Wilcoxon Rank Sum Test for Independent Samples

How to perform the Wilcoxon ranked sum non-parametric test for independent samples in Excel, a test used when the assumptions for the t test are violated. Skiptocontent Menu Whentherequirementsforthet-testfortwoindependentsamplesarenotsatisfied,theWilcoxonRank-Sumnon-parametrictestcanoftenbeusedprovidedthetwoindependentsamplesaredrawnfrompopulationswithanordinaldistribution. Forthistestweusethefollowingnullhypothesis: H0:theobservationscomefromthesamepopulation Fromapracticalpointofview,thisimplies: H0: ifoneobservationismadeatrandomfromeachpopulation(callthemx0andy0),thentheprobabilitythatx0>y0isthesameastheprobabilitythatx0115=Wcrit,wecannotrejectthenullhypothesis,andsoconcludethereisnosignificantdifferencebetweentheeffectivenessofthedrugandthecontrol. Example2:RepeatExample1withthelastdataelementforthegroupthattookthedrugremoved. WeagainusetheWilcoxonRank-Sumtest,butthistimethesamplesizesareunequal.ThetestisasinFigure4. Figure4–WilcoxonRank-SumTestforExample2 Theranksumsarecalculatedasinthepreviousexample,althoughsincesomeofthedatamaybeblank,weneedtouseaformulasuchas =IF(A6<>””,RANK_AVG(A6,$A$6:$B$17,1),””). Sincethesamplesizesaredifferent,abitmorecareisrequired.EssentiallyWrepresentsthelefttailstatisticandsoweneedtoalsoevaluatetherighttailstatisticW′,whichcanbeobtainedbyusingreverserankingasdescribedinFigure5: Figure5–CalculationofW′usingreverseranks ThevalueofW′ isthereforethesumoftheranksforthesmallersample,i.e.105.5.Fortunately,becauseofsymmetry,W’canmoreeasilybeobtainedviatheformula wheren1=11(thesmallersamplesize)andn2=12(thelargersamplesize).Thusweobtain W′ =11(11+12+1)–158.5=105.5(thevalueincellH11) Forthetwo-tailedtest,whichiswhatweusuallyrequire,wecomparethesmallerofWandW′ withWcrit.TofindthevalueofWcrit, weagainusetheWilcoxonRank-SumTable forα=.05(two-tail)wheren1 =11andn2 =12toobtainWcrit =99.Sincemin(W,W′)=min(158.5,105.5)=105.5>99=Wcrit ,onceagainwecannotrejectthenullhypothesis. Observation:Whenn1 =n2,thenW′ =R2,i.e.theranksumofthelargersample.ThusinExample1,W′ =180.5 Property1:Supposesample1hassizen1 andranksumR1 andsample2hassizen2 andranksumR2,thenR1 +R2 =n(n+1)/2wheren=n1 +n2. Property2:Whenthetwosamplesaresufficientlylarge(sayofsize>10,althoughsomesay20),thenthe WstatisticisapproximatelynormalN(μ,σ2)where Observation:Clickhere foraproofofProperty1or2. Observation:UsingProperty2,forsamplessufficientlylarge,wecantestWusingthetechniquesfromSamplingDistributions.NotethattheresultisthesamewhetherweuseWorW′. Observation:Sinceitcomparesranksums,theWilcoxonRank-Sumtestismorerobustthanthet-testasitislesslikelytoindicatespuriousresultsbasedonthepresenceofoutliers.Evenforlargesampleswheretheassumptionsforthet-testaremet,theWilcoxonRank-Sumtestisonlyalittlelessefficientthanthet-test. Example3:Theobjectiveofastudywastodeterminewhetherthereisasignificantdifferenceinthemedianlifeexpectancybetweensmokersandnon-smokers.38smokersand40non-smokerswerechosenatrandomandtheirageatdeathrecordedinFigure6. Figure6–Lifeexpectancyforbothgroups AtableofranksiscreatedandthevaluesofWandW′ arecalculatedasinExamples1and2.Sincethesamplesizesaresufficientlylarge,wecantestW(orW′)usingthenormaldistributionasdescribedinFigure7. Figure7–Wilcoxonrank-sumtestusingnormalapproximation Sincetherearefewersmokersthannon-smokers,W=theranksumforthesmokers=1227(cellU8).Wecalculatethemean(cellU14)andvariance(cellU15)forWusingtheformulas=U6*(T6+U6+1)/2and=U14*T6/6respectively.Thestandarddeviation(cellU16)isthengivenbytheformula=SQRT(U15)asusual. Wenowcalculatethep-value(cellU17)usingtheformula=2*NORM.DIST(U8,U14,U16,TRUE)sinceWW̄,wewouldusetheformula=2*(1–NORM.DIST(U8,U14,U16,TRUE)).Alternatively,wecouldhavecreatedthez-scoreandcalculatedthep-valueusingNORM.S.DIST. Sincep-value=.006161<.05 notethathadweusedw realstatisticsexcelfunctions:thefollowingfunctionsareprovidedintherealstatisticspack: rank_combined rank_sum wilcoxon wtest wcrit wprob ifinterp="TRUE(default)thentherecommendedinterpolationisusedifnecessaryinthetablelookup;otherwiselinearinterpolationisused." notethatthevaluesfor anyemptyornon-numericcellsinr1orr2areignored. observation:ifr1representsthefirstncolumnsofrangerandr2representstheremainingcolumnsinranger similarly observation:inexample2 also>.05=α,andsoonceagainwecan’trejectthenullhypothesis. SimilarlyinExample3,wecanusetheWILCOXONfunctiontoarriveatthesamevaluefortheminimumofW and W′,namelyWILCOXON(A6:H15,4)=WILCOXON(A6:D15,E6:H15)=1227,aswellasthesamep-value(assuminganormalapproximation),namelyWTEST(A6:H15,4)=WTEST(A6:D15,E6:H15)=0.003081.AlsoRANK_COMBINED(72,A6:D15,E6:H15,1)=37,RANK_SUM(A6:D15,E6:H15,1)=1854andRANK_SUM(E6:H15,A6:D15,1)=1227. Observation:TheeffectsizefortheWilcoxonRankSumtestisgivenbythecorrelationcoefficient (seeBasicConceptsofCorrelation). ThecorrelationcoefficientfortheWilcoxonRankSumtestisgivenbytheformula wherethez-scoreis ForExample3, andso AsdescribedinCorrelationinRelationtot-test,aroughestimateofeffectsizeisthatr= .5 representsalargeeffectsize,r=.3representsamediumeffectsizeandr=.1representsasmalleffect.Thus,forExample3wehaveamedium-sizedeffect. AlsoseeMann-WhitneyTest(includingFigure2)formoreinformationabouthowtocalculatetheeffectsizer inExcel. ExactTest Clickhereforadescriptionoftheexactversion oftheWilcoxonRank-SumExact Testusingthepermutationfunction. 81thoughtson“WilcoxonRankSumTestforIndependentSamples” CuáleselvalordenparaidentificarenlaTablaelestadísticodepruebadeWilcoxon,paran=10diferenciasdedosfilasdedatos,deloscualessolo6deellosquedanordenadosporrangosdelospruebasunasumaderangos?(esdecirquelosotrosquedaroniguales;parap=0.05,unilátero,esT=11paran=10?,oT=2paran=6?) Reply Francisco, FortheWilcoxonRankSumtest,therearetwosamplesandsotherearetwosamplesizesn1andn2.Theydon’thavetobeequal. Charles Reply HelloCharles, Idownloadedtherealstatsadd-inandcannotfindtheWCRITfunctionyoumentioned.Diditgetremoved? Reply Dillon, Itisstillavailableeventhoughyoudon’tseeitasyouenterthefunctioninaspreadsheet. ThereasonforthisisthatithasbeenreplacedbythefunctionMWINV(alpha,n1,n2,tails,False).Thisfunctiongivesanexactvalueinsteadofusingatablelookup. Charles Reply