ANOVA with post-hoc Tukey HSD Test Calculator with Scheffé ...

Tukey originated his HSD test, constructed for pairs with equal number of samples in each treatment, way back in 1949. When the sample sizes are unequal, we the ... One-wayANOVA(ANalysisOfVAriance)withpost-hocTukeyHSD(Honestly SignificantDifference)TestCalculatorforcomparingmultipletreatments ....nowmoooooovedtotheabovedomainnameastatsa.com! One-wayAnovawithpost-hocTukeyHSDCalculator,withScheffé, BonferroniandHolmmultiplecomparsonresultsalsoprovided. -TukeyHSDuseswithTukey-Kramerformulawhentreatments(samplegroups)haveunequalobservations(i.e.unbalancedobservations) -Selectthenumberoftreatments,thenenteryourobservationdatabytypingorcopy-paste,thenproceedtotheresults Selectthenumberofindependenttreatmentsbelow: Select\(k\),thenumberofindependenttreatments,sometimesalsocalled samples.Sincetheseareindependentandnotpairedorcorrelated,thenumber ofobservationsofeachtreatmentmaybedifferent. Thiswouldleadtoaninputscreenwith\(k\)columnstopasteyour observationdataonvarioustreatments.Thiscalculatorishard-codedfora maximumof10treatments,whichismorethanadequateformostresearchers. \(k=2\)* \(k=3\) \(k=4\) \(k=5\) \(k=6\) \(k=7\) \(k=8\) \(k=9\) \(k=10\) ←Check thisboxifyouwishtousethedemoexampledatawith\(k=4\)treatments. Youmay,ofcourse,overwritethedemoexampledata,but\(k=4\)wouldremain fixedinthedemoirrespectiveofyourselectionof\(k\). ↑Unchecktheaboveboxandselectyourappropriate\(k=\)numberof treatments,andsubsequentlyclicktheboxbelowtoenteryourtreatmentdata ←Clickhere tocontinuewiththenextstepofdataentry. *Notethatwhen\(k=2\)thereisonlyonepairof(independent)treatements/ samplestobecompared,sotheTukeyHSDTestforpairwisecomparisonof multipletreatments/samplesisnotconducted.Inthiscase,theone-wayANOVA isequivalenttoat-testwiththe\(F\)ratiosuchthat\(F=t^2\). Whatthiscalculatordoes: MicrosoftExcelcandoone-wayANOVAofmultipletreatments(columns)nicely. Butitstopsthereinitstracks.WithinExcel,followupofasuccessfulANOVA withpost-hocTukeyHSDhastobedonemanually,ifyouknowhowto!This self-containedcalculator,withflexibilitytovarythenumberoftreatments (columns)tobecompared,startswithone-wayANOVA.IfANOVA indicatesstatisticalsignificance,thiscalculatorautomatically performspairwisepost-hocTukeyHSD,Scheffé,BonferroniandHolmmultiple comparisonofalltreatments(columns).Excelhasthe necessarybuilt-instatisticalfunctionstoconductScheffé,Bonferroniand Holmmultiplecomparisonfromfirstprinciples.However,itlacksthekey built-instatisticalfunctionneededforconductingExcel-containedTukeyHSD. ContinuingeducationinStatistics101: Thehard-corestatisticalpackagesdemandacertainexpertisetoformat theinputdata,writecodetoimplementtheproceduresandthendeciphertheir 1970sOldSchoolMainframeEraoutput.Incontrast,whenspoutingoutTukey HSD,Scheffé,BonferroniandHolmmultiplecomparisonresults,thiscalculator alsotellsyouhowtoverifyandreproducetheiroutputandresultsmanuallyin Excel,byteachingyouhowtotaketheoutputofAnova(fromExcelorother package),enablingyoutoconductpost-hocTukeyHSD,Scheffé,Bonferroniand HolmmultiplecomparisonbyhandinExcel.YourautomaticAgraderesultsfrom wizardryinproducingpost-hocTukeyHSD,Scheffé,BonferroniandHolm pairwisemultiplecomparisonyourselfmanuallyinExcel,inwhichcaseyou wouldnolongerneedthiscalculator,norhavetostrugglewithharnessingthe oldschoolstatisticalpackages. AfterprovidingguidelinesonhowtoconductTukeyHSD,Scheffé,Bonferroni andHolmpairwisemultiplecomparisonbyhandinExcel,thissiteprovidesR codewithatutorialonhowtorepeatandreproducetheresultsprovidedin thiscalculatorusingR.UsersunfamiliarwiththeRstatisticalpackageare encouragedtofollowthistutorialandnotonlylearnsomebasicR,butalso becomegrandmastersofharnessingacomplexmodernstatisticalpackagetoconductTukey HSD,Scheffé,BonferroniandHolmpairwisemultiplecomparison. Thiscalculatorisdesignedtorelievebiomedicalscientistsfromthe travailsofcodingheavy-dutystatisticalpackages: Areyouabiomedicalorsocialscientist,whohasnarrowinterestinone-way ANOVAfollowedautomaticallybypost-hocTukeyHSD,Scheffé,Bonferroniand Holmmethods,butdonothavethepatienceandperseverencetohackcodeto harnessR,Stata,SPSS,SASorMatlab?Thisistherighttoolforyou!Itwas inspiredbythefrustrationofseveralbiomedicalscientistswithlearningthe softwaresetupandcodingoftheseseriousstatisticalpackages,almostlike