Research Article Analysis of Severe Injury Accident Rates on Interstate Highways Using a Random Parameter Tobit Model

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1 Hndaw Mathematcal Problems n Engneerng Volume 2017, Artcle ID , 6 pages Research Artcle Analyss of Severe Injury Accdent Rates on Interstate Hghways Usng a Random Parameter Tobt Model Mnho Park 1 and Dongmn Lee 2 1 Hghway & Transportaton Research Dvson, Korea Insttute of Cvl Engneerng and Buldng Technology, Gyeongg-do, Republc of Korea 2 Department of Transportaton Engneerng, Unversty of Seoul, Seoul, Republc of Korea Correspondence should be addressed to Dongmn Lee; dmlee@uos.ac.kr Receved 8 November 2016; Accepted 12 January 2017; Publshed 2 February 2017 Academc Edtor: Inmaculada T. Castro Copyrght 2017 Mnho Park and Dongmn Lee. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. In ths study, a random parameter Tobt regresson model approach was used to account for the dstnct censorng problem and unobserved heterogenety n accdent data. We used accdent rate data (contnuous data) nstead of accdent frequency data (dscrete count data) to address the zero cell problems from data where roadway segments do not have any recorded accdents over the observed tme perod. The unobserved heterogenety problem s also consdered by usng random parameters, whch are parameter estmates that vary across observatons nstead of fxed parameters, whch are parameter estmates that are fxed/constant over observatons. Nne years ( ) of panel data related to severe njury accdents n Washngton State, USA, were used to develop the random parameter Tobt model. The results showed that the Tobt regresson model wth random parameters s a better approach to explore factors nfluencng severe njury accdent rates on roadway segments under consderaton of unobserved heterogenety problems. 1. Introducton Over the last decade, numerous studes have been conducted to explore the factors that cause accdents on roadway segments; varous statstcal modelng technques have been employed, especally count models. Intally, smple lnear regresson models were employed. Although ths s the smplest and easest model, the data generally volate the basc assumpton of homoscedastcty, whch means that the varance ncreases as the varable ncreases. In addton, negatve accdent count values are predcted whch should be greater than or equal to zero n realty [1]. To address these problems that the lnear regresson model had, prevous nvestgators suggested a Posson regresson model wheren accdent frequency s translated as a dscrete random varable [2]. However, there was an mportant constrant n ths model, namely, that the mean must be equal to the varance; the standard errors wll be based when ths restrcton s not vald. Meanwhle, real accdent data were found to be overdspersed, meanng that the varance s greater than the mean [3, 4]. As a result, the Posson regresson model ncorrectly estmates the lkelhood of accdent frequences and, to overcome ths overdsperson problem, a negatve bnomal model was suggested, whch relaxes the constrant that the mean s equal to the varance. Negatve bnomal models had been shown to be more approprate than the Posson model for descrbng the relatonshps between accdent frequency and geometrc elements [5 8]. Besdes, a varety of attempts to analyze accdent frequences were made, and these resulted n a reducton n accdents and mproved accdent preventon (random effects models [9, 10], zero-nflated count models [11 13], and random parameters count models [14 18]). Asde from methodologes that address accdent frequency data, a Tobt regresson method that uses accdent rates as the dependent varable was also suggested [19]. Ths method employs contnuous varables, that s, the number of accdents per vehcle-mle traveled, nstead of accdent frequency (dscrete count data) data. In addton, the lkelhood

