Modeling crash frequency and severity using multinomial-generalized Poisson model with error components

ContentslistsavailableatSciVerseScienceDirect

Accident

Analysis

and

Prevention

jo u r n al hom e p a g e :w w w . e l s e v i e r . c o m / l o c a t e / a a p

Modeling

crash

frequency

and

severity

using

multinomial-generalized

Poisson

model

with

error

components

Yu-Chiun

Chiou

∗

,

Chiang

Fu

InstituteofTrafficandTransportation,NationalChiaoTungUniversity,4F,118,Sec.1,Chung-HsiaoW.Rd.,Taipei100,Taiwan

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received16October2011

Receivedinrevisedform25March2012 Accepted26March2012 Keywords: Crashfrequency Crashseverity Multinomial-generalizedPoisson Errorcomponents

a

b

s

t

r

a

c

t

Sincethefactorscontributingtocrashfrequencyandseverityusuallydiffer,anintegratedmodelunder themultinomialgeneralizedPoisson(MGP)architectureisproposedtoanalyzesimultaneouslycrash frequencyandseverity—makingestimationresultsincreasinglyefficientanduseful.Consideringthe substitutionpatternamongseveritylevelsandthesharederrorstructure,fourmodelsareproposed andcompared—theMGPmodelwithorwithouterrorcomponents(EMGPandMGPmodels, respec-tively)andtwonestedgeneralizedPoissonmodels(NGPmodel).Acasestudybasedonaccidentdatafor Taiwan’sNo.1Freewayisconducted.TheresultsshowthattheEMGPmodelhasthebestgoodness-of-fit andpredictionaccuracyindices.Additionally,estimationresultsshowthatfactorscontributingtocrash frequencyandseveritydiffermarkedly.Safetyimprovementstrategiesareproposedaccordingly.

© 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Toimprovetrafficsafety,numerousstatisticalmodelshavebeen developed that identifyfactors contributing tocrash frequency andseverity.Mostidentifyriskfactorsforeithercrashfrequency orseverityindependently.Whenmodelingcrashfrequency(the numberofaccidentsonroadwaysegmentsoratintersectionsover aspecifiedperiod),aconsiderablenumberofstudieshaveused variousmethodologicalapproaches.Duetothediscreteand non-negativeintegercharacterofaccidentcounts,count-datamodels suchasthePoissonmodel(e.g.,Jonesetal.,1991;Miaou,1994; Shankaret al.,1997), negativebinomialmodel(e.g.,Hadietal., 1995; Shankar et al.,1995; Poch and Mannering,1996; Milton and Mannering, 1998; Lord, 2006; Malyshkina and Mannering, 2010),Poissonlognormalmodel(e.g.,Miaouetal.,2005;Lordand Miranda-Moreno,2008),Gammamodel(e.g.,Ohetal.,2006), gen-eralizedPoissonmodel(e.g.,Dissanayakeetal.,2009;Famoyeetal., 2004)aswellaszero-inflatedmodelingandotherflexiblemodeling techniques(e.g.,Abdel-AtyandRadwan,2000;Wangand Abdel-Aty,2008;ParkandLord,2009;AnastasopoulosandMannering, 2009;seeLordandMannering,2010forelaborateandcomplete reviews)havebeenappliedtomodelcrashcounts.

Crash frequencies are commonly collected by severity on relativelyhomogenousroadwaysegments,supportingthe devel-opment of crash count models. Thus, crash data are typically classified according to severity (e.g., property damage only, injury, and fatality) or collision type (e.g., rear-end, head-on,

∗ Correspondingauthor.Tel.:+886223494940;fax:+886223494953. E-mailaddress:ycchiou@mail.nctu.edu.tw(Y.-C.Chiou).

sideswipe,and rightangle). Withthis datasegmentation, sepa-rateseverity–frequencymodelsaredeveloped foreachaccident severitylevel.Inthisway,aseriesofnegativebinomialaccident frequencymodelsweredevelopedforeachcrashseveritylevelto predictthenumberofaccidentsateachseveritylevelonroadway segments.Unfortunately,suchanapproachcangeneratea statis-ticalprobleminthatinterdependenceduetolatentfactorslikely existsacrosscrashratesatdifferentseveritylevelsforaspecific roadwaysegment(Maet al.,2008).Forexample,anincreasein numberofaccidentsthatareclassifiedashavingacertainseverity levelisalsoassociatedwithchangesinthenumberofaccidents thatareclassifiedwithotherseveritylevels,settingupa correla-tionamongvariousinjury-outcomecrashfrequencymodels(Lord andMannering,2010).

Considerableresearchefforthasfocusedonmodelingaccident severityfromanindividualperspectiveusingsuch methodolog-icalapproaches aslogisticregression(e.g.,Luiet al.,1988;Yau, 2004),bivariatemodels(e.g.,Saccomannoetal.,1996;Yamamoto andShankar,2004),themultinomialandnestedlogitstructures toevaluateaccident-injuryseverities(e.g.,Shankar etal.,1996; Changand Mannering,1999;Carson and Mannering,2001;Lee andMannering,2002;UlfarssonandMannering,2004;Khorashadi etal.,2005),andthediscreteorderedprobitmodel(e.g.,O’Donnell and Connor, 1996; Duncan et al., 1998; Renski et al., 1999; Kockelmanand Kweon, 2002;Khattak etal., 2002; Kweonand Kockelman,2003;Abdel-Aty,2003).Formoredetailsonaccident severitymodelsmayrefertoSavolainenetal.(2011).

Although these models have been applied by a number of researcherswithaconsiderablesuccess,Miltonetal.(2008) indi-catedthatthesestudiesreliedheavilyondetaileddatainindividual accidentreportsandtheyhavebeenprovedtobedifficulttouse 0001-4575/$–seefrontmatter © 2012 Elsevier Ltd. All rights reserved.

insafetyprogrammingbecausealargenumberofevent-specific explanatoryvariables needto beestimated toproduceuseable severityforecasts. Moreover,significant contributory factors in the model are usually not closely related to traffic manage-mentstrategies,roadwaygeometrics,andweather-relatedfactors; therefore,thecorrespondingcountermeasuresaredifficultto pro-poseaccordingly.Furthermore,asdifferentdatascalesareusedby frequencymodelsandtheseveritymodel,integrationisextremely difficult.

