MODELING THE PROBABILITY OF FREEWAY REAR-END CRASH OCCURRENCE. Initial submission: February 8, Revised submission: June 12, 2006.

Transcription

1 Manuscript Numbr: TE/2006/ MODELING THE PROBABILITY OF FREEWAY REAR-END CRASH OCCURRENCE by Joon-Ki Kim 1, Yinhai Wang 2, and Gudmundur F. Ularsson 3* Initial submission: Fbruary 8, Rvisd submission: Jun 12, Submittd to th Journal o Transportation Enginring 1 Graduat Rsarch Assistant, Dpartmnt o Civil Enginring, Campus Box 1130, Washington Univrsity in St. Louis, On Brookings Driv, St. Louis MO ; jk9@cc.wustl.du; tlphon ( ; ax ( Assistant Prossor, Dpartmnt o Civil and Environmntal Enginring, Box , Univrsity o Washington, Sattl WA ; yinhai@u.washington.du; tlphon ( , ax ( Assistant Prossor, Dpartmnt o Civil Enginring, Campus Box 1130, Washington Univrsity in St. Louis, On Brookings Driv, St. Louis MO ; gu@wustl.du; tlphon ( ; ax ( CE Databas Kywords: Collisions; Crash; Accidnt; Analysis; Modls; Highway Saty; Drivr Rspons Tim; Risk * Corrsponding author

2 1 ABSTRACT A microscopic modl o rway rar-nd crash risk is dvlopd basd on a modiid ngativ binomial rgrssion and stimatd using Washington Stat data. Compard with most xisting modls, this modl has two major advantags: 1 it dirctly considrs a drivr s rspons tim distribution; and 2 it applis a nw dual-impact structur accounting or th probability o both, a vhicl bcoming an obstacl ( P o and th ollowing vhicl s raction ailur ( P. Th rsults show or xampl that truck prcntag-mil-pr-lan has a dual impact, it incrass P o and dcrass P, yilding a nt dcras in rar-nd crash probabilitis. Urban ara, curvatur, o-ramp and mrg, shouldr width, and mrg sction ar actors ound to incras rar-nd crash probabilitis. Daily VMT pr lan has a dual impact, it dcrass P o and incrass P, yilding a nt incras, indicating or xampl that ocusing VMT rlatd saty improvmnt orts on rducing drivrs ailur to avoid crashs, such as crash-avoidanc systms, is o ky importanc. Undrstanding such dual impacts is important or slcting and valuating saty improvmnt plans or rways.

3 2 INTRODUCTION Approximatly 60% o rway traic congstion is causd by incidnts (Lindly, Incidnts can b classiid as ithr prdictabl vnts such as work zons, or unxpctd vnts such as accidnts. Rar-nd crashs ar th most common typ o crash in Washington Stat: rar-nd crashs (35.9%, ixd objct crashs (17.0%, and sidswips (10.7% (WSDOT, Whn rar-nd crashs occur, thy tmporarily rduc roadway capacity and caus congstion. According to th 2003 Urban Mobility Rport (Schrank and Lomax, 2003, th annual avrag dlay pr prson in th 75 survyd urban aras was 26 hours in 2001, a 371% incras compard to Congstion costs an avrag o $520 pr travlr in th survyd urban aras in Thror, through inding th actors which inlunc rar-nd crashs, w can idntiy controllabl actors which can improv highway dsign, lading to a dcras in th rquncy o rar-nd crashs. I succssul, this will hlp rduc numbr o injuris and rduc ovrall congstion, thus also saving tim and mony. This papr dscribs a numrical approach that can b usd to valuat rway rar-nd crash risk basd on known traic and roadway actors. LITERATURE REVIEW In rcnt yars, a signiicant amount o rsarch has bn prormd to undrstand crashs on rways using modling mthods such as linar rgrssion, Poisson rgrssion, and ngativ binomial rgrssion. Jovanis and Chang (1986 ound som undsirabl problms with th us o linar rgrssion in thir study. Miaou t al. (1992 usd a Poisson modl and ound that th Poisson constraint (th man and varianc o th crash rquncy hav to b qual was violatd. Th prormancs o th Poisson rgrssion and ngativ binomial rgrssion wr

4 3 compard and th ovrdisprsion o crash data was addrssd (Miaou, 1994 and Shankar t al., Poch and Mannring (1996 ound that th ngativ binomial modl was th appropriat modl or dtrmining crash rquncy at intrsctions du to ovrdisprsion in th data. Shankar t al. (1997 applid zro-inlatd Poisson (ZIP and zro-inlatd ngativ binomial (ZINB to handl data which violat th Poisson and ngativ binomial modl assumptions du to numrous obsrvations o sctions with no crashs in th obsrvd priod. Wang (1998 modld th man rats o rar-nd crashs at our-lggd signalizd intrsctions through multiplying traic volum by rar-nd crash probability. A common criticism o many prvious studis is that thy do not usually considr human actors. Massi t al. (1993, howvr, pointd out that th classical human actors approach ignord th problm associatd with classiying collisions and thir rlatd causs, b it human or othrwis, and aild to addrss th issu o hlping drivrs avoid collisions. By idntiying gomtric conditions that lnd thmslvs to producing crashs, ths conditions could b corrctd. Milton and Mannring (1998 stimatd annual crash rquncy on sctions o principal artrials with ngativ binomial rgrssion modls and ound numrous traic and gomtric charactristics to b important. Carson and Mannring (2001 idntiid signiicant spatial, tmporal, traic, roadway, and crash charactristics that inluncd ic involvd-crash rquncy and svrity. L and Mannring (2002 usd a nstd logit modl or run-o-roadway crash modling. Golob and Rckr (2003 applid linar and nonlinar multivariat statistical analyss to dtrmin how th typs o crashs occurring on havily usd rways in Southrn Caliornia ar rlatd both to th low o traic and to wathr and ambint lighting conditions. Ularsson and Shankar (2003 xplord th ngativ multinomial modl to prdict mdian crossovr crash