Charles–yourwebsiteandstatisticalpackageareterrific!I’vebeensearchingthroughmyoldcollegestatisticsbookandtonsofotherwebsites,andit’shandsdownthebestsource.Thankssomuchforthis! OntheWilcoxonRank-SumTest(one-tailed),Irealizeit’snormallyusedfordeterminingifthere’sastatisticaldifferencebetweentwogroups,butisitpossibletoexplicitlysayonegroupisgreaterthanorlessthantheother?Thewaythe“final”Wcanbe(1)thesmallerofthetwoW’s(ifsamesamplesizes)or(2)theWforthesmallersamplesizehasmesecond-guessingifthat’spossible.ForanappropriatelysmallW,youcansaythegroupsaredifferent,butisthereawayto“force”theWtobeforonegroup,soyoucanexplicitlysaywhichoneislessthantheother(perhapsdependingonthesignoftheZ-score)?I’mhopingtodothatwhenrunningmultiplecomparisonsandjusthavingasimple“Group1Group”,or“NotSignificantlyDifferent”output. Potentiallyrelatedtothat,onyourExcelfileexample(Real-Statistics-Examples-Non-Parametric-1)ontab“Wilcoxon4”,ifyouswitchthe“Smokers”and“Non-Smokers”data,thep-valuerisesfrom0.006to1.99,butwiththesameZ-scores.Shouldthatnotkeepthesamep-valuethatwouldrejectthehypothesis/couldtherebeadditionalcriteriaonthe“final”Wthatwouldcorrectforthat?Andfurtherrelatedtothat,shouldthattabhaveasimilarformulaforwhenthesamplesizesarethesametousethesmallerW? Thanksagain! Reply Thanksforthekindwordsaboutthewebsite. 1.WistheranksumforthelargersampleandW’istheranksumforthesmallersample.Ifthereisasignificantresult,youcanassignanordertothetwogroupsbasedontherank-sumdividedbythesamplesize.Ifthetwosampleshavesimilarshape(i.eitappearsthattheycomefrompopulationswiththesametypeofdistribution,thenthesamplewiththesmallerrank-sumdividedbysamplesizewouldcomefromthepopulationwiththesmallermedian. 2.Thep-valuecan’tbelargerthan1(itisaprobability)andsop-valuecan’tbeequalto1.99.Evenifyoureversetherolesofsmokersandnon-smokersthep-valueshouldbethesame,namely.006. Charles Reply IthinkI’veconfusedmyselfwithmyfirstpost,sorry! Intabs“Wilcoxon3”and“WilExact”ofthe“Non-Parametric1”file,Wiscalculatedas“theminimumranksum”ifthesamplesizesareequalandas“theranksumofthesmallersample”ifthesamplesizesaredifferent(cellH10inboth).Inthecaseofthedifferentsamplesizes,theW(andresultingW’)couldcomefromeitherofthesamples,andtheoutputwouldonlytellyouthey’resignificantlydifferentornot(youloseknowingwhichsampleiswhichthrutheIFstatement).Iwashopingforawaytooutputwhichsampleisthelesserorgreaterofthetwo(inaone-tailedtest). Potentiallyrelatedtothatontab“Wilcoxon4,”whenyouswitchthedatabetweenNon-smokersandSmokers,theWcalculationchangesfromSmokerstoNon-smokers(becausesamplesizeschange),andthatflowsthrutomakethep-value=1.99currently.Itseemslikethep-valuemightneedanIFstatementtoadjusttotheothersamplesize(butthatmaybethecurrentlayoutwouldprovideawayto“remember”theW,topotentiallyoutputwhichsampleislesserorgreaterthantheother(insteadofjustdifferent)likeIwashopingabove). I’mprobablytoofarintheweedswithmylackofknowledge,butthanksagainforyourhelp! Reply