Obviously,crashfrequency and severityaretwo key indices thatmeasureriskforaroadwaysegment.Eitheroneonly gener-atespartialinsightsforcrashrisk.Increasedscopeandin-depth insights cannot be obtained without considering both indices together.Thus,twopossibleintegratedmodelingapproacheswere attempted. The first approach uses a conventional frequency modeltopredicttotal numberof crashesand aseveritymodel, suchasthemultinomiallogitmodel,nestedlogitmodel,ordered probitmodel,ormixed logitmodel, topredictaggregate sever-ity probability (e.g., Yamamoto et al., 2008; Kim et al., 2008; Milton et al., 2008). However, the assumption that crash fre-quencyand severityare mutually independent still exists. The secondapproachappliesmultivariateregressionmodelsto pre-dictcrashfrequencies for differentseverity levels.Multivariate regressionmodelssimultaneouslydevelopcrashfrequency mod-elsbyseverity(Bijleveld,2005;MaandKockelman,2006;Song etal.,2006;ParkandLord,2007;Maetal.,2008;Aguero-Valverde andJovanis,2009;El-BasyounyandSayed,2009;Yeetal.,2009) toovercomethecorrelationproblemamongcrashfrequenciesat differentseveritylevels.However,thisapproachrequiresa com-plexestimation procedurecombinedwitha subjectivelypreset correlationmatrixofseveritylevels,makingfieldvalidationvery difficult.

Anotherdrawbackofthemultivariatemodelingapproachisits inabilitytograspassociatedchangesrelatedtoseverityand fre-quencyvariablesonly.Ifonefailstoobserveseparatelytheeffectsof factorscontributingtocrashfrequencyandseverity,the multivari-atemodelmaybepartlylimitedforpracticalprogramevaluation. Anappealingideaistoviewriskfactorsaccordingtotheiraccident descriptivecomponents(i.e.,severityandfrequency)individually under an integrated framework. However, expected difficulties arisewhenanalyzingsubjectsandprocedures.Consequently,using aconceptualmodelcombiningbothcrashfrequencyandseverity isworthwhile.

Thus, this paper aims to develop a novel multinomial gen-eralized Poisson (MGP) model to simultaneously model crash frequency(countdata)andseverity(ratiodata).Furthermore,the proposedmodelconsidersthesubstitutionpatternamong sever-itylevelsandconstructsasharederrorstructureasacorrelation matrixthrougherrorcomponentsspecifiedunderanintegrated modelframework.AcasestudyofTaiwanfreewaycrashdatais utilized to assess the applicabilityof theproposed model. The remainderofthispaperisorganizedasfollows.Section2presents theproposedMGPmodel.Section3addressesdatacollectionand descriptive statistics of theaccident datasetfor Taiwan’s No. 1 Freeway.Section4presentsmodelestimationresultsand compar-isons.Section5discussessafetyimplicationsbasedonestimation results.Section6givesconcludingremarksandsuggestionsto fur-therresearch.

2. Theproposedmodels

TheMGPmodelisanextensionofthemultinomial-Poisson(MP) regressionmodel(TerzaandWilson,1990).Inthecontextofcrash frequencyandseveritymodeling,weassumethataccidentscanbe classifiedintoafinitenumberofclustersaccordingtoseveritylevels

andthatthefrequencyofeachseveritylevelfollowsaconditional multinomialdistribution,whichisexpressedasfollows:

f

⎛

⎝

Y







J



j=1 yj=N

⎞

⎠

= N!



J j=1 yj j



J j=1yj! (1)

wheref(·)istheconditionalprobabilityofY;Y=[y1,y2,...yj,...,yJ]

and

J_j=1yj=N;yj=0,1,2,...,∞,forj=1,2,...,J,isarandomvector

representingtheobservedcrashcountsofsegmenttwithinagiven period(e.g.,1year)atseveritylevelj;Jisthetotalnumberof sever-itylevelsdeterminedinadvance; 1,2,...,Jaremultinomial

probabilitiesofseveritylevels1,2,...,J,respectively;j=yj/Nand

1+2+...+J=1;andNisthetotalnumberofaccidentsacross

differentseveritylevelsofsegmentmwithinagivenperiod.Thus, theconditionalmultinomialdistributioncanbeusedtodetermine crashfrequenciesatvariousseveritylevels,i.e.,y1,y2,...,yJ,given

totalnumberofaccidents,N.Furthermore,thejointprobabilityof thesecrashfrequenciesh(y1,y2,...,yJ)canbeexpressedasthe

productofconditionalprobabilityandmarginalprobability:

h(y1,y2,...,yJ)=f

⎛

⎝

Y







J



j=1 yj=N

⎞

⎠

·g

⎛

⎝



J j=1 yj=N

⎞

⎠

(2) whereg(·)=g

J_j=1yj=N



isthemarginalprobabilityofcrash counts.TerzaandWilson(1990)assumedthatthemarginal (uncon-ditional)probabilityhasthefollowingPoissondistribution:

g(·)=Nexp(_N!−) (3)

whereg(·)istheprobabilitythatNaccidentsoccurred,andisthe expectednumberofaccidents.Forestimationpurposes,isusually specifiedas

=exp(ˇX) (4)

whereXandˇarevectorsofexplanatoryvariablesandestimated parameters,respectively.Theformulationofthemultinomial Pois-son(MP)modelisthenderivedbysubstitutingEqs.(1)and(3)into Eq.(2).