5 4 rquncis. Golob and Rgan (2004 studid, by applying a multinomial logit modl, how various typs o truck crashs ar rlatd to traic low conditions and roadway charactristics on urban rways. Although a signiicant amount o rsarch has attmptd to study crashs basd on crash typ, location, and svrity, vry w studis hav bn conductd to modl rar-nd crashs on rways. Shankar t al. (1995 notd that sparat rgrssion modls ocusd on spciic crash typs hav gratr xplanatory powr than an ovrall rquncy modl. Thror, thr is nd or studying rar-nd crashs sparatly rom othr typs o crashs on rways. METHODOLOGY For modling rar-nd crashs on rways, w mploy a microscopic modling approach introducd by Wang (1998. This modling approach has bn succssully applid to intrsction saty studis (Wang t al., 2002 and Wang and Nihan, Th occurrnc o rar-nd crashs on rways is a combind rsult o a lad vhicl s tim-hadway rduction action and a ollowing vhicl s inadquat action or th ollowing vhicl s inctiv rspons. In this study, th occurrnc o crashs is considrd to b basd on two prmiss: on is that a lad vhicl bcoms an obstacl vhicl to a ollowing vhicl and th othr is that a ollowing vhicl ails to avoid a collision givn th obstacl vhicl. Whn a lad vhicl rducs th tim-hadway with rspct to th ollowing vhicl (such as by stopping, dclrating, or prorming a cut-in movmnt, it bcoms an obstacl vhicl to th ollowing vhicl. Th ollowing vhicl drivr may nd to ract to avoid a collision with th obstacl vhicl. Dpnding on th Manuvring Tim (Prcption-Rspons Tim (PRT plus vhicl rspons tim availabl to th drivr, th drivr s raction may or may

6 5 not b succssul. I unsuccssul, a rar-nd crash occurs. Thus, th probability o having a rarnd crash is dtrmind by: 1 th probability o a lading vhicl bcoming an obstacl, dnotd by P o ; and 2 th probability o th ollowing vhicl drivr s ailur to avoid th collision givn an obstacl vhicl, dnotd by P. Noting th conditional natur o P avoids problms rlatd to th dpndnc o th two drivrs dcision making whn both s a joint vnt that lads both to brak. Sinc th ollowing vhicl s ailur to avoid a crash is conditional on thr bing an obstacl vhicl, th total probability o a rar nd crash is th multiplication o th probability o an obstacl vhicl and th conditional ailur to avoid crashing. Thn th probability o having a rar-nd crash can b xprssd as th product o P o and P : P = P o P. (1 Not that dirnt rar-nd crashs ar assumd to b indpndnt vnts bcaus chain-raction crashs ar xcludd and only two-vhicl rar-nd crashs ar usd in this study. Thr wr a total o 8,452 rar-nd accidnts in th data and two-vhicl rar-nd accidnts accountd or about 64% (5,868. Chain-raction rar-nd crashs ar mor likly undr high volum conditions. Thror, xcluding th chain-raction rar-nd crashs may act th traic volum variabl. Th Probability o a Lading Vhicl bcoming an Obstacl ( P o An vnt that causs a lad vhicl to bcom an obstacl vhicl is calld a disturbanc. Disturbancs ar rar vnts, non-ngativ, and discrt. Also, th occurrncs o disturbancs ar indpndnt during non-ovrlapping tim intrvals and dirnt disturbancs ar indpndnt o ach othr. Thror, th occurrnc o disturbancs is assumd to ollow a

7 6 Poisson procss. Th intrvals btwn Poisson-distributd disturbancs ollow an xponntial distribution. Th probability dnsity unction (PDF o th xponntial distribution is η t j ( t, η = η, or t > 0, η > 0 j j j, (2 whr j is a disturbanc, η j is th occurrnc rat o disturbanc j, and t is th tim intrval. This lads to th probability o a disturbanc j occurring at last onc in t P t η jt j = η j dt = 1 0 η jt. (3 Sinc any o th disturbancs can caus th lad vhicl to bcom an obstacl vhicl, th probability o that, P o, is th sam as th probability that at last on disturbanc occurs in t xprssd as J P o = 1 (1 P j, (4 j= 1 whr 1 Pj is th probability that disturbanc j dos not occur, J is a thortical maximum numbr o disturbancs that can occur in tim intrval t (sinc w cannot hav an ininit numbr o disturbancs occur in a init tim, and disturbanc occurrd during tim intrval t. Substitut j ( 1 P is th probability that no j P j in (4 by (3 and P o can b writtn jη jt P o = 1. (5 To lt (5 dpnd on a st o xplanatory variabls such as gomtric aturs and traic low w paramtriz j η jt, noting it must b positiv sinc probability cannot b gratr than 1. A loglinar paramtrization satisis this condition:

