HeatingUpinNBAFreeThrowShootingPaulR.PudaiteJanuary12,2018

AbstractIdemonstratethatrepetitionheatsplayersup,whileinterruptioncoolsplayersdowninNBAfreethrowshooting.Myanalysisalsosuggeststhatfatigueandstresscomeintoplay.If,asseemslikely,allfouroftheseeffectshavecomparableimpactonfieldgoalshooting,theywouldjustifystrategicchoicesthroughoutabasketballgamethattakeintoaccountthe‘hothand.’Moregenerallymyanalysismotivatesapproachingcausalinvestigationofthevariationinthequalityofalltypesofhumanperformancebyseekingtooperationalizeandmeasuretheseeffects.Viewingthehothandasadynamic,causalprocessmotivatesanalternativeapplicationoftheconceptofthe‘hothand’:insteadoftryingtodetectwhichplayerhappenstobehotatthemoment,promotethatwhichheatsupyouandyourallies.PerspectiveontheHotHandInmuchoftheliteratureandincommonlore,the‘hothand’hasbeenconceptualized(whetherimplicitlyorexplicitly)asadirectlyobservablephenomenon.Gelman(2015)statesthat“thehothandeffectissubtletodetect”butmaystillbefound“…ifyoureanalyzethe…datacarefully”.WiththisinmindIdevelopmathematicalmodels,employingBayesiandataanalysiswhererequired. Furthermore,ratherthanviewingthehothandasadiscretestate,Iproposeinvestigatingthequalityofeachindividualhuman’sperformanceasadynamicprocesswithcontinuousvariationinmagnitude.Whencedoesthehothandarise?Isitastatethatmysteriouslyappearsandthenvanishes?Myproposalisthat‘hands’arecontinuallyheatinguporcoolingdown,andwhathasbeendubbed‘thehothand’correspondstoanimpreciseupperzonealongaspectrumofperformance.Factorsthatimproveperformancecauseentryintothiszone;factorsthatdegradeperformancecausetheexitorpreventitsoccurrence.Thisperspectiveofthehothandarisingfromacausalprocess,incontrasttoanevanescentstate,motivatesadifferentmethodofapplication.Ratherthanundertakethedifficulttaskofidentifyingwho’shot(whetherbytheoffensetogethimtheball,orthedefensetoshuthimdown),focusonfosteringthatwhichwillcauseyoutoheatup,andthatwhichhindersyouropponent’sabilitytodolikewise.

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IntroductionVariationinthequalityofperformanceisubiquitousinallhumanactivities.Iproposethatrepetition,interruption,fatigue,andstressmakesubstantialcontributionstothisvariation.Chang(2017)alsopresentsevidencethaterrorcorrectionmakesacontribution.Performancegenerallyimproveswithrepetition,buteventuallydeclinesasfatiguesetsin,andtypicallydropswheninterruptedorunderstress.Individualscanalsovaryintheirresponsetotheoutcomeoftheiractions:forexample,errorcorrectionproducesimprovementafterfailure.Humanactivitycanthusbeviewedasadynamicprocesssubjectto‘damping’(negativefeedbackfromfatigueanderrorcorrection)and‘driving’(positivefeedbackfromrepetitionandoutcomereinforcement),alongwithperturbation(interruption).Theseeffectsarealwayspresent,butmaynotbeeasytomeasure.Inadditiontonoisefromperturbationandintrinsicstochasticity,deterministicdynamicsofdampinganddrivingbythemselvescangeneratechaos,inwhichhighsensitivitytoongoingconditionsmakespredictiondifficult.IfindstrongevidencefortheimpactofrepetitionandinterruptiononNBAfreethrowshooting.Ialsopresentevidencesuggestiveoffatigueandstress,butfurtherinvestigationwillberequiredtobeconclusive.TheHotHandinFreeThrowShootingAsnotedbyGVT(p.304),freethrowsare“freefromthecontaminatingeffectsofshotselectionandopposingdefense.”Thiscontrolledsituationforwhichhundredsofthousandsofobservationsarenowavailablemakesitpossibletoaccuratelymeasurecontributionsfromrepetitionandinterruption,andalsodetectindicationsoffatigueandstress.InTable3of“TheHotHandinBasketball:OntheMisperceptionofRandomSequences”,Gilovich,ValloneandTversky(1985;hencefort‘GVT’)present“dataforallpairsoffreethrowsbyBostonCelticsplayersduringthe1980‐1981andthe1981‐1982seasons”(p.304).GVTfound“noevidencethattheoutcomeofthesecondfreethrowisinfluencedbytheoutcomeofthefirstfreethrow.”Wenowknowthatthisfailuretofindevidencearisesinpartfrompoordiscriminationbetween‘hot’and‘cold’states.See,forexample,Gelman(2015).Ironically,GVT’sTable3actuallycontainsawealthofinformationthatunveilsthepathtoeffectiveexplorationofthe‘hothand’.Tobeginwith,itrevealsrepetitionasafundamentalcauseof‘heatingup’.Anditsuggeststhatinterruptioncancause‘coolingdown.’Initialfailurestoobservevariationinhumanperformancesufferedprimarilyfromapplicationofinadequatestatisticaltools(forexample,Richardson1945;GVT).By

