Statistical issues in traffic accident modeling

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1 See dscussons, stats, and author profles for ths publcaton at: Statstcal ssues n traffc accdent modelng Artcle January 003 CITATIONS 8 READS 13 authors, ncludng: Tarek Sayed Unversty of Brtsh Columba - Vancouver 54 PUBLICATIONS 3,833 CITATIONS SEE PROFILE Some of the authors of ths publcaton are also workng on these related projects: Investgaton of mcroscopc pedestran walkng behavor Vew project All content followng ths page was uploaded by Tarek Sayed on 17 March 015. The user has requested enhancement of the downloaded fle.

2 Statstcal Issues n Traffc Accdent Modelng Zad Sawalha Research Assocate Department of Cvl Engneerng Unversty of Brtsh Columba Vancouver, B.C., Canada V6T 1Z4 Tarek Sayed Assocate Professor Department of Cvl Engneerng The Unversty of Brtsh Columba 34 Man Mall Vancouver, BC, Canada V6T 1Z4 Tel: (604) Fax: (604) E-mal: tsayed@cvl.ubc.ca TRB 003 Annual Meetng CD-ROM Paper revsed from orgnal submttal.

3 ABSTRACT Accdent predcton models are nvaluable tools that have many applcatons n road safety analyss. However, there are certan statstcal ssues related to accdent modelng that ether deserve further attenton or have not been dealt wth adequately n the road safety lterature. Ths paper dscusses and llustrates how to deal wth two statstcal ssues related to modelng accdents usng Posson and negatve bnomal regresson. The frst ssue s that of model buldng or decdng whch explanatory varables to nclude n an accdent predcton model. The study dfferentates between applcatons for whch t s advsable to avod model over-fttng and other applcatons for whch t s desrable to ft the model to the data as closely as possble. It then suggests procedures for developng parsmonous models,.e. models that are not overftted, and best-ft models. The second ssue dscussed n the paper s that of outler analyss. The study suggests a procedure for the dentfcaton and excluson of extremely nfluental outlers from the development of Posson and negatve bnomal regresson models. The procedures suggested for model buldng and conductng outler analyss are more straghtforward to apply n the case of Posson regresson models due to an added complexty presented by the shape parameter of the negatve bnomal dstrbuton. The paper, therefore, presents flowcharts detalng the applcaton of the procedures when modelng s carred out usng negatve bnomal regresson. The descrbed procedures are then appled n the development of negatve bnomal accdent predcton models for the urban arterals of the ctes of Vancouver and Rchmond located n the provnce of Brtsh Columba, Canada. TRB 003 Annual Meetng CD-ROM Paper revsed from orgnal submttal.

4 1. INTRODUCTION Statstcal modelng s used to develop accdent predcton models (APMs) relatng accdent occurrence on varous road facltes to the traffc and geometrc characterstcs of these facltes. These models have several applcatons such as estmaton of the safety potental of road enttes, dentfcaton and rankng of hazardous or accdent-prone locatons, evaluaton of the effectveness of safety mprovement measures, and safety plannng. Currently, generalzed lnear regresson modelng (GLM) s used almost exclusvely for the development of APMs snce several researchers (e.g. Jovans and Chang, 1986; Maou and Lum, 1993) have demonstrated that certan standard condtons under whch conventonal lnear regresson modelng s approprate (Normal error structure and constant error varance) are volated by traffc accdent data. Most safety researchers now adopt ether a Posson or a negatve bnomal error structure n the development of APMs. Several GLM statstcal software packages are avalable for the development of these models. Ths software allows the modelng of data that follow a wde range of probablty dstrbutons belongng to the exponental famly (among whch are the Posson and the negatve bnomal dstrbutons). The road safety lterature s rch wth APMs developed by Posson or negatve bnomal regresson. However, there are certan statstcal ssues related to accdent modelng that ether deserve further attenton or have not been dealt wth adequately n the road safety lterature. Ths paper dscusses and llustrates how to deal wth two statstcal ssues related to accdent modelng, namely the ssue of selectng the explanatory varables of an APM and the ssue of outler analyss. The procedures gven for dealng wth these ssues are then appled n the development of APMs for the urban arterals of the ctes of Vancouver and Rchmond located n the provnce of Brtsh Columba, Canada. TRB 003 Annual Meetng CD-ROM Paper revsed from orgnal submttal.