8 7 ln jt η =βoxo, (6 j η jt = xp( βoxo, (7 j whr β o is a vctor o stimabl paramtrs and x o is a vctor o xplanatory variabls. By combining (5 and (7, th probability o a lading vhicl bcoming an obstacl ( P o in a givn priod o tim is writtn P o βoxo =1. (8 Th Probability o Failur to Avoid a Collision givn an Obstacl Vhicl ( P On o th most important actors to avoid crashs on rways is a drivr s Prcption- Rspons Tim (PRT. PRT is dind as th PIEV tim in th Manual on Uniorm Traic Control Dvics (MUTCD, 2003, which can b summarizd as th total tim ndd to prciv and complt a raction. PRT is not a constant valu o all driving situations but dpnds on th complxity o th problm and th drivr s xpctation o a hazard (Bats, To modl th probability o a drivr s ailur to avoid a collision, two concpts ar considrd: Availabl Manuvring Tim (AMT and Ndd Manuvring Tim (NMT. AMT rrs to th actual tim availabl or a drivr to avoid a collision with an obstacl vhicl. NMT rrs to th minimum tim that a drivr nds to avoid a collision (PRT plus vhicl rspons tim. I NMT is gratr than AMT, a drivr cannot avoid a collision. To modl NMT and AMT with appropriat distributions, w nd to know th charactristics o PRT. Summala (2000 addrssd th ollowing points: 1 not all drivrs prorm th xpctd rspons in on-road studis, and th obtaind PRT stimats may b biasd

9 8 du to th drivrs who brak th slowst; 2 drivrs attntions dir btwn locations so that in crtain placs thy ar mor attntiv to thir task than in othrs; 3 although brak raction latncis appar to incras with availabl tim, string rspons latncis do not, at last within a crtain rang o tim; 4 th total PRT distributions do not dir at all or th two groups (18-40 yars and yars. This rsult was also notd by othr studis. For xampl, Olson and Sivak (1986 showd that both ag groups hav th 95 th prcntil PRT tim o about 1.6 s. Thy indicatd that whil oldr drivrs prcption tim is slowr than youngr drivrs, th brak raction (including oot movmnt and dcision procsss that ollows is astr in oldr drivrs. Lrnr (1993 also pointd out that although most o th astst obsrvd PRTs wr rom th young group, thr wr no dirncs in cntral tndncy (man = 1.5 s or uppr prcntil valus (85 th prcntil = 1.9 s among th ag groups. Whil AASHTO (2001 suggsts a consrvativ PRT o 2.5 sconds or highways, Mannring t al. (2005 mntiond that a drivr s PRT is a unction o a numbr o actors including th drivr s physical condition, motional stat, and complxity o th situation. As xplaind in th abov studis, PRT is not a constant valu, but a random variabl rlating to many actors such as drivrs skill, physical condition, traic condition, and gomtric aturs. In this study, w assumd that both th AMT and NMT ar Wibull distributd bcaus th Wibull distribution is a good approximation to th normal distribution (Plait, 1962 and or this modl it rsults in closd-orms, whras th lognormal and th loglogistic do not. Th Wibull distribution is a gnralizd orm o th xponntial distribution. Th Wibull distribution has two paramtrs, scal θ > 0 and shap > 0 th Wibull distribution is. Th dnsity unction or

10 9 1 ( θ ( t, θ, = θ t, t > 0. (9 t A hazard unction is a conditional probability that an vnt occurs btwn t and t + t givn that an vnt dos not occur until t. Th hazard unction or th Wibull distribution is h( t 1 = θ t, t > 0. (10 Whn th shap = 1, th Wibull distribution bcoms th xponntial distribution, and th hazard is constant ovr tim (duration indpndnc. Whn > 1, th hazard is monotonically dcrasing ovr tim (ngativ duration dpndnc, and whn < 1, th hazard is monotonically incrasing ovr tim (positiv duration dpndnc. Not that although th Wibull distribution provids a mor lxibl mans o capturing duration dpndnc than th xponntial distribution, it dos not allow th hazard to incras and thn dcras ovr tim bcaus it rquirs th hazard to b monotonic ovr tim. Th Log-normal and log-logistic distributions hav non-monotonic hazard unctions but ar computationally cumbrsom in this modl bcaus o non-closd orm solutions. W thror us th Wibull distribution hr in spit o its limitation. Th ailur probability is xprssd as P = P( AMT < NMT, as mntiond bor. Th probability distributions or AMT and NMT ar assumd as ollows: AMT = ( t, θ, = θ t a 1 ( θ ta or > 0, θ > 0, t > 0, (11 NMT = ( t n, λ, = λ t 1 ( λ tn or > 0, λ > 0, t > 0. (12 Hr, w mployd two assumptions as ollows: 1 Th shap paramtr,, is th sam or both AMT and NMT. This is a limitation but is ncssary to achiv a closd orm rsult. Th scal

11 10 paramtr is howvr allowd to vary. 2 AMT and NMT ar indpndnt manuvring tims. Thn, th drivrs probability o ailur to avoid a collision can b calculatd as,. 1 1, (,,,,, (,, (, ( ( 0 0 ( 1 0 ( 1 ( 0 θ λ θ λ θ λ θ λ θ λ θ θ θ θ λ + = + = = = = > = + + a a a a a t a t a a t a t t a n a n dt t dt t dt dt t t AMT NMT P P (13 Now, P is xprssd as a unction o λ, θ, and. Sinc 0 > λ and 0 > θ, θ λ / ( is gratr than 0. Thn, θ λ / ( can b rlatd to a st o xplanatory variabls by using th xponntial link unction: β x = θ λ, (14 whr β is a vctor o stimabl paramtrs and x is a vctor o xplanatory variabls. P can thn b xprssd as β x + = P 1 1. (15

