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ExtremeShootersintheNBA

ManavKantandLisaR.Goldberg

December24,20181.IntroductionItiswidelyperceivedthatstarshootersinbasketballhavehotandcoldstreaks.Thisperceptionwasfirstchallengedbyaninfluentialarticle,Gilovich,ValloneandTversky(1985),inwhichtheauthorsarguedthathotandcoldstreaksinbasketballareacognitiveillusionandcanbeexplainedbymerechance[1].Theyfurtherstatedthatthebreadthandpersistenceoftheperceptionthatsomethingexceptionalisgoingonillustratesjusthowpoorlyweunderstandrandomness.Gilovichetal.mainlyidentifiedhothandednessintwoways:asanelevatedprobabilityofscoringafterastreakofafixedlength,andasthetendencyofashootertoshootinruns,orclusters,ofhitsandmisses.

Usingtheseidentifications,anumberofsubsequentstudiescameoutwithdisparateanswerstothe“hothandquestion.”KoehlerandConley(2003)determinedthatdatafromtheNBALongDistanceShootoutcontestdidnotprovideevidenceforhotnessorforsequentialdependencies[2].Morerecently,however,MillerandSanjurjo(2018)foundthatthereis“asubtlebutsubstantialbias...inastandardmeasureoftheconditionaldependenceofpresentoutcomesonstreaksofpastoutcomesinsequentialdata,”anddeterminedthat“uponcorrectingforthebias,theconclusionsofprominentstudiesinthehothandfallacyliteraturearereversed”[3].However,Daks,DesaiandGoldberg(2018),accountingforthebiaspointedoutbyMillerandSanjurjowitha(nonparametric)permutationtest,concludedthattherewaslittleevidenceforhothandednessamongstarplayersoftheGoldenStateWarriorsteamthatwonthe2016-2017NBAchampionship[4].

Thebriefsummarygivenaboveis,ofcourse,bynomeansacomprehensivehistoryoftheliteratureonhothandednessinbasketball.Nevertheless,wehopeitprovidessomeinsightintothecontinueddebateonthesubtletiesinanalyzingstreakshooting.

Inthisstudy,weexploreanovelapproachtothehothandquestionbyconsideringthepossibilitythatthehumanperceptionofhothandednessmaybemorecloselyrelatedtothepercentageofshotsmadebyaplayerinagame(relativetotheexpectationssetbyhisseason-longhitpercentage)thantostreaksofshotsmadethroughoutthegame.Weworkwithsimpleformulationsof“hot”and“cold”thatrelyentirelyonfieldgoalpercentagesandnumberofshotstaken,usingdatafromtheofficialNBAAPIonallfieldgoalsattemptedinthe2015-2016,2016-2017,and2017-2018NBAseasonsbytopshooters.

Foranyplayer,weidentify“extremeshooting”inanyparticulargamebydeterminingtheprobabilitythattheplayerwouldhavescoredmoreoranequivalentnumberofshots(outofthosehetook)thanheactuallydidinthegame,giventhattheprobabilityoftheplayermakinganygivenshotwastheplayer’sseason-longshootingpercentage.Theplayerisdeemedan“extremeshooter”ifcertainhighorlowquantilesofthedistributionoftheseprobabilities,or“singlegamep-values,”throughoutaseasondiffersignificantlyfromthecorrespondingquantilesofadistributionthatassumestheplayer’ssinglegamep-valuesdeviatemodestlyfromwhatisexpectedbasedonhis

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season-longshotpercentage.Hotandcoldhandedness,inourstudy,aremerelytwodifferentcasesofextremeshooting.Aplayercanbehothanded,coldhanded,orboth.Ourconceptof"modestdeviation"isformulatedintermsofamixturebasedonbinomialdistributionsdeterminedbythenumberofshotstheplayertookineachgameandhisseason-longshootingpercentage.

Wedeterminethatcertainplayers—includingKlayThompson,whoisoftenconsideredtheprimeexampleofhothandedshooting—displaysignsofhotandcoldhandednessinourextremeshootingformulationofthehothandquestion.

Ourfindingscouldinfluencethewaythatcoachesselecttheirlineups(possiblyselectinghothandedplayersformatchesagainstmuchbetteropponents)andhowfansconnectwiththegameofbasketball(byconnectingpeople’sintuitionwithharddata).