5 . DATA DESCRIPTION A total of 58 arterals n the ctes of Vancouver and Rchmond were nvestgated for the purpose of developng APMs relatng the safety of urban arteral sectons to ther traffc and geometrc characterstcs. Geometrc data were drectly collected from the feld. The approach to geometrc data collecton conssted of dvdng each arteral nto sectons between consecutve sgnalzed ntersectons and gatherng feld nformaton for each secton separately. The geometrc data collected conssted of: (a) secton length, (b) between-sgnal number of lanes, (c) number of unsgnalzed ntersectons, (d) number of drveways, (e) number of bus stops, (f) number of pedestran crosswalks, (g) type of medan (no medan, flush medan, or rased-curb medan), (h) type of land use (resdental, busness, or other), and () percentage of secton length along whch parkng s allowed. The data on accdent frequences and traffc volumes along the arterals were obtaned from the ctes of Vancouver and Rchmond and covered the perod from 1994 to Accdents that occurred at sgnalzed ntersectons were excluded from the accdent data used to develop the models. Summary statstcs descrbng the data are provded n Table MODEL DEVELOPMENT The development of the APMs for the urban arteral sectons of Vancouver and Rchmond was carred out usng the GLIM4 statstcal software package (Numercal Algorthms Group, 1994). The followng elements were necessary for model development: approprate model form, approprate error structure, procedure for selectng the model explanatory varables, procedure for outler analyss, and methods for assessng model goodness of ft. The procedures for TRB 003 Annual Meetng CD-ROM Paper revsed from orgnal submttal.

6 selectng the model explanatory varables and conductng outler analyss are the statstcal ssues targeted by ths paper and are dscussed n the followng sectons. 3.1 Model Form The mathematcal form used for any APM should satsfy two condtons. Frst, t must yeld logcal results. Ths means that a) t must not lead to the predcton of a negatve number of accdents and b) t must ensure a predcton of zero accdent frequency for zero values of the exposure varables, whch, for road sectons, are secton length and annual average daly traffc (AADT). The second condton that must be satsfed by the model form s that, n order to use generalzed lnear regresson n the modelng procedure, there must exst a known lnk functon that can lnearze ths form for the purpose of coeffcent estmaton. These condtons are satsfed by a model form that conssts of the product of powers of the exposure measures multpled by an exponental ncorporatng the remanng explanatory varables. Such a model form can be lnearzed by the logarthm lnk functon. Expressed mathematcally, the model form that was used s as follows: ˆ a1 a E( Y ) = a0 L V exp b x (1) j j j where E ˆ( Y ) = predcted accdent frequency, L = secton length, V = secton AADT, x j = any varable addtonal to L and V, and 0, a1, a bj = the model parameters. a, TRB 003 Annual Meetng CD-ROM Paper revsed from orgnal submttal.

7 3. Error Structure As mentoned earler, the GLM approach to modelng traffc accdent occurrence assumes an error structure that s Posson or negatve bnomal. Let Y be the random varable that represents the accdent frequency at a gven locaton durng a specfc tme perod, and let y be a certan realzaton of Y. The mean of Y, denoted by Λ, s tself a random varable (Kulmala, 1995). For Λ = λ, Y s Posson dstrbuted wth parameterλ : y λ λ e P( Y = y Λ = λ) = ; E( Y Λ = λ) = λ; Var( Y y! Λ = λ) = λ (a,b,c) It s the usual practce to assume that the dstrbuton of Λ can be descrbed by a gamma probablty densty functon. Hauer (1997) examned many accdent data sets and the emprcal evdence he obtaned supported the gamma assumpton for the dstrbuton of Λ. If Λ s descrbed by a gamma dstrbuton wth shape parameter κ and scale parameter κ / µ, then ts densty functon s: κ κ 1 ( κ / µ ) λ ( κ / µ ) λ e µ f Λ ( λ) = ; E( Λ) = µ ; Var( Λ) = (3a,b,c) Γ( κ ) κ and the dstrbuton of Y around E (Λ) = µ s negatve bnomal (Hnde and Demetro, 1998; Hauer et al., 1988). Therefore, uncondtonally: κ Γ( κ + y) κ µ µ P( Y = y) = ; E( Y ) = µ ; Var( Y ) = µ + Γ( κ ) y! κ + µ κ + µ κ y (4a,b,c) As shown by equatons (4b,c), the varance of the accdent frequency s generally larger than ts expected value reflectng the fact that accdent data are generally over-dspersed. The only excepton s when κ, n whch case the dstrbuton of Λ s concentrated at a pont and the negatve bnomal dstrbuton becomes dentcal to the Posson dstrbuton. TRB 003 Annual Meetng CD-ROM Paper revsed from orgnal submttal.