The Poisson model assumes that variance equals mean. If observed data exhibit over-dispersion (under-dispersion), this assumptiondoesnot hold.Thisleadstoestimation inefficiency becauseinferencewasinvalidatedbyunreliableestimated stan-darderrors.Wecanrelaxthisassumptionusingthegeneralized Poisson(GP)model(Famoyeetal.,2004;Dissanayakeetal.,2009). Theprobabilityfunctionoftotalaccidentsatanysegment,N,can bewrittenasEq.(5): g(·)=

 1+



N (1+N)N−1 N! exp

_−(1+N) 1+



(5) whereisthedispersionparameter.If>0,theGPmodel indi-catestheover-dispersionfeatureintheempiricaldata.If=0,the probabilityfunctiondegeneratedtothePoissonmodel.Incontrast, if<0,theGPmodeldenotestheunder-dispersionfeatureinthe empiricaldata.AllotherinvolvedargumentsassociatedwithEq. (5)areasdefinedpreviously.ThemeanandvarianceofNare rep-resentedbyEqs.(6)and(7),respectively:

E(N|X)= (6)

V (N|X)=(1+)2 (7)

AccordingtoEq.(6),theprobabilityfunctioninEq.(5) degener-atesintotheoriginalPoissonmodelas=0.Hence,theGPmodelis ageneralizedPoissonmodel.InterestedreaderscanrefertoFamoye (1993)fordetailedproofs.Inaccordancewiththederivationby

TerzaandWilson(1990),formulationoftheMGPmodelcanbe derivedbysubstitutingEqs.(1)and(5)intoEq.(2)asfollows:

h(y1,y2,... yJ)=



J j=1 yj j



J j=1yj!

⎡

⎣

1+

⎛

⎝



J j=1 yj

⎞

⎠

⎤

⎦

J j=1yj−1 ×

 1+



J j=1yj exp

⎡

⎣

−

1+

J_j=1yj



1+

⎤

⎦

(8) where=

J

j=1jisexpectedtotalaccidents.j=jisexpected

crashcountatthejthseveritylevel.jistheprobabilityofseverity

levelj.

Weassumeprobabilitycanbedeterminedbythemultinomial logit(MNL)model: i= exp(si)

j j=1exp(sj) (9)

whereSj=Z+

v

jisalinearlyadditivefunctionformeasuringriskof

severitylevelj;Zisavectorofnon-randomexplanatoryvariables, suchasroadwaygeometrics,trafficfactors,landuse,andweather condition; isavectorofunknownparameters;

v

jisarandom

errorterm,whichweassumeisaGumbeldistributionacrossall observations(McFadden,1981).

Eq.(8) is a straightforwardequationfor integrating thetwo descriptive model components(i.e., thefrequency and severity model),andtheprobabilityofaccidentfrequencyatroadsegment yjistheweightedsumofcrashcountsofallseveritylevelsonthe

samesegmentoveraunittimespan.However,animportant prop-ertyoftheMNLmodelisitsindependencefromirrelevantalternate severityoutcomes.Thisindependencemaybeamajorconcernif somecrash-injuryseveritylevelsshareunobservedeffects.To over-comepartlysucharestriction,anestedlogit(NL)modelcangroup somepossiblelevelsthatshareunobservedeffectsintoconditional nests(KoppelmanandWen,1998).TheNLmodelpartitionsa sever-ityoutcomesetintoseveralnests,eachcontainingcorrelatedlevels. TheNLmodelcanbeexpressedas

i= exp(si/k)

j∈Bkexp(sj/k)



k−1

L l=1

j∈Blexp(sj/l)



l (10)

whereBkrepresentsnestk,whichisasubsetcontainingcorrelated

outcomeswithrespecttocrashseverity;(1−k)isacorrelation

measureofunobservedfactorswithinnestk;andkisintherange

of0–1.Asthevalueofkdecreases,thestrengthofthecorrelation

withinthenestincreases.Notably,kisalsocalledtheinclusive

valuerepresentingthedegreeofcorrelationamongalternate sever-itylevelswithinnestk.Ifk=1,theNLmodelbecomesanMNL

model.Ifkisequaltozero,perfectcorrelationisimpliedamong

theseveritylevelsinthenest,indicatingthattheprocessbywhich crashesresultinparticularseveritylevelsisdeterministic.

Sincerelatedstudies(Shankaretal.,1996;LeeandMannering, 2002;SavolainenandMannering,2007;Savolainenetal.,2011) have revealed that two nearer accident severitylevels suchas “propertydamageonly”and“possibleinjury”,or“disablinginjury” and“fatality”,maytendtohavestrongcorrelationsduetoordinal natureofcrashseveritydata.SuchproblemviolatedMNLmodel’s independenceofirrelevantalternatives(IIA)propertyresultedin biasedparameterestimates.Inthatcase,anNLmodelispreferred. Whenthenestedstructureexistsinj,theMGPmodelevolves

intoaflexiblenestedgeneralizedPoisson(NGP)model,solvingthe problemofsubstitutionpatternsamongseveritylevels.

BasedontheworkbyYeetal.(2009),specifyingapartialorfull errorcomponentsstructuremaybeaninnovativechoice compar-ingtotheformulationofcorrelationmatrix.Theerrorcomponents structureisconsideredintheexpectedfrequencyandseverity functionofthreeseveritylevelssj(j=1,2,3),whichincludefatality

(s1),injury(s2),andpropertydamageonly(s3).Inthisstudy,four

randomcoefficients(i)arespecifiedtothefrequencyfunction

andseverityfunctionsj(j=1,2,3)tomodelthefollowingpartial

covariancestructure:

=exp(ˇX+ε+1u) (11)

s1=Z+

v

1+2u (12)

s2=Z+

v

2+3u (13)

s3=Z+

v

3+4u (14)

where uis anindependentrandomvariable,which is normally distributed;andjaretheircorrespondingcoefficientstobe

esti-mated. To simplifythenumber of estimatedparameters in the errorcomponentsstructure,j−1standard deviationparameters are identified by subjectively setting one parameter equals to 1. We assume ε and

v

j have Generalized Poisson and Gumbel

distributions,respectively.Thus,thecumulativeprobability func-tionsconditionalonthisrandomvariableh(y1,y2,...,yJ|u)are

expressedas h(y1,y2,... ,yJ|u)=



J j=1j(Z,u)yj



J j=1yj!