2 2 Mathematcal Problems n Engneerng that segments/spots wll have no accdent records durng some perod can be taken nto account usng data that s left-censored at zero. Partcularly, accdent frequences that nvolve severe njury accdents (fatal and dsablng njures) are lower than those of other types of accdents such as EPDO, possble njures, and evdent njures, that s, zero records n many cases. Ths has made t dffcult to analyze severe njury accdents usng the tradtonal frequency method, whch s the reason why the Tobt model s a more approprate approach to analyze the causes of severe njury accdents than the tradtonal frequency models. In addton, there s another remanng problem, namely, that prevous methods mentoned above do not account for heterogenety. Snce aggregated accdent data for analyss do not have nformaton related to the resdual envronmental effects and socoeconomc, drver, and vehcle characterstcs, there s a possblty that unobserved heterogenety may occur n accdent data, whch creates varaton n the mpact of the effect of observed varables on accdent frequences [20]. Ths problem cannot be addressed n the tradtonal count models n whch the estmated parameters are fxed. Therefore, we must consder models that nclude random parameters, whch allow some or all parameters to vary randomly across observatons. Some relevant research has shown that the random parameters approach can account for varatons n the varables [14 16, 18, 21]. Here, we developed a random parameter Tobt model that allowed us to address the unobserved heterogenety n accdent data for severe njures and also compared the estmated results from a fxed parameter Tobt model on roadway segments except for nterchange segments on nterstates. To the best of the authors knowledge, ths s the frst attempt to model severe njury accdent case usng random parameters Tobt model method. And ths approach usng accdent rates could be appled n the process of select performance measures n HSM (Hghway Safety Manual) whose framework andmodelngarchtecturehavebeenntroducedn[22]. 2. Methodology The Tobt model s a regresson model proposed by Tobn (1958) n whch the dependent varable s ether left- or rght-censored. Here, left-censored means that the data are censored at a low threshold, whle rght-censored data are censored at a hgh threshold. In accdent data, the data wll be left-censored wth clusterng at a zero base snce vehcle crash frequences may not be observed on all/some segments durng the observaton perod. Usng ths nformaton, the Tobt model was constructed as follows: Y = β X +ε, =1,2,3,...,N, Y = { Y f Y >0 { 0 f Y { 0. Here, N s the number of observatons, Y s the dependent varable (severe njury accdent rate), β s a vector of estmable parameters, X s a vector of ndependent varables (e.g., traffc volumes and segment geometrcs), and ε s a normally (1) and ndependently dstrbuted error term wth zero mean and constant varance σ 2. Here, there s an mplct and stochastc ndex (latent varable) expressed as Y,whchsobserved only when the value of Y s greater than zero (postve). Hence, the lkelhood functon for the Tobt model over zero and postve observatons s as follows: L= [1 Φ ( βx 0 σ )] 1 σ 1 Φ[ (Y βx) ]. (2) σ Here, Φ refers to the standard normal dstrbuton functon and Φ s the standard normal densty functon. The tradtonal Tobt (fxed parameter) model s descrbed. However, t s dffcult to account for heterogenety (unobserved factors that may vary across observatons) n ths model. In order to account for heterogenety usng a random parameter, Greene [23] developed a smulated maxmum lkelhood estmaton procedure, whch has been showntobeanacceptablemethod[15,16,18,21]. Estmable parameters that allow for random parameters are as follows: β =β+φ, =1,2,3,...,N. (3) Here, β ndcates estmated parameters and φ s a randomly dstrbuted term. Unform, normal, lognormal, and other forms are consdered to be potental densty functons for random parameter estmaton. The latent varable mentoned n (1) becomes Y φ =βx +ε, and the lkelhood functon from (2) s as follows n log-lkelhood form: LL = ln g(φ )P(n φ )dφ. (4) φ Here, g refers to the probablty densty functon of φ. To estmate the random parameters, a