ACCOUNTING FOR SERIAL CORRELATION IN COUNT MODELS OF TRAFFIC SAFETY

Transcription

1 Journal of the Eastern Asa Socety for Transportaton Studes, Vol. 6, pp , 2005 ACCOUNTING FOR SERIAL CORRELATION IN COUNT MODELS OF TRAFFIC SAFETY Sttpan SITTIKARIYA PhD Canddate Department of Cvl and Envronmental Engneerng Pennsylvana State Unversty 104 Transportaton Research Buldng, Unversty Park, PA USA Fax: Emal: Mng-Bang SHYU PhD Canddate Department of Cvl and Envronmental Engneerng Pennsylvana State Unversty 104 Transportaton Research Buldng, Unversty Park, PA USA Fax: Emal: Venky N. SHANKAR, PhD, PE Assocate Professor Department of Cvl and Envronmental Engneerng Pennsylvana State Unversty 226C Sackett Buldng, Unversty Park, PA USA Fax: Emal: Songrt CHAYANAN, PhD DKS Assocates, Inc. 719 Second Avenue, Sute 1250 Seattle, WA 98104, USA Emal: Abstract: Ths research explores the mpact of nflated varances on statstcal sgnfcance and dentfcaton of varables correlated wth medan crossover accdent frequences. In roadway and accdent data that s comprsed of cross-sectonal panels, seral-correlaton effects across tme arse due to shared unobserved effects through random error terms. We use an emprcal technque to adjust for downward bas n standard errors n count models of traffc accdents, so that proper dentfcaton of varables s ensured. The dentfed model structures reflect the mpact of nflated sgnfcance values, and provde specfcatons that nclude truly sgnfcant geometrc, traffc and envronmental effects contrbutng to medan crossover accdents. It should be noted that the results from ths study are localzed and hence lmted to nferences from the medan crossover context n Washngton State n the Unted States. Further study s requred to ensure the transferablty of these fndngs to other contexts. Key Words: Medan crossover accdents, Correlated data, Negatve bnomal, Zero-nflated count models, Negatve multnomal, 1. INTRODUCTION There are some common modelng problems n estmaton of count models. Unobserved heterogenety s a common occurrence n accdent occurrence, leadng to the well-known overdsperson problem (Shankar et al., 1995). The negatve bnomal (NB) model s sutable for overdspersed accdent frequences, as show n several pror works (see for example Poch and Mannerng, 1996; Mlton and Mannerng, 1996). 3645

2 Journal of the Eastern Asa Socety for Transportaton Studes, Vol. 6, pp , 2005 Another common ssue that arses n accdent data contexts s the ssue of seral correlaton. Multple years of cross-sectonal data on hghway accdent occurrences are often avalable from publc domans, ncludng tme seres nformaton on traffc volumes, accdent counts and roadway geometrcs. In addton, weather nformaton s also avalable from natonal databases (see for example Shankar et al., 2004.) In the presence of seral correlaton, the effcency of parameter estmates comes nto queston. Smlar to the classcal lnear regresson model, one can expect parameter estmates to be neffcent n the presence of seral correlaton n count models (Greene, 2000.) A method to adjust for repeated observaton effects on parameter estmates s necessary to adjust for seral correlaton effects. One such method relates to the use of the negatve multnomal (NM) model (Ulfarsson and Shankar, 2003.) In that model, a jont lkelhood based on repeated observatons s constructed to modfy the tradtonal negatve bnomal lkelhood. As a result, parameter estmates reflect an adjustment n ther standard errors, wth much of the adjustment resultng from proxmate years. The second type of adjustment for seral correlaton nvolves the development of a random effects or fxed effects specfcaton, where segment-specfc correlaton s accounted for. In the traffc accdent context, the random effects approach has been shown to be reasonable (See for example, Shankar et al., 1998). The thrd common ssue that arses n traffc accdent contexts s one that pertans to partal observablty and the resultng problem of excess zero counts. For example, especally n events of potentally hgh severty such medan crossover accdents, the problem of excess zeros s a sgnfcant one. A hghway segment wth hstorcally excess zeros may appear to be a safe locaton, but there s no guarantee that no medan crossover accdents wll occur n the future. As