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applyingsatisfactorystatisticalmethodology,subsequentstudieshavedemonstrateda‘hothand’in,respectively,theoutbreakofwar(HouwelingandKuné1984;Pudaite1991)andinbasketballfieldgoalshooting(Bocskocsky,EzekowitzandStein2014;MillerandSanjurjo2016).Thehothandhasbeendetectedinfreethrowshootingbyincreasingsamplesize,hencethepowerofthestatisticaltestsemployed.JeremyArkes(2010)analyzeddatafortheentireNBA2005‐2006season.Viafixed‐effectlogitmodeling,heestimatedthatthedifferencein2ndfreethrowpercentageconditionedbythe1stfreethrowwasCD=2.9%(0.8%)(standarderrorinparentheses).1PreviewofKeyEvidenceHothandresearchtypicallyemployssometypeofstatisticthatisconditionedbypastoutcomes.Forpairsoffreethrows,theconditionislimitedtothebinaryoutcomeofthe1stshot.Gelman(2015)showsthatevenforanontrivialeffectsize,conditioningonasingleshothasaverylargevarianceincomparisonbecausepastoutcomesonlyweaklyidentifytheshooter’sstate.Fortuitously,freethrowshootingexhibitsalargecausaleffectthatdoesnotdependonpastoutcomes:theactofshootingthefirstfreethrow,regardlessofwhetherhitormissedcausesatypicalplayertohit5‐to6‐percentage‐pointshigheronhis2ndshotthanonhis1st.Thiseffectislargeenough(abouttwicethesizeoftheconditionaleffectArkesestimated)toemergeclearlyinthefollowinganalysisperformedonGVT’srelativelysmallsampleoffreethrowdata:2

1However,Chang(2017)foundthatLeBronJames’sfreethrowsinthe2016‐2017seasonexhibiterrorcorrection(adampingeffectpermyintroductionabove),hittingfreethrows2GVT’stable3includesenoughdatatopreciselyrecoverallofthe‘raw’data,enablingtheanalysispresentedinTable1above.(SeeAppendix1fordetails.)

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N H1 H2 Pct1 Pct2 Pct2-Pct1 StdErr z

2049 1473 1590 71.9% 77.6% 5.7% 1.4% 4.21

N: Number of pairs of free throws

H1: Number of 1st free throws hit

H2: Number of 2nd free throws hit

Pct1: Percentage of 1st free throws hit

Pct2: Percentage of 1st free throws hit

StdErr: Classical standard error of Pct2-Pct1

z: Classical standard score of Pct2-Pct1

Table13NineMembersoftheBostonCelticsduringthe1980‐1981and1981‐1982seasons

Observingthisincreasedaccuracyonthe2ndshotmotivatesthehypothesisthatNBAplayers‘heatup’duringtripstothefreethrowline.Becauseplayersfrequentlyreceivemorethanonetriptothefreethrowlineduringagame,Table1alsohintsthatplayers‘cool’downbetweentripstothefreethrowline.TabulatingfourteenseasonsofNBAconfirmsthis:

Situation N H1 H2 Pct1 Pct2 Pct2-Pct1 z

S1: first of 2+ Trips of 2+ Shots 79,771 58,226 62,436 73.0% 78.3% 5.3% 24.598

S2: second of 2+ Trips of 2+ Shots 79,771 59,176 62,411 74.2% 78.2% 4.1% 19.043

Pctk[S2]-Pctk[S1]: 1.2% 0.0%

Classical Standard Error: 0.2% 0.2%

Classical Standard Score: 5.395 -0.152

Table21233players,NBA2000‐2001through2013‐2014seasons

Althoughthe1stshotofaNBAplayer’ssecondtriptothelineinagamerepeatshisactionfromhisfirstfreethrowtripofthegame,weseeasubstantialdropinsuccessratecomparedtothelastshotofhisfirsttrip(74.2%vs78.3%).Butalsoobservethattheplayersstillexhibitsomerepetitionbenefit:their1stshotpercentageishigheronthesecondtripthanonthefirsttriptothefreethrowline(74.2%vs73.0%).Table2conclusivelyestablishesthatrepetitionandinterruptionarecapableofcausingvariationinhumanperformance.Butthedemonstrationislightonrigorbecauseofmyinformaldefinitionsofrepetitionasasequenceoffreethrowshotsbyoneplayerduringonetriptothefreethrowline,andinterruptionaswhatevertranspiresbetweenthe

3Noteonterminology:percentagereferstoobservedsuccessrate;probabilityreferstothestatisticalexpectationofsuccessrate,whicharenotdirectlyobservable.

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player’stripstothefreethrowline.Itwillbeimportantforfutureresearchtoinvestigatehowbesttooperationalizetheseconceptsinavarietyofsettings.FreeThrowShootingDataThepatternof‘heatingup’inGVT’sfreethrowdata(Table1above)alsoappearsinNBAplay‐by‐playdatafor2000‐2001through2013‐2014seasons,withmuchgreaterstatisticalsupport:Situation N H1 H2 H3 Pct1 Pct2 Pct3 𝛿 1,2 𝛿 2,3 Z(1,2) Z(2,3)