⎡

⎣

1+

⎛

⎝



J j=1 yj

⎞

⎠

⎤

⎦

J j=1yj−1 ×

(X,u) 1+(X,u)



J j=1yj exp

⎡

⎣

−(X,u)

1+

J_j=1yj



1+(X,u)

⎤

⎦

(15) TodistinguishitfromEq.(8),Eq.(15)iscalledthe multino-mialgeneralizedPoisson modelwitherrorcomponents(EMGP). Theunconditionalcumulativeprobabilityfunctionoftherandom multinomialcan thenbederived byintegrating theconditional cumulativeprobabilityfunctionoverthedistributionaldomainof thespecifiedrandomvariable:

H(y1,y2,...,yJ)=



h(y1,y2,...,yJ)r(u)du (16)

wherer(u)istheprobabilitydensityfunctionofu.Asthisintegral doesnothavea neatclosed-formexpression,theunconditional probabilityfunctionmaybeapproximatedbythefollowing sim-ulatedprobabilityfunctionHs_(y

1,y2,...,yJ): Hs_(y 1,y2,...,yJ)= 1 R R



r=1 h(y1,y2,...,yJ|ur) (17)

The estimation procedureof the EMGP model typically fol-lows the simulation-based maximum likelihood method,using Haltondraws,whichhaveamoreefficientdistributionofdraws fornumericalintegrationthanpurelyrandomdraws(Bhat,2003; Train,2003).Inmanyempiricalsettings,thenumberofdrawsfor simulationis determinedaccordingtothenumberofestimated variables,thecomplexityofmodelspecification,andsamplesizes. Foraccuracypurposes,theestimationresultsofproposedmodels arepresentedfor200Haltondraws.Theestimatedparametersdo notvarymarkedlyoncethenumberofreplicationsexceeds150in theempiricalcase(seeTrain(2003)forfurthertechniquedetails andsimulationissues).

Table1

Descriptivestatisticsof124segments.

Variable Description Mean SE Min Max

Crashcounts

Y3 Propertydamageonly(PDO) 70.4 63.7 3.0 284.0

Y2 Injury 4.1 3.5 0.0 17.0

Y1 Fatality 0.5 0.8 0.0 4.0

N Total 75.1 65.4 6.0 290.0

Crashcountsperkm(=crashcounts/segmentlength)

Z3 PDOcrashesperkm 16.4 16.3 1.2 83.5

Z2 Injurycrashesperkm 0.8 0.6 0.0 2.8

Z1 Fatalcrashesperkm 0.1 0.2 0.0 1.5

N Totalcrashesperkm 17.3 16.6 2.2 85.3

Freewaygeometrics

GL Segmentlength(km) 5.9 4.4 0.8 22.4

GN Numberoflanes 2.6 0.6 2.0 4.0

GC Curvature(‰) 0.7 1.1 0.0 7.1

GU Maximumupwardslope(%) 1.3 2.1 0.0 13.7

GD Maximumdownwardslope(%) 1.2 1.4 0.0 5.2

GO Clothoidcurvevalue(thousanddegrees) 0.9 0.9 0.0 3.2 GS Speedlimit(GS=1for110km,GS=0else) 0.5 0.5 – – Rainfall

RF Annualrainfall(hundredmillimeters) 21.1 7.5 11.1 38.9 Averagedailytraffic

TTV Totaltraffic(thousandpassengercarunits) 69.2 28.7 10.8 157.0 PSV Percentageofsmallvehicles(%) 51.4 10.1 31.9 70.5

PLV Percentageoflargevehicles(%) 23.8 4.2 15.5 34.2

PKV Percentageoftrailer-tractors(%) 24.8 8.7 9.2 41.0 Freewayfacilities(dummyvariables,yes=1;no=0)

PT Presenceoftollstation 0.2 0.4 – –

PR Presenceofrestarea 0.1 0.3 – –

PS Presenceofpostedspeedcamera 0.3 0.5 – –

Neighborhood(dummyvariables,yes=1;no=0)

AM Adjacenttometropolitan 0.5 0.5 – –

AP Adjacenttoairport,seaportorindustryarea 0.2 0.4 – –

3. Data

TheaccidentdatasetforTaiwan’sNo.1Freewayin2005was

collected.Datawerefromthreesources:(1)theaccidentdatabase;

(2)geometricdocuments;and(3)thetrafficdatabase.The

acci-dentdatabase,maintainedbytheNationalHighwayPoliceBureau

(NHPB),containsaccidentinformation,suchascrashseverity,

loca-tionand timeofanaccident,andnumberand typesofvehicles

involved.Geometricdataweredigitalizedaccordingtotheofficial

as-constructedfreewaydrawings,includingnumberoflanes,slope,

curvaturedegree,andclothoidcurvevalue.

Taiwan’sNo. 1 Freeway runsnorth–south, is 373.3km long,

andhas63interchanges.Tofacilitatemodelestimation,astudy

segmentisformedbytwoadjacentinterchanges.Byconsidering

north-andsouth-bounddirectionsseparately,124analytical

sam-plesareobtained.Sincethelengthsofsegmentsremarkablydiffer,

tobetterreflectthecrashrisk,thedependentvariableispresented

bythecrashcountsdividedbythesegmentlength(GL).The

traf-ficdatabase,maintainedbytheNationalFreewayBureau(NFB),

includestrafficvolume,speedandoccupancyofthreevehicletypes

detectedbyloopdetectorsonabasicsegmentoron-ramp(small

vehicles,largevehiclesandtrailer-tractors).Consideringthe

val-uesofpassenger carequivalent(pce)of threetypesofvehicles,

thetotaltrafficateachroadsegmentaremeasuredinpassenger

carunits.Table1givesdescriptivestatisticsforthesesegments.

Themeanandstandarddeviationofaccidentsdifferineithertotal accidentcasesorthosecasesatvariousseveritylevels,suggesting thatthepotentialproblemofover-orunder-dispersionmaycause inefficientmodelestimationandbias.

