative(TypeII)errorattheexpenseofhavingagreaterriskofmakingfalsepositive(TypeI)errors.Duncan'stestiscommonlyusedinagronomyandotheragriculturalresearch. Theresultofthetestisasetofsubsetsofmeans,whereineachsubsetmeanshavebeenfoundnottobesignificantlydifferentfromoneanother. ThistestisoftenfollowedbytheCompactLetterDisplay(CLD)methodologythatrenderstheoutputofsuchtestmuchmoreaccessibletonon-statisticianaudiences. Contents 1Definition 1.1Procedure 1.2Criticalvalues 2Numericexample 3Protectionandsignificancelevelsbasedondegreesoffreedom 4DuncanBayesianmultiplecomparisonprocedure 5Criticism 5.1Differentapproachestotheproblem 6References 7Externallinks Definition[edit] Assumptions: 1.Asampleofobservedmeans m 1 , m 2 , . . . , m n {\displaystylem_{1},m_{2},...,m_{n}} ,whichhavebeendrawnindependentlyfromnnormalpopulationswith"true"means, μ 1 , μ 2 , . . . , μ n {\displaystyle\mu_{1},\mu_{2},...,\mu_{n}} respectively. 2.Acommonstandarderror σ {\displaystyle\sigma} .Thisstandarderrorisunknown,butthereisavailabletheusualestimate s m {\displaystyles_{m}} ,whichisindependentoftheobservedmeansandisbasedonanumberofdegreesoffreedom,denotedby n 2 {\displaystylen_{2}} .(Moreprecisely, S m {\displaystyleS_{m}} ,hasthepropertythat n 2 ⋅ S m 2 σ m 2 {\displaystyle{\frac{n_{2}\cdotS_{m}^{2}}{\sigma_{m}^{2}}}} isdistributedas χ 2 {\displaystyle\chi^{2}} with n 2 {\displaystylen_{2}} degreesoffreedom,independentlyofsamplemeans). Theexactdefinitionofthetestis: Thedifferencebetweenanytwomeansinasetofnmeansissignificantprovidedtherangeofeachandeverysubsetwhichcontainsthegivenmeansissignificantaccordingtoan α p {\displaystyle\alpha_{p}} levelrangetestwhere α p = 1 − γ p {\displaystyle\alpha_{p}=1-\gamma_{p}} , γ p = ( 1 − α ) ( p − 1 ) {\displaystyle\gamma_{p}=(1-\alpha)^{(p-1)}} and p {\displaystylep} isthenumberofmeansinthesubsetconcerned. Exception:Thesoleexceptiontothisruleisthatnodifference betweentwomeanscanbedeclaredsignificantifthetwomeansconcerned arebothcontainedinasubsetofthemeanswhichhasanon-significant range. Procedure[edit] Theprocedureconsistsofaseriesofpairwisecomparisonsbetweenmeans.Eachcomparisonisperformedatasignificancelevel α p {\displaystyle\alpha_{p}} ,definedbythenumberofmeansseparatingthetwomeanscompared( α p {\displaystyle\alpha_{p}} for p − 2 {\displaystylep-2} separatingmeans).Thetestareperformedsequentially,wheretheresultofatestdetermineswhichtestisperformednext. Thetestsareperformedinthefollowingorder:thelargestminusthesmallest,thelargestminusthesecondsmallest,uptothelargestminusthesecondlargest;thenthesecondlargestminusthesmallest,thesecondlargestminusthesecondsmallest,andsoon,finishingwiththesecondsmallestminusthesmallest. Withonlyoneexception,givenbelow,eachdifferenceissignificantifitexceedsthecorrespondingshortestsignificantrange;otherwiseitisnotsignificant.Wheretheshortestsignificantrangeisthesignificantstudentizedrange,multipliedbythestandarderror. Theshortestsignificantrangewillbedesignatedas R ( p , α ) {\displaystyleR_{(p,\alpha)}} ,where p {\displaystylep} isthenumbermeansinthesubset. Thesoleexceptiontothisruleisthatnodifferencebetweentwomeanscanbedeclaredsignificantifthetwomeansconcernedarebothcontainedinasubsetofthemeanswhichhasanon-significantrange. Analgorithmforperformingthetestisasfollows: 1.Rankthesamplemeans,largesttosmallest. 