Exactly 1 80,940 59,039 72.9%

Exactly 2 382,031 279,703 297,207 73.2% 77.8% 4.6% 46.56

3+ 4,638 3,622 3,861 3,943 78.1% 83.2% 85.0% 5.2% 1.8% 6.28 2.16

Total 73.2% 77.9% 85.0%

N: Number of free throws in this situation

Hk: Number of kth free throws hit

Pctk: Percentage of kth free throws hit

𝛿 𝑗, 𝑘 : Pctk-Pctj

Z(j,k): Classical standard score of Pctk-Pctj

Table3

ResultsofSingleTripstotheFreeThrowLineNBA2000‐2001through2013‐2014seasons

Intheadditional“3+”row–singletripstothelineforthreeormorefreethrows–performancecontinuestoimprovewithadditionalrepetition,atleastwithinasingletrip.Tomoredramaticallyframetheoverallimprovementduringatriptothefreethrowline:whenNBAplayerswenttothelineforthreeormorefreethrows,theymissed46%more(1016vs.695)oftheir1stthan3rdattempts!The‘improvement’from2ndto3rdfreethrowappearsevengreaterinthe‘Total’row(7.1percentage‐points,from77.9%to85.0%).However,thislevelofdataaggregationgreatlyoverestimatestheimprovementofindividualplayersfrom2ndto3rdfreethrowsuccessrates.Tripsforthreeormoreshotsoccurwhenaplayerisfouledattemptingathree‐pointfieldgoal,orwhenatwoshotfouliscompoundedwithatechnicalfoul.Asresultpoorershootersreceivefarfeweroftheseopportunities.Ingeneral,moredetailedclassicalanalysisbecomesweakerbecauseaddingconditions(1)subdividesthedataintogeometricallysmallersubsamples(aka‘bins’),and(2)mayintroducenewconfounders.Attemptingtocontrolforpotentialconfoundersviabinningreducessamplesizestillfurther.ApplyingBayesianmethodswiththeplayerastheunitofanalysis(ratherthantheleague)ismoreeffective.

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Model1:BayesiananalysisofindividualplayerdataBayesiandataanalysisprovidessuperiorstatisticalcontrolbypermittingamodeltoincorporateotherwiseconfoundingvariablesthataredifficultorimpossibletoestimateclassically–inthiscase,individualfreethrowshootingability,whichvariessubstantiallyacrosstheplayersintheNBA.Explicitmodelingofindividualabilityenablesmoreaccurateestimatesoftheimpactsofrepetitionandinterruption,andrevealspossibleeffectsoffatigueandstress.IorganizedModel1asasequenceoffournestedcomponents.Startingfromtheinsideout:

Yij{ }j=1

Ni: player i free throw trip j outcomes

Ni : number of free throw trips by player i

Yij = Yijk( )k=1

nij

nij : number of free throws in trip j by player i

Model1,component1:sampledata4Makingthesimplifyingassumptionthataplayer’sprobabilityofmakingafreethrowdoesnotdependontheoutcomeofpreviousfreethrowsinthetriptotheline,wecanwrite:

Pij = Pijk( )k=1

nij

Yijk = B 1,Pijk( )B n, p( ) : binomial random variable for n trials with probability p

Model1,component2:binomialdistributionAlthoughArkes(2014)andChang(2017)havereportedevidencethat1stfreethrowoutcomesaffect2ndfreethrowpercentage,wewillnonethelessseethatModel1providesasatisfactoryaccountoftheimpactofconditioningthe2ndfreethrowontheoutcomeofthe1st(seeAppendix3).

4Unlessotherwisenoted,uppercaseromanlettersindicaterandomscalarvariables,andboldindicatesrandomvectors.

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Toacknowledgethevariabilityofperformancewithineachindividualplayer,assumethatPijisdrawnfromalogisticallytransformedmultivariatenormallydistributedrandomvariable.Asafurthersimplifyingassumption,assumethatthemeanandvarianceoftheserandomvariablesdonotdependonthetotalnumberoffreethrowsinatriptotheline.Wecanthenwrite:

Pijk =eXijk

eXijk +1

= f Xijk( )Xij = N µi,!i( )

µi "#n

!i "#n$n

n : maximum number of free throws in one trip to the line

Model1,component3:logisticdistribution(Intra‐individualvariability)

Becausefewerthan1%ofthetripstothelineareforthreeormorefreethrows(4638outof467,609),Ireducedthecomputationalrequirementsbyestimatingthedistributionofthemomentsonlyforthe1stand2ndfreethrows.Estimation‐maximizationofthismodelfortheNBAdataproducesΨ1,adiscretedistributionof(model1)profiles,i.e.,hypothesesover µ

i,!

i( ), µi"#2

,!i"#2 x2 :

µi,!

i( ) : drawn from "1

"1= !

1m,µ

1m,!

1m( ){ }m=1

M1

µ1m#$2

!1m#$2%2

!1m= Pr µ

i,!

i( ) = µ1m

,!1m( ) "1

&' ()

Model1,component4:hierarchicalBayes(Inter‐individualvariability)

8

Figure1depictsModel1asaprobabilisticgraphicalmodel:

Figure1

Thelineswitharrowsindicaterandomdraws.Plates(therectangularregions)indicateindependent,identicallydistributeddraws.Lineswithoutarrowsindicatedeterministicrelationships.

!"#$%&'&

(µ!,!!)&

)%*+$,&!-'".""#"

)%*+$,&!/0&1,$$&23,"4&2,56&$-'".""#!"

%!$&='((!$&7&

1,$$&.3,"4&&-'"."")!$"

*!$&=+('8%!$&7&

!1= !

1m,µ

1m,"

1m( ){ }m=1

M1

Xij= N µ

i,!

i( )

9

Figures2aand2bshow84%5confidenceregionsforeachoftheM1=56profilespresentinthesupportofΨ1;figure2bdepictstheprofile’spriorprobabilityintheverticalaxis.

Figure2a

5Ichose84%becausemostoftheconfidenceregionsaresoeccentricthattheyareclosetolinear(inlogitspace).Inonedimension,an84%confidenceregionextendsonestandarddeviationineachdirection.

