Dates July 2010, Revised November 2010, Final Revised March Total Words 7,462 (5,962 Words, 5 Tables, 1 Figure) *Corresponding author

Transcription

1 Investgatng the Effects of Underreportng of Crash Data on Three Commonly Used Traffc Crash Severty Models: Multnomal Logt, Ordered Probt and Mxed Logt Models Fan Ye* Graduate Research Assstant Zachry Department of Cvl Engneerng Texas A&M Unversty 3136 TAMU College Staton, TX Tel: (979) Emal: Domnque Lord Assocate Professor Zachry Department of Cvl Engneerng Texas A&M Unversty 3136 TAMU College Staton, TX Tel. (979) Fax. (979) Emal: Dates July 2010, Revsed November 2010, Fnal Revsed March 2011 Total Words 7,462 (5,962 Words, 5 Tables, 1 Fgure) *Correspondng author

2 Ye & Lord 1 ABSTRACT Although a lot of work has been devoted to developng crash severty models to predct the probabltes of crashes for dfferent severty levels, very few studes have consdered the underreportng ssue n the modelng process. Inferences about a populaton of nterest wll be based f crash data are treated as a random sample comng from the populaton wthout consderng the dfferent unreported rates for each crash severty level. The prmary objectve of ths study amed at examnng the effects of underreportng for three commonly used traffc crash severty models: multnomal logt (MNL), ordered probt (OP) and mxed logt (ML) models. The objectve was accomplshed va a Monte-Carlo approach usng smulated and observed crash data. The results showed that n order to mnmze the bas and reduce the varablty of the model, fatal crashes should be set as the baselne severty for the MNL and ML models whle, for the OP models, the rank for the crash severty should be set from fatal to propertydamage-only (PDO) n a descendng order. In addton, none of the three models was mmune to ths underreportng ssue. The results also showed that when the full or partal nformaton about the unreported rates for each severty level s known, treatng crash data as outcome-based samples n model estmaton, va the Weghted Exogenous Sample Maxmum Lkelhood Estmator (WESMLE), dramatcally mprove the estmaton for all three models compared to the result produced from the Maxmum Lkelhood estmator (MLE).

3 Ye & Lord 2 1. INTRODUCTION Over many years now, a lot of work has been devoted on the development and applcaton of statstcal models for analyzng motor vehcle crashes. It s generally agreed that these statstcal models are classfed nto two categores: crash count and crash severty models (see 1 and 2 for a thorough revew of exstng models). The former (e.g., Posson and Posson-gamma models) estmate the probablty of observng the number of crashes for dfferent severty levels. Crash severty models (e.g., dscrete outcome models such as logt or probt models), on the other hand, are ntended to estmate the probablty for a crash to fall nto one of the severty levels condtonal on the fact that the crash has occurred. Crash count and severty models are usually based on polce reported crash data and are used for nvestgatng crash occurrences that are related to hghway desgn features, envronmental condtons and traffc flow among others. However, t has been well documented that crashes are often unreported, partcularly those assocated wth lower severty levels (3, 4, 5). Ths underreportng ssue can yeld to sgnfcant bases when used to predct the probablty of crash severty (3). There are numerous studes that have nvestgated factors that nfluence the unreported rates for dfferent crash severty levels. However, very few studes have thoroughly nvestgated underreportng ssues related to crash model development. The prmary objectve of ths study was to examne the effects of underreportng on three commonly used traffc crash severty models: multnomal logt (MNL), ordered probt (OP) and mxed logt (ML) models. More specfcally, ths study nvestgated how each of these models performs for dfferent unreported rates. A secondary objectve conssted n quantfyng how the outcome-based samplng method, va the Weghted Exogenous Sample Maxmum Lkelhood Estmator (WESMLE), could account for specfc underreportng condtons when the transportaton safety analysts had a full or partal knowledge for dfferent severty unreported rates. The study objectves were accomplshed va a Monte- Carlo approach usng smulated and observed crash data. Ths paper s dvded nto fve sectons. The second secton provdes background nformaton about the underreportng ssue n crash data and related to crash severty modelng, as well as the model estmaton methods that can account for underreported data. The thrd secton descrbes the results for the three models for varous unreported rates usng smulated data. The fourth secton presents the modelng results for the three models usng observed crash data. The ffth secton summarzes the key fndngs of ths study. 2. BACKGROUND Ths secton brefly summarzes the lterature on underreportng ssues assocated wth crash data and crash severty modelng, and then presents model estmaton methods that can be used to account for underreported crash data. 2.1 Underreported crash data About twenty years ago, Hauer and Hakkert (3) ponted out that not all traffc crashes were reportable and not all reportable crashes were n fact reported. Ths can lmt the ablty to manage road safety, snce most of the analyses related to road safety are based on reported crash data. The analyss of underreported crash data would lead to a based estmate when crash predcton models are used, thus resultng n neffectve treatments when the models are