2.Foreach m i {\displaystylem_{i}} samplemean,largesttosmallest,dothefollowing: 2.1foreachsamplemean,(denoted m j {\displaystylem_{j}} ),forsmallestupto m ( i − 1 ) {\displaystylem_{(i-1)}} . 2.1.1compare m i − m j {\displaystylem_{i}-m_{j}} tocriticalvalue σ m ⋅ R ( p , α ) {\displaystyle\sigma_{m}\cdotR_{(p,\alpha)}} , P = i − j , α = α p {\displaystyleP=i-j,\alpha=\alpha_{p}} 2.1.2if m i − m j {\displaystylem_{i}-m_{j}} doesnotexceedthecriticalvalue,thesubset ( m j , m j + 1 , . . . , m I ) {\displaystyle(m_{j},m_{j+1},...,m_{I})} isdeclarednotsignificantlydifferent: 2.1.2.1Gotonextiterationofloop2. 2.1.3Otherwise,keepgoingwithloop2.1 Criticalvalues[edit] Duncan'smultiplerangetestmakesuseofthestudentizedrangedistributioninordertodeterminecriticalvaluesforcomparisonsbetweenmeans.Notethatdifferentcomparisonsbetweenmeansmaydifferbytheirsignificancelevels-sincethesignificancelevelissubjecttothesizeofthesubsetofmeansinquestion. Letusdenote Q ( p , ν , γ ( p , α ) ) {\displaystyleQ_{(p,\nu,\gamma_{(p,\alpha)})}} asthe γ α {\displaystyle\gamma_{\alpha}} quantileofthestudentizedrangedistribution,withpobservations,and ν {\displaystyle\nu} degreesoffreedomforthesecondsample(seestudentizedrangeformoreinformation). Letusdenote r ( p , ν , α ) {\displaystyler_{(p,\nu,\alpha)}} asthestandardizedcriticalvalue,givenbytherule: Ifp=2 r ( p , ν , α ) = Q ( p , ν , γ ( p , α ) ) {\displaystyler_{(p,\nu,\alpha)}=Q_{(p,\nu,\gamma_{(p,\alpha)})}} Else r ( p , ν , α ) = m a x ( Q ( p , ν , γ ( p , α ) ) , r ( p − 1 , ν , α ) ) {\displaystyler_{(p,\nu,\alpha)}=max(Q_{(p,\nu,\gamma_{(p,\alpha)})},r_{(p-1,\nu,\alpha)})} Theshortestcriticalrange,(theactualcriticalvalueofthetest)iscomputedas : R ( p , ν , α ) = σ m ⋅ r ( p , ν , α ) {\displaystyleR_{(p,\nu,\alpha)}=\sigma_{m}\cdotr_{(p,\nu,\alpha)}} . For ν {\displaystyle\nu} ->∞,atabulationexistsforanexactvalueofQ(seelink). Awordofcautionisneededhere:notationsforQandRarenotthesamethroughoutliterature,whereQissometimesdenotedastheshortestsignificantinterval,andRasthesignificantquantileforstudentizedrangedistribution(Duncan's1955paperusesbothnotationsindifferentparts). Numericexample[edit] Letuslookattheexampleof5treatmentmeans: Treatments T1 T2 T3 T4 T5 TreatmentMeans 9.8 15.4 17.6 21.6 10.8 Rank 5 3 2 1 4 Withastandarderrorof s m = 1.796 {\displaystyles_{m}=1.796} ,and ν = 20 {\displaystyle\nu=20} (degreesoffreedomforestimatingthestandarderror). UsingaknowntabulationforQ,onereachesthevaluesof r ( p , ν , α ) {\displaystyler_{(p,\nu,\alpha)}} : r ( 2 , 20 , 0.05 ) = 2.95 {\displaystyler_{(2,20,0.05)}=2.95} r ( 3 , 20 , 0.05 ) = 3.10 {\displaystyler_{(3,20,0.05)}=3.10} r ( 4 , 20 , 0.05 ) = 3.18 {\displaystyler_{(4,20,0.05)}=3.18} r ( 5 , 20 , 0.05 ) = 3.25 {\displaystyler_{(5,20,0.05)}=3.25} Nowwemayobtainthevaluesoftheshortestsignificantrange,bytheformula: R ( p , ν , α ) = σ m ∗ r ( p , ν , α ) {\displaystyleR_{(p,\nu,\alpha)}=\sigma_{m}*r_{(p,\nu,\alpha)}} Reaching: R ( 2 , 20 , 0.05 ) = 3.75 {\displaystyleR_{(2,20,0.05)}=3.75} R ( 3 , 20 , 0.05 ) = 3.94 {\displaystyleR_{(3,20,0.05)}=3.94} R ( 4 , 20 , 0.05 ) = 4.04 {\displaystyleR_{(4,20,0.05)}=4.04} R ( 5 , 20 , 0.05 ) = 4.13 {\displaystyleR_{(5,20,0.05)}=4.13} Then,theobserveddifferencesbetweenmeansaretested,beginningwiththelargestversussmallest,whichwouldbecomparedwiththeleastsignificantrange R ( 5 , 20 , 0.05 ) = 4.13. {\displaystyleR_{(5,20,0.05)}=4.13.