Statistical Temporal Analysis of Freight-Train Derailment Rates in the United States: 2000 to 2012

Transcription

1 Lu Statstcal Temporal Analyss of Freght-Tran Deralment Rates n the Unted States: 2000 to 2012 Xang Lu, Ph.D. Assstant Professor Department of Cvl and Envronmental Engneerng Rutgers, The State Unversty of New Jersey Phone: (848) Emal: xang.lu@rutgers.edu

2 Lu ABSTRACT Safety s the top prorty for every ral system n the world. A wdely used measure for ral safety s accdent rate, whch s defned as the number of tran accdents normalzed by traffc exposure. Of nterest n ral safety research s to understand the temporal trend of accdent rates, the sgnfcant factors affectng the trend, as well as how to predct future accdent rates. For ths purpose, ths paper presents a statstcal analyss of U.S. freght-tran deralment rates by ralroad, year and accdent cause from 2000 to 2012, usng a negatve bnomal regresson model. The accdent and traffc data used n the analyss came from the Federal Ralroad Admnstraton of the U.S. Department of Transportaton. The analyss leads to several observatons, ncludng: There s a sgnfcant temporal declne n freght-tran deralment rate (-5.8% per year), and the rate of declne s dentcal among the four largest freght ralroads n the U.S. The rate of change n accdent rate vares by accdent cause. Freght-tran deralment rates due to broken rals or welds and track geometry defects declned by 7% annually, respectvely; bearng-falure-caused deralment rate decreased by 11% annually; and deralment rate caused by tran handlng errors fell by 9% annually. Usng the regresson model, future tran deralment rates by accdent causes are projected and can be used to evaluate the safety beneft of potental accdent preventon strateges. Ths research ams to provde polcy makers and practtoners wth a statstcal method to analyze the temporal trend of tran accdent rate for local, regonal and natonal development of ral safety polcy and practce. Keywords: Ral Safety, Tran Accdent Rate, Temporal Trend, Statstcal Modelng, Regresson

3 Lu INTRODUCTION Ral offers a safe and effcent way to transport freght and passengers. Whle ral transportaton provdes substantal socetal benefts, tran accdent rsks must be mtgated to the maxmum extent feasble. Safety s crtcal for every ral system n the world. One commonly used metrc of assessng ral safety s accdent rate, whch s defned as the number of tran accdents normalzed by traffc exposure, such as tran-mles, car-mles, gross ton-mles or passenger-mles (1-7). In the Unted States, FRA-reportable accdent (an accdent whose damage cost to nfrastructure, rollng stock and sgnals exceeds a specfed monetary threshold) and traffc exposure data are reported by ralroads to the Federal Ralroad Admnstraton (FRA) of U.S. Department of Transportaton (USDOT). Usng these data, the FRA publshes annual tran accdent rates, whch have been extensvely used n development of ral safety polcy and practce. The FRA-publshed accdent rates were based on an emprcal approach usng reported (observed) accdent count data dvded by the correspondng traffc exposure (e.g., mllon tranmles). Ths emprcal approach provdes a hgh-level, prelmnary assessment of ral operatonal safety performance, however, t does not tell whether the change n accdent rate s statstcally sgnfcant. Generally, the emprcal accdent rate analyss s subject to a type of statstcal error called Regresson to the Mean (RTM) (8). The RTM refers to the tendency that a random varable that devates from the mean wll return to "normal" gven nothng has changed. In the context of ral safety, t mples that a hgh accdent rate n one year may be followed by a lower rate n the next year due to the random varaton, even f there s no actual safety change. For example, U.S. Class I ralroad manlne freght-tran deralment rate was per mllon tranmles n 2006, followed by per mllon tran-mles n In ths example, the emprcal tran deralment rate declned by 10 percent (( )/0.843). However, s ths accdent rate reducton statstcally sgnfcant to ndcate safety mprovement? More generally, how should the statstcal trend of tran accdent rates be modeled to understand the assocated safety mplcatons? Ths paper s developed to address both questons. 2 METHODOLOGY 2.1 Defnton of Transportaton Safety Transportaton safety research communtes wdely accept the followng noton of safety (8): Safety s the number of accdents by knd and severty, expected to occur on the entty durng a specfed perod. One hghlght of ths noton s the number of accdents that are expected to occur. The dfference between the observed and expected number of accdents represents the stochastc nature of accdent occurrence and severty. The followng sub-secton wll llustrate the theoretcal framework for modelng ral transportaton safety, measured by the expected number of accdents. If the safety s measured by other metrcs, the methodology can be adapted accordngly.