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Figure2b

Infigure2b,theverticalbluedashedstemshelplocateeachprofile’smeanvalue.Ineachfigure,thethickredcurvehighlights𝜃!,themodal(i.e.,highestpriorprobability)profile.Theperformancevariationweobserveforeachprofileisduetointra‐individualvariability(ofaplayer’sabilitytoshootfreethrows)acrosstripstothefreethrowlineforaplayer’sobservedcareerwithinthedata.ThisisexplainedfurtherinAppendix2usingtheexampleofLeBronJames(LeBronhasthehighestposteriorprobabilityofbelongingto𝜃!).Playersvaryintheirresponsetofreethrowrepetitionwithinasingletriptotheline.Mostoftheprofileslieroughlyparalleltothediagonalaxis,𝐸 𝑃!"! 𝜃! = 𝐸 𝑃!"! 𝜃! .Buttherearealsomanyprofileswhoseconfidenceregionsextendbelowthediagonal:forsometripstothelinebysuchplayers,repetitionhastheatypicaleffectofworseningperformance.Thereareeventwoprofilesforwhich𝜇!!! < 𝜇!!!,i.e.,theplayershoots2ndfreethrowsworsethan1stfreethrowsmostofthetime.Thickbluecurveshighlighttheseprofilesinbothfigures(oneishighlyeccentric,theotherresolvesasapoint).NotmanyNBAplayersfollowtheseprofiles;Pr 𝜇!!! < 𝜇!!! Ψ! =0.0090.

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Figure3showsposteriorprofileestimatesforLeBronJamesconditionalontheoutcomeofhis1stshotinatriptothefreethrowline:

Figure3

Theredandbluepoints,locatedat(73.0%,78.3%)and(70.6%,76.1%),respectively,showLeBron’sconditionalexpectedvalues.WeexpectLeBrontohit2ndfreethrows2.2percentage‐pointsmoreoftenifhemakeshis1stthanifhemisses(78.3%‐76.1%).Figure3illustratesGelman’sdiscussionofthedifficultyofidentifyinga‘hothand,’showingthatGVTcouldhavedetectedpositiveserialcorrelationwithsufficientdata.6Figure3alsoprovidesanexampleofthewayinwhichModel1accuratelyaccountsforArkes’sresultontheeffectofthe1stfreethrowonthe2nd.SeeAppendix3forfurtherdiscussionofserialcorrelationandthedifferencein2ndfreethrowpercentageconditionedon1stfreethrowoutcome.Model1accountsfortheempiricalvaluesofallofthesestatistics.Appendix3alsointroducesanintriguinganomalyforfutureinvestigation.6Forexampleifwesetanullhypothesistestthresholdatastandardscoreofz=2,givenLebron’sposteriorprofileestimate,itwouldtakeN=1487tripstothefreethrowlinetogenerate50%powerforthattest.

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Model2:Intra‐gameperformancevariationforeachtriptothelineModel2usesΨ1,theprofiledistributionestimatedforModel1.ThenewcomponentinModel2isΔ! ,thedisplacementfromaplayer’sModel1profileasafunctionofh,the‘intra‐gametripindex’ofthatplayer.Foreachvalueofh,Δ! isdrawnonceforallofthedatafromΨ!,thepriordistributionofthesedisplacements.7

Pijk =eZijk

eZijk +1

Zij = Zijk( )k=1

2

Zij =Xij +!h i, j( )

h i, j( ) : index of trip to the line within game for player i's career trip j

!h = !hk( )k=1

2

: drawn from "2

"2 = N0

0

#

$%

&

'(,)!

#

$%%

&

'((

)! * +2,2

Xij = N µi,)

i( )

µi,)

i( ) : drawn from "1

7Addinginter‐playervariabilitywouldprovidebettersamplingcontrol,butmodel2providessatisfactoryresultsforthispaper.Futureresearchwilllikelybenefitfromincorporatingthishierarchicallevel.

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Figure4depictsModel2asaprobabilisticgraphicalmodel:

Figure4

ViaexpectationmaximizationIobtainedΣ! = 0.0402 0.0080

0.0080 0.0346 ;estimatesof Δ! appearinthenextchart.Σ!correspondstoastandarddeviationof0.20logitunitsfor1stfreethrowsand0.19unitsfor2ndfreethrows,withacorrelationof𝜌 =0.21.ForatypicalNBAplayer(75.4%freethrowpercentage),thistransformstoa3.5percentagepointstandarddeviationinexpected1stand2ndfreethrowpercentageacrosstripstotheline.

!"#$%&'&

(%)*$+&!,-".""#"

(%)*$+&!/0&1+$$&23+"4&2+56&$,-".""#!"

1+$$&.3+"4&%,-".""&!$"

!1= !

1m,µ

1m,"

1m( ){ }m=1

M1 !2= N 0,"#( )

Xij= N µ

i,!

i( )

Zij=Xij +!h i, j( )

µi,!

i( )

Pijk = f Zijk( ) Yijk = B 1,Pijk( )

7)8$&.+56&59#$:&',-&."&''&

!h

14

Figure5

Weseeinfigure5thatplayersdefinitelyimproveontheirsecondtriptothefreethrowlineinagame,comparedtotheirfirsttrip, !