12 11 Rar-End Crash Risk Modl By rplacing Equation (1 with (8 and (15, th probability o a rar-nd crash o an individual vhicl, P, can b rwrittn as, P = P P o 1 = 1+ β o x o β x. (16 Th numbr o crashs ( N or a vhicl low ( v on sction i in a givn priod j, ollows a binomial distribution, v n v n P( N = n = P P n (1. (17 Th man or this binomial distribution is m = vp. Whn v and P 0, whil v P rmains constant, th numbr o crashs, N, is a Poisson distributd random variabl with th paramtr m, n m P( N = n m =. (18 n! m Whil th probability o crashs is a vry small valu, th traic volum ( v is a vry larg valu or a givn tim priod. Thror, th Poisson distribution can b a good approximation to th binomial distribution as provn abov. Givn data such as traic low and gomtric aturs, th xpctd rar-nd crashs ( m on sction i in priod j can b paramtrizd as ln m =β x, (19 whr x is a vctor o gomtric aturs, traic low, and so on or sction i in th givn priod j, and β is a vctor o stimabl coicints.

13 12 Poisson modls ar not suitabl or ovr-disprsd data. Howvr, crash data tnds to b ovr-disprsd. To ovrcom this limitation, th Poisson modl is gnralizd by introducing an unobsrvd ct, ε, into th xpctd rar-nd crashs paramtrization (Grn, 2003, ln m' = βx + ε = ln m = ln m u, + ln u, (20 whr or mathmatical simplicity w din ln u = ε and us th logarithmic rul ln ab = ln a + ln b to simpliy urthr. Thn, th distribution o n conditiond on u (i.. ε is ( m u n P( N = n u =. (21 n! m u Th unconditional distribution P N = n is th xpctd valu o P N = n u, ( ( P ( m u n mu ( n = g( u du n! 0. (22 For mathmatical convninc, a gamma distribution is assumd or u, i.. xp( ε. Whn E [ u ] is 1 and V [ u ] is δ, th dnsity g ( u can b xprssd as, g( u κ κ = Γ( κ κ u u, (23 κ 1 whr κ = 1/ δ and Γ ( is th gamma unction. Thn, th probability o th numbr o crashs is writtn,

14 13 P( N = n whr r = ( mu 0 κ κ m = Γ( n + 1 Γ( κ Γ( κ + n = r Γ( n + 1 Γ( κ m =, m + κ n! m 0 κ κ Γ( κ ( m + κ u κ κ m Γ( κ + n = Γ( n + 1 Γ( κ( m + κ n n n mu n (1 r = v P. u κ + n κ κ + n 1, κ u, u κ 1 du du,, (24 This distribution has conditional man m and conditional varianc E[ n ][1 + δe[ n ]] (whr κ = 1/ δ. Th ngativ binomial modl can b stimatd by maximum liklihood. Using (24, th log-liklihood unction or th ngativ binomial modl is, L( m = κ n I T Γ( κ + n = = κ m ln 1 j 1 Γ( n + 1 Γ( κ κ + m κ + m i, (25 whr m = v P = v 1 1+ β o x o β x, I is th total numbr o rway sctions, and T is th numbr o yars o crash data. This unction is maximizd to obtain coicint stimats or β ( β o and β and κ. I th stimatd κ is statistically signiicant (signiicantly dirnt rom zro, th ngativ binomial rgrssion modl is mor appropriat than th Poisson modl. DATA DESCRIPTION Data rom th Highway Saty Inormation Systm (HSIS wr mployd or dvloping th rlationships btwn rar-nd crashs and xplanatory variabls. Th crash data usd or this study ar two-vhicl rar-nd crashs that occurrd on I-5 in Washington Stat

15 14 rom 2001 to Th HSIS classiid roadway sctions wr usd as crash obsrvation units. Each roadway sction rprsnts a homognous link in trms o curvatur and cross-sctional charactristics, such as numbr o lans, lan width, mdian typ and width, and shouldr width. Traic actors such as traic volum and truck prcntag play an important rol in crashs. Unortunatly, traic data whn crashs occur was not availabl. Thror, Annual Avrag Daily Traic (AADT and prcnt trucks wr usd or calibration in our study and roadway sctions without AADT and truck prcntag data wr xcludd rom th quantitativ analysis. Traic variabls: AADT and Truck prcnt data. W gnralizd th AADT variabl by considring sction lngth and numbr o lans. Sinc th numbr o lans varis rom sction to sction and th chanc or a sction to hav a crash incrass with sction lngth, th AADT variabl must b gnralizd to satisy th rquirmnt o this microscopic approach. Th gnralizd AADT is calld Daily VMT pr lan. It was calculatd rom AADT, sction lngth, and numbr o lans as ollows: AADT Sctionlngth Daily VMT pr lan =. (26 Th numbr o lans 1000 Th divisor o 1000 was usd to xpdit th calculation spd o modl calibration. Similarly, truck data wr also gnralizd by considring sction lngth and th numbr o lans. Truck prcntag-mil pr lan is a unction o truck prcntag, sction lngth, and th numbr o lans. It was calculatd as ollows: Truck % Sction lngth Truck prcntag-mil pr lan =. (27 Th numbr o lans Hr, th variabl is dividd by lan to xplain th ct o th numbr o lans. For xampl, although a on lan road and a our lan road hav th sam truck prcntag, th ct will b dirnt. Not that although this variabl is standardizd pr lan, thr is still potntial