8 The decson on whether to use a Posson or negatve bnomal error structure was based on the followng methodology. Frst, the model parameters are estmated based on a Posson error structure. Then, the dsperson parameter ( σ d ) s calculated as follows: Pearsonχ σ d = (5) n p where n s the number of observatons, p s the number of model parameters, and s defned as: n [ y Eˆ( Y )] Pearson χ Pearsonχ = (6) = 1 Var( Y ) where y s the observed number of accdents on secton, E ˆ( ) s the predcted accdent frequency for secton as obtaned from the APM, and Var Y ) s the varance of the accdent frequency for secton. The dsperson parameter, ( Y σ d, s noted by McCullagh and Nelder (1989) to be a useful statstc for assessng the amount of varaton n the observed data. If σ d turns out to be sgnfcantly greater than 1.0, then the data have greater dsperson than s explaned by the Posson dstrbuton, and a negatve bnomal regresson model s ftted to the data. 3.3 Assessment of Model Goodness of Ft Several statstcal measures can be used to assess the goodness of ft of GLM models. The two statstcal measures used are those cted by McCullagh and Nelder (1989) for assessng model goodness of ft. These are a) the Pearsonχ statstc, defned n equaton (6), and b) the scaled devance. The scaled devance s the lkelhood rato test statstc measurng twce the dfference between the maxmzed log-lkelhoods of the studed model and the full or saturated model. The full model has as many parameters as there are observatons so that the model fts the data perfectly. Therefore, the full model, whch possesses the maxmum log-lkelhood achevable TRB 003 Annual Meetng CD-ROM Paper revsed from orgnal submttal.

9 under the gven data, provdes a baselne for assessng the goodness of ft of an ntermedate model wth p parameters. McCullagh and Nelder (1989) have shown that f the error structure s Posson dstrbuted, then the scaled devance s as follows: n y SD = y ln (7) = 1 Eˆ( Y ) whle f the error structure follows the negatve bnomal dstrbuton, the scaled devance s: n y + = y κ SD = y + ln ( y κ )ln (8) 1 Eˆ( Y ) Eˆ( Y ) + κ Both the scaled devance and the Pearsonχ have χ dstrbutons for Normal theory lnear models, but they are asymptotcally χ dstrbuted wth n p degrees of freedom for other dstrbutons of the exponental famly. 4. SELECTION OF MODEL EXPLANATORY VARIABLES There seems to be a belef among many safety researchers that the more varables n an APM the better the model. Some researchers have even reported models contanng varables wth hghly nsgnfcant parameters based on the belef that such varables would stll mprove model predcton. Such varables are hardly of any value for explanng the varablty of the specfc accdent data used n generatng the model much less of any value for predctng accdent frequences at new locatons not used n the model development. Explanatory varables that have statstcally sgnfcant model parameters, on the other hand, contrbute to the explanaton of the varablty of the accdent data, and ther ncluson n the model therefore mproves ts ft to ths data. Nevertheless, mprovement of a model s ft to the accdent data s not enough justfcaton TRB 003 Annual Meetng CD-ROM Paper revsed from orgnal submttal.

10 for retanng a varable n the model. Ths paper presents a detaled analyss on how to select whch explanatory varables to nclude n an APM. The procedure that s suggested n ths paper for selectng the explanatory varables to nclude n an APM depends on the locatons whose safety s to be studed by the model. Safety researchers and road safety authortes are not nterested only n the safety study of the lmted number of locatons used for model buldng. They want to be able to study the safety of other locatons n the same regon that have traffc and geometrc characterstcs smlar to those of the locatons used to buld a model. Varous reasons could prevent many locatons from beng ncluded n the development of a model. For example, accdent data mght not be avalable for some of these locatons at the tme of model development. Another reason mght be that some of these locatons smply dd not exst at that tme. Therefore generalty s a characterstc that s requred n a model that wll be used for the safety study of new locatons not ncluded n ts development. Model generalty requres that a model be developed n accordance wth the prncple of parsmony, whch calls for explanng as much of the varablty of the data usng the least number of explanatory varables. The dea behnd the prncple of parsmony s to avod overfttng. It s temptng to nclude many varables n a model n an effort to make t ft the observed data as closely as desred. In fact, f a number of statstcally sgnfcant varables equal to the number of observatons can be found, a perfect ft to the data can be acheved by ncludng these varables n the model. However, the queston that must be asked s whether a model havng a perfect or extremely close ft to a partcular set of observatons can produce relable predctons when appled to a dfferent set of locatons. The answer s that such a model s unstable and may perform poorly when appled to a new sample drawn from the same populaton. For the safety study of new locatons, more s not better, and t may be worse. The procedure that s proposed n ths paper for buldng parsmonous models s explaned n the next secton. TRB 003 Annual Meetng CD-ROM Paper revsed from orgnal submttal.

11 On the other hand, f an APM s to be used for studyng the safety of the partcular set of locatons used to develop t, then a more accurate study would result by usng a model that fts the accdent data as closely as possble. Ths best-ft model s acheved by ncludng all the avalable statstcally sgnfcant explanatory varables. The procedure for buldng best-ft models s also explaned n the next secton. 4.1 Procedures for Buldng Parsmonous and Best-Ft Models The procedure that s suggested n ths paper for developng parsmonous APMs s a forward procedure by whch the varables are added to a model one by one. The decson on whether a varable should be retaned n the model s based on two crtera. The frst crteron s whether the t-rato of ts estmated parameter (equal to the parameter estmate dvded by ts standard error and equvalent to the Wald statstc) s sgnfcant at the 95 percent confdence level (or any other level selected by the model developer). The second crteron s whether the addton of the varable to the model causes a sgnfcant drop n the scaled devance at the 95 percent confdence level. The second crteron represents an analyss of devance procedure for comparng two nested models. Ths procedure s equvalent to carryng out a lkelhood rato test to determne whether the model contanng the addtonal varable sgnfcantly ncreases the lkelhood of the observed sample of accdent data. As mentoned earler, the scaled devance s asymptotcally χ dstrbuted wth n p degrees of freedom, and therefore, owng to the reproductve property of the χ dstrbuton, ths second crteron s met f the addton of the varable causes a drop n scaled devance exceedng χ 0.05,1 (equal to 3.84). TRB 003 Annual Meetng CD-ROM Paper revsed from orgnal submttal.