Table2 presentsthecross-tabulationof crashfrequency and severity.Intotal,67(1%)fatalaccidentsoccurredin2005,and8735 (94%)accidentswereproperty-damage-onlyaccidents.Moreover,

all124segmentshad atleast onePDOaccident, whileonly 47 segments(38%)hadatleastonefatalaccident.

4. Results

AccordingtomodelformulationinSection2,fourpossible mod-elscanbeestimated:theMGPmodelwithorwithoutconsidering errorcomponents amongseverity levels,namely, theMGP and EMGPmodels.TwoNGPmodelsconsideringdifferentnested struc-tureamongseveritylevels,namely,theNGP1(nestingtwosevere severitylevels:fatalityandinjury)andNGP2(nestingtwominor severitylevels:PDOandinjury)aretested.Unfortunately, accord-ingtotheestimatedinclusivevaluesfortwoNGPmodels,wecould notfindanypossiblecorrelatednestingbetweentwosevere acci-dentlevels(i.e.,injuryandfatalitywitht-ratioofk=0.634)nor

twonon-severeaccidentlevels(i.e.,injuryandPDOwitht-ratioof k=0.299).

Tables 3and 4compare performance indicesand prediction accuracyamong models,respectively.InTable3,the goodness-of-fit indices, including number of significant variables, means and standard deviations of predicted accident counts,  value, log-likelihoodvalues,adjustedrho-square,andtheBayesian infor-mationcriterion(BIC)arecompared.Theestimationresultsshow thatthemodelwiththeerrorcomponent(i.e.,theEMGPmodel) performbetterthanthosemodelsthatdonotconsidererror com-ponent(i.e.,theMGPandNGPmodels),andtwonestedmodelsdo notperformbetterthanthemultinomialmodels(i.e.,theMGPand EMGPmodels)intermsofBICvalues.Additionally,accordingto theestimateddispersionparameter()ofEMGP,whichis decreas-ingfrom0.082to0.062,theassociatedasymptoticallytstatistics aresignificantlydifferentfromzeroaswell,indicatingthat empir-ical data have a slightly over-dispersion problem. In addition,

Table2

Cross-tabulationofcrashfrequencyandseverity.

Severitylevel Totalcrash

PDO Injury Fatality

Numberofcrashes

Crashcounts 8735(94%) 509(5%) 67(1%) 9311(100%)

Crashcountsperkm 2038(95%) 96(4%) 13(1%) 2147(100%) Numberofsegments

Withatleastonesuchcrash 124(100%) 110(89%) 47(38%) Withoutanysuchcrash 0(0%) 14(11%) 77(62%)

Total 124(100%) 124(100%) 124(100%)

Table3

Comparisonsofgoodness-of-fitamongthemodels.

Models Goodnessoffit

Ka _Crash(std.)b _c _{LL(ˇ) Adj-}2d _BICe

MultinomialgeneralizedPoisson(MGP) 26 18.33(14.53) 0.082 −1108.130 0.147 2341.588

NestedgeneralizedPoisson(NGP1) 27 18.33(14.53) 0.082 −1108.095 0.147 2346.339

NestedgeneralizedPoisson(NGP2) 27 18.33(14.52) 0.082 −1106.496 0.148 2343.139

MultinomialgeneralizedPoissonwitherrorcomponents(EMGP) 29 16.39(12.76) 0.062 −1037.935 0.201 2215.659

a_{K:numberofsignificantvariablesunder˛=0.1level.} b_{Crash:meanpredictedcrashcounts.}

c _{:dispersionparameter.}

d _Adj-2_{:rho-squareadjustedincompassionwithNullmodel(withthreeconstantsforcrashseverityandasingleconstantforgeneralizedPoissonmodel).} e_{BIC=−2×LL(ˇ)+K×LnN.}

Table4

Comparisonsofpredictionaccuracyamongthemodels.

Severity Accuracy Model Actual

MGP NGP1 NGP2 EMGP Fatality Crash(%) 0.12(0.82%) 0.12(0.83%) 0.12(0.82%) 0.10(0.76%) 0.10(0.82%) MAPE 0.319 0.321 0.333 0.307 – RMSE 0.200 0.200 0.205 0.198 – Injury Crash(%) 1.06(7.48%) 1.06(7.48%) 1.06(7.49%) 0.90(7.38%) 0.78(7.48%) MAPE 0.816 0.816 0.814 0.695 – RMSE 0.746 0.746 0.748 0.654 – PDO Crash(%) 17.15(91.70%) 17.15(91.70%) 17.14(91.69%) 15.39(91.86%) 16.44(91.70%) MAPE 0.698 0.698 0.698 0.617 – RMSE 13.451 13.450 13.447 13.111 –

Note:Thepercentagesofthepredictedcrashcountsatthreeseveritylevelsaregiveninparentheses.

specifyingtheerrorcomponentcanmitigatevariationin.Hence,

theestimatedoftheEMGPmodelislowerthanthatoftheMGP

andtwoNGPmodels,buttheinclusionoferrorcomponentscould

notperfectlyresolvetheover-dispersionproblem.

Table4comparesthepredictionaccuracyofthefourmodels bymeanabsolutepercentageerror(MAPE)androot-mean-square error(RMSE).TheEMGPmodelperformsbestincomparisonwith othermodels,althoughallfourmodelsachieverelativelyhigh pre-dictionaccuracy(Table4).

Forsimplicity,onlyestimatedparametersoftwoextreme mod-elsare reportedand compared in Tables 5 and 6, respectively. Thisstudysets˛=0.10asthevariableselectioncriteriontoavoid an excessive number of non-stable and insignificant variables adverselyaffectingefficiencyincalculatingnumericalvaluesand convergenceresults.Therefore,thepotentialvariables,annual rain-fall(RF)andpercentageoftrailer-tractors(PKV),areremoveddue totheirinsignificanteffects.Thisstudyalsotestsallpossible rela-tionshipsamongvariables,includinglinear,squared,exponential, andnaturallogrelationships.