16 15 inaccuracy, sinc trucks typically concntrat in th right and middl lans. To account or xposur to truck traic th prcnt trucks is multiplid by th sction lngth. Frway gomtric variabls: Shouldr width, horizontal curvatur, th numbr o ramps, and th numbr o lans. To rlct th cts o gomtric aturs on rar-nd crashs, th variabls mntiond abov wr combind or transormd. Total shouldr width (th sum o lt and right shouldr widths was considrd du to th high corrlation btwn lt and right shouldr widths. W also cratd a nw variabl calld dviation o shouldr width. It is dind as: Dviation o shouldr width = max{ 0, 18 Total shouldr width}. (28 Hr, w considrd 5.5 m (18 t as an idal total shouldr width. This variabl xplains th ct o th dviation o idal shouldr width or a particular road sction. Each road sction has on horizontal curvatur. Curvatur-pr-lngth was assumd to hav dirnt cts on th occurrnc o rar-nd crashs. Th variabl calld curvatur-prlngth is dind as: Dgro curvatur Curvatur- pr-lngth =. (29 Sctionlngth 10 Th divisor o 10 was usd to xpdit th calculation. To xplain th cts o mrging on rway rar-nd crashs, a variabl calld mrg sction is introducd. It is a binary variabl with valu 1 i a sction is within 0.8 km (0.5 mil upstram o a mrg point and with valu 0 othrwis. W assum that vhicls hav a tndncy to chang lans within 0.8 km (0.5 mil bor a mrg point. In sctions with o-ramps or on-ramps, vhicls ar likly to chang lans to xit or ntr into th mainlin o traic. Also, this phnomnon is mor likly to occur in sctions containing

17 16 both mrging lans and ramps. To rlct this act, a variabl calld o-ramp and mrg was dvisd (on-ramps wr xcludd hr bcaus thy did not turn out to hav statistical signiicanc in th modl. This variabl is dind as: O-ramp a nd mrg = th numbr o o-ramps in a sction Mrg ratio, (30 whr mrg ratio is dind as, Th numbr o lans in an upstram sction Mrg rati o =. (31 Th numbr o lans in a downstram sction Land us variabl: an indicator o land us, split hr simply into rural or urban. Frway sctions hav dirnt charactristics dpnding on whthr thy ar in an urban ara or rural ara. To includ this ct in th modl, th variabl calld urban ara was cratd. It is a binary variabl: urban ara = 1 whn th sction is in an urban ara; urban ara = 0 i th sction is in a rural ara. Traic control variabl: spd limit. Anothr important variabl in xplaining crashs on rways is th postd spd limit. Postd spd limits on rways can b xpctd to b corrlatd with travl spd, but this corrlation braks down during congstion as travl spds drop and spds ar govrnd mor by th congstion, not roadway gomtrics. During congstion, chain-raction accidnts ar mor likly to occur whras in this papr w xclud ths and ocus on rar-nd crashs btwn only two vhicls. Such accidnts ar not as closly tid to congstion as chain-raction rar-nd crashs. Th postd spd limits ar thror likly to b corrlatd with travl spd in our study and thy ar corrlatd with important unobsrvd roadway gomtrics, such as sight-distanc and intrchang dnsity, which can inlunc th liklihood o rar-nd crashs. Th postd spd limits thror captur unobsrvd roadway

18 17 gomtrics and spd cts. Th modl includs th variabl spd limit that taks th actual valu o th postd spd limit or th sction. In summary, th modl includs six continuous xplanatory variabls: daily VMT pr lan, truck prcntag-mil-pr-lan, dviation o shouldr width, curvatur-pr-lngth, o-ramp and mrg, and spd limit; and two binary variabls: mrg sction and urban ara. RESULTS Th rar-nd crash risk modl was stimatd by maximum liklihood. In total, twlv coicints (including intrcpts on ight xplanatory variabls wr ound statistically signiicant at th 90% lvl in th modl (iv xplanatory variabls or P o, iv xplanatory variabls or P, and th rciprocal o th ngativ binomial disprsion paramtr, θ. Modl stimation rsults ar shown in Tabl 1. Th sign o an stimatd coicint indicats th dirction o th impact o th variabl, i.. a variabl with a positiv coicint incrass th probability and a variabl with a ngativ coicint has a dcrasing ct. Th ρ 2 in this papr compars th log-liklihood at ( β = 0, κ = 1 to log-liklihood at convrgnc. Th Probability o a Lading Vhicl bcoming an Obstacl ( P o Two variabls wr ound to dcras th probability o a lading vhicl bcoming an obstacl and our variabls wr ound to incras th probability. Th daily VMT pr lan tnds to dcras th probability o a lading vhicl bcoming an obstacl. This is somwhat countrintuitiv as highr volums suggst gratr opportunitis or crashs. Howvr, with incrasing low, traic is compactd and mor vhicls ntr into car-ollowing mod, rsulting in incrasingly similar spds on th rway. Importantly, our study ocuss on rar-nd crashs

19 18 btwn two vhicls and omits chain-raction rar-nd crashs. Chain-raction crashs bcom mor likly with highr volum and th rlativ numbr o two-vhicl crashs will drop and caus a ngativ rlationship with incrasing volum. Thr may also b non-linar cts in this variabl which ar not capturd by th modl. Ths cts may contribut to th rducd probability o a vhicl bcoming an obstacl with highr volums. It should b notd, that th nt ct rom th modl dos indicat that thr is a highr probability o rar-nd crashs with highr volums as xpctd. That happns bcaus th probability o ailing to avoid a crash gos up with incrasing volum. Truck prcntag-mil-pr-lan was ound to incras th probability. Exampls that could xplain this ar as ollows: (a whn a lading vhicl is a truck, th ollowing drivr may b mor likly to switch lans and ovrtak th truck du to th rlativly slow spd o th truck, and (b a passngr car somtims cuts in ront o a truck without allowing suicint hadway or a ollowing truck, ignoring th act that a truck nds a longr hadway than a passngr car. Thror, a highr truck prcntag rsults in mor rqunt lan changs and such disturbancs could contribut to an incras in P o. Golob and Ragan s study (2004 indicats this tndncy. As th numbr o vhicls incrass, lan changs may b diicult and drivrs may stay in thir currnt lans. Frway sctions in an urban ara ar associatd with a highr probability o a vhicl ncountring an obstacl vhicl. This may b du to th highr dnsity o ntrancs and xits that crat mor rqunt lan changs (waving. This rasoning can b supportd by Golob t al. (2004 who ound that rar-nd crashs hav th highst liklihood o occurring in a waving sction.

