Unobserved Heterogeneity and the Statistical Analysis of Highway Accident Data

Transcription

1 Unobserved Heterogenety and the Statstcal Analyss of Hghway Accdent Data Fred L. Mannerng Professor of Cvl and Envronmental Engneerng Courtesy Department of Economcs Unversty of South Florda 4202 E. Fowler Avenue, ENC 3506 Tampa, FL Emal: Venky Shankar Professor of Cvl Engneerng 226C Sackett Buldng Pennsylvana State Unversty Unversty Park, PA Emal: Chandra R. Bhat Adnan Abou-Ayyash Centennal Professor n Transportaton Engneerng Department of Cvl, Archtectural and Envronmental Engneerng The Unversty of Texas at Austn 301 E. Dean Keeton St. Stop C1761 Austn, Texas Emal: bhat@mal.utexas.edu Aprl 2016

2 Abstract Hghway accdents are complex events that nvolve a varety of human responses to external stmul, as well as complex nteractons between the vehcle, roadway features/condton, traffc-related factors, and envronmental condtons. In addton, there are complextes nvolved n energy dsspaton (once an accdent has occurred) that relate to vehcle desgn, mpact angles, the physologcal characterstcs of nvolved humans, and other factors. Wth such a complex process, t s mpossble to have access to all of the data that could potentally determne the lkelhood of a hghway accdent or ts resultng njury severty. The absence of such mportant data can potentally present serous specfcaton problems for tradtonal statstcal analyses that can lead to based and nconsstent parameter estmates, erroneous nferences and erroneous accdent predctons. Ths paper presents a detaled dscusson of ths problem (typcally referred to as unobserved heterogenety) n the context of accdent data and analyss. Varous statstcal approaches avalable to address ths unobserved heterogenety are presented along wth ther strengths and weaknesses. The paper concludes wth a summary of the fundamental ssues and drectons for future methodologcal work that addresses unobserved heterogenety. Keywords: Hghway safety, unobserved heterogenety, accdent lkelhood, accdent severty, statstcal and econometrc methods; statstcal methods; accdent analyss

3 1. Introducton Accdents, and specfcally hghway-vehcle accdents, cost the lves of roughly one and a quarter mllon people worldwde every year. In addton, hghway-traffc njures are globally the leadng cause of death among people 15 to 29 years old wth over 300,000 deaths (World Health Organzaton, 2015). From a polcy and engneerng perspectve, perhaps the most challengng element of these numbers s ther persstence and the nablty of advanced vehcle safety features, advances n hghway desgn, and varous safety-countermeasure polces to drastcally lower these numbers. Wthout doubt, efforts to mprove hghway safety are complcated by the behavor of ndvdual vehcle operators whch can vary wdely across the populaton and can be nherently dffcult to predct and/or modfy. Ths s n contrast to other transportaton modes (such as ar and water transport) where fewer operators mean the human element can be more tghtly controlled through lcensng standards and other safety protocols. On hghways, ndvdual vehcle operators have a wde range of physcal and mental abltes, dfferent perceptons of rsk, dfferent reactons to external stmul, and ther operatng abltes may be further complcated by varyng degrees of self-nflcted mpared drvng (alcohol and drug consumpton). Engneerng a safe transportaton system wth ths level of behavoral varance s vrtually mpossble. Ths safety problem s one of the leadng factors n the current move toward autonomous (connected and automated) vehcles that can remove the human element, potentally leadng to huge advances n safety by makng safety largely a functon of engneered systems (hardware and software) where varance n performance, and ultmately safety, can be more tghtly controlled and predcted. But, even after the ntroducton of autonomous vehcles n manstream traffc (whch wll lkely take many years to acheve), whch unquestonably has the potental to substantally reduce varaton n human elements, there wll stll reman varatons n the effects of many other factors that nfluence the lkelhood and resultng njury severty of hghway accdents. For example, on any hghway n the world, one wll fnd consderable varaton n vehcle attrbutes ncludng mass, occupant protecton, safety features, vehcle accdent-energy dsspaton features, and so on. In addton, there are varatons n roadway characterstcs such as pavement frcton, proxmty and types of objects just off the roadway, medan desgn, guardral desgn, and other nfrastructure related elements. Fnally, there are varatons n envronmental condtons such as lghtng, temperature, and precptaton, all of whch wll affect both the lkelhood and resultng njury-severty of accdents. 1

4 The entre process s further complcated by the varance n ndvdual vehcle operators physologes and responses to vehcle characterstcs, roadway characterstcs and envronmental condtons. Exstng data bases, whch typcally extract data from polce accdent reports, local weather statons, and state hghway-asset-management databases, contan a wealth of nformaton, especally after an accdent has occurred, when njury-severty levels, safety-feature deployment, and many other factors are reported. However, these conventonal databases only cover a small fracton of the large number of elements that defne human behavor, vehcle and roadway characterstcs, traffc characterstcs, and envronmental condtons that determne the lkelhood of an accdent and ts resultng njury severty. Many other elements reman unobserved to the analyst. For example, weather and lghtng condtons change contnually over tme as do the drver reactons to these condtons. In conventonal databases, analysts wll not have access to these data. Once an accdent has occurred, the characterstcs of energy dsspaton through the vehcle structure and the resultng effect on ndvduals, whch may vary wdely based on whch of the vehcle safety features deployed as well as bone mass, overall health, physcal dmensons, and so on, wll be largely unknown to the analyst. 1 In lght of the nherent defcences of current data sources (and lkely defcences n future data sources), statstcal and econometrc methods have been developed to address ths ssue as unobserved heterogenety (varatons n the effect of varables across the sample populaton that are unknown to the analyst). The ntent of these heterogenety models s to allow analysts to make more accurate nferences by explctly accountng for observaton-specfc varatons n the effects of nfluental factors (whch we wll refer to n ths paper as unobserved heterogenety). Our paper begns wth a quck revew of the statstcal consequences of gnorng unobserved heterogenety n hghway accdent data (Secton 2). The paper then moves on to a presentaton and dscusson of varous statstcal/econometrc methods (heterogenety models) that have been appled n the accdent analyss lterature to date, ncludng random parameter models (Secton 3), latent class models (Secton 4), jont latent-class/random-parameters models (Secton 5), Markovswtchng models (Secton 6), unobserved heterogenety n multvarate models (Secton 7), and 1 New data sources, such as those from naturalstc drvng where many vehcle and human functons are montored contnuously, wll help provde addtonal nfluental data but wll stll not approach the detal of data needed to fully model the lkelhood and severty of accdents. 2