} Next,thedifferenceofthelargestandthesecondsmallestiscomputedandcomparedwiththeleastsignificantdifference R ( 4 , 20 , 0.05 ) = 4.04 {\displaystyleR_{(4,20,0.05)}=4.04} . Ifanobserveddifferenceisgreaterthanthecorrespondingshortestsignificantrange,thenweconcludethatthepairofmeansinquestionissignificantlydifferent. Ifanobserveddifferenceissmallerthanthecorrespondingshortestsignificantrange,alldifferencessharingthesameuppermeanareconsideredinsignificant,inordertopreventcontradictions(differencessharingthesameuppermeanareshorterbyconstruction).   Forourcase,thecomparisonwillyield: 4 v s .1 : 21.6 − 9.8 = 11.8 > 4.13 ( R 5 ) {\displaystyle4vs.1:21.6-9.8=11.8>4.13(R_{5})} 4 v s .5 : 21.6 − 10.8 = 10.8 > 4.04 ( R 4 ) {\displaystyle4vs.5:21.6-10.8=10.8>4.04(R_{4})} 4 v s .2 : 21.6 − 15.4 = 6.2 > 3.94 ( R 3 ) {\displaystyle4vs.2:21.6-15.4=6.2>3.94(R_{3})} 4 v s .3 : 21.6 − 17.6 = 4.0 > 3.75 ( R 2 ) {\displaystyle4vs.3:21.6-17.6=4.0>3.75(R_{2})} 3 v s .1 : 17.6 − 9.8 = 7.8 > 4.04 ( R 4 ) {\displaystyle3vs.1:17.6-9.8=7.8>4.04(R_{4})} 3 v s .5 : 17.6 − 10.8 = 6.8 > 3.94 ( R 3 ) {\displaystyle3vs.5:17.6-10.8=6.8>3.94(R_{3})} 3 v s .2 : 17.6 − 15.4 = 2.2 < 3.75 ( R 2 ) {\displaystyle3vs.2:17.6-15.4=2.2<3.75(R_{2})} 2 v s .1 : 15.4 − 9.8 = 5.6 > 3.94 ( R 3 ) {\displaystyle2vs.1:15.4-9.8=5.6>3.94(R_{3})} 2 v s .5 : 15.4 − 10.8 = 4.6 > 3.75 ( R 2 ) {\displaystyle2vs.5:15.4-10.8=4.6>3.75(R_{2})} 5 v s .1 : 10.8 − 9.8 = 1.0 < 3.75 ( R 2 ) {\displaystyle5vs.1:10.8-9.8=1.0<3.75(R_{2})} Weseethattherearesignificantdifferencesbetweenallpairsoftreatmentsexcept(T3,T2)and(T5,T1).Agraphunderliningthosemeansthatarenotsignificantlydifferentisshownbelow: T1T5T2T3T4 Protectionandsignificancelevelsbasedondegreesoffreedom[edit] ThenewmultiplerangetestproposedbyDuncanmakesuseofspecialprotectionlevelsbasedupondegreesoffreedom.Let γ 2 , α = 1 − α {\displaystyle\gamma_{2,\alpha}={1-\alpha}} betheprotectionlevelfortestingthesignificanceofadifferencebetweentwomeans;thatis,theprobabilitythatasignificantdifferencebetweentwomeanswillnotbefoundifthepopulationmeansareequal.Duncanreasonsthatonehasp-1degreesoffreedomfortestingprankedmean,andhenceonemayconductp-1independenttests,eachwithprotectionlevel γ 2 , α = 1 − α {\displaystyle\gamma_{2,\alpha}={1-\alpha}} .Hence,thejointprotectionlevelis: γ p , α = γ 2 , α p − 1 = ( 1 − α ) p − 1 {\displaystyle\gamma_{p,\alpha}=\gamma_{2,\alpha}^{p-1}=(1-\alpha)^{p-1}} where α p = 1 − γ p {\displaystyle\alpha_{p}=1-\gamma_{p}} thatis,theprobabilitythatonefindsnosignificantdifferencesinmakingp-1independenttests,eachatprotectionlevel γ 2 , α = 1 − α {\displaystyle\gamma_{2,\alpha}={1-\alpha}} ,is γ 2 , α p − 1 {\displaystyle\gamma_{2,\alpha}^{p-1}} ,underthehypothesisthatallppopulationmeansareequal. Ingeneral:thedifferencebetweenanytwomeansinasetofnmeansissignificantprovidedtherangeofeachandeverysubset,whichcontainsthegivenmeans,issignificantaccordingtoan α p {\displaystyle\alpha_{p}} –levelrangetest,wherepisthenumberofmeansinthesubsetconcerned. For