4 Lu Statstcal Theory for Modelng Tran Accdent Occurrence It s assumed that each tme a tran enters a track segment there s a probablty (p) that ths tran wll be nvolved n an accdent. Tran accdent probablty s affected by nfrastructure condtons, tran characterstcs, operatonal factors, envronmental factors and many other varables (1-3, 9, 10). Holdng these condtons constant, t can be assumed that accdent probablty s constant (assumng homogeneous track characterstcs, rollng stock, and operatonal condtons wthn the study perod). Under these assumptons, each tran pass can be vewed as a Bernoull experment (Bernoull probablty s denoted as p). The probablty theory tells that the sum of ndependent, dentcally dstrbuted Bernoull varables constructs a bnomal dstrbuton (11) (Equaton 1): N n P( X n) p (1 p) n N n (1) Where: n = number of tran accdents N = total number of tran passes on a gven segment durng the study perod p = probablty that a tran s nvolved n an accdent each tme t enters a segment Let p, gven a large number of tran passes (N s large) and relatvely low accdent N probablty, equaton (1) can be re-wrtten as: n N n n N exp( ) lm P( X n) lm 1 Posson( ) N N n N N n! (2) Equaton (2) ndcates that the number of tran accdents wthn traffc exposure can be approxmated by a Posson dstrbuton. Ths assumpton was wdely adopted n prevous studes (4-6, 12, 13, 14, 15), wthout an explct explanaton to the ratonale. In the Posson dstrbuton, the Posson mean (λ) represents the expected tran deralment count. Ths parameter needs to be estmated based on sample data. Let λ * represent an estmator of λ. λ * can be estmated as a functon wth a combnaton of predctor varables. The exponental functon s commonly used to ensure that the estmated accdent count s strctly non-negatve (4-6) (Equaton 3):

5 Lu k * expbp X p M p0 (3) Where: * λ b p X p M = estmated (expected) deralment count on the th segment = parameter coeffcent for the p th predctor varable = the p th predctor varable on the th segment = traffc exposure (e.g., gross ton-mles) on the th segment 2.3 Negatve Bnomal Regresson A number of studes have been performed to determne the best functons and estmators to quantfy the statstcal assocaton between the response varable (accdent count) and affectng factors (traffc exposure, nfrastructure, rollng stock and operatonal factors). Negatve bnomal regresson (also called Posson-gamma regresson) s prevalent n the lterature (e.g., 4-6, 16-19). Ths model allows for a large varance n accdent count data and has been shown to be adequate n many accdent rate analyses (e.g., 4-6, 8). In statstcs, a negatve bnomal dstrbuton can be nterpreted as the probablty dstrbuton of the number of successes n a sequence of ndependent and dentcally dstrbuted Bernoull trals before a specfed number of falures occurs (11). In the context of ral safety, ths may be nterpreted as the dstrbuton of the number of accdents wthn traffc exposure. Hlbe (2007) provdes techncal detals of the negatve bnomal regresson model, and compared t wth other types of regresson models (20). Ths paper starts wth ths commonly used regresson technque. If the negatve bnomal regresson model does not provde a good ft to the emprcal U.S. freght-tran data, other regresson models wll be used. 3 SCOPE OF THE ANALYSIS AND DATA SOURCES 3.1 Research Scope Ths research ams to address the followng questons: 1) Dd U.S. freght-tran deralment rate change from 2000 to 2012? 2) How dd ths change vary by ralroad and accdent cause? 3) What are the predcted future accdent rates? 4) What are the safety mplcatons of the results? All the analyses n ths paper were focused on freght-tran deralments on U.S. Class I ralroad man tracks. Each Class I ralroad has the operatng revenue exceedng $378.8 mllon (2009 dollars). Class I ralroads accounted for approxmately 68% of U.S. ralroad route mles, 97% of total ton-mles transported and 94% of the total freght ral revenue (21). Deralments are the most common type of FRA-reportable manlne tran accdents n the U.S. (22, 23). 3.2 Accdent Data The FRA requres all the ralroads operatng n the U.S. to submt detaled accdent reports for the accdents or ncdents that exceeded a specfed monetary threshold of damage cost to nfrastructure and rollng stock. The reportng threshold s perodcally adjusted for nflaton and ncreased to $10,500 n 2014 (24). FRA comples these accdent reports nto the ral equpment