5 omtted varable and transferablty ssues relatng to unobserved heterogenety (Secton 8). The paper concludes wth a summary and nsghts for future work (Secton 9). 2. The Need to account for Unobserved Heterogenety The statstcal analyss of accdent data typcally addresses the lkelhood of an accdent and ts resultng njury severty (see Lord and Mannerng, 2010, Savolanen et al., 2011, and Mannerng and Bhat, 2014 for revews of studes that have addressed the lkelhood and severty of an accdent). The lkelhood of an accdent s often analyzed by consderng the number of observed accdents occurrng on a defned spatal entty over a specfed tme perod; for example, the number of accdents per month occurrng over a specfed hghway segment (of known dstance) or at a hghway ntersecton. Once an accdent s observed, the njury severtes of nvolved ndvduals are often modeled as dscrete outcomes (for example, no njury, possble njury, evdent njury, dsablng njury, fatalty). Wth commonly collected data, some of the many factors affectng the lkelhood of an accdent and the resultng njury severty are not lkely to be avalable to the analyst. These factors (whch consttute unobserved heterogenety) can ntroduce varaton n the mpact of the effect of observed varables on accdent lkelhood and njury severty. For example, consder gender as an observed human element that affects njury severty outcomes. Whle there are clearly physologcal dfferences between men and women (justfyng the use of an ndcator varable such as 1 for male and 0 otherwse), there s also great varaton across people of the same gender, ncludng dfferences n heght, weght, bone densty and other factors that are generally unavalable to the analyst (and are not controlled for, even f other observed varables are ncluded). As another example of unobserved varaton, consder the effect of the nature of an accdent on njury severty. Assume for now that all accdents are ether angle accdents or head-on accdents (the same dscusson extends n a straghtforward way to the more realstc case that consders addtonal types of accdents). As suggested by Castro et al. (2013), some angle accdents may lead to njury severtes of those nvolved that may be far more severe than head-on accdents, even f the majorty of angle accdents lead to a lesser degree of njury severty. More generally, the vehcle-tovehcle knematc nteractons relatng to vehcle speed dfferences, dfferences n vehcle sze, varatons n vehcle mpact locatons, varatons n structural ntegrty of the vehcles, and varatons 3

6 n angle of mpact all comprse a sgnfcant porton of heterogenety n collson-type effects. Such nteractons are mpossble to measure n a comprehensve manner. As a thrd example, consder the effect of an observed bnary roadway lghtng ndcator varable (one f roadway lghtng s present and zero otherwse). Unobserved factors are lkely to nfluence the mpact of ths ndcator due to varatons across roadway segments n lghtng type, the ambent lghtng from land uses nearby, as well as the lght-output and types of lghtng used. Recent studes have demonstrated such heterogeneous effects (Venkataraman et al., 2011; Venkataraman et al., 2013; Venkataraman et al., 2014). Table 1 provdes a descrpton of the potental heterogeneous effects of some other commonly avalable explanatory varables for modelng the lkelhood and njury severty of hghway accdents. If unobserved heterogenety s gnored, and the effects of observable varables s restrcted to be the same across all observatons, the model wll be msspecfed and the estmated parameters wll, n general, be based and neffcent, whch could n turn lead to erroneous nferences and predctons. As an example, consder traffc volume and ts effect on the lkelhood of an accdent. As dscussed n Table 1, there are compellng reasons to beleve that the effect of traffc volume on accdent occurrences would vary from one roadway entty (hghway segment or ntersecton) to the next as a result of unobserved tme-varyng envronmental characterstcs and unobserved varatons n drver responses to traffc and these condtons. However, f the analyst were to gnore the possblty of a heterogeneous effect of ths varable across roadway enttes, multple ncorrect conclusons could be drawn from the resultng bas n parameter estmate such as belevng that the effect of traffc volume on accdent lkelhood s non-lnear (that s, ncreases n traffc volumes at hgher levels of congeston do not ncrease accdent lkelhoods at the same rate as traffc-volume ncreases at lower levels of traffc congeston). However, wthout explctly accountng for unobserved heterogenety, t s mpossble to dscern whether the effect of traffc volume on accdent lkelhood s truly non-lnear or f t just appears to be non-lnear due to gnorng unobserved heterogenety (that s, the apparent non-lnearty s actually trackng unobserved heterogenety n the data and not true non-lneartes). 2 2 It should be mentoned here that models that can account for unobserved heterogenety can usually be compared statstcally wth those that do not (for example by usng a lkelhood rato test). It s also true that the use of an napproprate functonal form for the effect of a varable can be pcked up, ncorrectly, as unobserved heterogenety. So, f traffc volume actually were to have a non-lnear effect on accdent occurrence, and the analyst faled to capture ths non-lnearty, t can show up ncorrectly as unobserved heterogenety. In many safety applcatons, even after specfyng the approprate functonal form for the effects of exogenous varables, there wll very lkely reman 4

