Markov switching multinomial logit model: an application to accident injury severities

Mark o v switc hing m ultinomial logit mo del: an application to accid en t injury sev erities Natal iy a V. Maly shk ina ∗ , F red L. Mannering Scho ol of Civil Engine ering, 550 Stad iu m Mal l Drive, Pur due University, West L afayette, IN 4790 7, Unite d Sta tes Abstract In this study , tw o-state Mark ov switc hin g m ultinomial logit mo dels are prop osed for statistic al mo deling of a ccident injury sev erities. These mo d els assume Mark o v switc hing in time b et we en t wo unobserved states of roadwa y s afety . The states are distinct, in the sense that in differen t states acciden t sev erit y outcomes are generated b y separate multinomial logit pro cesses. T o demonstrate the applicabilit y of the approac h presente d h er ein, t wo -state Mark o v switc h ing multinomia l logit mo dels are estimated for seve rit y outcomes of acciden ts o ccurring on Indiana roads o ver a four-y ear time in terv al. Ba ye s ian inf er en ce metho ds and Mark ov Chain Mon te Carlo (MCMC) sim ulations are used for mo del estimation. The estimate d Marko v switc hing mo dels result in a sup erior statisti cal fit relativ e to the standard (single- state) m u ltinomial logit mo dels. It is foun d that the more fr equen t state of r oadw a y safet y is correlated with b ette r w eather conditions. The less f r equen t state is foun d to b e correlated with adv erse weather conditions. Key wor ds: Acciden t in jury sev erit y; multinomial logit; Mark ov switc h ing; Ba y esian; MCMC 1 In tr o duction V ehicle acciden ts result in prop ert y damage, injuries and loss of p eople lives . Th us, researc h efforts in predicting acciden t sev erit y ar e clearly v ery imp or- tan t. In the pa st there ha s b een a large n umber of studies that fo cused o n mo d- eling a ccide nt sev erit y outcomes. Common mo deling approac hes of acciden t ∗ Corresp onding author. Email addr esses: nmalyshk@purd ue.edu (Nataliy a V. Malyshkina), flm@ecn. purdue.edu (F red L. Mannering). Preprint su bmitted to Acciden t Analysis and Preve n tion sev erit y include multinomial logit mo dels, nested logit mo dels, mixed logit mo dels and ordered probit mo dels (O’Donnell and Connor, 1996; Shank ar and Mannering, 1996; Shank ar et al., 1996; D uncan et al., 1998; Chang and Mannering , 1999; Carson and Mannering, 2001; Khattak, 2001; Khatta k et al., 2002; Ko ck elman and Kw eon, 2002; Lee and Mannering, 2002; Ab del-A ty, 2003; Kw eon and Ko c ke lman , 2003; Ulfarsson and Mannering, 2004; Y a mamoto and Shank ar, 200 4; Khorashadi et a l., 2005; Eluru and Bhat, 2007; Sav olainen and Mannering, 2007; Milton et al., 2008). All these mo dels in v olve nonlinear r egr ession of the observ ed acciden t injury sev erit y outcomes on v arious acciden t c haracteristics and related factor s (suc h as roa dw ay and driv er c haracteristics, en vironmen tal factors, etc). In our earlier pap er, Malyshkina et al. (20 08), whic h w e will refer t o as P a - p er I, w e presen t ed t w o-state Mark ov switc hing coun t data mo dels of acciden t frequencies. In this study , whic h is a con tin uation of our w o rk on Marko v switc hing mo dels, w e presen t tw o- state Mark o v switc hing mu ltinomial logit mo dels f or predicting acciden t sev erit y outcomes. These mo dels assume t hat there are t wo unobserv ed states of roadwa y safet y , roadw a y entities (road- w ay segmen t s) can switc h b etw een these stat es o ver time, and the switc hing pro cess is Mark ovian. The t w o states in tend to a ccoun t for p ossible hetero- geneit y effects in roadw a y safet y , whic h ma y b e caused b y v arious unpre- dictable, uniden tified, unobserv able risk factors tha t influence roadwa y safet y . Because the risk factors can in teract and c hange, roadw ay entities can switc h b et w een the t wo states o ver time. Tw o-stat e Mark ov switc hing m ultinomial logit mo dels assume separate m ultino mial logit pro ces ses for acciden t sev erity data generation in the t w o states and, therefore, allo w a researc her to study the heterogeneit y effects in ro a dw a y saf ety . 2 Mo del sp ecification Mark ov switch ing mo dels are parametric and can b e fully sp ecified b y a lik e- liho o d function f ( Y | Θ , M ), whic h is the conditional probabilit y distribution of the v ector of all observ a tions Y , giv en the v ector of all parameters Θ of mo del M . First, let us consider Y . Let N t b e the n um b er of acciden ts ob- serv ed during time p erio d t , where t = 1 , 2 , . . . , T and T is the tot a l n umber of time p eriods. Let there b e I discrete outcomes observ ed for acciden t sev er- it y (for example, I = 3 and these outcomes are fat a lit y , injury and prop erty damage only). Let us in tr o duce acciden t sev erity outcome dummies δ ( i ) t,n that are equal to unity if the i th sev erit y outcome is o bserv ed in the n th acciden t that o ccurs during time p erio d t , and to zero otherwise. Here i = 1 , 2 , . . . , I , n = 1 , 2 , . . . , N t and t = 1 , 2 , . . . , T . Then, our o bserv ations a r e the a ccide n t sev erit y o utcomes, and the vec tor of all observ a tions Y = { δ ( i ) t,n } includes all outcomes observ ed in a ll acciden ts that o ccur during all time p erio ds. Sec- ond, let us consider mo del sp ecific ation v ariable M . It is M = { M , X t,n } 2 and includes the mo del’s name M (for example, M = “multinomial logit”) and the v ector X t,n of all a ccide n t c haracteristic v ariables (w eather and env i- ronmen t conditions, v ehicle and driv er c haracteristics, roadw ay and pa veme n t prop erties, and so on). T o define the lik eliho o d function, w e first in tro duce an uno bserv ed (laten t) state v ariable s t , whic h determines the state of all ro adw a y en tities during time p erio d t . At eac h t , the state v ariable s t can assume o nly t w o v alues: s t = 0 corresp onds to one state and s t = 1 corresp onds to the other state ( t = 1 , 2 , . . . , T ). The state v ariable s t is assumed to follo w a stationar y tw o- state Mark ov c hain pro cess in time, 1 whic h can b e sp ecified b y time-indep enden t transition probabilities as P ( s t +1 = 1 | s t = 0) = p 0 → 1 , P ( s t +1 = 0 | s t = 1) = p 1 → 0 . (1) Here, for example, P ( s t +1 = 1 | s t = 0) is the conditional probability of s t +1 = 1 at t ime t + 1 , give n that s t = 0 at t ime t . T ra nsition probabilities p 0 → 1 and p 1 → 0 are unkno wn parameters to b e estimated from acciden t sev erity data. The statio nary unconditional probabilities of states s t = 0 and s t = 1 are ¯ p 0 = p 1 → 0 / ( p 0 → 1 + p 1 → 0 ) and ¯ p 1 = p 0 → 1 / ( p 0 → 1 + p 1 → 0 ) resp ectiv ely . 2 Without loss of generality , w e assume that (on av erage) state s t = 0 o ccurs more or equally frequen tly than state s t = 1. Therefore, ¯ p 0 ≥ ¯ p 1 , and w e obtain restriction 3 p 0 → 1 ≤ p 1 → 0 . (2) W e refer to states s t = 0 and s t = 1 as “more frequen t” and “less frequen t” states resp ectiv ely . Next, a tw o- state Mark ov switc hing m ultinomial logit (MSML) mo del assumes m ultinomial logit (ML) data- generating pro cesse s f or acciden t sev erity in eac h of the t w o states. With this, the probabilit y of the i th sev erit y outcome ob- serv ed in the n th acciden t during time p erio d t is 1 Mark o v pr operty means that the probabilit y distribution of s t +1 dep ends only on th e v alue s t at time t , but not on th e previous history s t − 1 , s t − 2 , . . . . Stationarit y of { s t } is in the statistic al sense. 2 These can b e fou n d from stationarit y conditions ¯ p 0 = (1 − p 0 → 1 ) ¯ p 0 + p 1 → 0 ¯ p 1 , ¯ p 1 = p 0 → 1 ¯ p 0 + (1 − p 1 → 0 ) ¯ p 1 and ¯ p 0 + ¯ p 1 = 1. 