5. Discussions

Accordingtoestimation resultsof theMGPand EMGP mod-els(Tables5and6),allsignificantvariablesarealmostthesame

witha relatively similarmagnitudes;however, variables of the EMGPmodeltypicallyhavemoresignificanteffectsintermsofthe tstatistic,againdemonstrating thesuperiorperformance ofthe EMGPmodel.Thus,onlyestimationresultsoftheEMGPmodelare discussedbelow.

Onlytwovariablesofmaximumdownwardslope(GD)and adja-centtometropolitan(AM)havesignificanteffects onbothcrash frequencyandseverity,whileothervariablesofcrashfrequency andseveritymodelcomponentsarealldifferent,suggestingthat thefactorscontributingtocrashfrequencyandseveritydiffer.

First, in terms of geometric variables, maximum downward slope(GD)are significantlytestedin bothfrequencyand sever-itymodelcomponents,whilenumberoflanes(GN),exponentialof maximumupwardslope(GU),clothoidcurvevalue(GO),andspeed limit(GS)onlysignificantlycontributetocrashseverityand curva-ture(GC)onlyaffectcrashfrequency.TheGNreducesPDOcrashes butresultsintomoreseverecrashes,indicatingthatmorenumber oflanesmaycausesevereaccidents.Theexp(GU)hasanegative coefficientonfatalcrashesbecausedriverstendtodriveatalower speed onan upward-sloped segmentand then largelymitigate theseverityofcrashes.Contrarily,bothGDandGOhavepositive coefficientsassociatedwiththefatalandinjurycrashes,implying thecrashesathighdownward-slopedsegmentsandcurved transi-tioncurvesaremoresevere.TheGShasapositiveeffectoninjury

Table5

ModelresultsofthemultinomialgeneralizedPoisson(MGP).

Variable Severitylevel

Fatality Injury PDO

Para. t-Stat Para. t-Stat Para. t-Stat Logitcrashseveritymodelcomponent

Constant – 1.407 2.147 5.330 8.113

Freewaygeometrics

GN Numberoflanes – – −0.259 −4.053

Exp(GU) Exponentialofmaximumupwardslope −0.466 −2.097 – – GD Maximumdownwardslope 0.306 4.034 0.183 6.861 –

GO Clothoidparameter 0.250 2.341 0.177 4.158 –

GS Speedlimit – 0.280 3.017 –

Trafficcharacteristics

TTV Totaltraffic −0.709 −2.501 −0.424 −4.169 –

PLV Percentageoflargevehicles 4.604 5.755 4.604 5.755 – Neighborhood

AM Adjacenttometropolitan – – 0.271 3.276

Freewayfacilities

PS Presenceofpostedspeedcamera – −0.685 −2.880 −0.489 −2.175 PR Presenceofrestarea −1.361 −2.990 −0.321 −2.546 –

Variable Para. t-Stat

GeneralizedPoissoncrashfrequencymodelcomponent(forallseveritylevels)

Constant 1.237 3.465

 0.082 8.407

Freewaygeometrics

GC Curvature 0.151 2.238

GD Maximumdownwardslope −0.555 −4.099

GD2 _{Squareofmaximumdownwardslope 0.054 1.876}

Trafficcharacteristics

PSV Percentageofsmallvehicles 3.050 4.291

Neighborhood

AM Adjacenttometropolitan 0.504 3.442

AP Adjacenttoairport,seaportorindustryarea 0.498 2.565

PT Presenceoftollstation −0.419 −2.568

Goodnessoffitmeasures

Log-likelihood(Nullmodel) −1298.488 Log-likelihood(Fullmodel) −1108.130

Adj-2 _0.147

Samples 124

Note:Nullmodel:withthreeconstantsforcrashseverity(marketshare)andasingleconstantforgeneralizedPoissonmodel.

crashesbecausedriverstendtoincrease theirspeedatthe

seg-mentswithahigherspeedlimit,increasingaccidentseverityonce

theaccidentoccurred.

TheGCaffectscrashfrequency,suggestingthatahighfreeway

curvaturecoefficientincreasesaccidentfrequency.TheGDhasa

polynomialeffect(anegativelineareffectandapositivesquared

effect)onaccidentfrequencyandalinearpositiveeffecton

acci-dentseverity(onlyforfatalityandinjury).Bytakingaderivative

termofa variable,a 1◦ increase inGD hasa marginal effectof

increasingcrashfrequencyby−0.448+0.072×GD,suggestingthat

aslightdownward slope mayreduceaccidentfrequency.

How-ever,onceaslope’sgradeexceeds6.22%,crashfrequencyincreases

rapidly,suggestingthatanabruptdownward slopesignificantly

contributes to a reduced number of PDO crashes and, in turn,

increasesaccidentseverity.Asslopeincreases,driverawareness

increases,reducingaccidentfrequencyforgentleslopes.However,

whenaslopeexceedsathreshold(6.22%inthisstudy),stopping

becomesincreasingly difficult,resultinginseverer accident

fre-quency.

Intermsoftrafficcharacteristics,theTTVhasnegativeeffectson

twosevereseveritylevels,implyingthecrashseveritycanbe

low-eredatthesegmentswithhightrafficflowbecauseoflowertravel

speedcausedbytrafficcongestion.However,thePLVhasrelatively

higheffectsontwoseverecrashes,suggestingthehigher

percent-ageoflargevehicles,themoresevereofthecrashes.Additionally,

thePSVhasapositiveeffectoncrashfrequency.Asthepercentage

ofsmallvehiclesincreases,thepercentageoflargevehiclesand

tractor-trailersisthendecreasedanddrivers’awarenessmaybe

reducedandtravelspeedisincreased,resultingintoahighcrash

potentialcondition.ThisresultissimilartothefindingsofHiselius

(2004)inSweden.

Thevariableofadjacenttometropolitan(AM)increasescrash frequencyandseverityforPDOonly,suggestingthatahighnumber ofaccidentsoccuronsegmentsclosetourbanareasand, fortu-nately,theseaccidentshavelowseverity.Thisis becausetraffic volumeonsegmentsneighboringurbanarterialsisusuallyheavy andvehiclestravelatarelativelyslowspeed,increasingthe poten-tialforPDOaccidents.Thevariableofadjacenttoairport,seaportor industrypark(AP)alsoincreasescrashfrequency.Itisbecausethere aremoretruckstravelingatthesegmentsnearairports,seaports andindustryparks,makingcrashpotentialhigh.