4 Ye & Lord 3 appled for such purpose. Havng realzed the underreportng ssue n crash data, some researchers began to study ths topc n greater depth (3-15). These studes revealed that crashes were underreported n all ndustralzed countres, but the unreported rate was worse n developng countres. The probablty of reportng was found to be nfluenced by the crash severty, age of the vctm, role of the vctm (whether the vctm s the drver, the passenger, or etc.), and number of vehcles nvolved (3). Underreported data tend to produce based estmatons for crash count models and crash severty models. However, underreportng s more crtcal for crash severty models because the reported rates for varous severty categores are dfferent. Crashes wth a lower severty such as property-damage-only (PDO) collsons are more lkely to be unreported whch leads to the over-representaton of crashes wth hgher severty and underrepresentaton of crashes wth lower severty. It has been wdely accepted that fatal crashes have the hghest reportng rate and PDO crashes have the lowest reportng rate. After revewng 18 studes n whch researchers examned polce, hosptal and nsurance sources for common entres, Hauer and Hakkert (3) concluded that the unreported rates were 5 percent for fatalty, 20 percent for njures requrng hosptalzaton and perhaps 50 percent for all njures. In a comprehensve meta-analyss, based on 49 studes n 13 countres, Elvk and Myssen (4) found values equal to 5 percent for fatal njures, 30 percent for serous njures, 75 percent for slght njures, and 90 percent for very slght njures. Accordng to Blncoe et al. (16), up to 25 percent of all mnor njures and almost 50 percent of PDO crashes were lkely to be non-reported because most drvers dd not want to have the polce nvolved (or other authortatve fgures) due to nsurance concerns or legal repercussons. Only a lmted number of studes have nvestgated the effects of underreportng n both the crash count model (17, 18) and crash severty model (19). As a result, some new approaches have been proposed to account for underreportng n tradtonal crash model analyses. The next secton dscusses prevous research on underreportng n crash severty modelng. 2.2 Underreportng n crash severty modelng The nconsstent unreported rates among dfferent severty levels leads to based results, whch can overestmate the probablty of hgher severty crashes and underestmate that of lower severty crashes, partcularly PDOs. In addton, underreportng causes based parameters whch skew the nferences on the effects of key explanatory varables n predcton models. So far, only one study has been found that deals wth modelng crash severty and underreported data. Yamamoto et al. (19) nvestgated the effects of underreportng on parameter estmaton for the ordered probt model and the sequental bnary probt model. In ther study, the results ndcated that the estmates of the explanatory varables and parameter elastctes of both models could be sgnfcantly based f underreportng was not consdered. In addton, the researchers regarded traffc crash data as response-based samples wth unknown populaton shares of the njury severtes, and used a pseudo-lkelhood functon (20, 21) to account for the effects of underreportng on parameter estmaton for both models. The populaton shares of each severty category were estmated for each model whch provded nsghts on the levels of underreportng for each crash severty level. However, the valdaton and effcency of the methods were not confrmed. Meanwhle, snce only one set of crash

5 Ye & Lord 4 data was appled to the models, there was no nformaton attaned about the model effects on dfferent combnatons of unreported rates for each crash severty category. 2.3 Model estmaton methods for underreported crash data Crash severty models are usually estmated based on random samplng wthout consderng the underreportng n crash data. However, because of the unque underreportng characterstcs n crash data (unreported rates are dfferent accordng to the crash severty category), crash data should be treated as outcome-based or choce-based samples nstead of random samples from the populaton. Wthout consderng the underreportng ssue for the model, model estmaton results would defntely be based (19). Though t s rare to treat crash data as outcome-based samples, choce-based samples are commonly used n other areas of research, such as transportaton economcs. Chocebased samples are usually collected by stratfyng the data to obtan better nformaton about alternatves that are nfrequently chosen n the populaton when a random sample wll not fnd enough samples for effectve statstcal analyss (22). There are several methods that have been developed by economsts snce 1977 to handle choce-based samples, as summarzed n Ye s dssertaton (23): Weghted Exogenous Sample Maxmum Lkelhood Estmator (WESMLE), Condtonal Maxmum Lkelhood Estmator, Full nformaton Maxmum Lkelhood Estmator, Weghted Generalzed Method of Moments, Bayesan WESMLE, Smoothed Maxmum Lkelhood Estmator, and Weghted Condtonal Maxmum Lkelhood Estmator. Among all the methods, though not completely effcent, WESMLE s consstent and easy to compute whch makes t the most wdely used method. In ths research, WESMLE wll be used for underreported crash data n three crash severty models. WESMLE s the maxmand of the weghted lkelhood functon where the weghts depend upon both the populaton share of each severty type (the fracton of each severty category n populaton) and the sample share of each severty type (the fracton of each severty level n an underreported dataset). By weghtng the observatons approprately, WESMLE makes the outcome-based samples behave asymptotcally as f they were random samples (24). The log-lkelhood for a WESMLE, as shown n Equaton (1), s equvalent to that of the maxmum lkelhood estmator (MLE) except that each traffc crash s weghted by the rato of the actual crash severty s populaton share Q, to the sample share H whch s the severty share for the underreported crash data. N Q Log-lkelhood for WESMLE = dn ( )ln P( xn, ) (1) H n1 Cn Where, N s the number of recorded crashes; C s the set of severty categores from whch ndvdual crash n belongs to, n the study, n C n = (K=fatal njury, A=ncapactatng njury, B=non-ncapactatng njury, C=possble njury, and O=PDO) whch wll be descrbed n secton 3; d n s an ndcator varable equal to 1 f ndvdual crash n belongs to severty level, and 0 otherwse;