nteractons between traffc volumes and geometrcs, and geometrcs and weather condtons exacerbate accdent proneness, one would expect that over tme, wth ncrease n traffc volumes and adverse weather condtons, non-zero count probabltes would be expected to rse. Lmted hstory of observaton may not adequately capture the accdent proneness of the hghway segment, as nteractons ncrease over tme. It s the combnaton of the second and thrd common problems, namely, seral correlaton and excess zeros, whch we address n ths paper. We choose the emprcal context of medan crossover accdent settngs on the hghway network n Washngton State n the USA. Overdsperson, the frst problem dscussed s also mplctly dscussed n ths study; although n ths partcular case, overdsperson arses prmarly through the occurrence of excess zeros, and not necessarly through hgh-valued counts that are best modeled by a negatve bnomal dstrbuton. The rest of ths paper s organzed as follows. Model prelmnares are presented frst, where a bref dscusson of the negatve multnomal s presented, along wth the preferred model of excess zeros, namely, the zero-nflated Posson model for medan crossover accdents. The emprcal settng s then dscussed, followed by a dscusson of model results, along wth adjustments of standard errors for the zero-nflated model, due to seral correlaton. Fnally, conclusons and recommendatons are offered. 2. MODELING PRELIMINARIES 2.1 The Negatve Multnomal Model for Seral Correlaton n Accdent Counts The negatve multnomal (NM) model s suggested as a proper estmator when seral correlatons are present n the dataset (Ulfarsson and Shankar, 2003). The dervaton of the 3646

3 Journal of the Eastern Asa Socety for Transportaton Studes, Vol. 6, pp , 2005 NM model begns wth the Posson probablty densty functon, whch s an expresson for the probablty of the frequency equalng a partcular count (Guo, 1996): P(Y t λt yt e λ t = yt ) = ; y t = 0,1,2,..., (1) y! t where Yt s the observed frequency of medan crossover accdents n secton at tme t, and λ t s the mean of the number of medan crossover accdents. The Posson model s estmated by defnng: where xt ln λ = x β, (2) t s a vector of secton- and tme-specfc explanatory varables; and β s a vector of coeffcents to be estmated. To account for the secton-specfc varaton, we proceed as s done for the NB model. A random error term s added to the expresson for the mean, t ln λ +, (3) t = xt β ε where ε s a secton-specfc (not observaton-specfc as n the NB model) random error term, exp( ε ) s assumed to be ndependently and dentcally dstrbuted gamma wth mean 1 and varance α = 1/ θ. The assumpton of mean 1 does not cause loss of generalty f (3) ncludes an ntercept term. The condtonal jont densty functon of all ndvdual counts for a partcular secton, gven that the ndvdual counts are dstrbuted by (1) and condtoned on ε, can now be wrtten as: t P( Y 1 = y 1,..., Yt = yt ε ) = P( Yt' = yt' ε ), (4) where t denotes the number of tme perods observed for secton. Ths assumes the accdent counts n dfferent sectons are ndependent. Ths s not unreasonable because these sectons are generally not next to each other and wll therefore only share mnmal unobserved effects. The uncondtonal jont densty functon for the negatve multnomal dstrbuton can now be derved by ntegratng (4) and by usng the assumed dstrbuton of exp( ε ) to gve: t' = 1 P( Y Γ( y + θ ) Γ( θ ) y!... y 1 t 1 t y Yt y 1 = 1,..., = t ) =..., (5) 1 t θ θ η! η + θ η + θ y η η + θ y x β where Γ s the gamma functon, ηt = e t, η = η ηt, and y = y yt. Recall that the varance of exp( ε ) s α = 1/ θ. The degenerate case, when each secton has only one observaton, (.e. there s no secton-specfc correlaton) yelds the negatve bnomal dstrbuton. The expected value, the varance and covarance for the NM model are: E( Yt t ) = η, Var( Yt ) = E( Yt )[1 + α E( Yt )], Cov( Y, ) t Y t ' = αηtηt'. (6) 3647