12 However, t s mportant to note at ths pont that the analyss of devance procedure for developng parsmonous models should be conducted n dfferent manners for Posson and negatve bnomal regresson models. In the case of Posson regresson models, the dfference n scaled devance between the model wth p + 1 varables and the model wth p varables s equal to the lkelhood rato test statstc, whch compares the maxmzed lkelhood functons of the sample of accdent data under the two models. Ths statstc s defned as follows: L p+ 1 LRTS = ln (9) L p where LRTS = the lkelhood rato test statstc L = the maxmum lkelhood of the accdent data under the model wth p + 1 varables p+1 L p = the maxmum lkelhood of the accdent data under the model wth p varables The lkelhood of the accdent data under a certan model s equal to the jont probablty of the accdent observatons. It s therefore a functon of the model parameters and the error structure assumed by the model. LRTS s a non-negatve statstc and the mnmum value t can have s zero n whch case the addtonal varable contrbutes nothng to ncrease the lkelhood of the sample of accdent data. LRTS s used to test the null hypothess H 0 : β = p+1 0, where β p+ 1 s the parameter of the addtonal varable. A statstcally sgnfcant LRTS leads to the rejecton of the null hypothess and the adopton of the model wth p + 1 varables. A statstcally nsgnfcant LRTS means falng to reject the null hypothess and concludng that the model wth p varables s perfectly satsfactory. Determnng the statstcal sgnfcance of LRTS requres knowledge of ts samplng dstrbuton. Atkn et al. (1989) state that f two nested models wth degrees of freedom df1 and df are correct, the samplng dstrbuton of ther LRTS s an asymptotc χ dstrbuton wth degrees of freedom equal to (df1-df). Therefore for the LRTS n equaton (10) to be statstcally TRB 003 Annual Meetng CD-ROM Paper revsed from orgnal submttal.

13 sgnfcant at the 95 percent confdence level t has to exceed χ 0.05,1 = The fact that, n the case of Posson regresson models, the dfference n scaled devance between the model wth p +1 varables and the model wth p varables s equal to ther LRTS means that ths dfference can be drectly used to assess the usefulness of the addtonal varable. A drop n scaled devance exceedng 3.84 can be taken as the bass for choosng the model wth p + 1 varables. In the case of negatve bnomal regresson, models wth dfferent varables have dfferent κ values. The scaled devances of two negatve bnomal regresson models, wth dfferent values of κ, cannot be drectly compared (Hnde, 1996). The reason behnd ths s that the dfference n scaled devance between the two models s not equal to ther lkelhood rato test statstc. In ths case, conductng a meanngful analyss of devance procedure for developng parsmonous models requres that the value of κ for the model wth p varables be mposed on the model wth p + 1varables. The drop n scaled devance s then compared wth χ 0.05,1 n order to assess the usefulness of the addtonal varable. The procedure for developng parsmonous negatve bnomal regresson models s llustrated n the next secton wth the end result beng a parsmonous APM for the urban arteral sectons of Vancouver and Rchmond. Now that the procedure for developng parsmonous APMs has been outlned, t s necessary to justfy ths procedure. In other words, t s necessary to state why the t-test alone s not suffcent for developng parsmonous Posson and negatve bnomal regresson models. It should be stated at the outset that the t-test and the lkelhood rato test, or analyss of devance, are completely equvalent for two nested Normal regresson lnear models dfferng by one varable. They can both be used to test the null hypothess H 0 : β 0. Ths s not the case for the rest of the GLM famly models. The followng explans why: p+1 = TRB 003 Annual Meetng CD-ROM Paper revsed from orgnal submttal.