3 Without any loss of generalit y , restriction (2) is introdu ced for the pu rp ose of a vo id ing the pr oblem of state lab el sw itc h ing 0 ↔ 1. This problem would otherwise arise b ecause of the symm etry of Eqs. (1)–(4) un d er the lab el switc hing. 3 P ( i ) t,n =              exp( β ′ (0) ,i X t,n ) P I j =1 exp( β ′ (0) ,j X t,n ) if s t = 0 , exp( β ′ (1) ,i X t,n ) P I j =1 exp( β ′ (1) ,j X t,n ) if s t = 1 , (3) i = 1 , 2 , . . . , I , n = 1 , 2 , . . . , N t , t = 1 , 2 , . . . , T , Here prime means tra nsp ose (so β ′ (0) ,i is the transp ose of β (0) ,i ). P arameter v ectors β (0) ,i and β (1) ,i are unknown estimable parameters of the tw o standard m ultinomial log it probability mass functions (W ashington et al., 2 003) in the t wo states, s t = 0 and s t = 1 r espectiv ely . W e set the first comp onen t of X t,n to unit y , and, therefore, the first comp onen ts of v ectors β (0) ,i and β (1) ,i are the in tercepts in the tw o states. In addition, without loss of generality , we set all β -parameters for the last sev erit y outcome to zero, 4 β (0) ,I = β (1) ,I = 0 . If acciden t eve n ts are assumed to b e indep enden t, the lik eliho o d function is f ( Y | Θ , M ) = T Y t =1 N t Y n =1 I Y i =1 h P ( i ) t,n i δ ( i ) t,n . (4) Here, because the state v ariables s t,n are unobserv able, the v ector of all es- timable parameters Θ m ust include all states, in addition to mo del para meters ( β -s) and transition probabilities. Th us, Θ = [ β ′ (0) , β ′ (1) , p 0 → 1 , p 1 → 0 , S ′ ] ′ , where v ector S = [ s 1 , s 2 , ..., s T ] ′ has length T and contains a ll state v alues. Eqs. (1)- (4) define the tw o-state Mark ov switch ing multinomial lo git (MSML) mo del considered here. 3 Mo del estimation metho ds Statistical estimation of Marko v switc hing mo dels is complicated by unobserv- abilit y of the state v ariables s t . 5 As a result, the traditional maxim um lik eli- ho o d estimation ( MLE) pro cedure is of v ery limited use for Mark o v switc hing mo dels. Instead, a Ba ye sian inference approac h is used. Give n a mo del M with lik eliho o d function f ( Y | Θ , M ), t he Bay es form ula is f ( Θ | Y , M ) = f ( Y , Θ |M ) f ( Y |M ) = f ( Y | Θ , M ) π ( Θ |M ) R f ( Y , Θ |M ) d Θ . (5) 4 This can b e done b ecause X t,n are assumed to b e indep endent of the outcome i . 5 Belo w we will ha ve 208 time p erio ds ( T = 208). In this case, there are 2 208 p ossible com binations for v alue of v ector S = [ s 1 , s 2 , ..., s T ] ′ . 4 Here f ( Θ | Y , M ) is the p osterior probability distribution of mo del parameters Θ conditional on the observ ed data Y and mo del M . F unction f ( Y , Θ |M ) is the join t pro babilit y distribution o f Y and Θ giv en mo del M . F unction f ( Y |M ) is the marginal lik eliho o d function – the pro ba bilit y distribution of data Y give n mo del M . F unction π ( Θ |M ) is the prior proba bilit y distribution of parameters that reflects prior kno wledge ab out Θ . The in tuition b ehind Eq. (5) is straightforw a rd: giv en mo del M , the p osterior distribution accoun ts for b oth the observ ations Y and our prior kno wledge of Θ . In our study (and in most practical studies), the direct application of Eq. (5) is not feasible b ecause the para meter v ector Θ contains to o man y comp onen ts, making integration ov er Θ in Eq. (5) extreme ly difficult. Ho wev er, the p oste- rior distribution f ( Θ | Y , M ) in Eq. (5 ) is kno wn up to its normalization con- stan t, f ( Θ | Y , M ) ∝ f ( Y | Θ , M ) π ( Θ | M ). As a result, w e use Marko v Chain Mon te Carlo (MCMC) sim ulations, whic h pro vide a con venie nt and practi- cal computational metho dology for sampling from a probabilit y distribution kno wn up to a constan t (the p osterior distribution in o ur case). Giv en a large enough p osterior sample of para meter v ector Θ , any p osterior exp ectatio n and v ariance can b e found a nd Ba yes ia n inference can b e readily applied. A reader in terested in details is referred to our P ap er I or to Malyshkina (200 8), where w e describ e our c hoice of t he prio r distribution π ( Θ |M ) and the MCM C sim- ulation algorithm. 6 Although, in this study we estimate a t w o - state Mark ov switc hing m ultinomial logit mo del for acciden t sev erity o utcomes a nd in P a- p er I w e estimated a tw o-state Marko v switc hing negativ e binomial mo del for acciden t frequencies, this difference is not essen tial for the Ba y esian-MCMC mo del estimation metho ds. In f act, the main difference is in the like liho o d function (m ultinomial lo g it as opp osed to nega tiv e binomial). So w e used the same our own nume r ical MCMC co de, written in the MA TLAB programming language, for mo del estimation in b oth studies. W e tested our co de on arti- ficial data sets of acciden t sev erity outcomes. The t est pro cedure included a generation of artificial data with a kno wn mo del. Then these data w ere used to estimate the underlying mo del b y means of our sim ulation co de. With this pro cedure w e found that the MSML mo dels, used to generate the artificial data, w ere repro duced successfully with our estimation co de. F o r comparison of differen t mo dels we use a formal Ba y esian approac h. L et there b e t wo mo dels M 1 and M 2 with par a meter v ectors Θ 1 and Θ 2 resp ec- tiv ely . Assuming that w e ha v e equal preferences of these mo dels, their prior probabilities are π ( M 1 ) = π ( M 2 ) = 1 / 2. In this case, the ratio of the mo dels’ p osterior probabilities, P ( M 1 | Y ) and P ( M 2 | Y ), is equal to the Bay es fac- tor. The later is defined as t he ratio of the mo dels’ marginal lik eliho o ds (see Kass and Raftery, 1995). Th us, w e ha ve 6 Our priors for β -s, p 0 → 1 and p 1 → 0 are flat or nearly fl at, wh ile the prior for the states S reflects the Mark o v pro cess prop erty , sp ecified b y Eq. (1). 5 P ( M 2 | Y ) P ( M 1 | Y ) = f ( M 2 , Y ) /f ( Y ) f ( M 1 , Y ) /f ( Y ) = f ( Y |M 2 ) π ( M 2 ) f ( Y |M 1 ) π ( M 1 ) = f ( Y |M 2 ) f ( Y |M 1 ) , (6) where f ( M 1 , Y ) and f ( M 2 , Y ) are the joint distributions of the mo dels and the data, f ( Y ) is the unconditional distribution of the data . As in P ap er I, to calculate the marg inal lik eliho o ds f ( Y |M 1 ) and f ( Y |M 2 ), w e use the harmonic mean form ula f ( Y |M ) − 1 = E [ f ( Y | Θ , M ) − 1 | Y ], where E ( . . . | Y ) means p osterior exp ectation calculated b y using the p osterior distribution. If the ratio in Eq. (6) is larger than one, then mo del M 2 is fa vored, if the ratio is less than o ne, then mo del M 1 is fav ored. An adv an ta ge of the use of Ba yes factors is tha t it has an inherent p enalt y for including to o man y parameters in the mo del and guar ds ag ainst ov erfitting. T o ev aluate the p erformance of mo del {M , Θ } in fitting the observ ed data Y , w e carry out the P earson’s χ 2 go o dness-of-fit test (Maher and Summersgill , 1996; Co w an , 19 98; W o o d, 2 002; Press et al., 2 0 07). W e p erform t his t est b y Mon te Carlo sim ulations to find the distribution o f the P earson’s χ 2 quan- tit y , whic h measures the discrepancy b et w een the observ ations and the mo del predictions ( Cow an, 1 9 98). This distribution is then used t o find the g o o dnes s- of-fit p-v alue, whic h is the probabilit y that χ 2 exceeds the observ ed v alue of χ 2 under the h yp othesis that the mo del is true (the observ ed v alue of χ 2 is calculated by using the observ ed data Y ). F or additional details, please see Malyshkina (2008). 4 Empirical results The sev erit y outcome of an a cciden t is determined by the injury lev el sustained b y the most injured individual ( if any) in v olv ed into the acciden t. In this study w e consider three acciden t sev erit y outcomes: “fa talit y”, “inj ur y” and “PDO (prop ert y damage only)”, whic h w e n um b er as i = 1 , 2 , 3 respectiv ely ( I = 3). W e use data from 811 720 acciden ts that we r e observ ed in Indiana in 2003- 2 006. As in Paper I, w e use w eekly time p erio ds, t = 1 , 2 , 3 , . . . , T = 208 in total. 