7 Of the varous approaches to account for unobserved heterogenety, perhaps the so-called random parameters approach has been the most wdely adopted. The dea wth a tradtonal random parameters approach s that the heterogenety from one data observaton to the next s accounted for by allowng potentally every estmated parameter n the model to vary across observatons accordng to an analyst-specfed contnuous dstrbuton (such as the normal dstrbuton used to llustrate the problems of gnorng unobserved heterogenety earler). The estmaton of a tradtonal random parameters model thus requres a parametrc assumpton (assumed dstrbuton for the varaton n parameters across observatons). Whle the ndvdual parameters estmated n the model can have dfferent dstrbutons, and a varety of dstrbutons can be tested to determne whch provdes the best overall statstcal ft, there are stll potental problems wth adoptng parametrc assumptons. For example, t may be dffcult for conventonal dstrbutons to track heterogenety n the populaton f there are groups of observatons wth smlar parameters, whch may result n a complex multmodal dstrbuton wth varyng skewness and kurtoss. Another popular approach for addressng heterogenety s to assume fnte mxtures (latent classes) where nstead of havng the heterogenety vary across ndvdual observatons, the estmaton approach seeks to dentfy groups of observatons wth homogeneous varable effects wthn each group. Ths approach s sem-parametrc because t does not mpose a parametrc assumpton on the dstrbuton of parameter heterogenety (the approach stll requres a parametrc model structure such as a negatve bnomal, logt, and so on). The dsadvantage of ths approach s that dentfyng the many groups that may exst n the data can be computatonally cumbersome 3 and the procedure makes the assumpton of parameter homogenety wthn the dentfed groups. A combnaton of the two above approaches has also been consdered n the lterature where the number of latent classes (mass ponts) are specfed and then the parameters are allowed to vary across observatons wthn each dentfed latent class. Ths combned approach allows a more sophstcated representaton of unobserved heterogenety because t can track varatons across groups of data and ndvdual observatons. There also exst temporal and spatal elements n accdent data that are often overlooked n accdent studes. That s, accdents are rare events and, to get a suffcent number of observatons, unobserved heterogenety effects of the varable. Proper specfcaton for the effects of observed explanatory varables and accountng for potental unobserved heterogenety are both needed for a correct model specfcaton. 3 In most applcatons, after specfyng more than 3 or 4 mass ponts (latent classes), the model can become extremely dffcult to estmate and convergence may become very dffcult. 5

8 they are often aggregated over tme (for example, accdents per month) and/or space (accdents over a length of roadway segment). Ths creates addtonal unobserved heterogenety ssues that may be tme or space dependent. Methods such as Markov-swtchng models have been used to address the unobserved heterogenety ssue over tme and more advanced correlaton structures have been used to lnk accdent observatons spatally. Table 2 presents categorzed common methodologcal approaches for addressng unobserved heterogenety wth regard to the lkelhood of an accdent, along wth a lst of research studes that have appled these approaches. Table 3 presents categorzed common methodologcal approaches for addressng unobserved heterogenety wth regard to an accdent s resultng njury severty, along wth a lst of research studes that have appled these approaches. 4 A bref presentaton of the more common methodologcal approaches presented n Tables 2 and 3 s provded n the followng sectons. 3. Random Parameters Formulatons 3.1. Random Effects versus Random Parameters Before proceedng to random parameters model formulatons, we frst clarfy termnology ssues related to random effects and random parameters. In many econometrc treatments of the subject, the entry way to random parameters models s to frst brng up panel data, dscuss the so-called fxedeffects and random-effects estmators, and then proceed to ntroduce random parameters models. However, whle the fxed-effects and random-effects models typcally necesstate panel data, ths s not the case wth random parameters models. In partcular, the fxed-effects and random effects approaches are two dfferent ways to ntroduce unobserved ndvdual-specfc heterogenety n the constant terms wth panel data. In a random-effects model, the unobserved ndvdual-specfc heterogenety s assumed to be completely unrelated to the explanatory-varable vector, whch s a rather strong assumpton. In a fxed-effects model, ths assumpton s relaxed, but the fxed effects model poses the ncdental parameters problem that renders the usual maxmum lkelhood estmator nconsstent because the number of observatons generated by the same entty (for example, accdents per some tme perod for the same roadway entty) s fxed and very few n number (see 4 It s mportant to menton here that the varous models lsted n these tables (to address unobserved heterogenety) often do not lend themselves to drect conventonal statstcal comparson. For example, random parameters and latent class approaches cannot be drectly compared wth a conventonal method such as a lkelhood rato test. Ths can often complcate the selecton of one approach over another. 6