14 1. The use of the t-dstrbuton when parameter estmates are compared wth ther standard errors s exact for the classcal Normal regresson lnear model, but s otherwse justfed only by asymptotc theory (Numercal Algorthms Group, 1994). No general results are known about the adequacy of ths approxmaton for all the other models covered by GLM, so the t- test must be regarded as only a general gude to the accuracy of the estmates.. Atkn et al. (1989) state that an asymptotc χ dstrbuton for the LRTS of two nested models s not used n the case of Normal regresson lnear models, snce exact results are avalable. The exact dstrbuton for the LRTS of two nested Normal regresson lnear models contanng p + g and p explanatory varables s the Fg, n p g dstrbuton wth g degrees of freedom n the numerator and n p g degrees of freedom n the denomnator. Therefore when g = 1, the dstrbuton of LRTS s F 1, n p 1 whch s equal to t n p 1. Ths means that, n the case of Normal regresson lnear models, the t-test and the lkelhood rato test are completely equvalent. For regresson models n whch the response varable s not normally dstrbuted, the value of t n p 1 may be substantally dfferent from the LRTS of two nested models dfferng by one varable. The procedure used n ths paper for developng APMs that ft the accdent data as closely as possble s relatvely straghtforward. It s also a forward procedure by whch varables are added to a model one by one. However, n ths case, the only condton for retanng a varable n the model s that the t-rato of ts estmated coeffcent s sgnfcant at the 95 percent confdence level. TRB 003 Annual Meetng CD-ROM Paper revsed from orgnal submttal.

15 4. Illustraton of the Procedure for Developng Parsmonous Negatve Bnomal APMs The development of parsmonous negatve bnomal APMs s best llustrated by the flowchart n Fgure 1. Ths flowchart shows the detaled sequence of steps necessary to develop such models accordng to the procedure outlned n the prevous secton. Followng s the defnton of the letter symbols used n the flowchart: j = an nteger varable that s used as an ndex for the explanatory varables. = an nteger varable that s used to refer to the model number. κ (j) = shape parameter of the negatve bnomal error structure under Reference Model(j). N = an nteger varable representng the number of explanatory varables, other than the exposure varables, that are under nvestgaton at any phase of the process. X() = explanatory varable. D() = drop n scaled devance resultng from the addton of varable X() to a model. S = a real varable used to store the value of the greatest drop n scaled devance durng a cycle of varable addtons to Reference Model(j). The flowchart n Fgure 1 shows that several models are run wth and wthout fxng the value of κ at varous phases of the model development process. The default method used by GLIM 4 to determne the vector of model parameters, β, and the shape parameter, κ, s the method of maxmum lkelhood. Ths method provdes maxmum lkelhood estmates of β and κ. These are the values of β and κ that maxmze the log-lkelhood functon l (β,κ ) gven by: n l (β,κ ) = ln Γ ( y = yˆ + Γ + κ κ ) ln ( κ ) ln( y +!) κ ln 1 y ln 1 (10) κ y 1 ˆ The log-lkelhood functon depends on β through the terms y ˆ = Eˆ( Y ), whch represent the model predctons or ftted values. The model form most commonly used n accdent modelng was gven n equaton (1), and t specfes the ftted values as ŷ = exp(x β) where x s the vector TRB 003 Annual Meetng CD-ROM Paper revsed from orgnal submttal.

16 of explanatory varables correspondng to the th observaton. The method of maxmum lkelhood uses the Newton-Raphson technque to obtan β and κ as the soluton to the equatons l / β = 0 and l / κ = 0. GLIM 4 also allows the user to pre-specfy the value of κ. The value of β that maxmzes the log-lkelhood functon s then calculated usng the Newton Raphson technque wth Fsher scorng. The Fsher scorng allows the calculaton of β through an teratvely re-weghted least squares (IRLS) procedure. The startng pont n the model development process s a basc model contanng the exposure varables only. The reason for ths start s that any accdent model should at least contan the exposure varables snce no accdents occur wthout exposure. The basc model acts as the frst reference model. The term reference model s used here to denote a model that serves as the base for generatng a new model contanng one more addtonal varable. Subsequently, secondary models are developed by addng the rest of the varables ndvdually to the basc model wth κ fxed to the value obtaned under that model. Snce κ s fxed at ths stage n the process, t s possble to compare the scaled devance of the model resultng from the addton of a certan varable to the scaled devance of the basc model. No further consderaton s gven to any varable whose parameter has an nsgnfcant t-rato or that causes an nsgnfcant drop n the scaled devance obtaned under the basc model. The drop n scaled devance caused by each of the other varables s noted and recorded. The next stage n the process conssts of developng a new reference model contanng the exposure varables and that explanatory varable whch caused the maxmum drop n scaled devance durng the prevous stage; ths model wll have a new κ value. The remanng varables that were not dscarded n the prevous stage are now ndvdually added to the second reference model wth κ fxed to the new value, and a procedure smlar to the one n the prevous stage s used to obtan the next reference model. The model buldng process contnues n ths manner usng the method of maxmum lkelhood to obtan the reference models and the fxed κ method TRB 003 Annual Meetng CD-ROM Paper revsed from orgnal submttal.