7 Th us, the state s t can change eve r y w eek. T o increase the predictiv e p o w er of our mo dels, we consider acciden ts separately for eac h combination of acci- den t t yp e (1-vehic le and 2-vehic le) and roadwa y class (inte rstate high wa ys, US routes, state r outes, count y roads, streets). W e do not consider a ccide n ts with more than t w o v ehicles in v olve d. 8 Th us, in total, there are ten roadwa y-class- acciden t-ty p e com binations that w e consider. F or eac h roadwa y-class-acciden t - 7 A w eek is from S unda y to Saturda y , there are 208 f ull weeks in the 2003-2006 time int er v al. 8 Among 811720 acciden ts 24101 1 (29.7%) are 1-v ehicle, 525035 (64.7%) are 2- v ehicle, and only 45674 (5.6%) are acciden ts with more than t wo v ehicles inv olv ed. 6 t yp e combination the following three types of acciden t frequency mo dels are estimated: • First, we estimate a standard multinomial logit ( ML) mo del without Mark ov switc hing b y maxim um likelihoo d estimation (MLE). 9 W e refer to this mo del as “ML-by-MLE”. • Second, w e estimate the same standard m ultinomial logit mo del by the Ba ye sian inference approac h and the MCMC simulations. W e refer to this mo del as “ML-by -MCMC”. As one exp ects, the estimated ML-by-MCM C mo del turned out to b e v ery similar to the corresp onding ML-b y-MLE mo del (estimated for the same roadw ay -class-acciden t-type com bina t ion). • Third, we estimate a tw o- state Mark ov switc hing multinomial lo git (MSML) mo del by the Ba yes ian-MCMC metho ds. In order to make comparison of ex- planatory v ariable effects in differen t mo dels straightforw ard, in the MSML mo del w e use only t ho se explanatory v ariables that en ter the corresp onding standard ML mo del. 10 T o obtain the final MSML mo del rep orted here, we also consecutiv ely construct and use 60%, 85% and 95% Bay esian credible in terv als f or ev aluation of the statistical significance of eac h β - parameter. As a result, in the final mo del some comp onen ts of β (0) and β (1) are re- stricted to zero or restricted to b e the same in the t wo states. 11 W e refer to this final mo del a s “MSML”. Note that the tw o states, and th us the MSML mo dels, do not ha v e to exist for ev ery roadw ay - class-accide n t-type com bination. F or example, they will not exist if all estimated mo del parameters turn out to b e statistically the same in the t w o states, β (0) = β (1) , (whic h suggests the t w o states are iden tical and the MSML mo dels r educe to the corresp onding standar d ML mo dels). Also, the t wo states will not exist if all estimated state v ar ia bles s t turn out to b e close to zero, resulting in p 0 → 1 ≪ p 1 → 0 [compare to Eq. (2)], then the less 9 T o obtain parsimonious s tand ard mo dels, estimated b y MLE, w e c ho ose the explanatory v ariables and their d ummies by u s ing the Ak aike Information Criterion (AIC) and th e 5% statistical significance lev el for the t wo-ta iled t-test. Minimization of AI C = 2 K − 2 LL , we r e K is the num b er of free con tinuous mo del p aramete r s and LL is the log-l ik eliho o d, ensur es an optimal c h oic e of explanatory v ariables in a mo del and a voi ds o verfitting (Tsa y , 2002; W ash ington et al., 2003). F or details on v ariable select ion, see Malyshkina (2006). 10 A formal Ba ye sian approac h to mo del v ariable selection is based on ev aluation of mo del’s marginal lik eliho o d and the Bay es fact or (6 ). Unfortunately , b ecause MCMC sim ulations are computationally exp ensive , ev aluation of marginal lik eli- ho o ds for a large num b er of trial mo dels is n ot feasible in our stud y . 11 A β -parameter is restricted to zero if it is s tatistically insignificant . A β -parameter is restricted to b e the same in the t wo states if the difference of its v alues in the t wo states is statistically insignifican t. A (1 − a ) credible interv al is chosen in such w ay that the p osterior probabilities of b eing b elow and ab o ve it are b oth equal to a/ 2 (w e use significance leve ls a = 40% , 15 % , 5%). 7 frequen t state s t = 1 is not realized and the pro cess sta ys in state s t = 0. T urning to the estimation results, the findings show that tw o states o f roadwa y safet y and the appropriate MSML mo dels exist for sev erit y outcomes of 1- v ehicle acciden ts o ccurring on all roadw ay classes (in terstate high wa ys, US routes, state routes, count y roads, streets), and for sev erity outcomes of 2 - v ehicle acciden ts o ccurring on streets. W e did not find t wo states in the cases of 2-v ehicle acciden ts o n inters t a te high w ays , US routes, state routes and coun t y roads (in these cases all estimated state v ariables s t w ere found to b e close to zero). The mo del estimation results for sev erity outcomes of 1-ve hicle a ccide n ts o ccurring on in terstate high w ay s, US routes and state routes are giv en in T ables 1 – 3. All contin uous mo del para meters ( β -s, p 0 → 1 and p 1 → 0 ) are give n together with their 95% confidence interv als (if MLE) or 9 5 % credible in t erv als (if Bay esian-MCMC), refer to the sup erscript and subsc ript n umbers adjacen t to parameter estimates in T a bles 1 – 3. 12 T able 4 giv es summary statistics of all roadw ay a ccide n t c haracteristic v ariables X t,n (except the in tercept). 12 Note that MLE assumes asymptotic normalit y of the estimates, resu lting in con- fidence in terv als b eing symmetric around the means (a 95% confidence inte r v al is ± 1 . 96 stand ard deviations around the mean). In con trast, Bay esian estimation do es not require this assumption, and p osterior distributions of parameters and Ba yesia n credible interv als are u sually non-symmetric. 8 T ab le 1 Estimation results for multinomial logit mo dels of s everit y outcomes of one-v ehicle accident s on Indiana in terstate high wa ys (the sup erscrip t and sub script num b ers to the righ t of individual parameter estimates are 95% confidence/credible interv als) MSML c V ariable ML-b y -MLE a ML-by-MCMC b state s = 0 state s = 1 fatali t y injury fata lity injury fatali t y i njury fa tality injury In tercept (constant term) − 11 . 9 − 10 . 1 − 13 . 7 − 3 . 69 − 3 . 53 − 3 . 84 − 12 . 4 − 10 . 6 − 14 . 5 − 3 . 72 − 3 . 56 − 3 . 88 − 12 . 2 − 10 . 5 − 14 . 4 − 3 . 98 − 3 . 79 − 4 . 17 − 12 . 2 − 10 . 5 − 14 . 4 − 3 . 22 − 2 . 98 − 3 . 45 Summer season (dummy) . 235 . 329 . 142 . 235 . 329 . 142 . 237 . 329 . 143 . 237 . 329 . 143 . 176 . 293 . 0551 . 176 . 293 . 0551 . 176 . 293 . 0551 . 615 . 959 . 282 Th ursday (dumm y) − . 798 − . 115 − 1 . 48 – − . 853 − . 206 − 1 . 59 – − . 872 − . 225 − 1 . 61 – − . 872 − . 225 − 1 . 61 – Construction at the acciden t lo cation (dummy ) − . 418 − . 213 − . 623 − . 418 − . 213 − . 623 − . 425 − . 224 − . 632 − . 425 − . 224 − . 632 − . 566 − . 319 − . 822 − . 566 − . 319 − . 822 − . 566 − . 319 − . 822 – Da ylight or street li gh ts are li t up if dark (dummy) − . 392 − . 0368 − . 748 . 137 . 224 . 0501 − . 387 − . 0301 − . 740 . 143 . 230 . 0568 − . 378 − . 0236 − . 729 . 139 . 226 . 0522 − . 378 − . 0236 − . 729 . 139 . 226 . 0522 Precipitation: r ain/freezing rain/snow/sleet/ hail (dumm y) − 1 . 38 − . 830 − 1 . 92 − . 361 − . 264 − . 457 − 1 . 41 − . 884 − 1 . 99 − . 363 − . 267 − . 460 − 1 . 54 − 1 . 03 − 2 . 10 − . 563 − . 404 − . 729 − 1 . 54 − 1 . 03 − 2 . 10 – Roadw ay surface is cov ered by snow/slush (dummy) − 1 . 28 − . 0917 − 2 . 46 − . 432 − . 280 − . 583 − 1 . 43 − . 328 − 2 . 84 − . 438 − . 288 − . 590 − . 0515 − . 361 − . 671 − . 0515 − . 361 − . 671 − . 0515 − . 361 − . 671 − . 0515 − . 361 − . 671 Roadw ay median i s driv able (dummy) . 571 . 929 . 213 – . 577 . 939 . 223 – . 566 . 930 . 211 – . 566 . 930 . 211 – Roadw ay is at curve (dummy) . 114 . 212 . 0165 . 114 . 212 . 0165 . 116 . 213 . 0186 . 116 . 213 . 0186 – – – – Primary cause of the acciden t i s driver-related (dumm y) 4 . 24 5 . 30 3 . 18 1 . 53 1 . 64 1 . 43 4 . 39 5 . 64 3 . 39 1 . 54 1 . 64 1 . 43 4 . 48 5 . 73 3 . 48 2 . 00 2 . 18 1 . 84 4 . 48 5 . 73 3 . 48 . 715 . 946 . 468 Help arri v ed in 20 minutes or less after the crash (dummy) . 790 . 887 . 693 . 790 . 887 . 693 . 790 . 891 . 691 . 790 . 891 . 691 . 785 . 886 . 684 . 785 . 886 . 684 . 785 . 