smulaton-based maxmum lkelhood usng Halton draws was employed whch s an effcent dstrbuton of draws for numercal ntegraton [24, 25]. In summary, the random parameter Tobt model could account for unobserved factors and at the same tme support the complete use of avalable data from leftcensored severe njury traffc accdent data. 3. Data Vehcle crash accdent data of roadway segments on nterstates n Washngton State (I-5, I-82, I-90, I-182, I-205, I- 405, and I-705) had been collected over 9 years (1999 to 2007) to nvestgate the effects of geometrcs and traffc flow condtons such as number of lanes, rght and left shoulder wdth, number of horzontal and vertcal curves, and traffc volumes on severe njury accdent rates per 100-mllon vehcle-mles traveled (VMT). Frstly, the collected data were dvded nto data on roadway segments and data on nterchange segments of the nterstate hghways. In ths study, only crash data on roadway segments were used because crashes on nterchange mght generally occur wthn varous effects ncludng traffc flow changes, weavng maneuvers, complex geometrcs, drver behavors by traffc sgns, and other dfferent condtons

3 Mathematcal Problems n Engneerng 3 Table 1: Descrptve statstcs of varables. Varable descrpton Mean Std. dev. Mnmum Maxmum Number of dsablng njury accdents Number of fatalty njury accdents Segment length (m) Average annual daly traffc volume 13,052 8, ,224 Number of lanes per drecton Left shoulder wdth (ft) Rght shoulder wdth (ft) Number of horzontal curves Number of vertcal curves Table 2: Model estmaton results. Fxed parameter model Random parameter model Parameter estmate t-rato Parameter estmate t-rato Constant Logarthm of segment length Standard devaton of parameter dstrbuton NA NA Logarthm of annual average daly traffc volume Standard devaton of parameter dstrbuton NA NA Number of lanes per drecton Standard devaton of parameter dstrbuton NA NA Left shoulder wdth Standard devaton of parameter dstrbuton NA NA Rght shoulder wdth Standard devaton of parameter dstrbuton NA NA Number of horzontal curves Standard devaton of parameter dstrbuton NA NA Number of vertcal curves Standard devaton of parameter dstrbuton NA NA Number of observatons 589 Log-lkelhood functon at zero 3, Log-lkelhood functon at convergence 3, , NA: not applcable. from roadway segments. Consequentally, over a contnuous perod of nne years, the 589 roadway segments whch were used for the analyss yelded a panel of 5,301. Accdent rate, the dependent varable, was calculated usng the followng equaton: accdent rates = n y=1 accdents y, [ n y=1 AADT y, L 365] /100,000,000. (5) Here, accdent rates s the total number of severe accdents per 100-mllon VMT on segment, y s the year of observed data, accdents y, s the number of severe accdents on segment n year y, AADT y, s the average annual daly traffc volume on segment n year y,andl s the length of segment. Snce we sought to determne the effects of geometrcs on severe accdents, the dependent varable s defned as the summaton of dsablng and fatal njures. The descrptve statstc values for the prmary varables are shown n Table 1. The average length of roadway segments was mles, and 13,052 vehcles s the mean value of the average annual daly traffc volume on the objectve segments durng the study perod. On average, 2.6 lanes per drecton exst wth a mnmum of one lane and a maxmum of fve lanes. The mean value of the shoulder wdth s 6.9 ft on both the left and the rght shoulder. In terms of curves, 1.8 horzontal curves and 3.2 vertcal curves exst on the roadway segments. 4. Model Estmaton Results Two types of modelng methods were used to estmate whether parameters are fxed (fxed parameters, left sde n Table 2) or they vary across observatons (random parameters, rght sde n Table 2). For random parameter estmatons, Halton draws were used, whch has been shown to produce accurate parameter estmatons [25]. The normal dstrbuton