6 Lu accdent (REA) database, whch contans nformaton regardng accdent locaton, speed, consst type, damage cost and other useful nformaton. The FRA REA database has been wdely used n prevous ral safety studes (1, 3, 9, 10, 13, 14, 22, 23, 25-28). 3.3 Traffc Data Tran-mle data are commonly used to analyze tran deralment rate (1, 3-6, 29). Ralroads report to the FRA ther monthly tran-mle data, whch are avalable through the FRA Operatonal Data database. 4 RESULTS 4.1 Varables There are two prncpal predctor varables (affectng factors) used n the statstcal analyss. The frst varable s year, representng the temporal change n accdent rate. The second predctor varable s ralroad (A, B, C and D) that anonymously represents the four largest freght ralroads n the Unted States. The two predctor varables construct three terms n the model, representng the margnal effect of each varable and ther combned effect. The model has the followng basc structure: exp( T ) M R R (4) where: µ R = expected number of freght-tran deralments n year for a specfc ralroad T = year (for example, T s 2000 for year 2000) M = mllon tran-mles n year α R, R = parameter coeffcents specfc to each ralroad (A, B, C, D) Ths type of exponental functon was used n several ral safety studes n European natons (4-6, 29). However, ths statstcal technque has not been wdely used to analyze U.S. tran accdent rates.

7 Lu TABLE 1 Freght-Tran Deralment and Traffc Data, Class I Manlnes, 2000 to 2012 Number of Tran Deralments Year Ralroad A B C D Mllon Tran-Mles Year Ralroad A B C D Parameter Estmator The parameter coeffcents ( 1, 2 ) were estmated usng a commercal software called Statstcal Analyss System (SAS). The software generates each parameter estmator and ts standard error usng the maxmum lkelhood method n a negatve bnomal model (Table 2). The last column n Table 2 s the P-value of a parameter estmator, whch represents the statstcal sgnfcance of a predctor varable usng the Wald Test (20). A generally acceptable rule s that f a predctor varable has ts P-value smaller than 5%, ths varable s sgnfcant. Some mportant observatons were made from the statstcal analyss:

8 Lu ) There s no statstcal dfference among the four Class I ralroads n terms of expected annual deralment rate because the dfferences n ralroad-specfc parameter coeffcents (e.g., α B - α A, and - ) are statstcally zero 2) The varable year (ts parameter coeffcent s ) s sgnfcantly negatve (P < ), ndcatng that there s a sgnfcant temporal declne n tran deralment rate for all the four Class I ralroads TABLE 2 Sgnfcance Test of Predctor Varables Estmate Standard Error Wald Ch-Square P-value α A (Ralroad A) <.0001 (Ralroad <.0001 α B - α A α C - α A α D - α A C D After removng nsgnfcant terms, the regresson model s re-ftted and the fnal model s (Table 3): TABLE 3 Regresson Analyss Results of the Fnal Model Parameter Estmate Standard Error Wald Ch- Square Pr > ChSq α < <.0001 Devance = 2.16; df = 50, P > Model Evaluaton The goodness-of-ft of a negatve bnomal model can be evaluated usng a statstcal crteron called devance (20). Statstcal theory tells that the devance asymptotcally follows a Ch Square dstrbuton (20). Based on ths property, the P-value n the devance test can be calculated. In general, f the P-value n the devance test s larger than 5%, the model appears to be an adequate ft to the emprcal data (4-6, 16). Through model dagnostcs, the expected

9 Lu number of Class I manlne freght-tran deralments s estmated as: exp( T ) M (5) Equaton (5) s mathematcally equvalent to exp( T ) M t (6) Defne Z M t (7) Where: Z = expected freght-tran deralment rate per mllon tran-mles n year From Equatons (6) and (7), the expected tran deralment rate at a specfc year s estmated as follows: Z exp( T ) (8) Based on Equaton (8), the annual reducton of deralment rate s: Z Z exp( T) exp( ( T 1)) 1 Z 1 exp( ( T 1)) exp( ) % (9) Where: = annual rate of change n tran deralment rate n year compared to the prevous year Equaton (9) ndcates that freght-tran deralment rate declned by an average of 5.8 percent annually from 2000 to If ths trend contnues n the future, tran deralment rates can be projected (Fgure 1).