886 . 684 . 785 . 886 . 684 The vehicle at fault is a motorcycle (dummy) 3 . 88 4 . 59 3 . 17 2 . 74 3 . 12 2 . 36 3 . 87 4 . 57 3 . 13 2 . 75 3 . 15 2 . 37 4 . 61 5 . 49 3 . 74 3 . 23 3 . 83 2 . 70 – 1 . 39 2 . 49 . 326 Age of the vehicle at f ault (in ye ars) . 0285 . 0370 . 0201 . 0285 . 0370 . 0201 . 0286 . 0370 . 0201 . 0286 . 0370 . 0201 – . 0286 . 0371 . 0200 – . 0286 . 0371 . 0200 Number of o ccupan ts in the ve hicle at fault . 366 . 463 . 269 . 123 . 159 . 0859 . 367 . 465 . 264 . 123 . 159 . 0861 . 366 . 464 . 263 . 124 . 161 . 0874 . 366 . 464 . 263 . 124 . 161 . 0874 Roadw ay trav eled by the vehicle at fault is multi-lane and divided tw o-wa y (dummy) 2 . 60 4 . 00 1 . 20 – 2 . 86 4 . 63 1 . 56 – 2 . 86 4 . 66 1 . 56 – 2 . 86 4 . 66 1 . 56 – At least one of the vehicles i n volv ed was on fire (dumm y) 1 . 24 2 . 12 − . 345 − . 0257 − . 665 1 . 18 2 . 02 . 206 − . 345 − . 0335 − . 669 1 . 66 2 . 56 . 621 − . 332 − . 0198 − . 659 – − . 332 − . 0198 − . 659 Gender of the driver at fault (dummy) – . 328 . 410 . 246 – . 331 . 413 . 248 – . 224 . 338 . 107 – . 479 . 637 . 328 9 T ab le 1 (Con tinued) MSML c V ariable ML-by-MLE a ML-by-MCMC b state s = 0 sta te s = 1 fatali t y injury fatali t y injury fata lity injury f atality injury Probability of severit y outcome [ P ( i ) t,n give n by Eq. (3)], a v eraged o ver all v alues of explanatory v ariables X t,n – – . 00724 . 176 . 0073 3 . 174 . 00672 . 192 Marko v transition probability of jump 0 → 1 ( p 0 → 1 ) – – . 151 . 254 . 0704 Marko v transition probability of jump 1 → 0 ( p 1 → 0 ) – – . 330 . 532 . 164 Unconditional probabilities of states 0 and 1 ( ¯ p 0 and ¯ p 1 ) – – . 683 . 814 . 540 and . 317 . 460 . 186 T otal n umber of fr ee model parameters ( β -s) 25 25 28 Po sterior av erage of the log-likelihoo d (LL) – − 8486 . 78 − 8480 . 82 − 8494 . 61 − 8396 . 78 − 8379 . 21 − 8416 . 57 Max( LL ): estimated max. log-likelihoo d (LL) for MLE; maximum observ ed v alue of LL for Ba ye si an-MCMC − 8465 . 79 (MLE) − 8476 . 37 (observ ed) − 8358 . 97 (observed ) Logarithm of marginal likelihoo d of data (ln[ f ( Y |M )]) – − 8498 . 46 − 8494 . 22 − 8499 . 21 − 8437 . 07 − 8424 . 77 − 8440 . 02 Goo dness-of-fit p-v alue – 0 . 255 0 . 222 Maximum of the p oten tial scale reduction factors (PSRF) d – 1 . 00302 1 . 00060 Multiv ariate p oten tial scale reduction factor (MPSRF) d – 1 . 00325 1 . 00067 Number of av ailable observ ations acciden ts = f ata lities + i njuries + PDOs: 19094 = 143 + 3369 + 15582 a Standard (conv ent ional) multinomial logit (ML) mo del estimated by maximum likelihoo d estimation (MLE). b Standard multinomial l ogit (ML) model estimated by Marko v Chain Mont e Carl o (MCMC) simulations. c Two -state Marko v switching multinomial logit (MSML) mo del estimated by Marko v Chain Monte Carlo (MCM C) simulations. d PSRF/MPSRF are calculated separately/jointly for all contin uous mo del parameters. PSRF and M PSRF are close to 1 for con verged MCMC chains. 10 T ab le 2 Estimation results for multinomial logit mo dels of s everit y outcomes of one-v ehicle accident s on Indiana US routes (the sup erscrip t and sub script num b ers to the righ t of individual parameter estimates are 95% confidence/credible interv als) MSML c V ariable ML-by-MLE a ML-by-MCMC b state s = 0 state s = 1 fatali t y injury f atality injury fatality i njury fatal it y injury In tercept (constant term) − 6 . 51 − 5 . 00 − 8 . 03 − 2 . 13 − 1 . 79 − 2 . 47 − 6 . 62 − 5 . 16 − 8 . 14 − 2 . 12 − 1 . 78 − 2 . 47 − 5 . 72 − 4 . 69 − 6 . 92 − 2 . 05 − 1 . 71 − 2 . 40 − 5 . 72 − 4 . 69 − 6 . 92 − 2 . 79 − 2 . 37 − 3 . 23 Summer season (dummy) . 514 . 894 . 134 . 200 . 305 . 0947 . 509 . 883 . 124 . 200 . 305 . 0951 . 190 . 300 . 0789 . 190 . 300 . 0789 . 190 . 300 . 0789 – Da ylight or street li gh ts are li t up if dark (dummy) − . 498 − . 142 − . 855 . 194 . 287 . 101 − . 492 − . 136 − . 848 . 203 . 296 . 110 − . 493 − . 136 − . 857 . 197 . 290 . 105 – . 19 7 . 290 . 105 Sno wi ng weather (dummy) − 1 . 17 − . 170 − 2 . 18 – − 1 . 30 − . 357 − 2 . 47 – − 1 . 10 − . 151 − 2 . 27 . 165 . 317 . 0115 − 1 . 10 − . 151 − 2 . 27 . 165 . 317 . 0115 No roadw ay j unct i on at the acciden t lo cation (dummy) . 701 1 . 25 . 149 . 217 . 335 . 0994 . 727 1 . 31 . 199 . 213 . 331 . 0968 . 787 1 . 36 . 259 . 214 . 332 . 0965 . 787 1 . 36 . 259 . 214 . 332 . 0965 Roadw ay is s tr aigh t (dumm y) − . 741 − . 383 − 1 . 10 − . 295 − . 191 − . 399 − . 739 − . 377 − 1 . 09 − . 296 − . 192 − . 399 − 7 . 37 − . 372 − 1 . 09 − . 294 − . 189 − . 398 − 7 . 37 − . 372 − 1 . 09 − . 294 − . 189 − . 398 Primary cause of the acciden t i s en vi ronmen t-related (dummy) − 3 . 45 − 2 . 72 − 4 . 18 − 1 . 89 − 1 . 78 − 1 . 99 − 3 . 51 − 2 . 81 − 4 . 32 − 1 . 89 − 1 . 79 − 2 . 00 − 3 . 59 − 2 . 89 − 4 . 40 − 2 . 09 − 1 . 96 − 2 . 24 − 3 . 59 − 2 . 89 − 4 . 40 − . 701 − . 263 − 1 . 16 Help arri v ed in 10 minutes or less after the crash (dummy) . 594 . 681 . 507 . 594 . 681 . 507 . 562 . 650 . 475 . 562 . 650 . 475 . 560 . 648 . 472 . 560 . 648 . 472 . 560 . 648 . 472 . 560 . 648 . 472 The vehicle at fault is a motorcycle (dummy) 2 . 62 3 . 47 1 . 78 3 . 20 3 . 55 2 . 86 2 . 57 3 . 38 1 . 65 3 . 21 3 . 56 2 . 87 3 . 22 3 . 58 2 . 88 3 . 22 3 . 58 2 . 88 3 . 22 3 . 58 2 . 88 3 . 22 3 . 58 2 . 88 Age of the vehicle at f ault (in ye ars) . 0363 . 0444 . 0283 . 0363 . 0444 . 0283 . 0367 . 0448 . 0287 . 0367 . 0448 . 0287 – . 0366 . 0447 . 0285 – . 0366 . 0447 . 0285 Speed li mit (used if known and the same f or all vehicles i n volv ed) . 0363 . 0631 . 00950 . 0121 . 0178 . 00640 . 0373 . 0643 . 0117 . 0118 . 0176 . 00616 . 0285 . 0495 . 0104 . 0102 . 0178 . 00635 – . 0120 . 0178 . 00635 Roadw ay trav eled by the vehicle at fault is tw o- lane and one-w ay (dummy) − . 216 . 0417 − . 391 − . 216 . 0417 − . 391 − . 223 . 0517 − . 398 − . 223 . 0517 − . 398 − . 224 . 0504 − . 401 − . 224 . 0504 − . 401 − . 224 . 0504 − . 401 − . 224 . 0504 − . 401 At least one of the vehicles i n volv ed was on fire (dumm y) 1 . 19 1 . 94 . 439 – 1 . 13 1 . 85 . 315 – 1 . 27 1 . 98 . 452 – 1 . 27 1 . 98 . 452 – Age of the driver at fault (in yea rs) . 0114 . 0213 . 00150 – . 0113 . 0211 . 00137 – . 0101 . 0200 . 000054 2 – – – W eekda y (Monda y through F r ida y) (dummy) – − . 104 . 0116 − . 196 – − . 104 . 0124 − . 196 – − . 125 . 0242 − . 227 – – Gender of the driver at fault (dummy) – . 27 2 . 362 . 183 – . 27 6 . 365 . 186 – . 280 . 369 . 190 – . 28 0 . 369 . 190 11 T ab le 2 (Con tinued) MSML c V ariable ML-by-MLE a ML-by-MCMC b state s = 0 sta te s = 1 fatali t y injury fatali t y injury fata lity injury f atality injury Probability of severit y outcome [ P ( i ) t,n give n by Eq. (3)], a v eraged o ver all v alues of explanatory v ariables X t,n – – . 00747 . 179 . 0082 3 . 183 . 0021 8 . 158 Marko v transition probability of jump 0 → 1 ( p 0 → 1 ) – – . 0767 . 157 . 0269 Marko v transition probability of jump 1 → 0 ( p 1 → 0 ) – – . 613 . 864 . 337 Unconditional probabilities of states 0 and 1 ( ¯ p 0 and ¯ p 1 ) – – . 887 . 959 . 770 and . 113 . 230 . 0409 T otal n umber of fr ee model parameters ( β -s) 24 24 25 Po sterior av erage of the log-likelihoo d (LL) – − 7406 . 39 − 7400 . 61 − 7414 . 03 − 7349 . 06 − 7335 . 46 − 7364 . 47 Max( LL ): estimated max. log-likelihoo d (LL) for MLE; maximum observ ed v alue of LL for Ba ye si an-MCMC − 7384 . 05 (MLE) − 7396 . 37 (observ ed) − 7318 . 21 (observed ) Logarithm of marginal likelihoo d of data (ln[ f ( Y |M )]) – − 7417 . 98 − 7413 . 72 − 7420 . 23 − 7377 . 49 − 7369 . 62 − 7380 . 00 Goo dness-of-fit p-v alue – 0 . 337 0 . 255 Maximum of the p oten tial scale reduction factors (PSRF) d – 1 . 00319 1 . 00073 Multiv ariate p oten tial scale reduction factor (MPSRF) d – 1 . 00376 1 . 