4 4 Mathematcal Problems n Engneerng Table 3: Margnal effect and elastcty values of fxed and random parameters Tobt model. Fxed parameter Random parameter Margnal effect Elastcty Margnal effect Elastcty Logarthm of segment length Logarthm of annual average daly traffc volume Number of lanes per drecton Left shoulder wdth Rght shoulder wdth Number of horzontal curves Number of vertcal curves of densty functonal forms gave the best statstcal results among the normal, unform, and lognormal dstrbutons mentoned n the Methodology. The overall log-lkelhood at convergence n the random parameter Tobt model ( 3,169.62) showed relatvely greater mprovement than the fxed parameter Tobt model ( 3,191.58). As descrbed n Table 2, a total of seven varables wth random parameters were derved to have an effect on the severe accdent rates. These parameters are segment length, average annual daly traffc volume, number of lanes, left/rght shoulder wdth, and number of horzontal/vertcal curves. A random parameter was used when both the mean and the standard devaton of the parameter dstrbuton were statstcally sgnfcant ( =0).Inthssense,aparametern whch standard devaton s not statstcally sgnfcant (=0) ndcates that the effects are fxed across all segments. All derved varables wth random parameters showed statstcally sgnfcant mean and standard devaton values. On the other hand, some varables wth fxed parameters were found to be statstcally nsgnfcant, whch shows the flexblty of the random parameters n that the effect of the covarates must be constant/fxed across all observatons [21]. The results of modelng, the margnal effect, and elastcty of the random parameters and fxed parameters models are presented n Tables 2 and 3, respectvely. The logarthms of segment length, traffc volumes, and number of lanes wereshowntohavestatstcallysgnfcantfxedandrandom parameters wth postve sgns. Ths s consstent wth the expectaton of ncreased frequency of severe njury accdents wth hgher exposure (longer length, hgher traffc volumes, and more lanes) on the roads. A random parameter of the segment length that s normallydstrbutedwthameanof1.033andstandarddevaton of ndcates that the effect of segment length decreases the severe njury accdent frequency rate on 14.83% of the observed segments and ncreases the rate on 85.17% of the observed segments. In terms of elastcty, a 1% ncrease n length contrbuted to a 1.015% (fxed parameter) and 1.033% (random parameter) ncrease n severe njury accdent rate; these are ndcatons of elastcty. We found that traffc volumes have a normally dstrbuted random parameter wth a mean of and a standard devaton of Gven ths dstrbuton, the effect of traffc volume decreases the severe accdent rate on 12.12% of segments and ncreases the severe accdent rate on 87.88% of segments. The number of lanes varable had a random parameter wth a normal dstrbuton wth a mean of and a standard devaton of Gven these dstrbuton values, 10.69% of segments showed a decrease n the severe njury rate, and 89.31% of segments showed an ncrease n the severe njury rates. Wth regard to shoulder wdth, a negatve sgn (severe njury rate decrease) was found for both fxed and random parameters. The left shoulder wdth had a normally dstrbuted random parameter wth a mean of and a standard devaton of These parameter values ndcated that ncreasng the shoulder wdth decreases the severe njury rate n 79.86% of segments and ncreases the severe njury rate n 20.14% of segments. The rght shoulder wdth varable shows a postve sgn and s not statstcally sgnfcant n the fxed parameter model. However, the rght shoulder wdth varable was found to have a normally dstrbuted random parameter wth a mean of and a standard devaton of n the random parameter model. These dstrbuton values mean that the rght shoulder wdth ncreases severe njury rates n 7.04% of the man lne segments and decreases severe njury rates n 92.96% of the man lne segments. These results are consstent wth prevous studes [15, 17], whch have shown that accdent probablty decreases as shoulder wdth ncreases. We estmated that the number of horzontal curves varable had a random