2,! 2ndtrip( )"! 1st trip( )

2=12.378 .8

Theimprovementappearstocontinuethroughthesixthorseventhtriptotheline,andtheremaybeadeclinefortheeighthandsubsequenttripstotheline.However,thesamplesizedeclinesgeometricallyashincreases.Inaddition,asmentionedinfootnote7(above),Model2ignoresthevariationacrossindividualplayers.Ashincreases, Δ! isestimatedfromaneversmaller,lessrepresentativesubsetofplayers.Asaresult,themeasurementerrorsaretoolargetomaketheseclaims(continuingimprovementfromthesecondthroughsixthorseventhtrip,andsubsequentdecline)withconfidence.

8TheMahalanobisdistancefork‐dimensionalrandomvectorW, !

k,W

2= !WVar W[ ]

"1W ,

followsa𝜒!distributionwithkdegreesoffreedom.

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Model3:Intra‐gameperformanceasafunctionofgametimeelapsed.ExtendingModel2topermitindividualvariationof Δ!! couldhelpalleviatethesamplingbias,butcan’tovercomethedecliningsamplesizewithrespecttoh.Tobetterexplorethepossibleroleoffatigueinfreethrowshooting,Iestimatedthefollowingmodel:

Zij =Xij +! h i, j( )tij( )

h i, j( ) =min(h i, j( ), 2)

tij : game time elapsed in player i's career trip j

!h t( ) = !hk t( )( )k=1

2

: drawn from "2

Figure6depictsModel3asaprobabilisticgraphicalmodel:

Figure6

Model3incorporatesthetwobestsupportedfindingsfromModel2:(1)thefirsttriptothefreethrowlinedifferssubstantiallyfromsubsequenttrips,and(2)Ψ!,thedistributionofintra‐gamefreethrowshootingdisplacementsfromtheplayer’scareerprofileestimated

!"#$%&'&

(%)*$+&!,-".""#"

(%)*$+&!/0&1+$$&23+"4&2+56&$,-".""#!"

1+$$&.3+"4&%,-".""&!$"

!1= !

1m,µ

1m,"

1m( ){ }m=1

M1 !2= N 0,"#( )

Xij= N µ

i,!

i( )

Zij =Xij +! h i, j( )tij( )

µi,!

i( )

Pijk = f Zijk( ) Yijk = B 1,Pijk( )

7)8$&258$&95:&'&

',-0.&."&;<.3&85:=.$&>&"?$+@8$"

!ht( )

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viaexpectation‐maximization.Model3introducesnonewparameters,hencenoadditionaldegreesoffreedom.Ibinnedobservationswithinregulationbytheminute,andcollectedallovertimeobservationsintoafinal49thbin.Figures7aand7bshow !

ht( ) ,binnedestimates,and

!!ht( ) ,Kalmanfilterestimatesfor,respectively,firstandsubsequenttripstothefreethrow

line.Overtimeestimatesareplottedsomewhatarbitrarilyatt=50.5minutes,themidpointofthefirstovertime.

Figure7a

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Figure7b

!!1t( )and !!2 t( ) bothdeclineoverthecourseofthegame,providingsomesupportforthe

hypothesisthatfatiguehampersperformance.Table4presentsMahanalobisdistancesforthesetrendsandadditionalsub‐trendsvisibleinthecharts.

Trend B0,1 B1,1 B0,2 B1,2 𝜒!,!! 𝜒!,!!

Decrease 1 49 1 49 3.525 2.864

Increase 1 5 1 7 2.527 0.624

Decrease 5 13 7 25 6.021 3.849

Increase 13 27 25 31 3.465 3.849

Decrease 27 49 31 49 3.999 1.905

Decrease 31 39 3.420

Increase 39 46 0.903

Decrease 46 49 1.259

B0,h: IndexofstartingbinfortrendinΔ! 𝑡 B1,h: IndexofendingbinfortrendinΔ! 𝑡 𝜒!,!! : MahalanobisdistancefortrendinΔ! 𝑡

Table4TrendStatisticswithin !!

ht( )

NBA2000‐2001through2013‐2014seasons

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Otherfactorscouldcomeintoplay.Forexample,itseemsplausiblethatthereismorestressthefirsttimeaplayergoestothefreethrowlinethedeeperintothegamethatoccurs;thiscouldcontributetothesteeperdeclineof !!

1t( ) comparedto !!2 t( ) .

Visually,therearefoursimilarsub‐trendssharedbythetwocharts.9Ineachcase,thedirectionchangesfor !!