2.Methodology2.1.MotivatingExampleConsiderahypotheticalplayerthattakes20shotsineachgameandmakes50%ofthoseshotsthroughoutthecourseoftheseason.Wedon’texpecthimtomakeexactly10of20shotsineachgame,andwedon’texpecthimtomakeall20shotsinhalfthegamesandnoshotsintheotherhalf.Ourexpectationsliesomewherebetweenthosetwoextremes.Wemightselectabinomialdistributionwith20trialsandmean0.5asamodelofhowwemightexpecthisgameaveragestobedistributed.Ifthegameaveragesareunusuallyconcentratedinthetailsofthedistribution,ourplayerisanextremeshooter.

2.2.IdentifyingExtremeShootinginaSingleGameGeneralizingfromthepreviousexample,aplayer’sshootingpercentageingamen,qn,isextremeifitisunusuallyhighorlowrelativetohisseasonaverage,q.Acomplicationisthatunlikeourhypotheticalplayer,arealplayerisboundtoattemptdifferentnumbersofshotsindifferentgames.Thus,simplyanalyzingtheplayer’shitpercentageinallgameswouldbeanoversimplificationoftheproblem.Aplayerwhomakes12shotsoutof16isamoreimpressiveshooterthanonewhomakes3shotsoutof4.This,however,wouldnotbereflectedinananalysisbasedpurelyonshotpercentages.Toputourobservationsonamorecommonfooting,wecomputethep-value,pn,ofqnwithrespecttothebinomialdistributionB(sn,q),wheresnisthenumberofshotsattemptedingamen.Thebinomialdistributionservesasamodelofamodestdistributionaroundtheseasonaverage,andthep-valueistheprobabilityofequallingorexceedingtheobservedvalueinthebinomialdistributionB(sn,q).Alowp-valuecorrespondstoahighgameaverageandconverselysince,

𝑝" =$ %𝑠"𝑘 ()*

+,)*-*𝑞+(1 − 𝑞))*3+ .

Inthepreviousexample,iftheplayer’sseason-longhitfractionwas0.5,thep-valuefortheformercase(12shotsoutof16)wouldbe0.0384,whereasthatforthelattercase(3shotsoutof4)wouldbe0.3125.Thisreflectsthefactthatitbecomesmoredifficulttoattainorexceedagivenhighshootingpercentageasthenumberofattemptedshotsincreases.2.3.EvaluatingaPlayerOvertheCourseofaSeason