10 Lu Freght-Tran Deralment Rate per Mllon Tran-Mles Year Projected Deralment Rate FIGURE 1 Emprcal (dot) versus estmated freght-tran deralment rates (lne), Class I manlnes, 2000 to 2012 (projected deralment rates from 2013 to 2017) The projected (expected) tran deralment rate n 2013 s per mllon tran-mles usng the regresson model based on the 2000 to 2012 trend, compared wth the observed (emprcal) deralment rate of based on the latest FRA tran safety data (6.7% dfference). Ths appears to ndcate a reasonable accuracy of the regresson model. Further analyss can be conducted to evaluate the uncertanty assocated wth statstcal predcton, and quantfy the confdence nterval of the projected accdent rate. The temporal change n the overall accdent rate s concevably a net result of changng accdent rate by accdent cause, whch s dscussed n the next sub-secton. 4.4 Accdent-Cause-Specfc Tran Deralment Rate The FRA specfes more than three hundred accdent causes accountng for a varety of crcumstances and condtons that may result n tran accdents (30). These causes are herarchcally organzed and classfed nto fve categores track, equpment, human factor, sgnal and mscellaneous (30). Wthn each of these major cause groups, the FRA organzes ndvdual cause codes nto subgroups of related causes such as roadbed, track geometry, etc. wthn the track group, and smlar subgroups wthn the other major cause groups (30). In ths paper, we used a varaton on the FRA subgroups developed by Arthur D. Lttle Inc. (ADL) n whch smlar cause codes were combned nto groups based on expert opnons (31). ADL s groupngs are smlar to the FRA s subgroups but are more fne-graned thereby allowng greater resoluton for certan causes. The ADL cause groups were used to analyze accdent-causespecfc deralment frequency and severty n prevous studes (e.g., 13, 23, 32, 33). Broken rals or welds (08T), track geometry defects (04T), bearng falures (10E) and tran handlng errors (09H) are leadng deralments causes on Class I manlnes (22, 23), so they were used as an example n ths paper to llustrate the methodology for analyzng accdent-cause-specfc

11 Lu deralment rate (Table 4). The methodology can be adapted to other accdent causes as well. TABLE 4 Selected Accdent Cause Group (32) ADL Cause Group FRA Cause Code Descrpton 08T (Broken Rals or Welds) T202 T203 T204 T207 T208 T210 T212 T218 T219 T220 T221 Broken Ral - Base Broken Ral - Weld (plant) Broken Ral - Weld (feld) Broken Ral - Detal fracture from shellng or head check Broken Ral - Engne burn fracture Broken Ral - Head and web separaton (outsde jont bar lmts) Broken Ral - Horzontal splt head Broken Ral - Pped ral Ral defect wth jont bar repar Broken Ral - Transverse/compound fssure Broken Ral - Vertcal splt head 04T (Track Geometry, Excludng Wde Gauge) T101 T102 T103 T104 T105 T106 T107 T108 T199 Cross level of track rregular (at jonts) Cross level of track rregular (not at jonts) Devaton from unform top of ral profle Dsturbed ballast secton Insuffcent ballast secton Superelevaton mproper, excessve, or nsuffcent Superelevaton runoff mproper Track algnment rregular (other than buckled/sunknk) Other track geometry defects (Provde detaled descrpton n narratve) 10E (Bearng Falures) E52C E53C Journal (plan) falure from overheatng Journal (roller bearng) falure from overheatng 09H (Tran Handlng, Excludng Brakes) H524 H599 Excessve horsepower Other causes relatng to tran handlng or makeup (n narratve) For each accdent cause group, a smlar exponental functon s developed usng the negatve bnomal regresson, as descrbed prevously:

12 Lu Z exp( T) c c c (10) Where: Z c = accdent-cause-specfc freght-tran deralment rate per mllon tran-mles n year c, β c = parameter coeffcents (dependent on accdent causes) T = year An ndvdual negatve bnomal regresson model was developed for each cause group. The devance test (20) shows that each regresson model s adequate (Table 5). TABLE 5 Parameter Estmates of Accdent-Cause-Specfc Freght-Tran Deralment Rate, Class I Manlnes, 2000 to 2012 Broken Rals or Welds Parameter Estmate Standard Error Wald Ch- Pr > ChSq Square α < <.0001 Devance = 0.28; df = 11; P > 0.5 Track Geometry Defects Parameter Estmate Standard Error Wald Ch- Pr > ChSq Square α <.0001 Devance = 0.85; df = 11; P > 0.5 Bearng Falures Parameter Estmate Standard Error Wald Ch- Pr > ChSq Square α < <.0001 Devance = 0.59; df = 11; P > 0.5 Tran Handlng Errors Parameter Estmate Standard Error Wald Ch- Pr > ChSq Square α < <.0001 Devance = 0.62; df = 11; P > 0.5 Based on Table 5, the followng models are used to estmate accdent-cause-specfc annual tran deralment rate on Class I manlnes:

13 Tran Deralments per Mllon Tran-Mles Lu Z, exp( T) ral Z, exp( T) geometry Z, exp( T) bearng Z, exp( T) handlng (11) (12) (13) (14) Where: Z, ral = expected broken-ral-caused tran deralment rate per mllon tran-mles Z, geometry = expected track-geometry-defect-caused tran deralment rate per mllon tran-mles Z, bearng = expected bearng-falure-caused tran deralment rate per mllon tran-mles Z,handlng = expected tran-handlng-error-caused tran deralment rate per mllon tran-mles The emprcal and estmated tran deralment rates for the four accdent causes were compared (Fgure 2) Broken Rals or Welds Track Geometry Defects (Excludng Wde Gauge) Bearng Falures 0.15 Tran Handlng Errors Year FIGURE 2 Freght-tran deralment rate by accdent cause, Class I manlnes, 2000 to 2012

14 Lu Broken rals had a hgher deralment rate than the other accdent causes, hghlghtng the mportance of broken ral preventon (14, 22, 23). The annual rate of change n accdent-causespecfc deralment rate s estmated as follows: Broken rals or welds: -7% Track geometry defects (excludng wde gauge): -7% Bearng falures: -11% Tran handlng errors (excludng brakng errors): -9% The analyss shows that bearng falures and tran handlng errors had a hgher percent reducton n annual deralment rate, compared to track geometry defects and broken rals wthn the same study perod. The temporal trends of accdent-cause-specfc deralment rates were compared (Fgure 3). Gven these trends, broken rals may contnue to be leadng deralment causes n the projected future. Bearng falures had a hgher deralment rate than tran handlng errors untl 2008, but the dfference became mnor recently, n part due to the faster declnng rate of bearng-falure-caused deralments (11% reducton per year) Tran Deralment Rate per Mllon Tran-Mles Bearng Falures Tran Handlng Errors Broken Rals or Welds Track Geometry Defects Projected Rates Year FIGURE 3 Temporal trend of tran deralment rate by accdent cause (projected deralment rates from 2013 to 2017)

15 Lu DISCUSSION In ths secton, mplcatons of ths research are dscussed wth respect to ral transportaton safety and rsk analyss. 5.1 Temporal Change n Ral Safety Ths research fnds that the overall freght-tran deralment rate on U.S. Class I ralroad manlnes declned by 5.8% annually from 2000 to Ths change may n part be due to contnued nvestment n nfrastructure and rollng stock, safety culture, operatons, tranng and educaton, research and other safety ntatves. Addtonally, ths analyss fnds that annual deralment rates are statstcally dentcal among the largest four freght ralroads. However, the change of deralment rate could vary by accdent cause. The top two tran deralment causes (broken rals and track geometry defects) had smlar declnng rates (approxmately 7% annual reducton), whereas bearng-falure-caused tran deralment rate had a more sgnfcant declne (11% annual reducton). 5.2 Implcatons for Transportaton Rsk Analyss Many tran safety and rsk analyses were based on the average accdent rate nformaton wthn a mult-year study perod. In vew of declnng accdent rate, usng the average accdent rate may fal to represent up-to-date ral operatonal safety. An adjustment factor may be needed to estmate the most recent accdent rate based on hstorcal trends, when no better nformaton s avalable. In the long run, any ral safety rsk analyss should be perodcally revsted and possbly revsed to reflect changes n accdent rate and other rsk factors. 6 ONGOING RESEARCH 6.1 Casual Analyss of Tran Deralment Rate The ntent of ths research s exploratory rather than explanatory. Put another way, ths work s focused on dentfyng what the temporal trend s, nstead of explanng why t s. The causal relatonshp between tran accdent rate and affectng factors calls for future research to better understand the causal factors of ral safety and how changng these factors may affect safety (25). 6.2 Tran Deralment Severty Analyss Ths paper focuses on tran deralment rate (lkelhood). In addton, tran deralment severty (e.g., number of cars deraled, property damage, casualtes) s also crtcal n ralroad safety and rsk analyss (22, 23). Tran deralment severty may vary by accdent cause, accdent speed, tran length and other factors (10). The next step of ths work s to ncorporate tran deralment severty nto a larger ral safety management framework. 6.3 Crude Ol Transportaton Rsk Analyss Ths paper ncludes all types of tran accdents. Of recent nterest s crude ol tran accdent rate. The negatve bnomal regresson model descrbed n ths paper can be used to model the temporal varaton n crude ol tran accdent rate, and thus evaluate the safety trend before and after the mplementaton of certan safety mprovement strateges.