2t( ) furtherintothegamethanfor !!1 t( ) .Thesimilaritiesencourage

furtherinvestigation.Whatcausestheimprovingperformanceatthestartandmiddleofthegame?Fatigueseemstobeaplausibleexplanationforthetwodecliningtrends,butwhydotheyoccurearlierforthefirsttriptotheline?Patternsofplayersubstitutionmayplayarole,whichwouldentailincorporatinginformationonhowmuchtimeeachplayerhasbeeninthegameateachtriptothefreethrowline.Oneofthesteeperdropsoccursattheendofthegame,fromthe45minutebinthroughovertime.Stressseemsparticularlylikelytoplayaroleattheendofthegame,particularlyifthescoreisclose.Itmaybepossibletoisolatethisaspectofstressbyaddingasuitableexplanatoryvariable,suchasthechangeintheprobabilityoftheplayer’steamwinningconditionalontheoutcomeofhistriptotheline.Wewouldcertainlyexpectthistovaryacrossplayers;inparticular,thisanalysiscouldidentify‘clutch’players:thosewhoperformbetterunderstressthantheircareerprofile.SomeImplicationsforBasketballStrategyCoacheshavesomeabilitytoharnessthebenefitsofrepetitiononoffenseandinterruptionondefense.1.Ifatechnicaliscalledinconjunctionwithacommonfoul,theremayoftenbesituationsinwhichtheteamshouldchoosetohavethecommon‐fouledteammembershootthreefreethrows,ratherthanswitchingtoa‘superior’freethrowshooterforthetechnical,particularlyiftheotherwisesuperiorshooterwouldbemakinghisfirstfreethrowattemptofthegame.Thestrategicalternativesboildowntoanticipatedsuccessratesofthe‘superior’shooter’s1stshotvs.thefouledshooter’s3rdshot.Typicallya6to8percentage‐pointadjustmentshouldbemade,andmayoftenbeevenlarger.2.Weknowthatcoachessometimesattemptto‘ice’afreethrowshooterbycallingatimeout.Itremainstobeseenwhethertheefficacyofthisinterruptionstrategycanbeevaluatedfromavailabledata.3.Theopposingteamcanalsointerruptafoulshooterbysubstitutinginaplayerbetweenhis1standlastfreethrowsinatriptotheline.Anecdotally,IhaveobservedmanyNBAcoachesdoingthis,whileNCAAcoachestypicallymaketheirsubstitutionsbeforefree9KeepinmindthattheMahalanobisdistancesoftheseintra‐gametrendsareinflatedduetoposthocselectionofthepeakandvalleytimes.

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throwsbegin,oraftertheshooterhitshislastfreethrow(substitutionisnotpermittedifthelastfreethrowismissed).SpeculativeStrategyImplicationsTherearelikelytobeevenmorevaluablestrategicopportunitiesforemployingrepetitionandinterruptioninfieldgoalshootingthanforfreethrows.Optimaldeploymentwillrequireaccurateidentificationandmeasurementoftheseandothercausalandpotentiallystrategicfactorsinfieldgoalshooting.Bocskocsky,Ezekowitz,andStein(2014)havenowperformedthemoredifficulttask(atleastcomparedtofreethrows)ofidentifyinghotfieldgoalshooters.Theirresearchmayhelpguidecausalstudy.4.Whenaplayer‘bricks’afieldgoalattemptinapickupbasketballgame,histeammatesmayneverpasshimtheballagainduringthegame.Becauseof(a)thewiderangeofabilityinsuchgamesand(b)thelackoffamiliarityamongsometeammates,thisreactionmaybejustifiedbyintuitiveBayesianinference.However,intheNBA,thecoachshouldhaveanaccurateappraisaloftheabilitiesofeachofhisplayers.Heshouldmakeatmostsmalladjustmentstothisappraisalinresponsetooutcomeswithinasinglegame(recallLeBron’sconditionalposteriorsinFigure3),andheneedstoensurethathisplayersalsounderstandthis.5.Aswithfreethrows,themereactoffieldgoalshootingmayimproveaplayer’sprobabilityofhittinghisnextfieldgoal,provideditissufficientlysimilarinlocationandtime.Thiseffectneedstobeaccuratelymeasured,ifindeeditevenexists.Anecdotally,PhilJackson’schampionshipChicagoBullsteamswerethefirstIsawbehaveinaccordwiththisprinciple,viz.,goingrightbacktoaplayerafteramissprovidedthathehadasimilaropportunityonthenextpossession.6.Conversely,preventopponentsfromgettingasequenceofsimilarshootingopportunities.Don’tnecessarilyincreasedefensivepressureonplayerswhohavemadefieldgoals.Instead,forceopponentstotakeasufficientlydifferentshot,regardlessofpreviousoutcome.7.The‘heatcheck’islikelyill‐advisedunlessitissufficientlysimilartopreviousshots.Again,‘sufficientsimilarity’remainstobequantified.PossibilitiesforFutureResearchModel1canbeadaptedtomeasureintra‐playerperformancevariationatdifferenttimescales.Forindividualplayers,estimateintra‐gamevariance,singleseasonvariance,andcareervariance.

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Studyintra‐playerperformanceasatimeseries.Overwhattimescalescanweidentify‘hot’vs‘cold’streaks?Canwelearnfromthishowplayerscanendcoldstreaks,andlengthenhotstreaks?Canclutchperformancebetrained?Isthereawaytohelpplayersavoidchoking?Fatiguemaybereducedwhenplaystopsduetofouls,timeouts,etc.Ideally,wewouldwanttoaddthetimeofdayasanadditionalexplanatoryvariable,butthismaybehardtoobtain.Distinguishbetweenphysicalandmentalfatigue.Mentalfatiguehasnumerousaspects,includingboredomandover‐stimulation.Improvetheimplementationofthemodelsinthispaper.Allofthecomponentscanbeestimatedsimultaneously(possiblyincreasingtheriskofoverfitting).Incorporateindividualvariationintheimpactofgametime.Commentsandideaswelcome!ReferencesArkes,Jeremy(2010)“RevisitingtheHotHandTheorywithFreeThrowDatainaMultivariateFramework.”J.QuantitativeAnalysisinSports6(1),Article2.Bocskocsky,Andrew,JohnEzekowitz,andCarolynStein(2014)“TheHotHand:ANewApproachtoanOld‘Fallacy’.”In8thAnnualMITSloanSportsAnalyticsConference,2014.Chang,JoshuaC.(July4,2017)“EvaluatingthehothandphenomenonusingpredictivememoryselectionformultistepMarkovChains:LeBronJames’errorcorrectingfreethrows.”arXiv:1706.08881v2[stat.ME]3Jul2017.Gelman,Andrew(October18,2015)http://andrewgelman.com/2015/10/18/explaining-to-gilovich-about-the-hot-hand/Gilovich,Thomas,RobertVallone,andAmosTversky(1985)“TheHotHandinBasketball:OntheMisperceptionofRandomSequences”.CognitivePsychology17,295‐314.Houweling,H.W.andJ.BKuné(1984)“DoOutbreaksofWarFollowaPoisson‐Process,”J.ofConflictResolution28(1):51‐61.Miller,JoshuaB.andAdamSanjurjo(November15,2016)“SurprisedbytheGambler’sandHotHandFallacies?ATruthintheLawofSmallNumbers”.Pudaite,Paul.R.(1991)ExplicitmathematicalmodelsforbehavioralsciencetheoriesUniversityofIllinoisatUrbana‐Champaign.