17 to obtan the secondary models. The process termnates when all the explanatory varables have been examned, and the fnal reference model wll be the parsmonous model sought. 4.3 Parsmonous and Best-Ft Models Developed The procedures that were descrbed for developng parsmonous and best-ft accdent predctons models were used to develop such models for the urban arteral sectons of Vancouver and Rchmond. These models are lsted n Table, and they are models for predctng total accdent frequency on a secton n a perod of 3 yrs. The symbols used n the table are: a) L = secton length n km, b) V = annual average daly traffc n vehcles per day, c) UNSD = unsgnalzed ntersecton densty n ntersecton per km, f) CROD = crosswalk densty n crosswalks per km, g) NL = between-sgnal number of lanes, h) IRES = ndcator varable for resdental land use (equal to 1 f land use s resdental; 0 otherwse), ) IBUS = ndcator varable for busness land use (equal to 1 f land use s for busness; 0 otherwse), j) IUND = ndcator varable for undvded cross secton (equal to 1 f secton has no medan; 0 otherwse). Both models have values of scaled devance and Pearson χ that are sgnfcant at the 95% confdence level ndcatng that the models have an acceptable ft to the data. The t-ratos of the parameter estmates of both models are also sgnfcant at the 95% confdence level. It should be noted that the parsmonous model contans fewer varables than the best-ft model that was developed usng only the sgnfcant t-rato crteron for retanng varables n a model. TRB 003 Annual Meetng CD-ROM Paper revsed from orgnal submttal.

18 5. Outler Analyss Most data sets contan a few unusual or extreme observatons called outlers. Outlers occur n a set of data ether because they are genunely dfferent from the rest of the data or because errors took place durng data collecton and recordng. Errors can be detected and corrected, but how should outlers that are genunely dfferent from the rest of the data be dealt wth? Ths paper proposes excludng outlers from model development only f they have extreme nfluence on the model equaton. Such excluson releves the model equaton of the senstvty to a very small percentage of the data and establshes more confdence n ts valdty. However, t should be emphaszed that excludng nfluental outlers from the development of an accdent model s not synonymous wth neglectng these outlers or removng them from the database. Rather, they should be nvestgated to determne what makes them dfferent and whether any nformaton can be extracted from them. Kulmala (1995) used the leverage statstc to dentfy outlers that should be excluded from the model development. In the Normal lnear regresson case, the leverage of data pont s a measure of the dstance of the p-dmensonal pont x from the centrod of the other varate ponts n the p-dmensonal desgn space, whch s the vector space spanned by the column vectors of an n p desgn or model matrx X. The desgn space wll hereafter be referred to as the X-space. Hgh-leverage data ponts are therefore outlers n the X-space, and they tend to pull the regresson hyper-surface towards them because they exst alone n ther regon of the X-space. The data pont leverage values are the dagonal elements of the n n hat matrx, whch s the matrx that multples the vector of observatons to yeld the vector of predcted values. For Normal lnear regresson models, the hat matrx s H = X(X X) -1 X, and the th dagonal element of H s h = x (X X) -1 x. h has a value between zero and unty. In the case of Normal lnear regresson models, a value of unty means that the value for response predcted by the model s TRB 003 Annual Meetng CD-ROM Paper revsed from orgnal submttal.

19 equal to the observed value (Jorgensen, 1993). Thus a large value of h s an ndcaton that the correspondng observaton may be nfluental n determnng the poston of the ftted model. The concept of leverage may be extended to other models of the GLM famly. For these other models, the hat matrx s based on the IRLS procedure for model fttng. Therefore, H = W 1/ X(X WX) -1 X W 1/ where W s the n n dagonal matrx of weghts: w = 1 dµ ( ) Var y dη (11) where µ = the mean of y and η = x β s the lnear predctor assocated wth y. In GLM, µ and η are related by a lnk functon whch s the log-lnk for Posson and negatve bnomal regresson models. Therefore, for these models, µ = exp( η ) and dµ dη = exp(x β). The th dagonal element of H n the case of non-normal GLM models s: h = w x (X WX) -1 x. It should be noted that n the case of non-normal GLM models a data pont at the extreme X- range wll not necessarly have hgh leverage f ts weght w s very small (McCullagh and Nelder, 1989). The sum of the leverage values h, namely trace(h), s equal to the number of model parameters p. The average value of the leverage s therefore equal to p/n, where n s the number of data ponts. Many sources (e.g. Hoagln and Welsch, 1978) consder that a hgh leverage s one that exceeds p/n and that data ponts wth leverage n excess of ths value should be subject to further consderaton. Hgh-leverage data ponts have potental for beng nfluental by vrtue of ther locaton n the X- space. However, hgh leverage alone s not suffcent to render a pont nfluental and warrant excludng t from the model development. Influence requres that a pont be extreme n ts value of the response varable n addton to beng an outler n the X-space. An approprate measure of nfluence s the Cook s dstance, whch s defned as follows: TRB 003 Annual Meetng CD-ROM Paper revsed from orgnal submttal.