4 Journal of the Eastern Asa Socety for Transportaton Studes, Vol. 6, pp , 2005 A lkelhood functon s wrtten usng (5) to gve (7): L( β, θ ) = n = 1 θ 1 Γ( y + θ ) θ η η 1 t... Γ( θ ) η + θ η + θ η + θ y yt, (7) where n s the total number of sectons. The estmable coeffcents β and α = 1/ θ are estmated by maxmzng the lkelhood functon (7), (Green, 2003). In medan crossover accdent contexts, t s mportant to keep n mnd that there are the possbltes of near-crossover cases n the roadway secton. For example, as medan wdth ncreases, the probablty that a potental crossover wll occur decreases, but may be enhanced wth ncreasng volumes and nteractons wth adverse weather condtons over tme. A nearcrossover roadsde encroachment usually takes the form of a rollover, fxed object or other smlar types of roadsde accdents that do not result n the vehcle crossng the medan and enterng opposng traffc. The segment hence may be characterzed as operatng n multple states. For smplcty, we can consder two states, ncludng a zero crossover state and a nonzero crossover state. Dependng on how nteractons vary over tme, the segment may swtch to a non-zero state. In such a case, a process that adequately captures the probablty of the two states s necessary. The tradtonal count models, such as the Posson or negatve bnomal, only address the non-zero state, thus gnorng the potental shft of excess zeros nto non-zero counts. A plausble model presented below to address the excess zero problem s the zero-nflated Posson or the zero-nflated negatve bnomal model. 2.2 Zero Inflated Posson Model of Medan Crossover Accdents Let Z represents the zero-crash count state of the medan crossover accdent ste, and Y * denote the crash count state for that roadway secton. Nether Z nor Y * s observed, but only the observed medan crossover count Y s, such that Y=Z*Y *. Determnng the latent components can then be vewed as a mxng dstrbuton problem, wth Z beng modeled as a dchotomous outcome probablty and Y * modeled as a count probablty. In vehcular accdent contexts, such dstrbutons have been found to be approprate (Shankar et al., 1997). In partcular these studes have hghlghted the mportance of roadway desgn devatons as a motvator for partal observablty effects. The effect of such devatons has been found to, at the least, cause partal observablty, and n certan desgn stuatons, overdsperson as well. In the medan crossover accdent context, desgn devatons are a sgnfcant ssue. To hghlght ths ssue, a bref dscusson of medan barrer warrants s necessary. Medan barrer warrants n Washngton State, and for that matter n the entre Unted States, are based on two smplfyng factors, namely, average daly traffc (ADT) and medan wdth. No account of nteractons between traffc volumes and geometrcs, or weather and geometrcs s consdered n the decson to nstall a medan barrer. Hence, potental nteractons arsng due to the presence of devatons from preferred geometrc desgn are gnored, leadng to the potental for the excess zero medan crossover problem. To formally address the excess zero problem, let Y be the annual number of accdent counts reported for secton, and let be the probablty that secton wll exst n the zero-count p state over ts lfetme. Thus 1- p s the probablty that secton wll operate n the non-zero accdent state. It s to be noted that n the non-zero count state, the probablty of zero 3648

5 Journal of the Eastern Asa Socety for Transportaton Studes, Vol. 6, pp , 2005 accdent counts stll exsts. For our mmedate purposes, we assume that ths count state follows a Posson dstrbuton. Gven ths, and, = 0 λ Y wth probablty ( ) p + 1 p e, (8) λ k e λ Y = k wth probablty (1 p )( ), (9) k! In (8) and (9), the probablty of beng n the zero-accdent state p s formulated as a logstc p dstrbuton such that log = G γ and λ s defned by log( λ ) = H β, where G and H 1 p are covarate vectors, and γ and β are coeffcent vectors. The covarates that affect the mean λ of the Posson state may or may not be the same as the covarates that affect the zeroaccdent state probablty (.e., p ). Alternatvely, vectors G and H may be related to each other by a sngle parameter τ. Ths s essentally a parametrc constrant n the sense that the explanatory varables are forced to be the same n both the zero and count states. In such a case, a natural parameterzaton s log(p /1 p ) = τ H β. It s useful n the accdent context to begn wth unconstraned parameter vectors γ and β. Ths allows dfferent effects to be correlated wth the zero state and count state respectvely. For example, n the medan crossover accdent context, t may be useful to consder the mpact of medan wdths alone