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Richards,L.F.(1945)“TheDistributionofWarsinTime.”J.oftheRoyalStatisticalSocietyCVII(NewSeries),III‐IV:242.250.__________(1960)StatisticsofDeadlyQuarrels.Chicago:BoxwoodPress.Appendix1:RawDataRecoveryfromGVTTable3Table3inGVT(p.305)reportstheobservedpercentagesofhittingasecondfreethrowconditionedoneachoutcomeofthefirstfreethrow,alongwiththenumberofshotstakenineachcondition,andthe(normalized)serialcorrelationforninemembersoftheBostonCelticsduringthe1980‐1981and1981‐1982seasons.Thesmallsamplesizesmakeitpossibletounambiguouslydeterminetheintegernumberofshotsmadeineachcondition,enablingfullrecoveryofthe‘raw’data:

Name N MM MH HM HHLarryBird 338 5 48 34 250CedricMaxwell 430 31 97 57 245RobertParish 318 29 76 49 165NateArchibald 321 14 62 42 203ChrisFord 73 5 17 15 36KevinMcHale 177 20 29 35 93M.L.Carr 83 5 21 18 39RickRobey 171 31 49 37 54GeraldHenderson 138 8 29 24 77

N: NumberofpairsoffreethrowsMM: Miss1st,Miss2nd

MH: Miss1st,Hit2ndHM: Hit1st,Miss2ndHH: Hit1st,Hit2nd

TableA1

NineMembers,BostonCeltics,1980‐1and1981‐2seasons

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Fromtherawdata,wecanobtainsuccessratesfor1stfreethrowsand2ndfreethrows:Name N H1 H2 Pct1 Pct2 Pct2‐Pct1 StdErr zBird 338 285 298 84.3% 88.2% 3.9% 2.6% 1.57Maxwell 430 302 342 70.2% 79.5% 9.3% 3.0% 3.15Parish 318 213 241 67.0% 75.8% 8.8% 3.6% 2.36Archibald 321 245 265 76.3% 82.6% 6.2% 3.2% 1.95Ford 73 51 53 69.9% 72.6% 2.7% 7.5% 0.37McHale 177 128 122 72.3% 68.9% ‐3.4% 4.8% ‐0.70Carr 83 57 60 68.7% 72.3% 3.6% 7.1% 0.51Robey 171 91 103 53.2% 60.2% 7.0% 5.4% 1.31Henderson 138 101 106 73.2% 76.8% 3.6% 5.2% 0.70

Total 2049 1473 1590 71.9% 77.6% 5.7% 1.4% 4.21

N: Numberofpairsoffreethrows H1: Numberof1stfreethrowshit H2: Numberof2ndfreethrowshit

Pct1: Percentageof1stfreethrowshit Pct2: Percentageof2ndfreethrowshit

StdErr: ClassicalstandarderrorofPct2‐Pct1 z: Standardscore

TableA2

NineMembers,BostonCeltics,1980‐1and1981‐2seasons

Appendix2:Model1ProfileInterpretationThethickredcurvecenteredat(72.8%,78.5%,6.6%)correspondsto𝜃!,themostcommonprofileofNBAfreethrowshooters.Inexpectation,about7%oftheplayersinthedata(81.8of1233)fallintothisprofile.Thecurveishighlyeccentric,indicatingthat1stand2ndfreethrowprobabilitiesco‐varytightly.

Butthecurvealsocoversawiderange:(59.9%,61.9%)atthelowerendofthe84%confidenceregionto(82.8%,89.2%)attheupperend.Onatriptothelinefortwofreethrows,theseplayersmissboth15%oftimewhen‘cold’,butlessthan2%ofthetimewhen‘hot.’…Andtheseplayersaremoreconsistentthanmost!

Fromtheperspectiveoffreethrowpercentage,theseplayersbenefitmorefromrepetitiononthe‘hot’endoftheconfidenceregion,witha6.4%increaseinfreethrowpercentagefrom1stto2ndcomparedtoa2.0%increaseinthe‘cold’end.Inlogitspace,therepetition

23

benefitisevenlarger:+0.08unitswhencold(0.40to0.48),+0.54units(1.57to2.11)whenhot.

TableA3showsthefiveplayerswiththehighest𝜃!profileposteriorprobability:10

Player Pr[𝜃!] SeasonsL.James 0.668 2003‐2004to2013‐2014T.McGrady 0.630 2000‐2001to2011‐2012P.Gasol 0.626 2001‐2002to2013‐2014T.Parker 0.610 2001‐2002to2013‐2014E.Brand 0.576 2000‐2001to2013‐2014

TableA3

FigureA1providesmoredetailon!