20 19 Th dgr o curvatur is dirctly rlatd to th radius ( R o th horizontal curv. Thror, as R dcrass, curvatur incrass. Carson and Mannring (2001 ound that crash rquncy dcrass as horizontal curv radius incrass. As shown in Tabl 1, curvatur-prlngth is idntiid to hav an incrasing impact on th probability o a lad vhicl bcoming an obstacl. Th o-ramp and mrg variabl incrass th probability o th lad vhicl bcoming an obstacl. Whn vhicls lan chang rquncy incrass, th liklihood o having a rar-nd crash grows highr. Jason t al. (1998 ound that rar-nd crashs involving trucks ar mor likly to occur in sctions with o-ramps than in sctions with on-ramps. An on-ramp variabl was originally includd in th modl, but rmovd rom th inal orm bcaus it was not signiicant. This may indicat that dirnt typs o crashs, such as sidswip, ar mor rqunt than rar-nd crashs nar on-ramps. Th Probability o Failur to Avoid a Collision givn an Obstacl Vhicl ( P Thr variabls wr ound to dcras th rar-nd crash probability and thr variabls wr ound to incras its probability. In th P modl, daily VMT pr lan has an incrasing impact and truck prcntag-mil-pr-lan has a dcrasing impact. Obviously, th impacts o ths two variabls ar opposit to thir cts in th P o modl. As daily VMT pr lan incrass, th traic dnsity incrass and th incras o traic dnsity mans th dcras o hadway distanc i all othr conditions ar th sam. As hadway distanc dcrass, AMT dcrass, and as a rsult, a drivr s probability o ailur to avoid a collision incrass. Whn ollowing a truck, drivrs tnd to kp longr gaps. Also, truck drivrs ar prossional drivrs. Thror, th incras o th prcnt truck mans an incras in th numbr o prossional

21 20 drivrs in th traic stram, and thy may b bttr abl to rspond to avoid crashs than rgular drivrs. This tndncy rsults in longr AMT and lowrs P. This rasoning can b supportd by Golob and Ragan (2004. In thir study, 45% o crashs not involving trucks wr rar-nd crashs, whras only 18% o truck-involvd crashs wr rar-nd crashs. Th P modl also ound that road sctions with a highr postd spd limit hav lowr drivr ailur rat. This is may b du to th corrlations btwn postd spd limits and unobsrvd actors such as travl spd, dsign spd, and roadway gomtrics. For xampl, roadway sctions with high postd spd limit hav gratr stopping sight-distanc; othr actors, such as rducd rquncy o intrchangs on sctions with a highr spd limit, would rduc waving manuvrs which can rduc rar-nd crash rquncis. Golob and Rckr (2003 drw a similar conclusion rom thir study: rar-nd crashs ar mor likly to occur at lowr spds and during highr variations o spd. Anothr variabl which incrass P is dviation o shouldr width. It has bn ound that a narrow shouldr width (total shouldr width is smallr than 5.5 m (18 t incrass th probability o a drivr s ailur to avoid a collision. On sctions with narrow shouldrs, drivrs hav lss room to avoid rar-nd crashs or tak corrctiv actions, which may xplain this rsult. Milton and Mannring (1998 also concludd that narrow shouldrs (including both th right and lt shouldrs tnd to incras crash rquncy. Th mrg sction variabl also incrass P. This indicats that a drivr has a gratr P whn driving in a mrg sction. This can b xplaind by two rasons: 1 cut-in vhicls signiicantly rduc AMT; and 2 othr vhicls movmnts can distract a drivr s attntion which dlays th prcption o an obstacl vhicl.

22 21 Finally, th t-statistic o th coicint stimat or th rciprocal o th ngativ binomial disprsion paramtr ( κ was , which mans that this coicint was statistically vry signiicant, and that it was corrct to rjct th Poisson modl. Th avrag o th probability o ncountring an obstacl vhicl ( P o was 32.88% and th avrag o th probability o th ollowing vhicl drivr s ailur to avoid a collision ( P was %. This rsult is consistnt with Wang t al. (2002. Thy rasond that whil traic low is rquntly intrruptd by disturbancs, th drivrs AMT is gnrally gratr than NMT and hnc allow th appropriat prcption and raction tim to accomplish an avoidanc manuvr. Elasticity Two variabls in this modl hav dual impacts with opposit dirctions on th rar-nd crash risk: daily VMT pr lan and truck prcntag-mil-pr-lan. To know th ovrall cts on probability o rar-nd crashs, lasticity was calculatd or thos variabls. Not that lasticity was calculatd or Equation (16, th probability o a rar-nd crash occurring, but not Equation (24 th probability o a crtain numbr o rar-nd crashs occurring in a sction. Th lasticitis o daily VMT pr lan and truck prcntag-mil-pr-lan wr about and 0.542, rspctivly. That is, as daily VMT pr lan incrass, th probability o rar-nd crashs incrass ( ; as truck prcntag-mil-pr-lan incrass, th liklihood o rar-nd crashs dcrass (