In terms of freeway facilities, posted speed cameras (PS) decreasethepotentialofnon-severecrashes(injuryandPDO)given thenumberofaccidentsunchanged,suggestingthatalthoughaPS

Table6

ModelresultsofmultinomialgeneralizedPoissonwitherrorcomponents(EMGP).

Severitylevel

Fatality Injury PDO

Para. t-Stat Para. t-Stat Para. t-Stat Logitcrashseveritymodelcomponent

Constant – 2.000 65.822 5.587 85.503

Freewaygeometrics

GN Numberoflanes – – −0.152 −6.421

exp(GU) Exponentialofmaximumupwardslope −0.496 −15.989 – – GD Maximumdownwardslope 0.342 11.068 0.185 6.316 –

GO Clothoidparameter 0.231 7.459 0.158 5.248 –

GS Speedlimit – 0.369 11.954 –

Trafficcharacteristics

TTV Totaltraffic −0.724 −23.387 −0.738 −25.559 – PLV Percentageoflargevehicles 6.488 89.411 6.488 89.411 – Neighborhood

AM Adjacenttometropolitan – – 0.200 6.447

Freewayfacilities

PS Presenceofpostedspeedcamera – −0.711 −22.936 −0.483 −15.604 PR Presenceofrestarea −1.382 −44.519 −0.579 −18.647 –

Errorcomponentincrashseveritymodel

S 1.000 – 0.262 8.459 0.824 26.527

Para. t-Stat

GeneralizedPoissoncrashfrequencymodelcomponent(forallseveritylevels)

Constant 0.982 31.684

 0.062 6.390

Freewaygeometrics

GC Curvature 0.152 4.925

GD Maximumdownwardslope −0.448 −14.797

GD2 _{Squareofmaximumdownwardslope 0.036 3.352}

Trafficcharacteristics

PSV Percentageofsmallvehicles 3.260 95.035

Neighborhood

AM Adjacenttometropolitan 0.428 13.763

AP Adjacenttoairport,seaportorindustryarea 0.534 17.217 Freewayfacilities

PT Presenceoftollstation −0.387 −12.458

Errorcomponentincrashfrequencymodel

GPM 0.246 7.923

Goodnessoffitmeasures

Log-likelihood(Nullmodel) −1298.488 Log-likelihood(Fullmodel) −1108.130

Adj-2 _0.201

Samples 124

Note:Nullmodel:withthreeconstantsforcrashseverity(marketshare)andasingleconstantforgeneralizedPoissonmodel.

doesnotreducecrashfrequency,itmayincreasecrashseverity.The

causeandeffectrelationshipmaybereversed.Thatis,postedspeed

camerasareusuallyinstalledatthesegmentswithhighpotential

forfatalcrashes.Ifasegmenthasarestarea(PR),ithasaneffect

incontrasttothatofPS,becausemerginganddiverging

maneu-versonthesesegmentsslowtrafficdownandreducepotentialfor

severeaccidents.Meanwhile,ifthesegmenthasatollstation,the

frequencyofaccidentsisreduced.Driverswouldbemorecareful

whiletraversingtollstationsatalowerspeedduetomore

compli-cateddrivingmaneuversrequiredthantravelingatothersegments,

sothecrashpotentialismitigated.

Theestimatedparametersoftheexplanatoryvariablesin

pro-posedmodelresultsdo notdirectlyshowthemagnitudeofthe

effects ontheexpected frequencyfor each leveland all

severi-ties.Moreover,someexplanatoryvariables(i.e.,AMandGD)donot

carrythesameeffectsandimplicationsoncrashfrequencymodel

andseveritymodelcomponents,respectively.Tobetterunderstand

theimpactofcontributoryfactors,Table7furtherreportsthe

elas-ticityeffectsofsignificantvariables onindividualseveritylevels (i.e.,PDO,injuryandfatality)andonaggregatelevel.Since calcula-tionformulasandimplicationsofdummyvariablesandcontinuous variables aredifferent,theycannotbecomparedand shouldbe describedrespectively.

Aggregatelevelelasticityvalues forcontinuous variablesare computedbasedontheestimatedEMGPmodelbyEq.(18): tjk=



∂E(yjt) ∂xjtk





xjtk E(yjt)



(18) whereE(yjt)=jt(xjtk)jt(xjtk);E(yjt)isexpectedfrequencyof

sever-ityleveljatsegmentt;andxjtkisthecontributoryvariablekof

acci-dentfrequencyatseverityleveljonsegmentt.Asx_jtkhaschanged, the accident frequency and severity are adversely affected, such that elasticity represents the effect of the corresponding

Table7

AggregateelasticityestimatesoftheEMGPmodel.

Variable Severitylevel Frequency

Fatality Injury PDO Continuousvariable

Freewaygeometrics

GN Numberoflanes 0.383 0.377 −0.024 0.000

GC Curvature 0.243 0.142 0.147 0.147

GU Maximumupwardslope −0.098 0.001 0.001 0.000

GD Maximumdownwardslope 0.226 −0.081 −0.244 −0.232

GO Clothoidcurvevalue 0.208 0.132 −0.009 0.000

Trafficcharacteristics

TTV Totaltraffic −0.659 −0.671 0.043 0.000

PSV Percentageofsmallvehicles 1.820 1.757 1.887 1.880 PLV Percentageoflargevehicles 1.422 1.440 −0.093 0.000 Dummyvariable

Freewaygeometrics

GS Speedlimit −0.623 10.790 −0.626 0.000

Neighborhood

AM Adjacenttometropolitan −0.313 3.877 −0.478 −0.238

AP Adjacenttoairport,seaportorindustryarea 39.476 46.454 33.394 34.147 Freewayfacilities

PS Presenceofpostedspeedcamera 22.259 −9.784 0.430 0.000 PT Presenceoftollstation −22.654 −23.631 −23.710 −23.699

PR Presenceofrestarea −61.650 −35.320 2.456 0.000

factoroncrashfrequencyateachseveritylevel.Additionally,

“elas-ticityeffects”ofdummyvariablesarecomputedbyalteringthe

valueofthevariableto“1”forthesubsampleofobservedsegments

forwhichthevariabletakesavalueof“0”,andto“0”forthe

sub-sampleforwhichthevariabletakesavalueof“1”.Wethensumthe

shiftsofexpectedfrequenciesinthetwosubsamplesafter

revers-ingthesignoftheshiftsinthesecondsubsample,andcomputean

effectivepercentagechangeinexpectedaggregatefrequency.Thus,

thedummyelasticityeffectcouldbeinterpretedasthepercentage

changeattheexpectedfrequencyofaninjuryseverityleveldue

tochangeinthedummyvariablefrom0to1(formoredetailssee

EluruandBhat,2007).