parameter wth a mean of and a standard devaton of Ths varable decreases severe accdent rates n 78.28% of the roadway segments and ncreases severe accdent rates n 21.72% of the roadway segments. The effect of the number of horzontal curves on the severe njury accdent rate n the fxed parameters model was postve for all nterstate roadway segments consdered, but no statstcally sgnfcant nfluence was derved; ths brngs about addtonal support to the use of the random parameter. Fnally, the varable for the number of vertcal curves was shown to be statstcally sgnfcant n both the fxed parameter model and the random parameter model under a normal dstrbuton wth a mean of and a standard devaton of Ths random parameter result ndcates that the effect of the number of vertcal curves decreases the lkelhood of severe njury rates n 85.49% of all observed segments and ncreases the lkelhood of severe njury rate n 14.51% of all observed segments. Results from these curverelated varables are smlar to a prevous study that showed

5 Mathematcal Problems n Engneerng 5 that some varatons n roadway geometrcs may mprove drver alertness, resultng n more careful drvng [26]. 5. Conclusons We used a Tobt regresson model wth fxed and random parameters to examne the geometrc factors that nfluence the rate of accdents that result n severe njury (fatal and dsablng njury). In the Tobt regresson model, a dependent varable (severe njury accdent rate) was appled as a contnuous varable and was left-censored at zero, whch s an alternatve to the tradtonal dscrete accdent frequency approach. Unobserved heterogenetes were consdered as well n the parameter estmaton processes by employng a random parameter, whch s dffcult to do n tradtonal fxed parameter estmaton. In other words, by usng the random parameter Tobt method, heterogenety from factors such as vehcle type, weather, ndvdual character, and other unobserved factors whch are not captured n the data collecton was accounted and corrected for n fndng sgnfcant factors on the severe njury accdent rates on the nterstates. Nne years of severe njury accdent and geometrcs data from seven nterstate man lnes n Washngton State, USA, were used to develop the models. Seven varables were found to have random parameters wth statstcally sgnfcant standard devaton values. The effects of these varables vary across observatons: segment length, annual average daly traffc volume, number of lanes, left and rght shoulder wdth, and numbers of horzontal and vertcal curves. Whle ths study s exploratory n nature, random parameters Tobt regresson model has the potental to provde a fuller understandng of the factors on severe accdent rates, whch showed the outstandng result related to fxed parameters model. Although the predctve power was mproved (as shown n log-lkelhood values), other possble varables that may affect the lkelhood of severe accdents such as pavement condtons were not consdered. In addton, nterchange segments havng more complex nfrastructures and more dverse traffc flow types (merge, dverge, and weave) were not consdered n ths study. We recommend that future studes nclude those varables and varous analyses for explorng addtonal causes of traffc accdents. Competng Interests The authors declare that there are no competng nterests regardng the publcaton of ths paper. References [1] P. P. Jovans and H. L. Chang, Modelng the relatonshp of accdents to mles traveled, Transportaton Research Record: Journal of the Transportaton Research Board,no.1068,pp.42 51,1987. [2] S. C. Joshua and N. J. Garber, Estmatng truck accdent rate and nvolvements usng lnear and posson regresson models, Transportaton Plannng and Technology,vol.15,no.1,pp.41 58, [3] V. Shankar, F. Mannerng, and W. Barfeld, Effect of roadway geometrcs and envronmental factors on rural freeway accdent frequences, Accdent Analyss and Preventon,vol.27,no.3,pp , [4] A. Vogt and J. Bared, Accdent models for two-lane rural segments and ntersectons, Transportaton Research Record: Journal of the Transportaton Research Board, vol.1635,pp.18 29, [5] J. Engel, Models for response data showng extra-posson varaton, Statstca Neerlandca,vol.38,no.3,pp ,1984. [6] J. F. Lawless, Negatve bnomal and mxed Posson regresson, The Canadan Statstcs, vol.15,no.3,pp , [7] M. J. Maher, New bvarate negatve bnomal model for accdent frequences, Traffc Engneerng and Control,vol.32,no.9, pp , [8] S.