23 22 Statistical Tsts o Tmporal Transrability and Coicint Stability W statistically tstd th modl or tmporal transrability and coicint stability. Tabl 2 shows th rsults o th tmporal transrability and coicint stability tsts. For th tmporal transrability tst, th null hypothsis is that th coicints ar transrabl btwn yars. W irst stimat th modl or th two yars (2001 and 2002 togthr, ctivly constraining th coicints to b qual or both yars. Thn, w stimat th modl or th yars individually using th sam modl structur and apply a liklihood ratio tst to compar th constraind modl to th two unconstraind modls. Th liklihood ratio tst rsults indicat a 2 χ valu o with 13 dgrs o rdom, which is smallr than th tabl χ 2 valu, , at th 95% conidnc lvl. W thror do not ind statistical vidnc to rjct th null hypothsis o transrability, sinc allowing th coicints to b dirnt did not rsult in a signiicant chang compard to th constraind modl. In th scond tst, or coicint stability, a similar liklihood ratio tst is prormd to tst th null hypothsis o coicint stability. Th modl was stimatd or th ntir two-yar priod. Thn, with a uniorm distribution, th total data wr randomly dividd into two sub data 2 sts having th sam numbr o obsrvations. Th liklihood ratio tst rsults indicat a χ valu 2 o with 13 dgrs o rdom, which is smallr than th χ tabl valu, , at th 95% conidnc lvl. Thr is thror no statistical vidnc to rjct th null hypothsis o stability and th modl spciication can b usd or sub data sts. CONCLUSION Following th microscopic modling approach dvlopd by Wang (1998, a rar-nd crash risk modl was dvlopd and stimatd using rway rar-nd crash data obsrvd in Washington Stat. Unlik most xisting crash modls, th modl dvlopd in this rsarch

24 23 considrd th occurrnc mchanism o rar-nd crashs on rways and was capabl o capturing th dual impacts o xplanatory variabls in th occurrnc o rar-nd crashs. Whn intrprting a modl with dual cts, th cts can contradict ach othr. In that cas it is ncssary to draw th intrprtation rom th ovrall modl rsult which will show which ct is strongr. Most otn, th rsults ar in harmony and it is not ncssary to look to th ovrall modl rsults, sinc th individual modls dirctly yild usul intrprtations that can lad to saty improvmnts. For xampl, th daily VMT pr lan variabl has dual impacts with opposit dirctions: It rducs th probability o a vhicl bcoming an obstacl ( P o as indicatd by th ngativ coicint, and it incrass th probability o a ollowing vhicl s raction ailur ( P, yilding a nt incras in probability o rar-nd crash with volum as indicatd by th avrag lasticity or th ovrall modl. Th dual procss in this modl thror suggsts th incrasd probability o rar-nd crashs with volum happns bcaus o th incrasing probability o drivrs ailing to avoid crashs but not bcaus o an incras in probability o a vhicl bcoming an obstacl vhicl. This indicats that ocusing saty improvmnt orts on rducing drivrs ailur to avoid crashs is o ky importanc. Potntial applications could b crash avoidanc systms that assist drivrs to avoid crashs,.g., hadway warning systms and smart cruis controls that rduc th PRT signiicantly. Th truck prcntag-mil-pr-lan variabl was also signiicant in both P o and P. A highr truck prcntag incrass P o but dcrass P. Whn considring th ovrall modl, this variabl rducs th probability o rar-nd crashs. Undrstanding such dual impacts o controllabl variabls is important or slcting saty improvmnt plans. This inding can or xampl b usd to improv saty on highways

25 24 through th us o inormation systms. Although truck prcntag-mil-pr-lan dcrass th probability o rar-nd crashs undr simultanous considration o both P o and P, this variabl incrass P o. Thror, ral-tim inormation o truck prcntag-mil-pr-lan may b usd in warning systms that inorm drivrs whn traic conditions ar mor likly to lad to rar-nd crashs. This modl mainly ocusd on th mchanism o rway rar-nd crash occurrnc. Traditional human actors such as ag, xprinc, halth condition, and gndr play an important rol in th mchanism but individual-spciic data cannot b usd in a rquncy modl bcaus crash inormation must b aggrgatd ovr a tim priod in ach sction. Thror, a distribution o drivrs rspons tim was mployd as a surrogat variabl or rlcting th impacts o human actors. A modiid ngativ binomial rgrssion approach was mployd to calibrat th risk modl using obsrvd rar-nd crash data and was succssully stimatd by maximum liklihood. Th stimatd ngativ binomial distribution paramtr was ound statistically signiicant, which indicats th data was ovrdisprsd, and that th Poisson modl would hav bn lss appropriat. For utur study, w rcommnd th us o micro-scal traic data, such as 5 minut volums at th tim o ach crash to bttr xplain th ct o traic low on rar-nd crashs in a microscopic modl. In summary, this study dmonstratd that a microscopic modling approach can b applid to rway rar-nd crashs and it producd rasonabl rsults. This typ o microscopic crash rquncy modling adds to th undrstanding o th rlationships btwn th risks o rway rar-nd crashs and causal actors. It can also hlp dcision-makrs slct ctiv