IthinkIunderstandyourresponse#1abovenow:Afteryou’vegottenasignificantlydifferentresult,youcanthendividetherank-sums(nottheWorW’)byrespectivesamplesizes,andthesamplewiththesmallerofthosewouldbesignificantlysmaller(orsimilarly,thelargeronewouldbesignificantlylarger)? SoinExample2/tab“Wilcoxon3”(assumingahighersignificancelevelthatmadethemdifferent),youcoulddividethetherank-sumsof117.5and158.5by12and11,toget9.8and14.4,showingthefirstsampleissmallerthanthesecond.AndinExample3/tab“Wilcoxon4,”youcoulddivide1854and1227by40and38,toget46.4and32.3,showingthesecondsampleissmallerthanthefirst? Thanksagain! Reply Ibelievethatwhatyousaidiscorrect.YoucanprobablygetthesameresultbyusingW,W’andthesamplesizesbutIhavenotlookedintothis. Charles Seemyresponsetoyourlatercomment. Charles Reply Charles, Forexample3,IwouldthinkthatwewillneedtousetheWTESTfunctionwith2-tailedtest.However,thep-valueobtainedusingthe2-tailedfunctionWTEST(R1,R2,2)gavea2timesbiggerp-valuethanthep-valueobtainedusingthenormalapproximation.Why? Inyourobservationparagraph,youhaveused1-tailedWTESTfunction(ie,WTEST(A6:D15,E6:H15)=0.003081.),whichmatchesthep-valuebasedonthenormalapproximation. Reply HelloSun, Thanksforpointingoutthiserror.Sinceweareconductinga2-tailedtest,thep-value=0.006161(twicethevalueindicated).Ihavenowcorrectedthewebpage.Youhavebeenextremelyhelpfulinidentifyingquiteafewerrors,forwhichIamverygrateful. Charles Reply Charles, Youareverywelcome.Thereisonemoreareatobecorrectedinthebodyofthetextshownbelow–pleasechange“onetailtest”to“twotailtest”: “Sincep-value=.006161<.05 reply hisun thanksagain.ihavejustchangedthetextto charles pleasedisregardmyqaboutwtest myapology asiwasnotabletosendtheentireqsofmyoriginalqs hereisthesecondone.itisaboutthep-valueobtainedbasedonwprob. forthewprobformula iusedthesmallest forexample itcamebackwiththevalueof1.999 idoappreciateyourguidanceonhowtocorrectthis. icameacrossafewcontradictoryvaluesinproducingp-valuesusingwtestandwprob. forthefirstexample pleaseadvisemehowicancorrecttheerror. thanks thankyoucharles whatdoyouseewhenyouenter="VER()intoanycell?" itisreturningthereusult sunitha thismeansthatyouhavenotinstalledtherealstatisticssoftware.youneedtogobacktothewebpagefromwhereyoudow nloadedthesoftwareandfollowtheinstallationinstructions. isthereanythingelseineedtodotoinstallitcorrectly ihavenotseenthisproblembeforeanddon whichlanguageareyouusing hicharles iuninstalledthesoftwareandinstalleditagainanditisworkingfinenow. thankyouforallthehelpwiththeinstallation. regards sunitha. thankyouforputtingtogetherthiswonderfulwebsite ihaveaquickquery.forthewilcoxontest thankyouverymuch. gladyoulikethewebsite. thewilcoxonranksumtableinthewebsitegoesupto25x25 sincethewilcoxonranksumcanbecalculatedfromthemann-whitneystatistic thankyousomuchcharlesforaquickreply.canyoupleaseexplainwhatyoumeanby sorrythatigaveyousuchacrypticresponse.itisactuallyquitestraightforward.ifyouhavethecriticalvaluefrom themann-whitneytableyoujustneedtoaddm thankyoucharles.idonotseetherealstatisticsfunctionmcrit.ihavedownloadedandaddedtherealstatisticsinex celaddins.also thisisnotadataanalysistool.youneedtosimplytypetheformula="MCRIT(9,27,.05)inanycell.WilcoxonRankSumonlyworksforindependentsamples.Forpairedsamples,youcanuseWilcoxon’sSignedRanksTest." plswhenyouhavenegativeobservationsinwilcoxonranksumtest somethinglikethis francis therankingisdoneinthesameway