6 Ye & Lord 5 x n s the vector of contrbutng factors assocated wth ndvdual crash n at severty category C n ; s a vector of the estmable parameters assocated wth contrbutng factors x n ; P, ) s the probablty of severty level belongng to gven the contrbutng factors, ( x n x n, and estmates. Dfferent models have dfferent probablty functons: For the MNL model, exp X n P( xn, ) (2) exp( X n ) For the ML model, exp X n P( xn, ) f ( ) d (3) exp( X n ) For the OP model, P( 1 xn, ) ( 1 X n ) P( xn, ) ( X n ) ( 1 X n ) (4) P( 5 xn, ) 1 ( 1 X n ) More detals of the model structures and probablty functons for all three models can be found n Ye s dssertaton (23). 3. ANALYSIS USING SIMULATED DATA In order to study the effects of underreportng on three models and verfy the effectveness of WESMLE for underreported data, a Monte-Carlo approach was developed to examne the underreportng usng smulated and observed crash data. By repeatng the samplng to produce estmates more clustered around the true values, a Monte-Carlo approach s an deal way to verfy the underreportng effects on the three models snce we create the data wth the knowledge of true values of estmators and true response functons. In addton, varous data wth dfferent unreported rates could be created enough tmes. Thus, the bas can be evaluated by comparng the model estmaton wth the true values. 3.1 Smulaton desgn Snce the crash data have fve severty categores, the number of parameters to nvestgate s very large. The crash data are categorzed, n ths study, as K, A, B, C, and O. To smplfy the analyss, one covarate randomly generated from the standard normal dstrbuton was ntroduced for all three models. In addton, fve outcomes (denoted as levels 1 to 5) were used to replcate the fve severty categores. In addton, covarates were kept the same values no matter what crash severty the target observaton was snce all the varables ncluded n a crash severty model are observaton-related rather than outcome-related (25). The parameter values chosen for the three models were based on the assumpton that the results would not be affected sgnfcantly by dfferent values of the parameters. For the MNL model, the parameters of the covarate were kept the same wth a value equal to 1 for each level, =1. The constant parameter was equal to 0, 0.5, 1, 1.5 for