9 Greene and Hensher, 2010, page 60 for a dscusson of ths ssue). In contrast to the fxed and random effects models, random parameters models can be estmated even wth cross-sectonal data as well as panel data. Wth panel data, one can allow random parameters not only n the response to explanatory varables (as n cross-sectonal data), but also ncorporate the typcal panel random effect. Other possbltes exst as well, such as the flexblty to estmate an ndvdual-specfc as well as an observaton-specfc random parameter vector on the explanatory varables (see, for example, Bhat and Sdharthan, 2011). In the rest of ths paper, we wll motvate much of the dscusson on random parameters from the standpont of a cross-sectonal notaton set-up, though the concepts are readly extendble to panel data. 3.2 Random Parameters Accdent Lkelhood Models The lkelhood of an accdent can be studed usng a number of statstcal technques ncludng tradtonal count-data models, zero-nflated count data models (whch consder the possblty of a two-state process, one a near safe zero-accdent state and the other a normal count state wth nonnegatve ntegers), duraton models (reframng observed accdent counts nto the tme between accdents occurrng on a specfed roadway segment), generalzed count models (through reframng count data as orgnatng from a generalzed ordered model set-up), or tobt regresson models (arrvng at a censored contnuous varable by convertng accdent counts nto accdent rates by dvdng observed accdent counts over some tme perod by the traffc over that tme perod tme the length of roadway beng consdered). The applcaton of tradtonal count-data nvolves determnng the number of accdents that occur over some predetermned space (a roadway entty such as an ntersecton or a segment of specfed length) and tme (such as a month or a year). 5 Ths results n a non-negatve nteger that s well suted to tradtonal count-data models. The most popular count-data approach s based on Posson regresson or ts dervatves whch nclude the negatve bnomal and zero-nflated models (see Washngton et al., 2011). For the basc Posson model, the probablty P(n ) of road entty (for example, and ntersecton or hghway segment) havng n accdents s, n e Prob ( n ), (1) n! 5 Other methods consder the tme between accdents nstead of counts over some pre-specfed tme perod (Mannerng, 1993). 7

10 where s the Posson parameter for hghway entty. The Posson regresson specfes the Posson parameter (whch s also the expected number of accdents for entty ) as a functon of explanatory varables by typcally usng a log-lnear functon, where exp( bx ), (2) x s a vector of explanatory varables (now ncludng a constant) and b s a vector of estmable parameters (Washngton et al., 2011). Dependng on the data, a Posson model may not always be approprate because the Posson dstrbuton restrcts the mean and varance to be equal (E[n ] = VAR[n ]). If ths equalty does not hold, the data are sad to be underdspersed (E[n] > VAR[n]) or overdspersed (E[n] < VAR[n]), and the standard errors of the estmated parameter vector wll be ncorrect and ncorrect nferences could be drawn. To account for the possblty of overdsperson (whch s more commonly encountered n accdent count data), the negatve bnomal model s derved by rewrtng, where exp( ) exp( b x ), (3) s a Gamma-dstrbuted error term wth mean 1 and varance. 6,7 The addton of the exp( ) term allows the varance to dffer from the mean as VAR[n] = E[n][1+ E[n]] = E[n]+ E[n] 2. The negatve bnomal probablty densty s, 1/ 1/ (1/ ) n Pn ( ), (4) (1/ ) (1/ ) n! (1/ ) where Γ(.) s a gamma functon. The Posson regresson s a lmtng model of the negatve bnomal regresson as approaches zero. Thus, f s sgnfcantly dfferent from zero, the negatve bnomal s approprate and f t s not, the Posson model s approprate (Washngton et al., 2011). To account for unobserved heterogenety n response to the non-constant explanatory varables n count models, random parameters approaches have been developed and are avalable n n 6 Although uncommon, t s possble for the data to be underdspersed n whch case the negatve bnomal s not approprate and other models must be used (see Lord and Mannerng, 2010, for a full dscusson of ths pont as well as methodologcal alternatves). 7 Note that we are able to accommodate a random-effects type specfcaton n a cross-sectonal count data model because of the functonal form adopted for count models. Ths s easest seen n the reframng of a count model as a generalzed ordered-response model, where the λ term (whch ncludes the error term ε n the negatve bnomal model) appears n the threshold part, whle the orgnal error term leadng to the probablty expresson n any count model orgnates n the typcal latent regresson part (see Bhat, 2015). 8

11 standard software packages (see, for example, Greene, 2012). 8 To allow for such random parameters n count-data models, each estmable parameter on explanatory varable l n the vector x can be wrtten as, l bl, (5) l where s the parameter on the lth explanatory varable for observaton, l b s the mean parameter l estmate across all observatons for the lth explanatory varable, and s a randomly dstrbuted l scalar term that captures unobserved heterogenety across observatons, and the term can assume an analyst-specfed dstrbuton (such as the normal dstrbuton or others). Wth Equaton (5), the analyst can test for random parameters, usng a specfed dstrbuton, across all observatons for each ncluded explanatory varable (varous dstrbutons can be specfed to determne the best statstcal ft such as normal, lognormal, trangular, unform and Webull dstrbutons). If the varance of the chosen dstrbuton s not sgnfcantly dfferent from zero, t suggests that a conventonal fxed parameter (one parameter estmate for all observatons) s statstcally approprate. Thus the model s lkely to consst of a combnaton of fxed and random parameters across the ncluded explanatory varables. It s also mportant to note that random parameters models can be readly structured to account for heterogenety among analyst-specfed groups of observatons nstead of ndvdual observatons. 9 For example, nstead of estmatng separate parameter vectors for accdents on the ndvdual approaches to an ntersecton, a sngle parameter vector may be estmated for all approaches to a specfc ntersecton (see Wu et al., 2013). Ths s done smply by re-wrtng Equaton (5) as b group g l l gl, where gl s now the group-specfc random term that generates unobserved heterogenety across groups n response to the lth explanatory varable. These analyst specfed groups can account for forms of both spatal and temporal effects. The estmaton of random parameters models s typcally acheved wth maxmum smulated lkelhood (for more on ths technque, see Bhat, 2001, 2003; Tran, 2009). However, Bhat (2012) has more recently proposed a maxmum approxmate composte margnal lkelhood approach that 8 An alternatve to a random parameters approach n the negatve-bnomal case would be to allow the dsperson parameter α to vary as a functon of the mean, λ (see Cameron and Trved, 1986). However, ths would be more restrctve n terms of ts ablty to account for heterogenety across observatons. 9 Such groupng of observatons often forms the bass of what are commonly called multlevel models n the lterature. Multlevel model termnology smply refers to a modelng approach that parttons the data and potentally accounts for heterogenety wthn these parttons. 9