1Lebron, LeBron’sposteriorprofileestimate:

FigureA1

10Notationalhumor: argmaxPr !1 Y

ij{ }j=1

Ni!"#

$%&= Lebron James .

24

!1Lebron combinesmeasurementerror(uncertaintyinidentifyingLeBron’sspecific

profile)andtheactualvariabilityinhisfreethrowshootingacrosshiselevenseasonsinthedata.Becauseoftheprevalenceofhighlyeccentricprofilesin!

1thatareroughlyparallel

toy=x,wecaninferthatthemajoraxesoftheconfidenceregionsinFigureA1primarilyrepresentshisactualvariability,whilemeasurementerrorlengthenstheminoraxesandreduceseccentricityof!

1Lebron comparedto𝜃!.

SinceLeBronhasmadethousandsoftripstothefreethrowline,!

1Lebron assertsthatitis

highlylikelythathehashadmany‘cold’tripstotheline(say,E PLebron, jk!" #$< 0.60 ),andonthe

otherextreme,comparablymanytrips‘inthezone’(say,E PLebron, jk!" #$> 0.85 ).Appendix3:CorrelationStatisticsThefollowingstatisticisanunbiasedestimatorofthecovarianceoftheprobabilitiesofmakingthefirsttwofreethrowsinatriptotheline:

Ri : unbiased estimator of serial correlation of player i's first two free throws

=1

Ni2 !1Yij1 !Yi1( ) Yij2 !Yi2( )

nij!2

"

Yik = Yijkj:nij!2

" Ni2

Nih = 1j:ni #j !h

"

E Ri$%

&'=Cov Pij1,Pij2$% &'

=!12

Var Ri!"

#$=

Pij1 1%Pij1( )Pij2 1!Pij2( )N

25

Serialcorrelationandconditionaldifferenceareeffectivelyscaledversionsof𝑅!:

Ri: (normalized) serial correlation of player i's first two free throws

=1

Ni2si1si2

Yij1 !Yi1( ) Yij2 !Yi2( )

j:nij"2

#

E Ri[ ] $

Ni2!12

Ni2 !1( ) E P

i1[ ] 1!E Pi1[ ]( )E P

i2[ ] 1!E Pi2[ ]( )

CDi : difference of player i's 2nd free throw percentage,

conditional on 1st free throw outcome

Niqr=

Yij2j:nij"2,Yij1=1

#

1j:nij"2,Yij1=1

#!

Yij2j:nij"2,Yij1=0

#

1j:nij"2,Yij1=0

#

E CDi[ ] $!12

E Pi1[ ]

Theexpectedvalueformulasareapproximateduetothepresenceofrandomvariablesintheirdenominators.𝑅! bestisolatestheprimaryquantityofinterest,covariationofexpected1stand2ndfreethrowprobabilities.Italsosimplifiesthestatisticalassessment.ThenexttablereportstheexpectedvaluesofthesestatisticsassumingΨ1,alongwithobservedvaluesfortheNBAdata.Toavoiddivisionbyzeroinanyofthestatistics,Ionlyincludedplayersforwhomtherewasatleastoneoccurrenceeachofthefourpossibleoutcomesina2+shottriptotheline,reducingthenumberofplayersinthesamplefrom1233to992.

𝜙 𝐸 𝜙 Ψ! Average StdErr z Wtd Avg Wtd StdErr Wtd z

𝑅! 0.0059 0.0051 0.0009 6.980 0.0040 0.0003 15.315

Ri 0.031 0.027 0.004 8.116 0.026 0.002 16.444

CDi 2.93% 2.47% 0.40% 7.412 2.18% 0.15% 14.408

TableA4

CorrelationStatistics992players,NBA2000‐2001through2013‐2014seasons

Icomputedinformation‐weightedvalues(lastthreecolumnsofTableA4),i.e.,Iusedthereciprocalofthesamplestatistic’svarianceastheweight.Playerswithmoretripstothefreethrowlinereceivemoreweight.

26

Theuniformly‐weightedaveragesarefairlycloseto𝐸 𝜙 Ψ! .However,theinformation‐weightedaveragesarelower,andfurtherfrom𝐸 𝜙 Ψ! .Thisissurprisinginthecontextofperformancevariationatdifferenttimescales(seefirstparagraph,‘PossibilitiesforFutureResearch’section).Define

!it,"t( ) = E !

i#1,Yi t,"t( )$

%&'

Yit,"t( ) : observations of trips by player i between times t and t +"t

Wewouldexpect !

it,"t( ) toincreasewithrespectto!t ifthedynamicprocessgoverning

playeri’sfreethrowshootingabilityovertimefollowsarandomwalk.If,forexample,therewereasystematicdeclineinfreethrowshootingatcareer’send,thiswouldfurtherincrease !

it,"t( ) .

Possibleexplanationsofthisanomalyinclude:

(1) Playerswithmorestablefreethrowshootingabilitytendtohavelongercareers.(2) Errorcorrectingautoregressioncomponentinthedynamicprocessgoverning

individualfreethrowperformance.Thesearenotmutuallyexclusivehypotheses.Theycouldevenbemutuallyreinforcing.Errorcorrectingautoregressioncanbegeneratedbyputtinginmorepractice,orperhapsjustbyapplyingmorementalfocus,whenaplayermissesmorefreethrowsthanusual.Forexample,playerswhoaremorediligentaboutthis,orwithgreateraptitudeforcorrectingflaws,mighttendtohavelongercareers.