α = 0.05 {\displaystyle\alpha=0.05} ,theprotectionlevelcanbetabulatedforvariousvalueofrasfollows: Protectionlevel : γ p , α {\displaystyle:\gamma_{p,\alpha}} probabilityoffalselyrejecting H 0 : α p {\displaystyleH_{0}:\alpha_{p}} p=2 0.95 0.05 p=3 0.903 0.097 p=4 0.857 0.143 p=5 0.815 0.185 p=6 0.774 0.226 p=7 0.735 0.265 NotethatalthoughthisproceduremakesuseoftheStudentizedrange,hiserrorrateisneitheronanexperiment-wisebasis(aswithTukey's)noronaper-comparisonsbasis.Duncan'smultiplerangetestdoesnotcontrolthefamilywiseerrorrate.SeeCriticismSectionforfurtherdetails. DuncanBayesianmultiplecomparisonprocedure[edit] Duncan(1965)alsogavethefirstBayesianmultiplecomparisonprocedure,forthepairwisecomparisonsamongthemeansinaone-waylayout. Thismultiplecomparisonprocedureisdifferentfortheonediscussedabove. Duncan'sBayesianMCPdiscussesthedifferencesbetweenorderedgroupmeans,wherethestatisticsinquestionarepairwisecomparison(noequivalentisdefinedforthepropertyofasubsethaving'significantlydifferent'property). Duncanmodeledtheconsequencesoftwoormoremeansbeingequalusingadditivelossfunctionswithinandacrossthepairwisecomparisons.Ifoneassumesthesamelossfunctionacrossthepairwisecomparisons,oneneedstospecifyonlyoneconstantK,andthisindicatestherelativeseriousnessoftypeItotypeIIerrorsineachpairwisecomparison. Astudy,whichperformedbyJulietPopperShaffer(1998),hasshownthatthemethodproposedbyDuncan,modifiedtoprovideweakcontrolofFWEandusinganempiricalestimateofthevarianceofthepopulationmeans,hasgoodpropertiesbothfromtheBayesianpointofview,asaminimum-riskmethod,andfromthefrequentistpointofview,withgoodaveragepower. Inaddition,resultsindicateconsiderablesimilarityinbothriskandaveragepowerbetweenDuncan'smodifiedprocedureandtheBenjaminiandHochberg(1995)Falsediscoveryrate-controllingprocedure,withthesameweakfamilywiseerrorcontrol. Criticism[edit] Duncan'stesthasbeencriticisedasbeingtooliberalbymanystatisticiansincludingHenryScheffé,andJohnW.Tukey. DuncanarguedthatamoreliberalprocedurewasappropriatebecauseinrealworldpracticetheglobalnullhypothesisH0="Allmeansareequal"isoftenfalseandthustraditionalstatisticiansoverprotectaprobablyfalsenullhypothesisagainsttypeIerrors.AccordingtoDuncan,oneshouldadjusttheprotectionlevelsfordifferentp-meancomparisonsaccordingtotheproblemdiscussed.TheexamplediscussedbyDuncaninhis1955paperisofacomparisonofmanymeans(i.e.100),whenoneisinterestedonlyintwo-meanandthree-meancomparisons,andgeneralp-meancomparisons(decidingwhetherthereissomedifferencebetweenp-means)areofnospecialinterest(ifpis15ormoreforexample). Duncan'smultiplerangetestisvery“liberal”intermsofTypeIerrors.Thefollowingexamplewillillustratewhy: Letusassumeoneistrulyinterested,asDuncansuggested,onlywiththecorrectrankingofsubsetsofsize4orbelow.Letusalsoassumethatoneperformsthesimplepairwisecomparisonwithaprotectionlevel γ 2 = 0.95 {\displaystyle\gamma_{2}=0.95} .Givenanoverallsetof100means,letuslookatthenullhypothesesofthetest: Thereare ( 100 2 ) {\displaystyle100\choose2} nullhypothesesforthecorrectrankingofeach2means.Thesignificancelevelofeachhypothesisis 1 − 0.95 = 0.05 {\displaystyle1-0.95=0.05} Thereare ( 100 3 ) {\displaystyle100\choose3} nullhypothesesforthecorrectrankingofeach3means.Thesignificancelevelofeachhypothesisis 