12 he shows to be much more computatonally effcent and even more accurate than tradtonal maxmum smulated lkelhood approaches for most random parameters models. Table 2 lsts random parameters count models that have been successfully appled n accdent studes. Ths basc random parameters formulaton can be readly extended to other accdent-lkelhood modelng methods such as zero-nflated count models (Shankar et al., 1997), duraton models (Mannerng, 1993), and tobt regressons (Anastasopoulos et al., 2012). A relatvely recent development n count models that facltates the ntroducton of unobserved heterogenety and many other generalzatons s the nsght that any count data model structure can be recast as a restrcted verson of a generalzed ordered-response model (see Castro el al, 2012, Bhat et al., 2014a,b). 3.3 Random Parameters Injury Severty Models Along smlar lnes to those above, njury severty models (whch seek to study the probablty of dscrete njury outcomes such as no njury, possble njury, evdent njury, dsablng njury and fatalty) can address unobserved heterogenety wth parameters that vary across observatons. A common example of such a model s the random parameters multnomal logt model (also referred to as the mxed logt model). To see ths model, defne a functon S k that determnes the probablty that accdent wll result n njury-severty level k as, Sk ~ βx ; β b β, (6) k k k where s a constant specfc to njury-severty level k (wth one of them set to zero for k dentfcaton), x s an ( L1) -column vector of exogenous attrbutes specfc to accdent and k njury-severty level k, β s an accdent-specfc ( L 1) -column vector of correspondng parameters that vares across accdents based on unobserved accdent-specfc attrbutes, and s k assumed to be an ndependently and dentcally dstrbuted (across njury severty levels and accdents) standard extreme-value error term. If β b, ths mples no accdent-specfc unobserved heterogenety, and the resultng model form s the standard multnomal logt (McFadden, 1981). However, f accdent-specfc unobserved heterogenety s allowed, and the β vector has a contnuous densty functon Prob f β β β, where φ s a vector of parameters characterzng the chosen densty functon (such as the locaton and scale). The resultng random 10

13 parameters multnomal logt njury-severty probabltes are (see Bhat, 1998, McFadden and Tran, 2000; Tran, 2009), where k βx k e P k f d m β β, (7) β xm e m P k s the probablty of njury severty k. As noted above wth count-data models, f the elements related to scale n the vector φ are determned to be sgnfcantly dfferent from zero, there wll be accdent-specfc varatons of the effect of one or more elements of the explanatory varable vector x on njury severty. As wth other random parameters models, maxmum smulated k lkelhood (MSL) s typcally used to estmate mxed logt models. 10 Bhat (2011) and Bhat and Sdharthan (2011) have shown how the maxmum approxmate composte margnal lkelhood (MACML) estmaton of a normally mxed multnomal probt model offers substantally more computatonal effcency as well as superor accuracy n recoverng parameters relatve to the maxmum smulated lkelhood (MSL). They demonstrate ths through the estmaton of a normally mxed multnomal logt model, and ths opens up a new drecton for estmatng random parameters multnomal models n the safety area. In addton to the random parameters multnomal model dscussed above, random parameters can be readly ntroduced n other models that have been hstorcally used to analyze accdent-njury severtes, ncludng ordered probablty models (models that account for the orderng of severty levels from lower to hgher njury levels). Further, n these ordered probablty models, unobserved heterogenety can be ntroduced n both the latent regresson as well as n the thresholds, as n Eluru et al. (2008). Savolanen et al. (2011), Castro et al. (2013) and Mannerng and Bhat (2014) are good sources of revew of ths lterature. 3.4 Random Parameters Models wth Correlated Parameters Almost all research n the accdent feld to date has assumed that the unobserved heterogenety captured by random parameters are ndependent. That s, there s no allowance for correlaton among 10 In ths case, logt probabltes shown n Equaton (11) are approxmated by drawng values of β from f(β φ) for gven values of φ. Research by Bhat (2000) and Bhat (2001) has shown that an effcent way of drawng to compute logt probabltes s to use a Halton sequence approach (for more on the Halton sequence, see Halton, 1960)). As wth count-data models, a varety of functonal forms can be consdered ncludng normal, lognormal, trangular, unform and Webull dstrbutons. 11

14 the dstrbuton of two or more random parameters n the model. In realty, there may be correlaton among random parameters. As an example, consder unobserved heterogenety caused by weather events and drvers heterogeneous responses to these events. In ths case, one mght expect the effect of precptaton to nfluence the lkelhood and severty of accdents dfferently across observatons as drvers respond dfferently, and one mght also expect the effect of pavement condton (coeffcent of frcton or roughness) to do the same. However, there s lkely a correlaton between these two sources of heterogenety due to the nteractve effects of precptaton and pavement condton. Accountng for correlaton among random parameters can be acheved, for example, by assumng a multvarate normal dstrbuton for where β b C, β and wrtng, β s a vector of random parameters correspondng to explanatory varables for observaton, b s the mean parameter estmate across all observatons, C s a lower trangular matrx that engenders correlaton among the elements of the parameter vector β, and s a randomly and ndependently dstrbuted uncorrelated vector term. Allowng for correlaton among random parameters can complcate the nterpretaton of results, but explctly consderng correlaton among random parameters can provde addtonal nsghts. 11 (8) 3.5 Random Parameters Models wth Means (and Varances) as Functons of Explanatory Varables As shown n Equatons (5) and (8), the most common applcaton of random parameters models s to assume that there s a sngle mean ( b l n Equaton (5) and b n Equaton (8)) across the populaton (but see later). Equaton (8) can be generalzed to allow for the possblty that the mean may vary from one observaton to the next as a functon of observed explanatory varables (we use Equaton (8) nstead of Equaton (5) to contnue to allow for the possblty of correlated random parameters). To allow the means of random parameters to vary as a functon of explanatory varables, Equaton (8) can be re-wrtten as, 11 Ths ssue s mportant n the case of multple random parameters where the parameters are not all necessarly normally dstrbuted. It must be noted that emprcally, t s rare to see a non-correlated model perform as well as a correlatedparameters random parameters model n safety applcatons. The correlated-parameters approach also has a hgh degree of senstvty to the sparse ndcator-varable problem (where ndcator varables wth low denstes are used n the model). However, ths ssue can be mtgated by omttng sparse ndcators n order to make estmaton and convergence feasble. 12