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WelookatwhetheraplayershootsinanextrememannerbyexaminingP,thecollectionofaplayer’ssingle-gameshootingpercentagep-valuesrealizedoverthecourseofaseason.Considerfirst,thecumulativedistributionfunction(CDF)associatedwithP,denotedF.Roughlyspeaking,anunexceptionaldistributionofgameshootingpercentageswoulddriveFtowardtheCDFoftheuniformdistributionoftheunitinterval,denotedU,whichisa45degreeline.However,thisisnotexactlyright,asabinomialdistributionisdiscreteandthesingle-gamep-valuesareconstructedfromdifferentbinomialdistributions.Consequently,amoreaccuratenullhypothesisisamixtureofdiscretedistributions.LetMbetheCDFofamixtureofthep-valuedistributionsofbinomialdistributions,wheretheweightofthep-valuedistributionofB(s,q)inthemixtureisthefractionofgamesintheseasoninwhichtheplayerattemptedsshots.Inwhatfollows,wewillusesimulationsofseasonsbasedonthenullhypothesisconstructedabovetodeterminewhich,ifany,quantilesofFareexceptionalandwhattheseexceptionalquantilesindicateaboutthedistribution.Forexample,anunusuallylow10%quantilewouldcorrespondtoapreponderanceofunusuallyhigh-percentagegames(indicatinghothandedshooting),andanunusuallyhigh90%quantilewouldcorrespondtoapreponderanceofunusuallylow-percentagegames(indicatingcoldhandedshooting).Ingeneral,unusuallylowlowerquantiles(quantilesbelow50%)indicatehothandedness,andunusuallyhighupperquantiles(quantilesabove50%)indicatecoldhandedness.Oursimulationisbasedon30,000replicationsofastarplayer’sseason.Ineachreplication,wedrawashootingpercentageatrandomfromeachbinomialdistribution,B(sn,q)andnoteitsquantileintheempiricaldistributionofsinglegamep-valuesassociatedtothedraws.Finally,forallQsuchthatQisamultiple5ontheinterval[10,90],wecomputethep-valueoftheQ%quantileoftheplayer’ssinglegamep-valuedistributionwithrespecttothedistributionofQ%quantilesofthe30,000seasonsimulations.IfQ>50andthis“Q%quantilep-value”isgreaterthan0.9,thenwedeemtheplayercoldhanded.IfQ<50andtheQ%quantilep-valueislessthan0.1,thenwedeemtheplayerhothanded.Forpracticalpurposes,wedefinetheQ%quantileofanydistributionconsistingofnvaluestobethevalueatthe⌈(Q/100)n⌉thpositionwhenthevaluesareplacedinascendingorder.2.4.CriteriaforPlayerEvaluationInordertoreducetheeffectsofthenoisethatispresentinsmalldatasets,wesetcriteriaforevaluationbasedonthenumberofgamesplayedinaseasonandtheaveragenumberoffieldgoalsattemptedpergamebyanyplayer.Welookataplayeronlyifheplayed60ormoregames(outofapossible82)andattemptedanaverageofatleast14fieldgoalspergameduringtheseasonbeingstudied.Byensuringthateachplayerplayedareasonablenumberofgames,wereducetheprobabilityofmakingaTypeIerror;itismorelikelythat,bysheerchance,thelowest3p-valuesinadistributionconsistingof30p-valuesarelowenoughtoindicatehothandednesswhenthereisnone,thanthatthelowest8p-valuesinadistributionconsistingof80p-valuesareassuch.Ourcriteriononaveragenumberoffieldgoalsattemptedaccomplishesthesamepurpose,butforsinglegamep-valuesratherthanfortheseason-longdistributionofp-values.Ouruseofaplayer’sseason-longshootingpercentagetosetexpectationsforhisshotpercentageinanygivengameintroducesanelementoflook-aheadbiastoouranalysis.Thisisbecausetheplayer’sseason-longhitpercentageisdeterminedfromanaggregateofthedatafromallofthegamesheplayed,manyofwhichmayhavebeenplayedafterthegamebeingstudied.Inessence,wemeasuretheplayer’sperformanceagainstanulldistributionthatisbasedonavaluethatwasnot

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fixedatthetimeofthegame,andthatinfact,thegamebeingstudiedplayedaroleindetermining.Inordertoreducetheeffectsoflook-aheadbiasonouranalysis,wedeterminedthat,tobedeclaredanextremeshooter,aplayermustnothaveacleartrendinhishitpercentagethroughouttheseason.Forexample,ifaplayerwhoattempted20shotsineachgame(forthesakeofsimplicity)consistentlyshotataround20%forthefirsthalfoftheseasonandataround80%forthesecond,thenaseason-longhitpercentageofabout50%maybeanincorrectexpectationforeitherhalfoftheseason,astheplayerlikelysimplygotbetterathittingshotsaroundhalfwaythroughtheseason.Then,neithera20%hitpercentageinagameatthebeginningoftheseasonnoran80%hitpercentageinagameattheendoftheseasonwouldbeconsideredextremeperformances,astheywouldbothcorrespondtotheexpectedperformancesfortheirrespectiveportionsoftheseason.3.ResultsWefirstappliedourmethodologytothoseplayersinthe2016-2017seasonthatsatisfiedourplayerevaluationcriteria(seesection2.4),andthenanalyzedthemembersofthatgroupthatalsosatisfiedthosecriteriainthe2015-2016and2017-2018seasons.3.1.2016-2017SeasonResultsForthe2016-2017season,wefound4players--KlayThompson,DeMarDeRozan,GordonHayward,andJimmyButler--tobehothanded,coldhanded,orboth(seeTableI).

TableI.Quantilep-valuesforDeMarDeRozan,KlayThompson,JimmyButler,andGordonHaywardinthe2016-2017season.SeeTableVforquantilep-valuesforallplayersinthe2016-2017season.