20 c h p(1 h PS ' = ( r ) (1) ) where r PS ' s the studentzed resdual of data pont. In the case of Posson and negatve bnomal regresson models, the studentzed resdual s gven by: r PS ' y ˆ y = (13) (1 h ) Var( y ) Thus Cook s dstance s made up of a component that reflects how well or how poorly the model fts the th observaton y, namely ŷ, and a component that measures how far the data pont s from the rest of the ponts n the X-space, namely h. Data ponts havng a hgh Cook s dstance are nfluental ponts that play a sgnfcant role n determnng the model parameter estmates. Stuatons n whch a relatvely small percentage of the data has a sgnfcant mpact on the model may not be acceptable to model users. Generally, they are more content assumng that a regresson equaton s vald f t s not overly senstve to a few observatons. Unfortunately, there s no rule for how hgh Cook s dstance has to be to make a data pont extremely nfluental. Therefore, a procedure s needed by whch to decde whether a data pont s extremely nfluental wth a hgh degree of confdence. Ths paper proposes the followng procedure for dentfyng extremely nfluental outlers: 1. The data are sorted n descendng order accordng to the c values.. Startng wth the data pont havng the largest c value, data ponts are removed one by one, and the drop n scaled devance s observed after the removal of each pont. 3. Ponts causng a sgnfcant drop n scaled devance are extremely nfluental outlers. The same analyss of devance procedure used to develop parsmonous APMs can be used n the dentfcaton of extremely nfluental outlers. As mentoned before, the scaled devances of two negatve bnomal regresson models, wth dfferent values of κ, cannot be drectly compared. Performng outler analyss for a negatve bnomal regresson case requres that the value of κ TRB 003 Annual Meetng CD-ROM Paper revsed from orgnal submttal.

21 for the model wth n data ponts be mposed upon the model wth n 1 data ponts; then the dfference n scaled devance can be compared to χ α,1 n order to assess whether the removed data pont s an extremely nfluental outler. The followng secton llustrates the procedure that s proposed by ths paper to perform outler analyss for negatve bnomal regresson models. 5.1 Illustraton of the Procedure for the Identfcaton and Removal of Influental Negatve Bnomal Model Outlers The flowchart n Fgure gves a complete descrpton of the procedure that s proposed by ths paper for the dentfcaton and removal of extremely nfluental negatve bnomal model outlers. Followng s the defnton of the letter symbols used n the flowchart: n j = the total number of data ponts avalable. = an nteger varable that s used as an ndex for the data ponts. = an nteger varable that s used to refer to the model number. κ(j) = shape parameter of the negatve bnomal dstrbuton under Reference Model(j). D() = drop n scaled devance resultng from the removal of pont. The term reference model s used here to denote a model that serves as the base for generatng a new model usng one less data pont (the dentfed nfluental outler). The procedure outlned n Fgure was employed to dentfy and remove the extremely nfluental outlers of the parsmonous model n Table. It resulted n the removal of seven data ponts, and the fnal parsmonous APM for the urban arterals of Vancouver and Rchmond s lsted n Table 3. TRB 003 Annual Meetng CD-ROM Paper revsed from orgnal submttal.

22 6. SUMMARY Ths study presents a detaled dscusson of two statstcal ssues related to accdent modelng usng Posson and negatve bnomal regresson. These are the ssue of model buldng,.e. selectng from the avalable safety database whch explanatory varables to nclude n an accdent predcton model, and the ssue of the dentfcaton and removal of extremely nfluental model outlers. The study suggests procedures for model buldng and conductng outler analyss and presents enough justfcaton n defense of the valdty of these procedures. The procedures are straghtforward to apply n connecton wth Posson regresson models. However, n the case of negatve bnomal regresson models, there s an added complexty arsng from the fact that the scaled devances of two negatve bnomal regresson models wth dfferent values of the shape parameter, κ, cannot be drectly compared. The study therefore presents flowcharts detalng the applcaton of the procedures n connecton wth negatve bnomal regresson models. The procedures are then appled n the development of accdent predcton models for the urban arterals of the ctes of Vancouver and Rchmond n the provnce of Brtsh Columba, Canada. ACKNOWLEDGEMENTS Fnancal support for ths study was provded by the Natural Scences and Engneerng Research Councl of Canada and the Insurance Corporaton of Brtsh Columba. REFERENCES Atkn, M., Anderson, D., Francs, B., Hnde, J., Statstcal Modellng n GLIM. Oxford Unversty Press, New York. TRB 003 Annual Meetng CD-ROM Paper revsed from orgnal submttal.