-P. Maou and H. Lum, Modelng vehcle accdents and hghway geometrc desgn relatonshps, Accdent Analyss and Preventon,vol.25,no.6,pp ,1993. [9] V. N. Shankar, R. B. Albn, J. C. Mlton, and F. L. Mannerng, Evaluatng medan crossover lkelhoods wth clustered accdent counts an emprcal nqury usng the random effects negatve bnomal model, Transportaton Research Record,no.1635, pp , [10] H. C. Chn and M. A. Quddus, Applyng the random effect negatve bnomal model to examne traffc accdent occurrence at sgnalzed ntersectons, Accdent Analyss and Preventon, vol.35,no.2,pp ,2003. [11] V. Shankar, J. Mlton, and F. L. Mannerng, Modelng accdent frequences as zero-altered probablty processes: an emprcal nqury, Accdent Analyss & Preventon, vol. 29, no. 6, pp , [12] J. Lee and F. Mannerng, Impact of roadsde features on the frequency and severty of run-off-roadway accdents: an emprcal analyss, Accdent Analyss and Preventon, vol.34, no. 2, pp , [13] N. Malyshkna and F. L. Mannerng, Zero-state Markov swtchng count-data models; an emprcal assessment, Accdent Analyss and Preventon,vol.42,no.1,pp ,2010. [14] P.Ch.AnastasopoulosandF.L.Mannerng, Anoteonmodelng vehcle-accdent frequences wth random parameter count models, Accdent Analyss and Preventon,vol.41,no.1,pp , [15] N.S.Venkataraman,G.F.Ulfarsson,V.Shankar,J.Oh,andM. Park, Model of relatonshp between nterstate crash occurrence and geometrcs: exploratory nsghts from random parameter negatve bnomal approach, Transportaton Research Record, no. 2236, pp , [16] N. Venkataraman, G. F. Ulfarsson, and V. N. Shankar, Random parameter models of nterstate crash frequences by severty, number of vehcles nvolved, collson and locaton type, Accdent Analyss and Preventon,vol.59,pp ,2013. [17] M. Park, Relatonshp between nterstate hghway accdents and heterogeneous geometrcs by random parameter negatve bnomal model a case of nterstate hghway n Washngton State, USA, JournaloftheKoreanSocetyofCvlEngneers,vol. 33,no.6,pp ,2013. [18] N. Venkataraman, V. Shankar, G. F. Ulfarsson, and D. Deptuch, A heterogenety-n-means count model for evaluatng the effects of nterchange type on heterogeneous nfluences of nterstate geometrcs on crash frequences, Analytc Methods n Accdent Research,vol.2,pp.12 20,2014.

6 6 Mathematcal Problems n Engneerng [19] P. C. Anastasopoulos, A. P. Tarko, and F. L. Mannerng, Tobt analyss of vehcle accdent rates on nterstate hghways, Accdent Analyss and Preventon,vol.40,no.2,pp ,2008. [20] F. L. Mannerng, V. Shankar, and C. R. Bhat, Unobserved heterogenety and the statstcal analyss of hghway accdent data, Analytc Methods n Accdent Research, vol. 11, pp. 1 16, [21] P.C.Anastasopoulos,F.L.Mannerng,V.N.Shankar,andJ.E. Haddock, A study of factors affectng hghway accdent rates usng the random-parameters tobt model, Accdent Analyss and Preventon, vol. 45, pp , [22] N. S. Venkataraman, G. F. Ulfarsson, and V. N. Shankar, Extendng the Hghway Safety Manual (HSM) framework for traffc safety performance evaluaton, Safety Scence,vol.64,pp , [23] W. L. Greene, Verson 9.0, Econometrc Software Inc, Planvew, NY, USA, [24] J. H. Halton, On the effcency of certan quas-random sequences of ponts n evaluatng mult-dmensonal ntegrals, Numersche Mathematk,vol.2,no.1,pp.84 90,1960. [25] C. R. Bhat, Smulaton estmaton of mxed dscrete choce models usng randomzed and scrambled Halton sequences, Transportaton Research Part B: Methodologcal, vol.37,no.9, pp , [26] C. Wnston, V. Maheshr, and F. Mannerng, An exploraton of the offset hypothess usng dsaggregate data: the case of arbags and antlock brakes, Rsk and Uncertanty,vol.32,no. 2, pp , 2006.

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