operatingheavybulldozermachinerytoswatanirritatingmosquito.Forcode grandmasters,fullyworkingcodeandsetupinstructionsareprovidedfor replicationoftheresultsintheseriousacademic-research-gradeopen-source (andhencefree)Rstatisticalpackage. FormulaeandMethodology: Theone-wayANOVAstartingpointofthiscalculatorreproduces theoutputofMicrosoftExcel'sbuilt-inANOVAfeature.Thefollow-uppost-hocTukeyHSDmultiplecomparison partofthiscalculatorisbasedontheformulaeandproceduresattheNIST EngineeringStatisticsHandbookpageonTukey'smethod.Tukeyoriginated hisHSDtest,constructedforpairswithequalnumberofsamplesineachtreatment,waybackin1949.When thesamplesizesareunequal,wethecalculatorautomaticallyappliestheTukey-KramermethodKramer originatedin1956.Adecentwriteupontheserelevantformulaeappearin theTukeyrangetest Wikientry.TheNISTHandbookpagementionsthismodificationbutdooes notprovidetheformula,whiletheWikientrymakesadequatelyspecifiesit. TheScheffé,BonferroniandHolmmethodsofmultiplecomparisonappliesto contrasts,ofwhichpairsareasubset.TheNIST EngineeringStatisticsHandbookpagedefinescontrasts.However,this calculatorishard-codedforcontraststhatarepairs,andhencedoes notpestertheuserforadditionalinputthatdefinesgeneralizedcontrast structures.Thepost-hocScheffémultiplecomparisonoftreatmentpairs bythiscalculatorisbasedontheformulaeandproceduresattheNIST EngineeringStatisticsHandbookpageonScheffé'smethodthatwaspublishedby Schefféin1953. TheBonferroniandHolmmethodsofmultiplecomparisondependsonthenumberof relevantpairsbeingcomparedsimultaneously.Thiscalculatorishard-codedfor BonferroniandHolmsimultaneousmultiplecomparisonof(1)allpairsand(2)onlya subsetofpairsrelativetoonetreatment,thefirstcolumn,deemedtobethe control.Ontheotherhand,Scheffé'smethodisindependentofthenumberof contrastsunderconsideration.Thepost-hocBonferronisimultaneousmultiple comparisonoftreatmentpairsbythiscalculatorisbasedonthe formulaeandproceduresattheNIST EngineeringStatisticsHandbookpageonBonferroni'smethod.Theoriginal BonferronipublishedpaperinItaliandatingbackto1936ishardtofind ontheweb. AsignificantimprovementovertheBonferronimethodwasproposedbyHolm(1979).Amongthe manyreviewsofthemeritsoftheHolmmethodanditsuniformsuperiorityover theBonferronimethod,thatofAickinandGensler(1996) isnotable.ThispaperisthealsosourceofouralgorithmtomakecomparisonsaccordingtotheHolmmethod.AllstatisticalpackagestodayincorporatetheHolmmethod. RelativemeritsofTukey,Scheffé,BonferroniandHolm Methods: Thereiswideagreementthateachofthesethreemethodshavetheirmerits. Therecommendationontherelativemeritsandadvantagesofeachofthese methodsintheNIST EngineeringStatisticsHandbookpageoncomparisonofthesemethodsare reproducedbelow: "ComparisonofBonferroniMethodwithSchefféand TukeyMethods Noonecomparisonmethodisuniformlybest-eachhasitsuses Ifallpairwisecomparisonsareofinterest, Tukeyhastheedge.Ifonlyasubsetofpairwisecomparisonsare required,Bonferronimaysometimesbebetter. Whenthenumberofcontraststobeestimatedis small,(aboutasmanyastherearefactors)Bonferroniisbetterthan Scheffé.Actually,unlessthenumberofdesiredcontrastsisatleast twicethenumberoffactors,Schefféwillalwaysshowwiderconfidence bandsthanBonferroni. Manycomputerpackagesincludeallthree methods.So,studytheoutputandselectthemethodwiththesmallest confidenceband. Nosinglemethodofmultiplecomparisonsis uniformlybestamongallthemethods." UniformsuperiorityoftheHolmMethodovertheBonferronimethod: ThefollowingexcerptsfromAickinandGensler(1996) makesitclearthattheHolmmethodisuniformlysuperiortothe Bonferronimethod: "Publichealthresearchersaresometimes requiredtomakeadjustmentsformultipletestinginreportingtheir results,whichreducestheapparentsignificanceofeffectsandthus reducesstatisticalpower.TheBonferroniprocedureisthemostwidely recommendedwayofdoingthis,butanotherprocedure,thatofHolm,is uniformlybetter.......Aswehaveshown,Holm(ed)Pvaluesareeasyto compute.Consequently,theredoesnotappeartobeanyvalidreasonto continueusingtheBonferroniprocedure." InadditiontothewisdomoftheNISTscientistsasabove,wehaveobserved raresituationswhereone-wayANOVAproducesap-valueabove0.05,producing human(thoughnotcomputer)disappointment,butBonferronicomparionoffewer contrasts(pairs)discernsasubsetofcontrasts(pairs)thataresignificantly different. Attribution 2016NavenduVasavada