1 − ( 0.95 ) 2 = 0.097 {\displaystyle1-(0.95)^{2}=0.097} Thereare ( 100 4 ) {\displaystyle100\choose4} nullhypothesesforthecorrectrankingofeach4means.Thesignificancelevelofeachhypothesisis 1 − ( 0.95 ) 3 = 0.143 {\displaystyle1-(0.95)^{3}=0.143} Aswecansee,thetesthastwomainproblems,regardingthetypeIerrors: Duncan’stestsisbasedontheNewman–Keulsprocedure,whichdoesnotprotectthefamilywiseerrorrate(thoughprotectingtheper-comparisonalphalevel) Duncan’stestintentionallyraisesthealphalevels(TypeIerrorrate)ineachstepoftheNewman–Keulsprocedure(significancelevelsof α p ≥ α {\displaystyle\alpha_{p}\geq\alpha} ). Therefore,itisadvisednottousetheprocedurediscussed. DuncanlaterdevelopedtheDuncan–WallertestwhichisbasedonBayesianprinciples.ItusestheobtainedvalueofFtoestimatethepriorprobabilityofthenullhypothesisbeingtrue. Differentapproachestotheproblem[edit] Furtherinformation:Multiplecomparisonsproblem§ Controllingprocedures,andFamily-wiseerrorrate§ Controllingprocedures Ifonestillwishestoaddresstheproblemoffindingsimilarsubsetsofgroupmeans,othersolutionsarefoundinliterature. Tukey'srangetestiscommonlyusedtocomparepairsofmeans,thisprocedurecontrolsthefamilywiseerrorrateinthestrongsense. AnothersolutionistoperformStudent'st-testofallpairsofmeans,andthentouseFDRControllingprocedure(tocontroltheexpectedproportionofincorrectlyrejectednullhypotheses). Otherpossiblesolutions,whichdonotincludehypothesistesting,butresultinapartitionofsubsetsincludeClustering&HierarchicalClustering.Thesesolutionsdifferfromtheapproachpresentedinthismethod: Bybeingdistance/densitybased,andnotdistributionbased. Needingalargergroupofmeans,inordertoproducesignificantresultsorworkingwiththeentiredataset. References[edit] Duncan,D.B.(1955)."MultiplerangeandmultipleFtests".Biometrics.11(1):1–42.doi:10.2307/3001478.JSTOR 3001478. Shaffer,JulietPopper(1999)."Asemi-BayesianstudyofDuncan'sBayesianmultiplecomparisonprocedure".JournalofStatisticalPlanningandInference.82(1–2):197–213.doi:10.1016/S0378-3758(99)00042-7. Berry,DonaldA.;Hochberg,Yosef(1999)."Bayesianperspectivesonmultiplecomparisons".JournalofStatisticalPlanningandInference.82(1–2):215–227.doi:10.1016/S0378-3758(99)00044-0. Parsad,Rajender."MultiplecomparisonProcedures".I.A.S.R.I,LibraryAvenue,NewDelhi110012.{{citejournal}}:Citejournalrequires|journal=(help) TablesfortheUseofRangeandStudentizedRangeinTestsofHypotheses H.LeonHarter,Champaigne,IL;N.Balakrishnan,McMasterUniversity,Hamilton,Ontario,Canada;Hardback-PublishedOct27,1997 Externallinks[edit] CriticalvaluesforDuncan'smultiplerangetests Retrievedfrom"https://en.wikipedia.org/w/index.php?title=Duncan%27s_new_multiple_range_test&oldid=1108649173" Categories:StatisticaltestsMultiplecomparisonsHiddencategories:CS1errors:missingperiodical Navigationmenu Personaltools NotloggedinTalkContributionsCreateaccountLogin Namespaces ArticleTalk English Views ReadEditViewhistory More Search Navigation MainpageContentsCurrenteventsRandomarticleAboutWikipediaContactusDonate Contribute HelpLearntoeditCommunityportalRecentchangesUploadfile Tools WhatlinkshereRelatedchangesUploadfileSpecialpagesPermanentlinkPageinformationCitethispageWikidataitem Print/export DownloadasPDFPrintableversion Languages 日本語Polski Editlinks