00085 Number of av ailable observ ations acciden ts = f ata lities + i njuries + PDOs: 17797 = 138 + 3184 + 14485 a Standard (conv ent ional) multinomial logit (ML) mo del estimated by maximum likelihoo d estimation (MLE). b Standard multinomial l ogit (ML) model estimated by Marko v Chain Mont e Carl o (MCMC) simulations. c Two -state Marko v switching multinomial logit (MSML) mo del estimated by Marko v Chain Monte Carlo (MCM C) simulations. d PSRF/MPSRF are calculated separately/jointly for all contin uous mo del parameters. PSRF and M PSRF are close to 1 for con verged MCMC chains. 12 T ab le 3 Estimation results for multinomial logit mo dels of s everit y outcomes of one-v ehicle accident s on Indiana state routes (the sup erscrip t and sub script num b ers to the righ t of individual parameter estimates are 95% confidence/credible interv als) MSML c V ariable ML-by-MLE a ML-by-MCMC b state s = 0 state s = 1 fatali t y injury fatali t y injury fatali t y injury fata lity injur y In tercept (constant term) − 3 . 98 − 3 . 66 − 4 . 30 − 1 . 67 − 1 . 53 − 1 . 80 − 4 . 03 − 3 . 71 − 4 . 36 − 1 . 71 − 1 . 58 − 1 . 85 − 3 . 44 − 3 . 10 − 3 . 79 − 1 . 68 − 1 . 54 − 1 . 81 − 4 . 96 − 4 . 15 − 5 . 96 − 1 . 68 − 1 . 54 − 1 . 81 Summer season (dummy) . 232 . 307 . 156 . 232 . 307 . 156 . 232 . 307 . 157 . 232 . 307 . 157 . 238 . 314 . 163 . 238 . 314 . 163 . 238 . 314 . 163 . 238 . 314 . 163 Roadw ay t ype (dummy: 1 if urban, 0 if rural ) − . 390 − . 302 − . 478 − . 390 − . 302 − . 478 − . 395 − . 306 − . 483 − . 395 − . 306 − . 483 – − . 385 − . 296 − . 474 − 2 . 05 − . 954 − 3 . 62 − 3 . 85 − . 296 − . 474 Da ylight or street li gh ts are li t up if dark (dummy) − . 646 − . 408 − . 884 . 193 . 261 . 125 − . 641 − . 404 − . 879 . 199 . 267 . 132 − . 689 − . 448 − . 931 – − . 689 − . 448 − . 931 . 277 . 378 . 177 Precipitation: r ain/freezing rain/snow/sleet/ hail (dumm y) − . 854 . 466 − 1 . 24 – − . 868 − . 494 − 1 . 27 – − . 829 − . 448 − 1 . 24 – − . 829 − . 448 − 1 . 24 – Roadw ay median i s driv able (dummy) − . 583 − . 225 − . 940 – − . 596 − . 250 − . 964 – − . 589 − . 241 − . 960 – − . 589 − . 241 − . 960 – Roadw ay is s tr aigh t (dumm y) − . 284 − . 214 − . 353 − . 284 − . 214 − . 353 − . 283 − . 214 − . 352 − . 283 − . 214 − . 352 − . 117 − . 0184 − . 214 − . 117 − . 0184 − . 214 − . 117 − . 0184 − . 214 − . 465 − . 360 − . 573 Primary cause of the acciden t i s en vi ronmen t-related (dummy) − 4 . 23 − 3 . 59 − 4 . 86 − 1 . 83 − 1 . 76 − 1 . 91 − 4 . 28 − 3 . 67 − 4 . 97 − 1 . 84 − 1 . 76 − 1 . 91 − 4 . 40 − 3 . 79 − 5 . 10 − 2 . 30 − 2 . 16 − 2 . 44 − 4 . 40 − 3 . 79 − 5 . 10 − 1 . 41 − 1 . 26 − 1 . 55 Help arri v ed in 20 minutes or less after the crash (dummy) . 840 . 917 . 762 . 840 . 917 . 762 . 863 . 945 . 781 . 863 . 945 . 781 – . 861 . 944 . 778 1 . 64 2 . 64 . 856 . 861 . 944 . 778 The vehicle at fault is a motorcycle (dummy) 3 . 10 3 . 31 2 . 89 3 . 10 3 . 31 2 . 89 3 . 10 3 . 31 2 . 89 3 . 10 3 . 31 2 . 89 3 . 37 3 . 66 3 . 09 3 . 37 3 . 66 3 . 09 3 . 37 3 . 66 3 . 09 2 . 82 3 . 19 2 . 47 Number of o ccupan ts in the ve hicle at fault . 0557 . 0850 . 0265 . 0557 . 0850 . 0265 . 0565 . 0858 . 0276 . 0565 . 0858 . 0276 . 0942 . 138 . 0528 . 0942 . 138 . 0528 . 0942 . 138 . 0528 – At least one of the vehicles i n volv ed was on fire (dumm y) 1 . 90 2 . 45 1 . 33 . 456 . 780 . 133 1 . 87 2 . 42 1 . 28 . 447 . 768 . 124 1 . 87 2 . 43 1 . 28 . 461 . 782 . 137 1 . 87 2 . 43 1 . 28 . 461 . 782 . 137 Age of the driver at fault (in yea rs) 14 . 6 21 . 4 7 . 80 × 10 − 3 − 2 . 80 − . 800 − 4 . 70 × 10 − 3 14 . 5 21 . 3 7 . 67 × 10 − 3 − 2 . 71 − . 723 − 4 . 69 × 10 − 3 14 . 5 21 . 4 7 . 63 × 10 − 3 − 2 . 46 − . 469 − 4 . 44 × 10 − 3 14 . 5 21 . 4 7 . 63 × 10 − 3 − 2 . 46 − . 469 − 4 . 44 × 10 − 3 Gender of the driver at fault (dummy) − . 496 − . 211 − . 780 . 279 . 344 . 214 − . 505 − . 225 − . 794 . 278 . 343 . 213 − . 473 − . 192 − . 764 . 283 . 348 . 218 − . 473 − . 192 − . 764 . 283 . 348 . 218 Age of the vehicle at f ault (in ye ars) – . 0334 . 0392 . 0276 – . 0335 . 0393 . 0277 – . 0332 . 0390 . 0274 – . 0332 . 0390 . 0274 license s tate of the veh i cle at fault is a U.S. state except Indiana and its neighboring states (IL, KY, OH, MI)” indicator v ariable – − . 449 − . 217 − . 681 – − . 444 − . 217 − . 679 – − . 436 − . 208 − . 671 – − . 436 − . 208 − . 671 13 T ab le 3 (Con tinued) MSML c V ariable ML-by-MLE a ML-by-MCMC b state s = 0 sta te s = 1 fatali t y injury fatali t y injury fatali t y injury f atality injury Probability of severit y outcome [ P ( i ) t,n give n by Eq. (3)], a v eraged o ver all v alues of explanatory v ariables X t,n – – . 0089 . 179 . 00951 . 180 . 00804 . 179 Marko v transition probability of jump 0 → 1 ( p 0 → 1 ) – – . 335 . 465 . 216 Marko v transition probability of jump 1 → 0 ( p 1 → 0 ) – – . 450 . 610 . 313 Unconditional probabilities of states 0 and 1 ( ¯ p 0 and ¯ p 1 ) – – . 574 . 681 . 504 and . 426 . 496 . 319 T otal n umber of fr ee model parameters ( β -s) 22 22 28 Po sterior av erage of the log-likelihoo d (LL) – − 1 3867 . 40 − 13861 . 92 − 13874 . 73 − 13781 . 76 − 13765 . 02 − 13800 . 89 Max( LL ): estimated max. log-likelihoo d (LL) for MLE; maximum observ ed v alue of LL for Ba ye si an-MCMC − 13846 . 60 (MLE) − 13 858 . 00 (observed) − 13745 . 61 (observe d) Logarithm of marginal likelihoo d of data (ln[ f ( Y |M )]) – − 13877 . 89 − 13874 . 24 − 13880 . 38 − 13820 . 20 − 13808 . 85 − 13821 . 73 Goo dness-of-fit p-v alue – 0 . 515 0 . 445 Maximum of the p oten tial scale reduction factors (PSRF) d – 1 . 00027 1 . 00029 Multiv ariate p oten tial scale reduction factor (MPSRF) d – 1 . 00041 1 . 00045 Number of av ailable observ ations acciden ts = fatalities + i njuries + PDOs: 33528 = 302 + 6018 + 27208 a Standard (conv ent ional) multinomial logit (ML) mo del estimated by maximum likelihoo d estimation (MLE). b Standard multinomial l ogit (ML) model estimated by Marko v Chain Mont e Carl o (MCMC) simulations. c Two -state Marko v switching multinomial logit (MSML) mo del estimated by Marko v Chain Monte Carlo (MCM C) simulations. d PSRF/MPSRF are calculated separately/jointly for all contin uous mo del parameters. PSRF and M PSRF are close to 1 for con verged MCMC chains. 14 The top, middle and b ottom plots in Figure 1 sho w w eekly p osterior pro ba - bilities P ( s t = 1 | Y ) of the less frequen t state s t = 1 fo r the MSML mo dels estimated for sev erit y of 1-vehic le a ccide nts o ccurring on interstate high w ay s, US ro utes and state ro utes resp ectiv ely . 13 Because of space limitations, in this pap er w e do not rep ort estimation results for sev erity of 1 -v ehicle acciden t s on coun ty roads and streets, and fo r sev erity of 2-vehic le acciden ts. Ho we ver, b e- lo w w e discuss our findings for all roa dwa y-class-acciden t-ty p e com binations. F o r unrep orted mo del estimation results see Malyshkina (2008). W e find that in all cases when the t w o states and Mark o v switc hing m ulti- nomial logit (MSML) mo dels exist, these mo dels are strongly f a v ored by the empirical data o ver the corresponding standard m ultinomial logit (ML) mo d- els. Indeed, from lines “marginal LL ” in T ables 1 – 3 we see that the MSML mo dels provide considerable, ranging from 40 . 5 to 61 . 4 , impro veme nts of the logarithm of the marginal lik eliho o d of the data as compared to the corre- sp onding ML mo dels. 14 Th us, from Eq. (6) w e find that, g iv en the acciden t sev erit y data, the p osterior probabilities of the MSML mo dels are larger than the probabilities of the corresp onding ML mo dels b y factors ra nging from e 40 . 5 to e 61 . 4 . In the cases of 1-v ehicle acciden t s on coun ty roads, streets a nd the case of 2-ve hicle acciden ts on streets, MSML mo dels (not rep orted here) are also strongly fa vored by the empirical da t a o v er the corresp onding ML mo dels (Malyshkina, 2008). Let us no w consider the maxim um like liho o d estimation (MLE) of the standard ML mo dels and an imaginary MLE estimation of the MSML mo dels. W e find that, in this imaginary case, a classical statistics approa c h for mo del comparison, based on the MLE, w ould also fav ors the MSML mo dels ov er the standard ML mo dels. F o r example, refer to line “max( LL )” in T able 1 g iv en for the case of 1-v ehicle acciden ts on in terstate high wa ys. The MLE gav e the maximum log- lik eliho o d v alue − 8465 . 