16 Lu CONCLUSION A statstcal methodology s developed to model the temporal trend of freght-tran deralment rates on U.S. Class I manlnes from 2000 to Wthn the study perod, the analyss shows that the natonal freght-tran deralment rate decreased by 5.8% per year, and there s no sgnfcant dfference among the four largest U.S. Class I freght ralroads n annual deralment rates. Broken rals or welds were the leadng deralment cause, and ts deralment rate declned by 7% per year. Track geometry defects, bearng falures and tran handlng errors all had declnng tran deralment rates, among whch the deralment rate reducton due to bearng falures was more substantal, at a 10.5% reducton per year. In 2017, the projected overall tran deralment rate s below 0.4 per mllon tran-mles (a 64% reducton compared to year 2000) f the current safety trend contnues. The tme-varyng accdent rate should be better understood and taken nto account n tran safety and rsk analyses and decson makng. ACKNOWLEDGEMENT I am very grateful to four anonymous revewers for ther constructve comments that have sgnfcantly mproved the qualty of ths paper. I also thank Mr. Bryan Schlake, lecturer of Ral Transportaton Engneerng at Pennsylvana State Unversty at Altoona, for hs careful edtoral revew. However, I am solely responsble for all the analyses and results presented heren. REFERENCES 1. Nayak, P.R., D.B. Rosenfeld, and J.H. Hagopan. Event Probabltes and Impact Zones for Hazardous Materals Accdents on Ralroads. Report DOT/FRA/ORD-83/20. FRA, U.S. Department of Transportaton, Washngton D.C., Trechel, T.T. and C.P.L. Barkan. Workng Paper on Manlne Freght Tran Accdent Rates. Research and Test Department, Assocaton of Amercan Ralroads, Washngton D.C., Anderson, R.T. and C.P.L. Barkan. Ralroad Accdent Rates for Use n Transportaton Rsk Analyss. Transportaton Research Record, No. 1863, 2004, pp Evans, A.W. Ral Safety and Ral Prvatzaton n Great Brtan. Accdent Analyss and Preventon, Vol. 39, No. 3, 2007, pp Evans, A.W. Ral Safety and Ral Prvatzaton n Japan. Accdent Analyss and Preventon, Vol. 42, No. 4, 2010, pp Evans, A.W. Fatal Tran Accdents on Europe s Ralways: Accdent Analyss and Preventon, Vol. 43, No. 1, 2011, pp Slla, A. and V.P. Kallberg. The Development of Ralway Safety n Fnland. Accdent Analyss and Preventon, Vol. 45, 2012, pp Hauer, E. Observatonal Before-After Studes n Road Safety. Pergamon Press, Elsever Scence Ltd., Oxford, England, Bagher, M., F.F. Saccomanno, S. Chenour, and L.P. Fu. Reducng the Threat of In- Transt Deralments Involvng Dangerous Goods through Effectve Placement Along the Tran Consst. Accdent Analyss and Preventon, Vol. 43, 2011, pp Lu, X., M.R. Saat, and C.P.L. Barkan. Analyss of U.S. Freght-Tran Deralment Severty Usng Zero-Truncated Negatve Bnomal Regresson and Quantle Regresson. Accdent Analyss and Preventon, Vol. 59, 2013, pp

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