n the specfcaton of the zero-state probablty functon, whle ncludng desgn, traffc and envronmental nteractons as well n the count state functon. Equatons (8) and (9) combned provde the zero-nflated Posson (ZIP) model. We refer to a model of unconstraned parametrc vectors such as the one dscussed above s referred to as the ZIP-Full model. The maxmum lkelhood estmaton usng the gradent/lne search approach proposed by Green (1996) s performed to estmate the β and γ. A lkelhood functon s gven by (10): L(β, γ,) = n = 1 (1 p λ yt e λ ) y! + Z p, (10) where Z = 1 when Y = s observed to be zero and 0 otherwse. An alternate method to estmate ZIP parameters s to employ the expectatons maxmzaton technque. Greene (1996) argues that the method shown n (10) s relable. 2.3 Statstcal Valdaton of the ZIP Medan Crossover Model In statstcally valdatng the ZIP model, one has to dstngush the base count model (such as the Posson model for medan crossover counts) from the zero-nflated probablty model (such as the ZIP). The reasonng behnd ths test s that the Posson model does not adequately capture the entre zero densty, and that the ZIP structure s more relable. A statstcal test for ths has been proposed by Vuong n The Vuong test s a t-statstcbased test wth reasonable power n count-data applcatons (Green, 1994). The Vuong statstc (V-statstc) s computed as: 3649

6 Journal of the Eastern Asa Socety for Transportaton Studes, Vol. 6, pp , 2005 V = m N, (11) S m where m s the mean wth m = log [f 1 (.) /f 2 (.)], (wth f 1 (.) beng the densty functon of the ZIP dstrbuton and f 2 (.) s the densty functon of the parent-posson dstrbuton), and S m and N are the standard devaton and sample sze respectvely. The advantage of usng the Vuong test s that the entre dstrbuton s used for comparson of the means, as opposed to just the excess zero mass. A value greater than 1.96 (the 95 percent confdence level for the t- test) for the V-statstc favors the ZIP-Full model whle a value less than favors the parent-posson (values n between 1.96 and mean that the test s ndecsve). The ntutve reasonng behnd ths test s that f the processes are statstcally not dfferent, the mean rato of ther denstes should equal one. To carry out the test, both the parent and zeronflated dstrbutons need to be estmated and tested usng a t-statstc. Studes (Greene 1994) have shown that a Vuong statstc has reasonable power and hence s qute relable. Greene ndcates that a sgnfcant Vuong statstc n favor the ZIP model wll also favor the ZIP model over the negatve bnomal model, whch would otherwse be the default model for overdsperson from excess zeros. 2.4 The Relevance of the Negatve Multnomal to Medan Crossover Accdents It s mportant to keep n mnd that the estmated parameters from the ZIP-Full model would be effcent f seral correlaton dd not exst. Such s not the case n medan crossover contexts where multple years of observatons are avalable. In fact, as n the classcal lnear regresson model, one would expect standard errors to be downward based f seral correlaton s not accounted for. To correct the varances n a dual-state dstrbuton (such as the ZIP) wth temporal correlaton, the negatve multnomal model offers an ntutve approach. We showed that the negatve multnomal captures seral correlaton across years wthout basng the parameters of the negatve bnomal dstrbuton. Emprcally, ths characterstc s of some use n our current ZIP-Full context. We can benchmark standard errors from a negatve multnomal model aganst a naïve negatve bnomal model frst, when repeated observatons are the prmary source of seral correlaton. The pont of mportance to note here s that the negatve multnomal model adjusts the standard errors upward to account for seral correlaton. We call ths upward adjustment the loadng factor. In other words, the standard errors of the negatve bnomal model are factored up by loadng factors derved from the negatve multnomal model. As an example, consder the vectors of parameters estmated as β by a negatve bnomal model. We assume here that β has dmensonalty k. Consder the negatve multnomal model vector of parameters β ~ also of dmensonalty k, wth exactly the same set of regressors as those used for β. Then, the ~ loadng factor for β s a vector µ of wth dmensonalty k, where µ k = s.e.