hi alsoforarighttailtest hellorichard yes dearcharles thanksforintroducingthisnewtest. inpractice ifyes ismyunderstandingandstepstomakeconclusioncorrect hj youcanstillusethettestevenwhenthepopulationisnotnormallydistributedprovidedthedataisnottoofarfromnor mally youshouldalsomakesurethatthetwosamplesareindependent.ifnottheninsteadofusingthewilcoxonranksumtest providedyouhavetwoindependentsample thankyouforyourexellentwebsite.forashortcomment kylekim thewicoxonranksumtest hicharlesthanksforclearexplanationofthestage.myqoustionishowtointerthosedataonspsssoftwares tariku sorry hello firstofallthanksforthisveryclearexplanation idohavea>20….howcanIfindmycriticalWfora=0.05tocomparetomyWleftandW’right? Thanksalot Kindregards Reply Lara, ThelargeesttableIhaveseenonlygoupton1=40andn2=20,butwithsamplessolargeyoucansafelyusethenormalapproximationinsteadofthetablesofcriticalvalues.Thisapproachisdescribedonthereferencedwebpage. Charles Reply hi!CharlesdoyouaboutWILCOXONMANN-WHITNEYTEST?andtogettheU-statistics? Reply Jesmae, Seethefollowingwebpage: Mann-WhitneyTest Charles Reply HiCharles: Ihaveaquestionforexample2(unequalsamples). n1=12;R1=117,5;R1’=170,5 n2=11;R2=158,5;R2’=105,5 Ws=min(158,5;105,5)=105,5 Idon’tknowif158,5ischoosenbecauseisthebiggervalueoflefttailorbecauseifthevalueofthesmallersampleandnomatterifisthebiggervalueornot. Bestregard Felix Reply Felix, Thesmallersampleischosen. Charles Reply MayIuseaWilcoxonsinged-ranktestwhenthevairancesarenotsimilarbetweenthetwogroupscompared? Thanksforyourhelp Reply Alberto, TheWilcoxonSignedRankstestoperatesonthedifferencesbetweenthedataitemsandsothevarianceswon’tmatter.ThesituationisdifferentfortheWilcoxonRankSumtest. Charles Reply hi Ihaveaquestioninordertomodifydatabyusingwilcoxonrank-sumnon-parametricrank.supposeIhavearatingfor1parameterswhichIhavenitrateconcentratesaswell.Iamgoingtomodifyratingrespecttonitrateconcentration.HowwouldIbeabletomodifyratingbyWilcoxontest? forexample: ratenitrateconcentrationmodifiedrate 41.3? 52? 818.5? Reply Sorry,butIdon’tknowwhatyoumeanby“modifydatausingwilcoconrank-sumnon-parametricrank”. Charles Reply DearSir, IamusingWilcoxonranksumtestformyresearchresults.Ihaveresultsoftwoalgorithmsfor30functionsthatmeansn1is30andn2isalso30.Icalculatedpvalueandusedsignificantlevel.05.Now,Iwanttofindwhichvaluesofn1(outof30)issignificantlydifferentfromn2.Iftheanyofthevalueissignificantlydifferentthenwhichoneisbetter. Thankyouinanticipation. Reply Parul, Sorry,butIdon’tunderstandyourquestion. Charles Reply Hello,IamsearchingforthesignificancelevelsofaWilcoxonranksum(Mann-Whitney)test.Iusedstatatogeneratethepvaluesbutiamwonderingatwhichleveldoisaythefiguresaresignificantate.g0.01,0.050r0.20?Isthereawayicouldselectthelevelofsignificanceinstata? Reply Peter, Thesignificancelevelreallydependsonyou.ItsimplystatesthelevelofTypeIerroryouarewillingtoacceptforthetest.Thetypicalvalueis.05(i.e.onetypeIerrorevery20tests).Youcansetitlowerifyoulike.SeeNullandAlternativeHypothesisfordetails. Charles Reply Kindlyhelpwiththis,itsveryurgent.Whatstudydesigncanbeusedforsigntest,wilcoxonsign-rankedtest,mediantestandmannwhitneytest.Thanksinanticipation. Reply