23 Hauer, E., Ng, J.C.N., Lovell, J., Estmaton of safety at sgnalzed ntersectons. Transportaton Research Record 1185, Hauer, E., Observatonal before-after studes n road safety: estmatng the effect of hghway and traffc engneerng measures on road safety. Pergamon Press, Elsever Scence Ltd., Oxford, U.K. Hnde, J., Macros for fttng overdsperson models. Glm Newsletter 6, Hnde, J., Demetro, C.G.B., Overdsperson: models and estmaton. Computatonal Statstcs & Data Analyss 7, Hoagln, D.C., Welsch, R.E., The hat matrx n regresson and ANOVA. Am. Statst. 3 (1), 17-. Jorgensen, B., The Theory of Lnear Models. Chapman and Hall, New York. Jovans, P.P., Chang, H.L., Modelng the relatonshp of accdents to mles traveled. Transportaton Research Record 1068, Kulmala, R., Safety at rural three and four-arm junctons: Development and applcaton of accdent predcton models. VTT Publcatons, Espoo, Fnland. McCullagh, P., Nelder, J.A., Generalzed Lnear Models. Chapman and Hall, New York. Maou, S., Lum, H., Modelng vehcle accdent and hghway geometrc desgn relatonshps. Accdent Analyss & Preventon 5 (6), Numercal Algorthms Group, The GLIM System Release 4 Manual, The Royal Statstcal Socety, Oxford, UK. TRB 003 Annual Meetng CD-ROM Paper revsed from orgnal submttal.

24 Intalze = 1, j = 1, S = 0 Start wth the basc model contanng only the exposure varables. Ths s Reference Model(1). Add X() to Reference Model(j) wth k fxed to k(j) = + 1 No Is t-rato of X() sgnfcant? Yes N = 0? No Yes N = N - 1 Dscard X() No Is D() sgnfcant? Stop. Reference Model(j) s the fnal model. Yes j = j + 1 No Is D() > S Add the X() whose D() = S to Reference Model(j-1) wthout fxng k. Obtan Reference Model(j). N = N - 1 Yes S = D() = + 1 Yes N = 0? Yes = N? No No Stop. Reference Model(j) s the fnal model. Renumber remanng varables from 1 to N Reset to 1 and S to 0 Fgure 1 Flowchart Demonstratng the Process of Developng a Parsmonous Negatve Bnomal Regresson Model TRB 003 Annual Meetng CD-ROM Paper revsed from orgnal submttal.

25 Intalze = 1, j = 1. Develop a model usng all n data ponts. Ths s Reference Model(1). Obtan the Cook's dstance of each data pont as computer output for Reference Model(1). Arrange the data ponts n order of decreasng Cook's dstance. Number the ponts from 1 to n. Remove pont() and run a model wth k fxed to k(j). No Is D() sgnfcant? Stop. Reference Yes Model(j) s the fnal model. j = j + 1 Develop Reference Model(j) usng the remanng (n-) ponts wthout fxng k. = + 1 Fgure Flowchart Demonstratng the Procedure for Identfcaton and Removal of Negatve Bnomal Regresson Model Outlers TRB 003 Annual Meetng CD-ROM Paper revsed from orgnal submttal.

26 Table 1 Safety Data Base Summary General Statstcs Number of Street Sectons By Medan Type Vancouver Cty Rchmond Total Rased-Curb Medan Two Way Left Turn Lane (TWLTL) Undvded Cross Secton Total: Number of Street Sectons By Land Use Resdental Busness Other Total: Total Length By Medan Type Rased-Curb Medan km km km Two Way Left Turn Lane (TWLTL) 0 km km km Undvded Cross Secton km km km Total: km km km Accdent Data ( ) Cty Vancouver Rchmond Total Fataltes Injures Property Damage Only Accdents (PDO) Total: (PDO) Accdents as Percentage of Total Accdents Range of Database Geometrc and Traffc Data Geometrc Data % 61.1% 7.% Cty Vancouver Rchmond Number of through traffc lanes -6-4 Secton length, meters Number of drveways per km (two-way total) Number of unsgnalzed ntersectons per km (two-way total) Number of pedestran cross walks per km (two-way total) Number of bus stops per km (two-way total) Traffc Data Average daly traffc, vpd TRB 003 Annual Meetng CD-ROM Paper revsed from orgnal submttal.

27 Table Parsmonous and Best-Ft Accdent Predcton Models Parsmonous Model: Acc per 3 yrs = Scaled Devance L V df e UNSD CROD NL IRES Shape Parameter κ Pearson χ χ 0.05, Varable Constant L V UNSD CROD NL IRES Coeffcent t-rato Best-Ft Model: Acc per 3 yrs = L V e UNSD CROD NL BUSD IUND DRID. IBUS Scaled Devance df Shape Parameter κ Pearson χ χ 0.05, Varable Constant L V UNSD CROD NL BUSD IUND DRID.IBUS Coeffcent t-rato TRB 003 Annual Meetng CD-ROM Paper revsed from orgnal submttal.

28 Model: Table 3 Parsmonous Accdent Predcton Model After Outler Analyss Acc per 3 yrs = Scaled Devance L df 377 V e UNSD CROD NL IRES Shape Parameter κ Pearson χ χ 0.05, Varable Constant L V UNSD CROD NL IRES Coeffcent t-rato TRB 003 Annual Meetng CD-ROM Paper revsed from orgnal submttal. Vew publcaton stats