( βk )/s.e.(β k ). The loadng factor for any gven parameter β k vares dependng on the effect of seral correlaton n the varance-covarance matrx. The loadng factors are then used to adjust upward the standard errors obtaned for a naïve ZIP-Full model. The adjusted standard errors are then used to determne parameter sgnfcance under seral correlaton. To llustrate the development of loadng factors for standard errors emprcally, we present fndngs from a negatve bnomal/negatve multnomal analyss of medan crossover accdents n Washngton State. Pror to presentng these fndngs, we dscuss the emprcal settng used for the development of the negatve bnomal and negatve multnomal models. 3650

7 Journal of the Eastern Asa Socety for Transportaton Studes, Vol. 6, pp , EMPIRICAL SETTING OF MEDIAN CROSSOVER ACCIDENTS IN WASHINGTON STATE In ths dscusson, nformaton regardng the data, framework, and crtera for model analyss and evaluatons s presented. The Washngton State hghway system contans over 7,000 centerlne mles of state hghways. The annual records n the database developed by the Washngton State Department of Transportaton (WSDOT) contan almost every roadway and roadsde ncdent that occurs durng the year. The accdent records from ths database are the prmary source of data for the current medan crossover accdents study. The panel data n ths research conssts of fve years ( nclusve) of annual accdent counts for 275 roadway sectons n Washngton State. The panel s balanced, wth all sectons havng a full fve-year hstory. Ths panel represents all sectons (longer than 2,624 ft) wthout medan barrers on dvded state hghways. The reasons why only sectons longer than 2,624 ft are selected are that about 95% of shorter sectons on dvded hghways have barrers, and that the shorter sectons are more affected by access controls and ntersectons (Ulfarsson and Shankar, 2003). In addton to medan crossover accdent nformaton, other components of data extracted from the database ncluded roadway geometrcs, medan characterstcs and traffc volumes. The database dd not provde any weather nformaton requred for the study. The GIS program ArcVew 3.2 was used to match the roadway sectons to ther weather attrbutes stored n the hstorcal weather database provded by the Western Regonal Clmate Center. The mappng crtera nvolved lnkng the non-medan barrer roadway sectons to the nearest correspondng weather statons. Each weather staton provded clmate data ncludng daly, monthly and annual measurements of temperature, precptaton and snowfall ncludng snow depth, wth the records datng back to The selected roadway sectons and all weather statons n Washngton State are llustrated n fgure 1. Table 1 provdes descrptve statstcs of key varables n the medan crossover accdent dataset. * Blue (darker color) = The Selected Roadway Sectons, ** Green (lghter color) = The Weather Statons Fgure 1. Selected Roadway Sectons and Weather Statons n Washngton State 3651

8 Journal of the Eastern Asa Socety for Transportaton Studes, Vol. 6, pp , 2005 Table 1. Descrptve Statstcs of Key Medan Crossover Accdent Related Varables Varable Mean Std. Dev. Mn Max Number of crossover accdents n secton AADT 37,355 36,975 3, ,560 AADT per lane 7,445 5, ,690 Sngle truck percentage Double truck percentage Truck-tran percentage Percentage of AADT n the peak hour Speed lmt Maxmum medan shoulder wdth n feet Mnmum medan shoulder wdth n feet Percent medans narrower than 40 feet Percent medans between 40 feet and 50 feet n wdth Percent medans between 50 feet and 60 feet n wdth Percent medans wder than 60 feet n wdth Percent medans that are paved Length of the roadway secton n mles The number of nterchanges n secton The number of horzontal curves n secton The number of horzontal curves per mle Maxmum horzontal central angle n degrees Mnmum radus of horzontal curve n feet ,400 Average annual snow depth n nches * Average annual precptaton n nches Number of grade changes Average roadway wdth * Weather staton data for mountanous secton 3.1. Development of Loadng Factors for Standard Error Adjustment Table 2 presents a comparatve analyss of standard errors for sgnfcant parameters n the negatve bnomal and negatve multnomal models of medan crossover accdents. It s to be