Pleaselookatthewebpagesforeachoftheseteststogettheinformationthatyouarelookingfor. Charles Reply DearDr.Charles, Ihavetwomethods.Eachmethodistestedon8samplesandforeachsamplewehavePrecision,Recall,F-score.ThemethodXhashigheraverageF-scorethanmethodY.However,thedifferenceissmall.Iamaskedtocalculatethep-valueofthedifference. IstheWilcoxonranksumtestthecorrectway,orIshouldthinkinanotherdirection? Howtocalculatethep-valueofthedifference?ShouldIlistthearrayF-scoreforXandarrayF-scoreforYinMatlabandusethecommandranksum? Pleaseadvice. Thanksalot Reply Ahmed, Ican’ttellfromyourdescriptionwhatPrecision,RecallandF-scorerepresent.ArePrecisionandRecallthetworandomvariables?IsF-scoretheFstatistic? IamnotfamiliarwithMatlab’sranksumcommandandsocan’tcommentonthat. Charles Reply MyN1isonly16,butN2is5035.HowamIsupposetofindalphathen? Reply Bessie, Youwon’tbeabletousetheWilcoxonRankSumTablewithsuchahighvalueforN2.Insteadyouusethenormalapproximation,whichdoesn’trelyonthetable,asdescribedinExample3ofthereferencedwebpage. Alsothetabledoesn’tgiveyoualpha.Itgivesyouthecriticalvalues. Charles Reply Thanks! Iamactuallystillconfusedhere.Myn1setofdataisn’tnormal.andN2sinceithassuchahighnumber,weassumeittobenormal.Myproblemistocomparethemeanofthistwosetofdataseeiftheyaresignificantlydifferentfromeachother. N2isactuallymypopulation Reply Bessie, TheWilcoxonRankSumTestdoesn’tcomparethetwodatasets,itcomparestheranksofthevaluesinthedataset.Thesewillbeapproximatelynormallydistributed(eveniftheoriginaldataisnotnormallydistributed).Ifonesetisasamplefromthesecondset(i.e.thepopulation),thenyouareviolatingtheindependenceassumptionoftheWilcoxonRankSumTest;infacttheWilcoxonRankSumTestisreallytestingwhetherthetwodatasetscomefromthesamepopulation,whichinthiscasewouldclearlybetruesinceoneofthesetsisthepopulationfromwhichtheotherisderived. Charles Reply Thanksverymuch! Hello,thankyouforthewebsite.Ithashelpedalotintranslatingalotoftheformulasfortheseteststoexcel. Iwasjustwonderingaboutthecalculationofthevarianceinexample3.YourformulaforvariancereadsU14*T6/6.Iwasjustwonderingwherethe6camefrom. Reply Asyoucanseefromthereferencedwebpagetheformulaforthevarianceisn1*n2*(n1+n2+1)/12.Buttheformulaforthemeanisn1*(n1+n2+1)/2.Usingsimplealgebra,thismeansthatanalternativeformulaforthevarianceismean*n2/6. Charles Reply HelloagainDr.Charles, IaminabitofapredicamentasIhavesomesurveydatainwhichIhavesampledthesameindividualsbothbeforeandafter,butIdon’thaveanywaytolinktheirbeforeandafterresultstooneanother(asthesurveyitselfwasanonymous).Inaddition,thebeforeandaftergroupshavedifferentnumberofresponses.ThedataisfromLikertitems(notscales)soIassumenonparametrictestswouldbethewaytogo.MyonlyquestioniswoulditbeappropriatetousetheWilcoxonSumranktesteventhoughIcannotassumeindependentsamples?Thelossinpowerwouldgivemoreconservativeresults,butIwaswonderingifanothertestwouldbemoreappropriate. Reply IassumethatyouaretryingtoseewhetherthereisasignificantdifferencebetweenBeforeandAfter.IamnotsurehowyouwouldtestsuchdatasincetheWilcoxonRankSumtestrequiresindependentsamples.Ican’tthinkofanothertest,butfranklyIhaven’thadenoughtimetoreallythinktoomuchaboutthesituationthatyouhavedescribed. Charles