Inouranalysis,wefoundthatthequantilep-valuesforthe30%,35%,and40%quantilesofKlayThompson’ssinglegamep-valuedistributionwere0.052,0.065,and0.023respectively(seeTableI).ThesevaluesgiveusreasontobelievethatThompsonisahothandedshooter,astheyindicatethathisshootingfollowsadistributionthathasagreaterconcentrationofsinglegamep-valuesatitslowerendthandoesthenull.Aslowerp-valuesresultfromhighershotpercentages(seesection2.2),thisimpliesthatThompsonhasaproclivitytoget“hot”(withrespecttotheexpectationssetbyhisseason-longhitfraction)inadisproportionatelylargenumberofgames.AvisualizationofthisempiricalresultispresentedinFigure1.1(A).Atthe30%,35%,and40%quantiles(demarcatedbythedashedyellowlines),theCDFofKlayThompson’ssinglegamep-valuedistributionisclearlytotheleftofthenulldistribution.However,atthe45%quantileandabove,Thompson’sdistributionsticksfairlyclosetothenull.Finally,lookingatFigure1.1(B),wedeterminedthatKlayThompson’sshootingpercentagesthroughoutthe2016-2017seasondidnotseemtofollowanycleartrend.Thus,look-aheadbiaslikelydidnotplayalargeroleintheoutcomesfromouranalysis.

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(A) (B)

Figure1.1.ThegraphontheleftdisplaystheCDFofKlayThompson’ssinglegamep-valuedistributioninthe2016-2017season(inred)ascomparedtothatofthenulldistributionforKlayThompsonthatseason(ingreen).ThegraphontherightisatimeseriesgraphofThompson’shitfractionineachgamethroughoutthe2016-2017season.

WefurtherfoundDeMarDeRozanandGordonHaywardtobecoldhanded(incontrastwithKlayThompson)asillustratedinFigure1.2andFigure1.3.

Figure1.2.DeMarDeRozan’ssinglegamep-valuedistributionforthe2016-2017seasonascomparedtothenulldistribution,andthetimeseriesgraphofDeRozan’shitfractionsthroughoutthatseason.

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Figure1.3.GordonHayward’ssinglegamep-valuedistributionforthe2016-2017seasonascomparedtothenulldistribution,andthetimeseriesgraphofHayward’shitfractionsthroughoutthatseason.

Finally,wefoundJimmyButlertobeageneralextremeshooter(showingsignsofhotandcoldhandedness)inthe2016-2017season,with25%,85%,and90%quantilep-valuesof0.054,0.917,and0.933respectively(seeTableI).Thesevaluesindicatethathisshootinginthe2016-2017seasonfollowedadistributioninwhichsinglegamep-valuesareconcentratedatbothtails.ThiscanbeseeninthegraphoftheCDFassociatedwithButler’ssinglegamep-valuedistributionandthetimeseriesgraphofhiscrudeshootingpercentages,thelatterofwhichmakesapparentthevariabilityassociatedwithButler’sshooting(seeFigure1.4).

Figure1.4.JimmyButler’ssinglegamep-valuedistributionforthe2016-2017seasonascomparedtothenulldistribution,andatimeseriesgraphofButler’shitfractionsthroughoutthatseason.

3.2.2015-2016SeasonResultsForthe2015-2016NBAseason,wefoundevidencethatoneplayer--DeMarcusCousins--wasacoldhandedshooter,asillustratedbyTableIIandFigure2.

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TableII.Quantilep-valuesforDeMarcusCousins’singlegamep-valuedistributioninthe2015-2016season.SeeTableIVforquantilep-valuesforallplayersinthe2015-2016season.

Figure2.DeMarcusCousins’singlegamep-valuedistributionforthe2015-2016seasonascomparedtothenulldistribution,andthetimeseriesgraphofCousins’hitfractionsthroughoutthatseason.

3.3.2017-2018SeasonResultsWefound4players--RussellWestbrook,AndrewWiggins,KembaWalker,andKlayThompson--tobehothanded,coldhanded,orbothduringthe2017-2018NBAseason(seeTableIII).Westbrookinparticularshowedclearsignsofhothandedness,withquantilep-valuesof0.026,0.087,and0.058athis10%,15%,and20%quantilesrespectively(seeTableIIIandFigure3.1).