Specifically,for continuousvariablesofPSV andPLV havean estimatedelasticity>1forsevereaccidenttypes,suggestingthat theyarekey factorstomoresevereaccidents.Accordingtothe estimationresultsoftheEMGPmodel(Table6),anincreaseinPSV significantlyincreasesthenumberofaccidentsbutnotcrash sever-ity.By elasticityestimates,PSV wasactuallyidentified asa key factorcontributingtocrashfrequencywiththesimilarmarginal effectsoneachseveritylevel.Itisworthofnotingthatcrashesat thesegmentswithhighheavytraffictendtobelesssevere.

Comparingtothehighelasticityeffectsoftrafficcharacteristics, geometricvariableshaverelativelylowereffectsoncrashseverity andfrequency.Itisbecausethegeometricdesignstandardfor free-waysisusuallyhigherthansurfaceroadways,makinghighlycurved andslopedfreewaysegmentsbarelyexisted.However,accordingto thecomputedelasticityeffects,somegeometricvariablesstillaffect crashseverityandfrequency.Generally,toocurvedandtoomany lanesfreewayshouldbeavoidedinfreewayplanninganddesign. Itisinterestingtonotethatthemaximumdownwardslopehave apositiveelasticityeffectonfatalcrashesbutanegativeelasticity effectoncrashfrequency,becausedriverswouldbemorecarefully whiletravelingatthedownwardslopedsegments,butonceacrash occurs,theseveritywouldbelargelyincreasedduetothedifficulty inbraking.

Theelasticityeffectsofdummyvariablesarerelativelylarger thanthoseofcontinuousvariablesbecauseoftheirdifferent formu-las.Therefore,itismeaninglesstocomparetheeffectsofcontinuous anddummyvariables.However,amongalldummyvariables,AP hasthelargestpositiveelasticityeffectsoncrashesatallseverity

levels.Toinstallwarningsignsandtoproperlyconfine overtak-ingbehaviorsatthesegmentsnearairports,seaportsandindustry parkscouldeffectivelyreducecrashesatallseveritylevels. Con-versely,PT andPR havenegativeeffects onseverecrashes.The presenceoftollstationandrestareacanslowvehiclespeedand reducethenumberofsevereaccidents.Notably,thepresenceof restareacanlargelyreducesevereaccidentsbutslightlyincreases PDOaccidents.

6. Conclusions

Thisstudycontributestoliteratureinseveralways.First,this studyintegratesanaccidentfrequencymodelwithaseveritymodel undertheMGParchitecture,andusestheintegratedmodelto ana-lyzeaccident data—countdata(crash frequency)and ratiodata (severity)—suchthattheMGPmodelismoreefficientin evaluat-ingandpresentingaccidentdata.Notably,accordingtoestimation results,thefactorscontributingtoaccidentfrequencyandseverity differmarkedly.Generally,trafficrelatedfactorshavelargereffects oncrashseverityandfrequencythangeometricfactors.

Additionally,four modelsaredeveloped and compared.This studyadoptedthesharederrortermtoconstructcommonerrors andcovariancestructuresoastoimprovemodelexplanatory capa-bilityandreliability.TheestimationresultsshowthattheEMGP modelperformsbest,asthismodelspecifiestheerrorcomponent inthecrashfrequencyandseveritymodelbyallowingdifferent errorsincrashfrequencyandseverity.Thus,theestimationresults showthattheproposedcovariancestructurecanfurtherenhance themodelperformance.

Basedontheproposedframework,futurestudiescanintroduce moreflexiblemodelsinthecontextoffrequencymodeling,such asPoissonlog-normal,random-parametersandothermixed dis-tributioncountmodels.Formodelingseverityoutcomes,ordered probit,mixedlogit(alsocalledtherandomparameterslogitmodel) andmorecompatiblegeneralizedextremevaluemodels(GEV fam-ilymodel)likegeneralizednestedlogit(GNL)arerecommended. Additionally,there is nosegmentwithzerocrashcountdueto thespatialsegmentationusedinthisstudy,whichmightleadto biasedestimationparameters.Morerefinedspatialsegmentation orothercensoredmodels(e.g.,TobitregressioninAnastasopoulos

etal.,2012)onaccidentratescanbeconsidered.Furthermore,this studyuses thesharederrorcomponent tohandlethe common errortermandcovariancestructure.Thecovariancestructurecan bederived toenhancemodelperformance further.Additionally, italsodeservestocomparepredictionperformancesamong dif-ferentmodelingframeworksinthecontextofcrashseverityand frequency,suchasmultivariatePoissonlog-normal(MPLN) mod-els,whichaimstosimultaneouslymodelingcrashfrequenciesat allseveritylevels.Last,additionalexplanatoryvariablescanbe uti-lizedtoinvestigatetheireffectsonaccidentfrequencyandseverity togeneratemoreeffectivesafetyimprovementstrategies.

Acknowledgements

Theauthorsareindebtedtothreeanonymousreviewersfortheir insightfulcommentsandconstructivesuggestions,whichhelp clar-ifyseveralpointsmadeintheoriginalmanuscript.Thisstudywas financiallysponsored bytheROCNationalScienceCouncil(NSC 97-2628-E-009-035-MY3).

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