Reply Charles, Thisisbrilliant.Thankyouforallyoureffort. UnfortunatelyIamhavingproblemswithusingyourfunctionswitharrayformulas.Atypicalsamplecodewouldlooklikethis. {=WTEST(IF($D$28:$D$30=F$21,$C$28:$C$30),IF($D$21:$D$27=F$22,$C$21:$C$27),2)} Haveyouheardofsimilarproblems?Doyouknowwhatcouldcausetheseproblems? Thankyouverymuchinadvance. Regards, Nicolas Reply Nicolas, Manyofthefunctionswereintendedtoreferencespecificrangesandnotformulasthatoutputarraysthatareequivalenttomatrices.Ihavebegunchangingthesefunctionssothattheyworkinarrayformulasofthetypethatyouhavedescribed. IhavealreadyrevisedtheWTESTfunction,althoughIbelievetherevisedversionwillbeinthenextreleaseofthesoftware.Itisimportanttorecallthatalthoughtheformulayouhavewrittenoutputsasinglevalue,ithasanembeddedarrayformulaandsoyoumustpressCtrl-Shft-Enterforittowork. Charles Reply (prisshorthandforprobability) IshouldnotetheChiSquarewassignificantforthistest.. Reply Itseemsmymessagewasn’tuploadedcorrectly,SASgeneratesthisforthenegativeWvalue: prlessthanZ=.00001 Reply Hi, SupposeIhavetwoverylargesamplesofseveralthousandobservationseach.Onesampleisafewthousandlargerthantheother.Withunevensamples,IwouldusethesmallerWvalue,andrefertothecriticalvalueofthelefttail.IfW-smallersampleislargerthantheW-criticalvalue,Icannotrejectthenullhypothesis.Isthatcorrect? Nowlet’ssayIamusingSAStoperformthewilcoxontest.Forthiswilcoxontest,SASgeneratesthisforaNEGATIVEWvalue: prZ=.00001. WouldthismeanthatIcannotrejectthenullhypothesis? Reply IfW(smaller)“”,RANK_AVG(A6,$A$6:$B$17,1),””). Charles Reply supposeIhavetwosampleswithunequalsizes,howcanIcomparethemusingwithWilcoxonranksum? Reply Kembo, Examples2and3onthereferencedwebpagecomparetwosamplesofunequalsize.Isuggestthatyoulookatthese. Charles Reply Firstofall,congratulationswithyoursite. IhaveaquestionrelatedtotheuseoftheWscoreintheWilcoxonranksumtest. IfyoudefineWasthesmallestofR1andR2,whydoyouuseatwo-tailedtestandnotjustaonetailed? Reply Jean-Pierre, Ifn1=n2,youwillgetthesametestresultwhetheryouuseR1orR2.IfIremembercorrectlyoneshouldbecomparedwiththeleftcriticalvalueandtheotherwiththerightcriticalvalue.Thesmalleronecorrespondstotheleftcriticalvalue,whichcanbecomparedwiththevaluesinthecriticalvalues. Thisverysimilartothettestwherenegativetvalueiscomparedwiththeleftcriticalvalueandthepositivetvalueiscomparedwiththerightcriticalvalue.Givensymmetrytodoatwo-sidedtestyoujustpickonesideandcomparewiththet-criticalvaluedeterminedbyhalvingthevalueofalpha.AsimilarthinghappensintheWilcoxonRankSumtest. Charles Reply Charles: Indeed,wellexplained,butIamstillnotsurewhywecannotrejectthenullhypothesis(asopposetot-test)becauseW=119.5and115=W-crit.Accordingtoyourearilertutorial“HypothesisTesting”,myunderstandingistorejectthenullhypothesissinceW-valueiswithinthecriticalregion. Reply Forthisandothernon-parametricteststhecriticalregionisthearealessthanthecriticalvalue.YoucanthinkofW-critasthecriticalvalueonthelefttail. 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