TableIII.Quantilep-valuesforRussellWestbrook,AndrewWiggins,KembaWalker,andKlayThompsoninthe2017-2018season.SeeTableVIforquantilep-valuesforallplayersinthe2017-2018season.

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Figure3.1.RussellWestbrook’ssinglegamep-valuedistributionforthe2017-2018seasonascomparedtothenulldistribution,andthetimeseriesgraphofWestbrook’shitfractionsthroughoutthatseason.

WealsofoundevidencethatbothAndrewWigginsandKembaWalkerwerehothandedinthe2017-2018season(seeFigure3.2andFigure3.3).

Figure3.2.AndrewWiggins’singlegamep-valuedistributionforthe2017-2018seasonascomparedtothenulldistribution,andthetimeseriesgraphofWiggins’hitfractionsthroughoutthatseason.

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Figure3.3.KembaWalker’ssinglegamep-valuedistributionforthe2017-2018seasonascomparedtothenulldistribution,andthetimeseriesgraphofWalker’shitfractionsthroughoutthatseason.

Finally,wefoundevidencethatKlayThompsonshowedsignsofbothhothandednessandcoldhandednessinthe2017-2018season.ThisisevidentinthegraphoftheCDFofhissinglegamep-valuedistributionforthatseason--thegraphseemstobetotheleftofthenulldistributionatthelowerquantiles,andtotherightofthenullattheupperquantiles(seeFigure3.4).

Figure3.4.KlayThompson’ssinglegamep-valuedistributionforthe2017-2018seasonascomparedtothenulldistribution,andthetimeseriesgraphofThompson’shitfractionsthroughoutthatseason.

4.ConclusionTheperceptionofthe“hothand”inbasketballiscommonamongfans.However,manypreviousstudiesthatfocusedonstreakshootingwithingameslargelydidnotfindthestatisticalevidenceforthisphenomenon.

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Bylookingatshotpercentagesthroughoutaseasonratherthanshotstreakswithinagame,wedeterminedthatthereisreasontobelievethatanumberofNBAplayersareextremeshooters--playersthatshootaccordingtosinglegamep-valuedistributionsthatdifferatthetailsfromwhatcanbeexpectedbasedontheirseason-longhitfractionsandthenumberoffieldgoalstheyattemptedineachgame.Someoftheseplayers,mostvisiblyKlayThompson,arewidelyperceivedtobeplayersthat“gethot”incertaingames,achievingresultsthatnormallyseemfaroutoftherangeofpossibilities.WedidinfactfindThompsontobeahot-handedshooterinoneseasonoutofthethreewestudiedandanextremeshooterinanother.Thisindicatesthatourmethodologymaybebetteralignedwithfanpsychologythantraditionalstreakshooting-basedmethods.5.AcknowledgementsWethankAlexPapanicolaouforprovidinguswithshotdatafromtheNBAAPIPHPLibrary.WearegratefultoNishantDesaiandJoshMillerfortheirhelpfulcommentsintheinitialstagesofourstudy.6.ResourcesCodehostedat:https://github.com/kantmanav/Reproducibility-for-Extreme-Shooters-in-the-NBA

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References[1] Gilovich,ThomasRobertVallone&AmosTversky(1985).Thehothandhandin

basketball:Onthemisperceptionofrandomsequences.CognitivePsychology17,295-314.

[2] Koehler,J.J.&Conley,C.A.(2003)The“hothand”mythinprofessionalbasketball.JournalofSports&ExercisePsychology,25(2),253-259.

[3] Miller,J.&Sanjurjo,A.(2018).Surprisedbythegambler’sandhothandfallacies?atruthinthelawofsmallnumbers.Econometrica86(6),2019-2047.

[4] Daks,A.,Desai,N.andGoldberg,L.R.(2018).DotheGoldenStateWarriorshavehothands?TheMathematicalIntelligencer40(4),1-10.

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AppendixBelowarecompletetablesofquantilep-valuesforallplayersanalyzedinthe2015-2016,2016-2017,and2017-2018NBAseasons.

TableIV.2015-2016playerquantilep-values.

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TableV.2016-2017playerquantilep-values.

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TableVI.2017-2018playerquantilep-values.