15 β b Θ z C, (9) ~ where z s a ( L 1) -vector of explanatory varables from observaton that nfluence the mean of ~ the random parameter vector, Θ s an ( L L) matrx of estmable parameters (each row of Θ corresponds to the loadngs of a specfc element of the β vector on the z vector; f a specfc column entry n a row of Θ s zero, t mples that there s no shft n the mean of the correspondng row element of the β vector due to the row element of the z vector correspondng to the column under consderaton). Note that such a specfcaton s equvalent to smply ncludng an approprate nteracton term wthn the systematc specfcaton of the model. For example, n the njury severty model of Equaton (6), substtutng Equaton (9) for β s equvalent to havng a random parameter vector wth a fxed mean on the varable vector x k along wth approprate nteractons of exogenous varables. In general, the analyst should always consder the varatons n the effect of a varable due to observed factors before consderng unobserved heterogenety. There have been several emprcal studes that have addressed ths ssue n the accdent lterature. For example, Km et al. (2013) found that, whle newer vehcles reduced njury severty probabltes n sngle-vehcle crashes, ths reducton was less for men than for women (they explan that ths could be because men drve more aggressvely). Ths s an example of the newer vehcle varable beng nteracted wth the gender of the drver to shft the mean of the effect of a newer vehcle (relatve to an older vehcle ) on njury severty. However, dong so does not nfluence the level of varaton n the amount of unobserved heterogenety tself. Ths can be noted from the fact that the unobserved heterogenety porton C n Equaton (9) remans unaffected when the mean s beng shfted. But, n the example above, t s possble that when women drve newer vehcles, there s less varaton (due to unobserved heterogenety) n the njury severty sustaned. In contrast, among men, ths varaton may be much hgher because of a larger range of varance n aggressveness. An approach to accommodate a shft n the varance (of unobserved heterogenety) n responsveness to newer vehcles across men and women s to wrte the standard devaton of the error terms n correspondng to the newer vehcle varable as a functon of gender Such a varance shft has seldom been pursued n the accdent lterature, though the concept has been appled n nonaccdent contexts (see, for example, Bhat, 1997a and Bhat and Zhao, 2002). 13

16 In addton to njury-severty analyss, heterogenety n the mean of a random parameter has also been explored n accdent-lkelhood contexts. For example, Venkataraman et al. (2014) explore a multtude of heterogeneous mean effects on the lkelhood of accdent occurrence. 4. Latent Class (Fnte Mxture) Models As dscussed n Secton 2 of ths paper, there are potental drawbacks of random parameters models n capturng unobserved heterogenety n that the analyst must assume a dstrbuton for the parameters across the populaton and the possblty that parameters may vary across unobserved groups of observatons nstead of across ndvdual observatons. The approach to latent class models s the same for models addressng the lkelhood of an accdent as well as ts resultng severty. As an example, consder a model where the probablty of belongng to a latent class s specfed by a multnomal logt model wth (Greene and Hensher, 2003), where P c P c γz e γ c z, (10) g e g s the probablty of observaton belongng to latent class c, z c s a vector of explanatory varables specfc to observaton and latent class c (ncludng a constant for all latent classes except one) and γ s a vector of estmable parameters. Wth Equaton (10), models of both the lkelhood of the accdent and ts resultng severty can readly be wrtten and estmated. For example, f an njury severty model s estmated as a multnomal logt model the condtonal severty model would be, where kc bx c k e P k c, (11) mc bx c m e m P k c s the probablty of an accdent njury-severty level k for accdent f accdent were a member of unobserved class c, s a constant specfc to njury severty level k for latent kc class c (wth set to zero for one of the alternatves n each class c for dentfcaton), kc x s as k defned n the context of Equaton (6), and b c s a class-specfc set of fxed parameters. The uncondtonal probablty for a specfc accdent resultng n njury severty k would then be, Pk Pc Pk c. (12) c 14