noted here that only sgnfcant effects appearng n the negatve multnomal model are used to develop the loadng factors. It may be that varables sgnfcant n the negatve bnomal may not be sgnfcant n the negatve multnomal due to nflaton n the standard error. As s noted n table 2, the loadng factor vares by parameter, from a mnmum adjustment of for the precptaton-horzontal curves nteracton varable to for the length varable for medans narrower than or equal to 40 feet. The standard error adjustment for the alpha parameter (overdsperson effect) s gnored n our dscusson, snce the alpha parameter does not appear n the ZIP model of medan crossover accdents. The ZIP model s dscussed next, consderng the loadng factors developed n table

9 Journal of the Eastern Asa Socety for Transportaton Studes, Vol. 6, pp , 2005 Table 2. Load Factors Developed from NB and NM Models for Adjustng Standard Errors n Zero-Inflated Posson Model of Medan Crossover Accdents Negatve Bnomal Negatve Multnomal Varable Model Model Load Estmated Standard. Estmated Standard Factor * Coeffcent Error Coeffcent Error Constant Per-lane AADT ndcator (1 f perlane AADT <= 5000 vehcles, 0 otherwse) Length of the roadway secton on medan wdths less than or equal to 40 feet Length of the roadway secton on medan wdths greater than or equal to 41 feet and less than or equal to 60 feet Length of the roadway secton on medan wdths greater than 60 feet The number of nterchanges n the secton Interacton 1 between average monthly precptaton ndcator and the number of horzontal curves per mle ndcator (1 f average monthly precptaton <= 1.5 nches and the number of horzontal curves per mle <= 0.5, 0 otherwse) Interacton 2 between average monthly precptaton ndcator and the number of horzontal curves per mle ndcator (1 f average monthly precptaton > 4.0 nches and the number of horzontal curves per mle > 0.5, 0 otherwse) α N.A. θ N.A. Restrcted log-lkelhood (All -1, , parameters = 0, α = 0) Log-lkelhood at convergence Adjusted ρ Number of observatons 1,375 *** * The load factor was the proporton of the S.E. of NM to the S.E. of NB ** θ = 1/α *** 5 years of data for 275 separate non-medan barrer sectons 3653

10 Journal of the Eastern Asa Socety for Transportaton Studes, Vol. 6, pp , AN EMPIRICALLY ADJUSTED SERIAL CORRELATION ZERO INFLATED POISSON MODEL OF MEDIAN CROSSOVER ACCIDENTS A ZIP-Full model of medan crossover accdents was estmated and loadng factors from table 2 were used to adjust for seral correlaton n the fve-year medan crossover dataset. Medan crossover accdent counts were the dependent varable n ths estmaton. The ZIP-Full model was estmated and valdated by the Vuong statstc. The result revealed a Vuong statstc of suggestng that the ZIP-Full as the favorable model compared to the basc Posson model. In estmatng the sgnfcance of key varables n the ZIP-Full model, standard errors estmated usng the lkelhood functon shown n equaton 10 were adjusted through multplcaton wth the loadng factors developed n table 2. By so dong, we emprcally account for seral correlaton present n the dataset. Table 3 shows the emprcally adjusted seral correlaton ZIP-Full model of medan crossover accdents. The results show that before accountng for seral correlaton, all the varables ncluded n the ZIP-Full model were hghly sgnfcant and thus have a hgh level of confdence (exceedng 95%). However, after accountng for seral correlaton, the level of sgnfcance for all varables decreased. The length varable for medans wder than 60 feet had the lowest t-statstc of It s also noted that the parameters n non-zero accdent state always negatvely correlate wth the same parameters n zero accdent state (e.g. opposte sgns). For example, the length varable for medans narrower than 40 feet postvely correlates wth medan crossover accdents, whle t negatvely correlates wth the probablty of a secton beng n a zero count state n ts lfetme. In the non-zero count state, factors postvely correlatng wth medan crossover accdents ncluded the nteracton varables between length and three categores of medan wdths,, as well as the number of nterchanges n the secton. The three categores of the medan wdth nteracted wth the length of the secton were 1) less than or equal to 40-foot medan wdth, 2) between 