79 for the standard ML mo del. The maxim um lo g-lik eliho o d v alue o bserv ed during our MCMC simulations for the MSML mo del is equal to − 835 8 . 97. An imaginary MLE, at its conv ergence, w ould give a MSML log-lik eliho o d v a lue that would b e ev en larger than this observ ed v alue. Therefore, if estimated by the MLE, the MSML mo del w ould pro vide la r g e, at least 106 . 82 impro v ement in the maxim um log- lik eliho o d v alue ov er the corresp onding ML mo del. This impro v emen t w ould come with only mo dest increase in the num b er of free con tinuous mo del parameters ( β - s) tha t en ter the lik eliho o d function (refer to T able 1 under “# free par.” ). 13 Note that these p osterior probabilities are equal to the p osterior exp ecta tions of s t , P ( s t = 1 | Y ) = 1 × P ( s t = 1 | Y ) + 0 × P ( s t = 0 | Y ) = E ( s t | Y ). 14 W e use the harm onic m ean f orm ula to calculate the v alues and the 95% confidence in terv als of the log-marginal-lik eliho o ds giv en in lines “marginal LL ” of T ables 1–3. The confidence inte r v als are cal culated by b o otstrap sim u lations. F or details, see P ap er I or Malyshkina (2008). 15 Jan−03 Jul−03 Jan−04 Jul−04 Jan−05 Jul−05 Jan−06 Jul−06 0 0.2 0.4 0.6 0.8 1 Date P(S t =1|Y) Jan−03 Jul−03 Jan−04 Jul−04 Jan−05 Jul−05 Jan−06 Jul−06 0 0.2 0.4 0.6 0.8 1 Date P(S t =1|Y) Jan−03 Jul−03 Jan−04 Jul−04 Jan−05 Jul−05 Jan−06 Jul−06 0 0.2 0.4 0.6 0.8 1 Date P(S t =1|Y) Fig. 1. W eekly p osterior probabilities P ( s t = 1 | Y ) for the MSML mo dels estimated for seve r it y of 1-v eh icle acci d en ts on in terstate high w ays (top p lot), US routes (mid- dle plot) and state routes (b ottom plot). Similar argumen ts hold for comparison of MSML and ML mo dels estimated for other roadw ay-class-accide nt-t yp e com binations (see T ables 2 and 3). T o ev aluate the go o dness-of-fit fo r a mo del, w e use the p osterior (or MLE) estimates of all con tinu ous mo del pa r a meters ( β -s, α , p 0 → 1 , p 1 → 0 ) and generate 10 4 artificial data sets under the h yp othesis that the mo del is true. 15 W e find the distribution of χ 2 and calculate the go o dness-of-fit p-v alue fo r the observ ed v alue of χ 2 . F or details, see Malyshkina (2008). The resulting p-v alues for our mo dels are giv en in T ables 1 – 3. These p-v alues ar e around 00–100%. Therefore, all mo dels fit the data w ell. 15 Note that the state v alues S are generated by u sing p 0 → 1 and p 1 → 0 . 16 No w, refer to T able 5. The first six row s of this table list time-correlation co ef- ficien ts b et wee n p osterior probabilities P ( s t = 1 | Y ) for the six MSML mo dels that exist a nd are estimated for six roadwa y-class-acciden t-t yp e com bina t ions (1-v ehicle acciden ts on in terstate highw ays , US routes, state routes, coun t y roads, streets, a nd 2-v ehicle acciden ts on streets). 16 W e see that the states for 1-v ehicle acciden ts on all high- speed roads (in terstate highw a ys, US routes, state routes and coun t y roads) a re corr elat ed with eac h other. The v alues of the corresponding correlatio n co efficien ts are p ositiv e and range from 0 . 263 to 0 . 688 (see T able 5). This result suggests an existence of common (unobserv- able) factors that can cause switc hing b et w een states of roadw ay safety for 1-v ehicle acciden ts o n all high- sp eed roads. The remaining row s of T able 5 show correlation co efficien ts b et wee n p oste- rior probabilities P ( s t = 1 | Y ) and w eather- condition v ariables. These cor- relations w ere found b y using daily and hourly historical we a ther data in Indiana, av ailable a t the Indiana St a te Climate Office at Purdue Univ ersity (www.agry .purdue.edu/climate). F or these correlations, the precipitation and sno wfall amoun ts are daily amounts in inche s av eraged o v er the w eek and across Indiana weather observ ation stations. 17 The temp erature v ariable is the mean da ily air temp erature ( o F ) a veraged ov er the w eek and a cross the w eather stations. The wind gust v ariable is the maximal instantaneous wind sp eed (mph) measured during the 10 -min ute p erio d just prior to the obser- v ationa l time. Wind gusts are measured ev ery hour and a v erag ed o ve r the w eek and across the w eather stations. The effect of fo g/frost is captured by a dumm y v ariable that is equal to one if and o nly if t he difference b et w een air and dewpoint tempera t ures do es not exceed 5 o F (in this case frost can form if the dewp oin t is b elow the freezing p oint 32 o F , and fog can form otherwise). The fog/frost dummies are calculated for ev ery hour and are a ve ra ged o v er the we ek and across the w eather stations. Finally , visibilit y distance v ariable is the harmonic mean of hourly visibility distances, whic h are measured in miles ev ery hour and a re av erag ed ov er the w eek and across the w eather stations. 18 F r om the results give n in T able 5 w e find t ha t for 1- vehic le acciden ts o n a ll high-sp eed roads (in terstate highw ays, US routes, state routes and coun t y roads), the less frequen t state s t = 1 is p ositiv ely correlated with extreme temp eratures (low during win ter and high during summer), ra in precipitations and sno wfalls, strong wind gusts, fo gs and frosts, lo w visibility distances. It 16 Here and b elo w we calculate we ighted correlation co efficien ts. F or v ariable P ( s t = 1 | Y ) ≡ E ( s t | Y ) we use weigh ts w t in versely p rop ortional to the p osterior standard deviations of s t . That is w t ∝ min { 1 / std( s t | Y ) , median[1 / std( s t | Y )] } . 17 Sno w fall and precipitation amoun ts are w eakly related with eac h other b ecause sno w densit y ( g/cm 3 ) can v ary by more than a factor of ten. 18 The harmonic mean ¯ d of distances d n is calculated as ¯ d − 1 = (1 / N ) P N n =1 d − 1 n , assuming d n = 0 . 25 miles if d n ≤ 0 . 25 m iles. 17 is reasonable to expect that roa dwa y safet y is differen t during bad w eather as compared to b etter w eather, resulting in the tw o-state nature of roadwa y safet y . The r esults of T able 5 suggest that Mark ov switc hing for road safety on streets is v ery differen t from switc hing on all other roadw ay classes. In particular, the states of roadw a y safet y on streets exhibit lo w correlation with states on other ro ads. In addition, only streets exhibit Mark ov switc hing in the case of 2-v ehicle acciden ts. Finally , states of roadw a y safety on streets sho w little correlation with w eat her conditions. A p ossible explanation of these differences is that streets are mostly lo cated in urban areas and they hav e traffic mov ing at speeds lo wer that t ho se on other roads. Next, w e consider the estimation r esults for the stationar y unconditional prob- abilities ¯ p 0 and ¯ p 1 of states s t = 0 a nd s t = 1 fo r MSML mo dels (see Section 2). In the cases of 1-v ehicle acciden ts on in terstate highw ays , US routes and state routes these transition probabilities are listed in lines “ ¯ p 0 and ¯ p 1 ” of T ables 1– 3. In the cases of 1-v ehicle acciden ts on coun ty roads and 1- and 2- v ehicle acciden ts on streets refer to Malyshkina (2008). W e find that the ra t io ¯ p 1 / ¯ p 0 is appro ximately equal to 0 . 46, 0 . 1 3 , 0 . 74, 0 . 25, 0 . 65 a nd 0 . 