17 Estmaton of latent class models generally requre the analyst to specfy the number of classes (mass ponts) so, much lke explanatory varable selecton, the approprate number of classes needs to be determned as part of the model-estmaton process. As shown n Tables 2 and 3, latent class models have become an ncreasngly popular method of accountng for unobserved heterogenety the study of the lkelhood and severty of an accdent. 5. Latent Class Models wth Random Parameters wthn Classes Both latent class and random parameters models have ther drawbacks. For example, random parameters models requre dstrbutonal assumptons and may have dffculty trackng groups of observatons wth shared unobserved heterogenety. Latent class models may have dffculty n accountng for unobserved heterogenety wthn the dentfed latent classes. An approach that generalzes the latent class models to allow random parameters wthn each class can easly be envsoned. For example, n the case of njury severty, the multnomal logt model could readly be replaced wth the random parameters logt model so Equaton (11) becomes (wth Equatons (10) and (12) stll applyng as before), where kc βc xk e Pk c f cd mc c βc β β x c, (13) m e m s a class-specfc vector of moment parameters characterzng the chosen densty c functon. From an estmaton perspectve, allowng for the possblty of random parameters wthn each latent classes can serously complcate the estmaton process. In fact, due to the complexty of the estmaton process Bayesan methods are typcally used requrng a Markov Chan Monte Carlo (MCMC) algorthm wth samplng provsons for model dentfcaton (see Xong and Mannerng 2013) for an applcaton of ths jont latent class/random parameters approach to accdent njury severty). Buddhavarapu et al (2016) demonstrate a smlar applcaton to the crash lkelhood context accountng for spatal dependences of crash counts. 6. Temporal Heterogenety and Markov Swtchng Models Hghway accdents are relatvely rare events and thus an accumulaton over tme s often used n analyss. For example, accdent lkelhoods on a specfed segment of hghway may be modeled as count data n the form of observed accdents per week or month. Ths ntroduces the potental for temporal heterogenety where unobserved factors may vary from one tme perod to the next. Ths 15

18 unobserved temporal heterogenety could nclude factors such as weather-related factors that may not be observable to the analyst. Statstcally, the presence of tme-varyng unobserved heterogenety could lead to based parameter estmates and erroneous nferences when varablty over tme s present (Xong et al., 2014). One way of addressng ths temporal heterogenety s usng hdden-state Markov swtchng models whch can account for unobserved heterogenety across tme perods by assumng that the lkelhood of accdent occurrence and the njury severtes of observed accdents transton between two or more states over tme. Theoretcally, there are a number of reasons why multple hdden states could exst and manfest tself as temporal unobserved heterogenety, ncludng varatons n drvers responses to weather condtons (not necessarly observed by the analyst) that change over tme. The transton from one state to the next s often assumed not to depend on explanatory varables, although the transton probabltes could theoretcally be made to be some functon of observable varables. Recently appled Markov-swtchng models n accdent research (Malyshkna et al., 2009; Malyshkna and Mannerng, 2009; Malyshkna and Mannerng 2010; Xong et al., 2014) assume that temporal heterogenety follows a statonary multple-state Markov chan process. For example, f two hdden states are assumed to exst ( st 0 and st 1 ) the tme-dependent transton probabltes can be wrtten as, 1 0 and 0 1 P s s p, P s s p, (14) t1 t 01 t1 t 10 where P s s 1 1 t 0 s the condtonal probablty of s 1 1 at tme t+1 gven that the t observaton s n state 0 t s at tme t, P s s t 1 0 t 1 s the condtonal probablty of s 1 0 at t tme t+1 gven that the observaton s n state st 1 at tme t, and the transton probabltes p0 1 and p1 can be estmated from the accdent data. 13 Estmaton of Markov-swtchng models can be 0 complex, and typcally requres Bayesan Markov Chan Monte Carlo (MCMC) methods to sample the hdden states. However, the potental to track temporal unobserved heterogenety n data that are typcally vewed as cross-sectonal makes Markov-swtchng models a very powerful tool that can t 13 As mentoned n the text and emphaszed agan here, exstng applcatons of Markov swtchng models n accdent analyss have not consdered the transton probabltes as a functon of explanatory varables. Whle the modelng approach can be readly extended to allow transton probabltes to be a functon of explanatory varables, addtonal complextes n model estmaton and dentfcaton can be problematc. 16

19 yeld mportant new nsghts nto the lkelhood of accdents and ther resultng njury severtes. Markov swtchng models can also be combned wth other methods of heterogenety modelng to arrve at a more complete characterzaton of unobserved heterogenety. For example, Xong et al. (2014) estmate a Markov swtchng ordered probablty model for accdent njury severty wth random parameters across observatons. 7. Unobserved Heterogenety and Multvarate Models Multvarate models can be encountered when studyng the lkelhood of an accdent and/or ts resultng severty. Multvarate models can result from correlatons that emerge from a varety of sources. For example, n consderng the lkelhood of accdents resultng n dfferent njury-severty levels, one may speculate that the factors that affect the lkelhood of accdents resultng n severe njures are fundamentally dfferent than those that generate the lkelhood of accdents resultng n no njures. Ths may be due to how specfc roadway-desgn characterstcs nteract wth the lkelhoods of specfc njury-severty levels. If ths s the case, one may consder estmatng separate accdent lkelhood models (such as separate count-data models) for each dscrete severty outcome (such as no njury, possble njury, evdent njury, dsablng njury, fatalty). However, estmaton of separate models n ths case can be problematc because unobserved factors are lkely to mpact multple accdent counts, of dfferent severty levels, smultaneously for each roadway entty beng consdered (for example, counts by roadway segment or ntersecton). In addton, f accdent count data are collected on specfc roadway enttes over multple tme perods (for example months or years), unobserved factors wll result n a temporal correlaton n the number of accdents at the roadway entty over tme. Ths temporal dependency can be combned wth spatal dependences (correlaton n observed factors among spatally adjacent roadway enttes) to produce multvarate models of very large dmenson (see, for example, Narayanamoorthy et al., 2013 and Bhat et al., 2014). Wth regard to njury-severty data, multvarate ssues can also arse wth vehcle accdents that nvolve multple occupant njures from the same accdent. In such cases, the dfferent occupants may experence dfferent levels of njury severty, but the unobserved factors nfluencng these njury levels (such as energy dsspated durng the accdent, structural features of the vehcle(s) nvolved, and so on) would be correlated (see, for example, Abay et al., 2013, Eluru et al., 2010, Yasmn et al., 2014 and Russo et al., 2014). 17