40 to 60-foot medan wdth, and 3) greater than 60-foot medan wdth. Dfferent ranges of the medan wdth were expermented but the result showed that these three categores, when nterrelated wth the secton length, had the greatest mpact on the crossover accdent lkelhood. The magntudes of the coeffcents of the length varables suggest that all for a gven length, the lkelhood of medan crossover accdents ncreases the greatest on sectons wth medan wdths between 40 and 60 feet wde n comparson to the other wdth categores. On the other hand, sectons wth medan wdth wder than 60 feet have the least contrbuton on the lkelhood of medan crossover frequency. In fact, n the presence of seral correlaton, ther mpact s statstcally margnal. Three factors negatvely correlate wth medan crossover accdent frequences. These ncluded the ndcator varable f average annual daly traffc was less than 5,000, the nteracton varable between the number of horzontal curves less than or equal to 0.5 per mle n the secton and the average monthly precptaton beng less than or equal 1.5 nches, and the nteracton varable between the number of horzontal curves greater than 0.5 per mle n the secton and the average monthly precptaton beng greater than 4 nches. 3654

11 Journal of the Eastern Asa Socety for Transportaton Studes, Vol. 6, pp , 2005 Table 3. Zero-Inflated Posson Model of Medan Crossover Accdents wth Seral Correlaton Adjusted Standard Errors Varables Zero-Inflated Posson Model Adjusted Adjusted Load Estmated Standard. T- Standard T- Factor Coeffcent Error statstc Error Statstc Non-zero accdent Posson probablty state Constant Per-lane AADT ndcator (1 f per-lane AADT <= 5000 vehcles, 0 otherwse) Length of secton where medans are less than 40 feet wde Length of secton where medans are between 40 feet and 60 feet wde Length of secton where medans are wder than 60 feet Number of nterchanges n secton Interacton 1 between average monthly precptaton ndcator and the number of horzontal curves per mle ndcator (1 f average monthly precptaton <= 1.5 nches and the number of horzontal curves per mle <= 0.5, 0 otherwse) Interacton 2 between average monthly precptaton ndcator and the number of horzontal curves per mle ndcator (1 f average monthly precptaton > 4.0 nches and the number of horzontal curves per mle > 0.5, 0 otherwse) Zero accdent probablty state as logstc functon Constant Length of secton where medans are less than 40 feet wde Length of secton where medans are between 40 feet and 60 feet wde Length of secton where medans are wder than 60 feet Restrcted log-lkelhood * (constant only) Log-lkelhood at convergence Vuong statstc * Ths restrcted log-lkelhood was obtaned from the restrcted log-lkelhood computed by the Posson model. 3655

12 Journal of the Eastern Asa Socety for Transportaton Studes, Vol. 6, pp , 2005 The weather effect appearng n the form of the nteracton varables played a sgnfcant role n medan crossover lkelhood. In a secton where the average number of horzontal curves was less than or equal to 0.5 per mle, the medan crossover counts were expected to decrease f the average monthly precptaton was less than or equal to 1.5 nches. Crossover accdent frequency also decreases when average monthly precptaton exceeds 4 nches on sectons wth greater than 0.5 horzontal curves per mle. Both effects pont to the range of nteractons between precptaton and horzontal curves on medan crossover frequences. As a fnal thought, t should be noted that the mpact of medan wdths on the probablty of a medan crossover accdent occurrng n a secton s lfetme (zero versus non-zero state) follows a trend that s dfferent from ther mpacts on postve crossover frequences (the Posson state.) In the zero state, for a gven length, medan wdths n excess of 60 feet have the hghest odds rato of beng n a non-zero accdent state, whle the count state effects suggests they wll have the least number of postve crossover counts. 