36 in the cases of 1- v ehicle acciden ts on in terstate high w ay s, US routes, state routes, coun ty roads, streets, and 2- v ehicle acciden ts on streets resp ectiv ely . Th us for some roadw ay-class-accide nt-t yp e comb inations (for example, 1-ve hicle acciden ts on US routes) the less frequen t state s t = 1 is quite rare, while for other combi- nations (for example, 1-ve hicle acciden t s on state routes) state s t = 1 is only sligh tly less fr equen t than state s t = 0. Finally , w e set mo del parameters ( β - s) t o their p osterior means, calculate the probabilities of fatalit y and injury outcomes by using Eq. (3) and av erage these probabilities o ve r all v a lues of the explanatory v ariables X t,n observ ed in the data sample. W e compare these proba bilities across the t wo states of roadw ay safet y , s t = 0 and s t = 1, f or MSML mo dels [refer to lines “ h P ( i ) t,n i X ” in T ables 1 – 3 and to Malyshkina (2008)]. W e find that in man y cases these a ve raged pro ba bilities o f fatality and injury outcomes do not differ ve r y signif- ican tly across the t w o states of roadw a y safet y (the only significan t differences are for fatality probabilities in the cases of 1-v ehicle acciden ts on US routes, coun ty roads a nd streets). This means that in many cases states s t = 0 and s t = 1 are appro ximately equally dangerous as far as acciden t sev erity is con- cerned. W e discuss this result in the next section. 18 5 Conclusions In this study w e found that t wo states of roadw ay safety and Mark ov switc h- ing m ultinomial logit (MSML) mo dels exist for sev erit y of 1- v ehicle a cciden ts o ccurring on high-sp eed roads (in terstate high w ays , US rout es, state routes, coun ty roads), but not f or 2-vehic le acciden ts on high-sp eed roads. One o f p os- sible explanations of this result is that 1- and 2-v ehicle a ccide nts ma y differ in their nature. F or example , on one hand, sev erity of 1- v ehicle acciden ts may frequen tly be determined b y driver-related factors (sp eeding, falling a sleep, driving under the influence, etc). Driv ers’ b eha vior mig ht exhibit a tw o-state pattern. In particular, drivers might b e o v erconfident and/or ha ve difficulties in a djustmen ts to bad we a ther conditions. On the other ha nd, sev erit y of a 2-v ehicle acciden t migh t crucially dep end on the actual ph ysics inv olv ed in the collision b etw een the t wo cars (for example, head-o n and side impacts are more dangerous than rear-end collisions). As far as slo w-sp eed streets are con- cerned, in this case b oth 1- and 2-v ehicle acciden t s exhibit t wo-state natur e for their sev erit y . F urther studies are needed to understand these results. In this study , the imp ortant result is that in all cases when tw o states of roadw a y safet y exist, the tw o-state MSML mo dels provid e muc h sup erior statistical fit for acciden t sev erity outcomes as compared to the standard ML mo dels. W e found that in many cases states s t = 0 and s t = 1 are appro ximately equally dangerous as f ar as acciden t sev erity is concerned. This result holds despite the fact that state s t = 1 is correlated with adv erse weather conditions. A lik ely and simple explanation of this finding is that during bad weather b oth n umber of serious a ccide n ts (fatalities and injuries) and num b er of minor acciden ts (PDOs) increase, so that their relative fraction sta ys appro ximately steady . In addition, most drivers a re rational and they a r e lik ely take some precautions while driving during bad w eather. F rom the results presen ted in P ap er I w e kno w that the total num b er of a cciden ts significan t ly increases during adv erse w eather conditions. Th us, driver’s precautions are pro bably not sufficien t to av oid increases in acciden t rates during bad weather. References Ab del-A t y , M., 2003. Analysis of drive r injury sev erit y lev els at multiple lo- cations using ordered probit mo dels. Journal of Safet y Researc h 34(5 ), 597 - 603. Carson, J., Mannering, F .L ., 2001 . The effect o f ice w arning signs on ice- acciden t frequencies and sev erities. Acciden t Analysis and Prev en tion 33(1), 99-109. Chang, L.- Y., Mannering, F.L., 1999. Analysis of injury sev erit y and v ehicle 19 o ccupancy in truc k- and non-t ruc k-in v olv ed acciden ts. Accide nt Analysis and Prev ention 31(5) , 579- 592. Co w a n, G., 1998. Stat istical Data Analysis. Clarendon Press, Oxford Univ. Press, USA Duncan, C., Khatta k, A., Council, F., 19 9 8. Applying the ordered probit mo del to injury sev erity in truck-passe ng er car rear-end collisions. T ransp ortation Researc h Record 16 35, 63-71 . Eluru, N., Bhat, C., 2007. A joint econometric analysis of seat b elt use and crash-related injury sev erity . Acciden t Analysis and Prev ention 39 (5), 103 7- 1049. Kass, R.E., Raftery , A.E., 1995. Bay es F actors. Journal of t he American Sta- tistical Asso ciat io n 9 0(430), 773-795. Khattak, A., 2001. Injury sev erit y in m ulti-vehic le r ear-end crashes. T rans- p ortation R esearch R ecord 17 46, 59 -68. Khattak, A., P a wlovic h, D., Souleyrette, R., Hallmark and, S., 2002. F actors related to more sev ere older drive r traffic crash injuries. Journal o f T rans- p ortation Engineering 128(3), 243-24 9. Khorashadi, A., Niemeier, D., Shank ar V., Mannering F .L., 2005 . Differences in rural and urban driver-injury sev erities in acciden ts in volving large truc ks: an exploratory analysis. Acciden t Analysis a nd Preve ntion 37(5), 910-9 21. Ko c k elman, K., Kw eon, Y.-J., 20 02. Driv er Injury Sev erit y: An application of ordered probit mo dels. Acciden t Analysis and Prev en tio n 34( 3 ), 313 - 321. Kw eon, Y.-J., Ko ck elman, K., 2003. Ov erall injury risk to different driv ers: com bining exp osure, frequency , and sev erity mo dels. Acciden t Analysis a nd Prev en tion 35(4), 414-45 0. Lee, J., Mannering, F.L., 2002 . Impact o f ro adside features on the f r equency and sev erity o f run-o ff-roadw ay acciden ts: an empirical analysis. Acciden t Analysis and Prev en tion 34(2), 149-1 6 1. Maher M. J., Summersgill, I., 1996. A comprehensiv e metho dology for the fitting of predictiv e acciden t mo dels. Accid. Anal. Prev. 28(3), 281-296 . Malyshkina, N.V., 2006. Influence of sp eed limit on roadwa y safet y in Indiana. MS thesis, Purdue Univ ersit y . Malyshkina, N. V., 2008. Mark o v switc hing mo dels: an application of to r o ad- w ay safet y . PhD thesis, Purdue Univ ersit y . h ttp://arxiv.org/abs/080 8.1448 Malyshkina, N.V., Mannering, F.L., T arko, A.P ., 2 008. Mark ov switc hing mo d- els: an application to Mark ov switc hing negativ e binomial mo dels: an appli- cation to v ehicle acciden t frequencies . Accepted for publication in Acciden t Analysis and Prev en tion. h ttp:// a rxiv.org/abs/0811.1606 Milton, J., Shank ar, V., Mannering, F.L., 2008. High w ay acciden t sev erities and the mixed log it mo del: an explorato ry empirical analysis. Acciden t Analysis and Prev en tion 40(1), 260-2 6 6. O’Donnell, C., Connor, D., 1996. Predicting the sev erit y of moto r v ehicle ac- ciden t injuries using mo dels of ordered m ultiple choice. Acciden t Analysis and Prev ention 28(6) , 739- 753. Press, W. H., T euk olsky , S. A., V etterling, W. T., Flannery B. P ., 2007. Nu- 20 merical Recip es 3 rd Edition: The Art of Scientific Computing. Cam bridge Univ. Press, UK. Sa volainen, P ., Mannering, F.L., 2007. Probabilistic mo dels of motorcyclists’ injury sev erities in single- and m ulti-ve hicle crashes. Acciden t Analysis and Prev en tion 39(5), 955-96 3. Shank ar, V., Mannering, F.L., 199 6. An exploratory m ultinomial logit ana lysis of single-v ehicle motorcycle acciden t sev erity . Journal of Sa fet y Researc h 27(3), 183-194. Shank ar, V., Mannering, F.L., Barfield, W., 1996. Statistical analysis of ac- ciden t sev erity on rura l freew ays. Acciden t Analysis a nd Prev en t ion 28(3), 391-401 . Tsa y , R. S., 20 02. Analysis of financial time series: financial econometrics. John Wiley & Sons, Inc. Ulfarsson, G ., Mannering, F .L., 2004. Differences in male and female injury sev erities in sp ort- ut ility vehic le, miniv an, pic kup and passe ng er car acci- den ts. Acciden t Analysis and Prev en tion 36( 2 ), 135 - 147. W ashington, S.P ., Ka rlaftis, M.G., Mannering, F.L., 2003. Statistical and econometric metho ds for transp ortation data a nalysis . Chapman & Hall/CR C. W o o d, G . R., 2002. Generalised linear acciden t mo dels and go o dness of fit testing. Accid. Anal. Prev. 34, 417-427. Y amamot o, T., Shank ar, V., 2004. Biv aria te ordered-respo nse probit mo del of driv er’s and passenger’s injury sev erities in collisions with fixed ob jects. Acciden t Analysis and Prev ention 36(5), 869- 876. 