20 Accountng for unobserved heterogenety (such as usng random parameters and potentally latent-class approaches) n a multvarate framework can complcate the error-term structure and estmaton process. Stll, as shown n Tables 2 and 3, a few studes have consdered random parameters n multvarate models. A partcularly appealng way to combne unobserved heterogenety effects wth a multvarate outcome context (wth the outcomes beng of dfferent types, ncludng contnuous, count, nomnal, ordered, and grouped outcomes) s based on dentfyng stochastc latent constructs (for example, unobserved drver-specfc psychologcal factors). These factors can be vewed as havng an nfluence on multple safety-related varables. Bhat proposes such a formulaton and refers to ths as a generalzed heterogeneous data model (GHDM). Recent applcatons of ths approach to the accdent lterature nclude Bhat and Dubey (2014) and Laver et al. (2016). The approach also provdes a convenent way to ncorporate varable endogenety n multvarate models wth unobserved heterogenety, offerng the opportunty to extend earler work n the feld such as Abay et al. (2013) and Palet et al. (2010). 8. Unobserved Heterogenety, Omtted Varables Bas and Transferablty A major concern n safety analyss (and other felds as well) s that detaled data relatng to the many factors that are lkely to affect the lkelhood and severty of an accdent are often not avalable to the analyst. In the absence of complete data the analyst may estmate models that obvously exclude mportant explanatory varables whch wll produce an omtted varables bas whch s lkely to result n based and nconsstent parameter estmates. Wth statstcal approaches that account for unobserved heterogenety, these omtted explanatory varables become part of the unobserved heterogenety. Whle random parameters, latent class, and other unobserved heterogenety approaches wll mtgate the adverse mpacts of omttng sgnfcant explanatory varables, the resultng model estmates wll not be able to track the unobserved heterogenety as well as when havng the sgnfcant omtted varables ncluded n the specfcaton. Thus leavng out mportant explanatory varables stll remans a problem even wth advanced approaches to account for unobserved heterogenety. A crtcsm often leveled aganst the estmaton of models that account for unobserved heterogenety, such as random parameters models, s that the results wll not be transferable to dfferent locatons snce the ndvdual parameter vector assocated wth each observaton s unque to that observaton. Ths s true, but fndng sgnfcant random parameters (parameters that produce 18

21 statstcally sgnfcant standard devatons for the analyst specfed dstrbutons) means that unobserved heterogenety s present on ndvdual observatons. If a fxed-parameters model s used for such data, the unobserved heterogenety does not smply dsappear. In fact, the fxed-parameters model wll be estmated wth a persstent bas and transferablty wll be problematc snce ths bas wll be a functon of unobserved heterogenety. Fndng sgnfcant random parameters suggests spatal transferablty problems regardless of the estmaton method used. 9. Summary and Conclusons Due to the complexty of hghway accdents (whch nvolve complex nteractons among human, vehcle, roadway, traffc and envronmental elements), t s mpossble to have access to all of the data that could potentally determne the lkelhood of a hghway accdent or ts resultng njury severty. Ths presents a problem wth the conventonal statstcal analyses of accdent data that can result n bas and neffcent model estmaton, and erroneous nferences and predctons. Ths n turn can lead to the mplementaton of neffectve and potentally counter-productve safety polces and countermeasures. As dscussed at length n the current paper, relatvely recent advances n statstcal and econometrc methods have allowed analysts to study conventonal and emergng accdent-data sources n new ways by addressng ssues relatng to unobserved heterogenety. Table 4 summarzes the unobserved heterogenety methods dscussed n the current paper, along wth a bref descrpton of ther strengths and weaknesses. As shown earler n Tables 2 and 3, a number of recent accdentanalyss research efforts have appled these methods to address unobserved heterogenety, thus allowng mportant new nsghts to be extracted from exstng accdent data. However, statstcal approaches that address unobserved heterogenety tend to be relatvely more complex from a model-estmaton perspectve, though the recent maxmum approxmate composte margnal lkelhood (MACML) approach proposed by Bhat (2011) should substantally allevate ths estmaton burden. Also, the varous models that address unobserved heterogenety are often not nested, so drect conventonal statstcal comparson between models s often not possble (for example, random parameters and latent class approaches cannot be drectly compared wth a conventonal method such as a lkelhood rato test). Ths often presents the analyst wth dffcult decsons that wegh model complexty and assocated computatonal ssues aganst the potental mprovements n statstcal ft. 19

22 As can be seen n Table 4, and from the dscussons n ths paper, no one approach to addressng unobserved heterogenety s necessarly clearly superor. In addton, any rgorous comparson between two or more approaches s lkely to be data-specfc because dfferent patterns of heterogenety are captured better by dfferent heterogenety modelng approaches, and these heterogenety patterns are lkely to vary from one data set to the next. There are substantal opportuntes for applyng exstng methods that address unobserved heterogenety as well as developng new methods that may be combnatons of random parameters, latent class, Markovswtchng, and possbly new approaches. Because complex approaches are needed to account for complex unobserved heterogenety, whch are often present n accdent data bases, contnung advances n estmaton technques and computatonal power wll be needed to contnue emprcal advances n addressng unobserved heterogenety n accdent data. Snce accdent data are composed of both tme varyng and tme nvarant heterogenety components, estmaton technques provdng nsghts nto the dstnctve effects of these components wll be requred. 20