5. CONCLUSIONS AND RECOMMENDATIONS Ths paper presents an emprcal technque to adjust for seral correlaton effects on standard errors n medan crossover accdent models. Smlar to the classcal lnear regresson model, standard errors n count models are downward based as observed n emprcal analyses on medan crossover accdents n ths study. To develop a technque for adjustng standard errors, we used the negatve multnomal specfcaton to account for correlaton across tme. Usng the rato of standard errors from the NM model to the nave NB model, we developed loadng factors whch represented the level of nflaton n standard errors requred to account for seral correlaton. We then developed a ZIP-Full model of medan crossover accdent counts to account for the presence of excess zeros n the dataset. Ths excess zero s unque to medan crossover accdents, especally due to the fact that only fve years of observaton are avalable. Usng the loadng factors developed on the bass of the NM model, we adjusted the standard errors of the ZIP-Full model to dentfy key factors affectng medan crossover accdent frequences. It s our belef that the emprcally standard errors n the ZIP-Full model are closer to effcent estmates that would be theoretcally derved. An obvous recommendaton from ths study s to compare theoretcally derved standard errors of a ZIP- Full model wth emprcal fndngs from ths study. Such a study would shed sgnfcant lght on the mpact of seral correlaton n a varety of count contexts. To summarze our emprcal fndngs, we conclude that the ZIP-Full model s the most approprate model among zero-altered probablty processes for predctng the number of medan crossover accdents. The man effects such as average daly traffc, and the number of nterchanges were nvestgated to have statstcally sgnfcant mpact on the frequency of medan crossover accdents. The nteracton varables of length of the roadway secton and the medan wdth as well as the nteracton between monthly precptaton and the number of horzontal curves per mle are also presented as mportant contrbutors to medan crossover lkelhoods. REFERENCES Chayanan, S., Shankar, V.N., Sttkarya, S., Ulfarsson, G.F., Shyu, M.B., and Juvva, N.K. (2004) Medan crossover accdent analyses and the effectveness of medan barrers. Fnal 3656

13 Journal of the Eastern Asa Socety for Transportaton Studes, Vol. 6, pp , 2005 Research Report, WA-RD 580.1, Washngton State Department of Transportaton, Washngton. ESRI (2002) ArcVew GIS Verson 3.2, Envronmental Systems Research Insttute, Inc., Calforna. Greene, W.H. (1994) Accountng for excess zeros and sample selecton n Posson and negatve bnomal regresson models. Workng papers EC-94-10, Stern School of Busness, New York Unversty, New York. Greene, W.H. (2004) LIMDEP, Verson 8.0, User s Manual. Econometrc Software Inc., New York. Greene, W.H. (2003) Econometrc Analyss. Ffth Edton, Prentce Hall Inc., New Jersey. Guo, G. (1996) Negatve multnomal regresson models for clustered event counts, Socologcal Methodology, Vol. 26, Mlton, J.C. and Mannerng, F.L., (1996) The relatonshp between hghway geometrcs, traffc related elements and motor vehcle accdent. Fnal research report, WA-RD 403.1, Washngton State Department of Transportaton, Washngton. Poch, M. and Mannerng, F.L. (1996) Negatve bnomal analyss of ntersecton-accdent frequences, Journal of Transportaton Engneerng, Vol. 122, No. 3, Shankar, V.N., Albn, R.B., Mlton, J.C. and Mannerng, F.L. (1998) Evaluatng medan cross-over lkelhoods wth clustered accdent counts: an emprcal nqury usng the random effects negatve bnomal model, Transportaton Research Record, Vol. 1635, Shankar, V.N., Chayanan, S., Sttkarya, S., Shyu, M.B., Juvva, N.K. and Mlton J.C. (2004) The margnal mpacts of desgn, traffc, weather, and related nteractons on roadsde crashes, Transportaton Research Record, Vol. 1897, Shankar, V.N., Mannerng, F.L. and Barfeld, W. (1995) Effect of roadway geometrcs and envronmental condtons on rural accdent frequences, Accdent Analyss and Preventon, Vol. 27, No.3, Shankar, V.N., Mlton, J.C. and Mannerng, F.L. (1997) Modelng statewde accdent frequences as zero-altered probablty processes: an emprcal nqury, Accdent Analyss and Preventon, Vol. 29, No.6, Ulfarsson, G. F. and Shankar V.N. (2003) An accdent count model based on mult-year cross-sectonal roadway data wth seral correlaton, Transportaton Research Record, Vol. 1840, Voung, Q. (1989) Lkelhood rato tests for model selecton and non-nested hypotheses, Econometrca, Vol. 57, No. 2, Western Regonal Clmate Center (2002) Hstorcal Clmate Informaton, Desert Research Insttute, Retreved from