21 T ab le 4 Summary statistics of roadw a y acciden t c h aracte r istic v ariables V ariable Description h P ( i ) t,n i X probability of i th sev er i t y outcome a veraged ov er al l v alues of explanatory v ariables X t,n p 0 → 1 Marko v transition probability of jump from state 0 to state 1 as time t increases to t + 1 p 1 → 0 Marko v transition probability of jump from state 1 to state 0 as time t increases to t + 1 ¯ p 0 and ¯ p 1 unconditional pr obabilities of states 0 and 1 # free par. tot al num b er of fr ee model co efficien ts ( β -s) a veraged LL posterior av erage of the log-li k eliho od (LL) max( LL ) for M LE it is the maximal v alue of LL at conv ergence; for Bay esian-MCM C estimation it is the maximal observ ed v alue of LL during the M CMC simulations marginal LL logarithm of marginal likelihoo d of data, ln[ f ( Y |M )], given mo del M max(PSRF) maximu m of the p oten tial scale reduction factors (PSRF) calculated separately for all con tinuous m odel parameters, PSRF is close to 1 for conv erged MC M C ch ains MPSRF m ultiv ariate PSRF calculated jointly f or all par ameters, close to 1 for conv erged MC M C accept. r ate a verage rate of acceptance of candidate v alues dur i ng Metrop olis-Hasting MCM C draws # observ. n umber of observ ations of acciden t severit y outcomes av ailable in the data sampl e age0 ”age of the driver at fault is < 18 yea rs” indicator v ariable (dumm y) age0o ”age of the oldest driver inv olve d into the acciden t is < 18 years” indicator v ariable cons ”const ruction at the acciden t l ocation” indicator v ariable curv e ”r oad is at curv e” indicator v ariable dark ” dar k time with no street l igh ts” indicator v ariable darklamp ”dark AN D street lights on” i ndicat or v ariable da y ”da yli gh t” indicator v ariable da yt ” da y hours: 9:00 to 17:00” indicator v ariable driv ”road median is driv able” indicator v ariable driver ”primar y cause of the acciden t is driver-related” indicator v ariable dry ”roadw ay surface is dry” indicator v ariable en v ”primary cause of the accident is en vi ronmen t-related” indicator v ariable fog ”fog OR smoke OR smog” indicator v ariable hl10 ”help arr iv ed in 10 mi n utes or less after the crash” indicator v ar iable hl20 ”help arr iv ed in 20 mi n utes or less after the crash” indicator v ar iable Ind ”license state of the ve hicle at fault is Indiana” indicator v ariable int ercept ”consta n t term (intercept )” quan titativ e v ariable jobend ”after work hours: from 16:00 to 19:00” indicator v ariable light ”da yli gh t OR street lights are l i t up if dark” indicator v ariable maxpass ”the l argest num b er of o ccupa nts in all v ehicles i n volv ed” quantita tive v ari able mm ”t wo male dri v ers are inv olved ” indicator v ariable (used only if a 2-vehicle accident ) morn ”morning hours: 5:00 to 9:00” indicator v ariable moto ”the vehicle at f ault is a motorcycle” indicator v ariable 22 T ab le 4 (Con tinued) V ariable Description nigh ”late nigh t hours: 1:00 to 5:00” indicator v ar iable nocons ”no construction at the acciden t lo cation” indicator v ariable no jun ”no road j unct i on at the acciden t lo cation” i ndicato r v ariable nonroad ”non-roadw ay crash (parking l ot, etc.)” indicator v ariable nosig ” no any traffic con tr ol device f or the ve hi cle at fault” i ndicato r v ariable olddrv ”the driver at fault is older than the other driver” indicator v ar. (if a 2-vehicle accident ) oldv age ”age (in y ears) of the oldest veh icle inv olved ” indicator v ariable othUS ”license state of the vehicle at fault is a U. S. state except Indiana and its neighboring states (IL, KY, OH, MI)” indicator v ar iable precip ”precipitation: r ain OR snow OR sl eet OR hail OR freezing rain” i ndicato r v ariable priv ”road trav eled by the vehicle at fault is a pri v ate drive” indicator v ariable r21 ”road trav eled by the vehicle at fault i s tw o-lane AND one-wa y” indicator v ariable rmd2 ”road trav eled by the ve hi cle at fault is multi-lane AND divided tw o-wa y” indicator v ar. singSUV ”one of the tw o v ehicles inv olved is a pickup OR a v an OR a sp ort utility ve hi cle” indicator v ar iable (used only i f a 2-vehicle acciden t) singTR ”one of the t wo vehicles is a truck OR a tractor” indicator v ar . (if a 2-vehicle accident ) slush ”roadwa y surface is co vered by sno w/sl ush” indicator v ariable sno w ”snowing weath er” indicator v ariable str ”road is straight” indicator v ariable sum ”summ er season” indicator v ar iable sund ”Sunda y” indicator v ariable thda y ”Thu rsday” indicator v ar i able v age ”age (in y ears ) of the ve hicle at fault” quantitativ e v ariable v eh ”primary cause of acciden t is vehicle-related” i ndicat or v ariable v oldg ”the vehicle at fault is m ore than 7 years old” indicator v ariable v oldo ”age of the ol dest vehicle inv olved is more than 7 years” indicator v ariable wa ll ”road median is a wall” indicator v ariable wa y4 ”acciden t l ocation is at a 4-wa y intersection” indicator v ariable wint ”win ter season” indicator v ar i able X 12 ”road type” indicator v ariable (1 if urban, 0 if rural) X 27 ”n umber of o ccupa nt s in the veh i cle at fault” quan titative v ariable X 29 ”speed l i mit” quan titativ e v ar. (used if known and the same for all ve hicles inv olved ) X 33 ”at least one of the vehicles i n volv ed was on fire” i ndicat or v ariable X 34 ”age (in yea rs) of the dri v er at fault” quan titative v ariable X 35 ”gender of the driver at fault” i ndicato r v ariable (1 if female, 0 if m ale) 23 T ab le 5 Correlations of the p osterior pr ob ab ilities P ( s t = 1 | Y ) with eac h other and with w eather-condition v ariables (for the MSML m o del) 1-ve hi cle, 1-vehicle, 1-vehicle, 1-v ehicle, 1- vehicle, 2-v ehicle, int erstates US r oute s state r oute s count y roads streets streets 1-ve hi cle, interstate s 1 0 . 418 0 . 293 0 . 606 − 0 . 013 − 0 . 173 1-ve hi cle, US routes 0 . 41 8 1 0 . 263 0 . 688 − 0 . 070 − 0 . 155 1-ve hi cle, state routes 0 . 293 0 . 263 1 0 . 409 − 0 . 047 − 0 . 035 1-ve hi cle, count y roads 0 . 606 0 . 688 0 . 409 1 − 0 . 022 − 0 . 051 1-ve hi cle, streets − 0 . 013 − 0 . 070 − 0 . 047 − 0 . 022 1 0 . 115 2-ve hi cle, streets − 0 . 173 − 0 . 155 − 0 . 035 − 0 . 051 0 . 115 1 All year Precipitation (inc h) − 0 . 139 − 0 . 060 0 . 096 − 0 . 037 0 . 067 0 . 146 T emp erature ( o F ) − 0 . 606 − 0 . 439 − 0 . 234 − 0 . 665 0 . 231 0 . 220 Sno wf all (inch) 0 . 479 0 . 635 0 . 31 9 0 . 723 0 . 003 − 0 . 100 > 0 . 0 (dumm y) 0 . 695 0 . 412 0 . 382 0 . 695 − 0 . 142 − 0 . 131 > 0 . 1 (dumm y) 0 . 532 0 . 585 0 . 328 0 . 847 − 0 . 046 − 0 . 161 Wind gust (mph) 0 . 108 0 . 100 0 . 087 0 . 206 0 . 164 0 . 051 F og / F rost (dumm y) 0 . 093 0 . 164 0 . 19 3 0 . 167 0 . 047 0 . 119 Visibili t y distance (mile) − 0 . 228 − 0 . 221 − 0 . 172 − 0 . 298 − 0 . 019 − 0 . 081 Win ter (Nov ember - Mar c h) Precipitation (inc h) − 0 . 134 − 0 . 037 0 . 027 − 0 . 053 0 . 065 0 . 356 T emp erature ( o F ) − 0 . 595 − 0 . 479 − 0 . 397 − 0 . 735 − 0 . 008 0 . 236 Sno wf all (inch) 0 . 439 0 . 592 0 . 37 5 0 . 645 0 . 157 − 0 . 110 > 0 . 0 (dumm y) 0 . 596 0 . 282 0 . 475 0 . 607 0 . 115 − 0 . 142 > 0 . 1 (dumm y) 0 . 445 0 . 518 0 . 370 0 . 789 0 . 112 − 0 . 210 Wind gust (mph) 0 . 302 0 . 134 0 . 122 0 . 353 0 . 237 0 . 071 F rost (dummy) 0 . 537 0 . 544 0 . 44 0 0 . 716 0 . 052 − 0 . 225 Visibili t y distance (mile) − 0 . 251 − . 304 − 0 . 249 − 0 . 380 − 0 . 155 − 0 . 109 Summer (May - September) Precipitation (inc h) 0 . 000 0 . 006 0 . 259 0 . 096 0 . 047 − 0 . 063 T emp erature ( o F ) 0 . 179 0 . 149 0 . 113 0 . 037 0 . 062 0 . 155 Sno wf all (inch) – – – – – – > 0 . 0 (dumm y) – – – – – – > 0 . 1 (dumm y) – – – – – – Wind gust (mph) − 0 . 126 − . 009 0 . 164 0 . 029 0 . 121 0 . 034 F og (dummy) 0 . 203 0 . 193 0 . 27 5 0 . 101 − 0 . 076 − 0 . 011 Visibili t y distance (mile) − 0 . 139 − 0 . 124 − 0 . 062 − 0 . 009 0 . 077 − 0 . 094 24