26 25 countrmasurs against rway rar-nd crashs, spcially in th ralm o dsign o roadways (.g., roadways can b dsignd with crtain shouldr widths and lss curvatur. ACKNOWLEDGEMENTS Th authors thank th Washington Stat Dpartmnt o Transportation and th Highway Saty Inormation Systm, Fdral Highway Administration or thir hlp in providing th data usd in this study. Th authors also thank th anonymous rviwrs, whos constructiv commnts signiicantly improvd th papr. REFERENCES AASHTO (2001. A policy on Gomtric Dsign o Highways and Strts, Amrican Association o Stat Highway and Transportation Oicials, Washington, D.C. Bats, J. T. (1995. Prcption-Raction Tim. ITE Journal, 65(2, Carson, J., and Mannring, F. (2001. Th ct o ic warning sings on ic-accidnt rquncis and svritis. Accidnt Analysis and Prvntion, 33, FHWA. (2003. Manual on Uniorm Traic Control Dvics (MUTCD, Fdral Highway Administration, U.S. Dpartmnt o Transportation. Golob, T. F., and Ragan, A. C. (2004. Traic conditions and truck accidnts on urban rways, Univrsity o Caliornia at Irvin, UCI-ITS-WP Golob, T. F., and Rckr, W. W. (2003. Rlationships among urban rway accidnts, traic low, wathr, and lighting conditions. Journal o Transportation Enginring, 129(4,

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29 28 Summala, H. (2000. Brak Raction Tims and Drivr Bhavior Analysis. Transportation Human Factors, 2(3, Ularsson, G. F., and Shankar, V. (2003. An accidnt count modl basd on multi-yar crosssctional roadway data with srial corrlation. Transportation Rsarch Rcord 1840, Wang, Y. (1998 Modling Vhicl-to-Vhicl Accidnt Risks Considring th Occurrnc Mchanism at Four-Lggd Signalizd Intrsctions. Ph.D. Dissrtation, Th Univrsity o Tokyo. Wang, Y., and Nihan, N. (2003. Quantitativ analysis on angl-accidnt risk at signalizd intrsctions. World Transport Rsarch, Slctd Procdings o th 9th World Conrnc on Transport Rsarch. Prgamon Prss, Oct. 2003, Soul, Kora. Wang, Y., Ida, H., and Mannring, F. (2002. Estimating Rar-End Accidnt Probability at Signalizd Intrsctions: An Occurrnc-Mchanism Approach. Journal o Transportation Enginring, 129(4, 1 8. Washington Stat Dpartmnt o Transportation. ( Washington Stat highway Accidnt Rport, < (Sp. 15, NOTATION Th ollowing symbols ar usd in this papr: E [ ] = xpctd valu = xponntial unction

30 29 = unction g ( = dnsity unction o a gamma distribution h = hazard unction I = total numbr o rway sctions J = maximum numbr o disturbancs L ( = log-liklihood unction m = Poisson distribution paramtr N = numbr o crashs n = numbr o crashs on a givn sction in a givn tim priod P = probability p = probability valu or p -valu T = numbr o yars o crash data t = tim V [ ] = varianc v = vhicl low x = vctor o xplanatory variabls or a singl obsrvation = shap paramtr o th Wibull distribution β = vctor o stimabl paramtrs Γ ( = gamma unction

31 30 δ = varianc o gamma distributd rror trm ε = unobsrvd rror trm η = paramtr o th xponntial distribution θ = scal paramtr o th Wibull distribution or Availabl Manuvring Tim κ = disprsion paramtr o th ngativ binomial modl λ = scal paramtr o th Wibull distribution or Ndd Manuvring Tim 2 ρ = rho-squard statistic 2 χ = chi-squard distributd liklihood ratio tst statistic Th ollowing subscripts ar usd in this papr: a = availabl manuvring tim = ollowing vhicl ailur i = rway sction j = disturbanc typ or givn priod n = ndd manuvring tim o = obstacl vhicl

32 31 TABLE 1 Modl Estimation Rsults Variabl Estimatd Coicint t -statistic Variabls acting th probability o bcoming an obstacl vhicl ( P o Constant ( Daily VMT pr lan (0.107 * Truck prcntag-mil-pr-lan (0.139 * Urban ara (0.133 * Curvatur pr lngth ( O-ramp and mrg ( Variabls acting th probability o ollowing vhicl's drivr ailur ( P Constant (0.749 * Daily VMT pr lan (0.113 * Truck prcntag-mil-pr-lan (0.146 * Spd limit (0.009 * Dviation o shouldr width (0.006 * Mrg sction (0.100 * Rciprocal o ngativ binomial disprsion paramtr (κ (0.064 * Log-liklihood at zro (β = zro, κ = 1-63, Log-liklihood at κ only ( β = 0-6, Log-liklihood at constants and κ (othr β = 0-3, Log-liklihood at convrgnc -3, ρ Standard rrors ar in parnthss. Lvl o signiicanc: all gratr than 90% and * > 99.9%. Coicints that wrn t signiicant at th 90% lvl wr rstrictd to zro and omittd rom th tabl. 2 ρ was calculatd by comparing th log-liklihood at zro (β = zro, κ = 1 to log-liklihood at convrgnc.

33 32 TABLE 2 Transrability and Stability Tst Rsults LL ( β t = LL ( β a = LL ( β b = χ Tmporal transrability Tst = -2(LL ( β t - LL ( β a - LL ( β b = -2( ( ( = (13.36 < , with 95% conidnc and 13 dgrs o rdom Transrabl Coicint Stability Tst LL ( β t = LL ( β a = LL ( β b = χ = -2(LL ( t β - LL ( a β - LL ( b β = -2( ( ( = (12.31< , with 95% conidnc and 13 dgrs o rdom Transrabl