7 Ye & Lord 6 levels 1 to 4 (level 5 was the baselne outcome wth ). The ndependent varable x for each level was drawn from a normal dstrbuton wth mean equal to -2 and a varance equal to 1. The error term for each level was drawn ndependently from a Type I extreme value dstrbuton by obtanng draws from a unform random dstrbuton and applyng the followng transformaton ln[ ln( u)], where u was a random number drawn from the unform dstrbuton between 0 and 1. Thus, they gave the followng proportons 5.7%, 9.4%, 15.4%, 25.4%, and 44.1% for levels 1 to 5 respectvely n the populaton. For the OP model, the varable parameter was equal to 1 for each level, x was drawn from a normal dstrbuton wth a mean equal to 2.2 and a varance equal to 1, and threshold varable was 0, 0.8, 1.5, 2.4 for levels 1 to 4 (for keepng the populaton ratos of each outcome as close as those for the MNL model). The error term was normally dstrbuted for each level. Thus, they gave the followng proportons 6.0%, 10.1%, 15.0%, 24.6%, and 44.3% for levels 1 to 5, respectvely. For the ML model, the steps for generatng the dataset were very smlar to those used n generatng the dataset for the MNL model. The only dfference was that the ndependent varable was assumed to have randomness n the parameter for level 1, whch followed a normal dstrbuton (mean=1, varance=1). The populaton ratos for each level were 14.1%, 8.7%, 14.3%, 23.6%, and 39.3% for levels 1 to 5. The datasets generated for three models based on the true parameters were treated as the complete datasets,.e., the populaton. The underreported dataset were replcated by randomly removng some data accordng to the desgned unreported rates. In order to generate suffcent samples even after the random removal of some data, the orgnal sample sze was set to be 50,000. In other word, the complete datasets had 50,000 observatons for three models and all the removed observatons were consdered to be the unreported ones. Datasets for each model were repeatedly drawn 100 tmes for each unreported rate desgnated accordng to the desgned true parameter values of the model. Based on the 100 estmated models, the bas of each parameter was calculated as Bas E( ˆr ) - baselne, where r was the number of replcatons (r=100), and represented each parameter n the model (both constant parameters and varable parameters). The root-mean-square-error (RMSE) of 2 each parameter n a model was calculated usng the equaton RMSE Bas Var, and the total RMSE of all the varable parameters for each model was used to measure the underreportng effects snce t comprsed both the bas and varablty. As a summary, the whole process descrbed above about the Monte-Carlo analyss on underreportng for smulated data s shown n Fgure Smulaton results Scenaro 1 For Scenaro 1, the change n bas and varablty wth the ncrease of the unreported rates (fve unreported rates 5%, 10%, 20%, 40%, and 80% were smulated n each level) for the three models was examned n order to verfy how the number of unreported observatons nfluences these two tems. (Note: for the complete datasets, based on the desgned data for the MNL and OP models, the number of observatons for the outcome ncreased from levels 1 to 5; whle for the ML model, the number of observatons ranked from low to hgh: levels

8 Ye & Lord 7 2, 1, 3, 4 and 5, respectvely.) In addton, for each underreported dataset, WESMLE was used to verfy whether t could provde a good model estmaton based on the known unreported rates. After 100 repettons, summary statstcs such as mean and standard devaton of each parameter for a model could be calculated. Due to the space constrant, the results were not ncluded n the paper. In addton, the total RMSEs were compared across dfferent unreported rates for each level and for each model (see Table 1). There are four key fndngs for Scenaro 1: (1) For all three models, wth larger unreported rate, the total RMSE ncreased usng the MLE method. However, when WESMLE s used to take account of the underreportng ssue, consderng the varaton caused by the randomness n the ML model, the total RMSE remaned relatvely constant gven the change n unreported rate. (2) When the MLE was used for model estmaton (.e., wthout consderng the underreportng ssue n the data), the unreported data dd not show any clear effects on the total RMSE. Instead, for ether the MNL model or the ML model, wth the same unreported rate, smlar total RMSE values were observed for the parameters from levels 1 to 4, and whle a much larger value of total RMSE was found when level 5 contaned underreported data. Ths s reasonable snce level 5 was used as the baselne outcome n both the MNL and ML models. The probabltes for other levels (levels 1 to 4) are based on the baselne outcome, so underreportng of baselne outcome would cause more bas n the lkelhood functon than other levels and accordngly t leads to more bas n the model estmaton (see more explanatons n Ye s dssertaton (23)). Ths ndcates that when the MNL and ML models are used for model estmaton wth the MLE method, the analysts should avod choosng an outcome wth large unreported rate as a baselne level. (3) The OP model has a dfferent result (the largest total RMSE exsted when level 1 was underreported) from the other two models when outcomes were setup n an ascendng order (the outcomes were ranked from levels 1 to 5). In order to verfy whether the same unreported rate n the level wth the lowest order produces the largest total RMSE, the same generated datasets for the OP model were estmated agan, but n a descendng order ths tme (from levels 5 to 1). The total RMSE for each unreported rate was shown n Table 2. From ths table, wth the same unreported rate when level 5 was underreported, the total RMSE acheved the largest whch supports the dea that underreportng for the outcome wth the lowest rank caused the largest total RMSE. Thus, when the OP model s used for underreported data wth the MLE method, the analysts should avod rankng the outcomes n an order wth the lowest order havng the largest unreported rate. (4) The WESMLE method worked well no matter how large the unreported rates and unreported data were for each level for all three models. It gave a more accurate estmaton of parameter to the true value wth the total RMSE dramatcally decreased from that by MLE method Scenaro 2 Though WESMLE performs well for varous underreportng stuatons, the prerequste for usng the method s that the analysts know the actual unreported rate for each outcome, whch usually s not fully known for crash data. As shown n Equaton (1), WESMLE ncludes weghts n the log-lkelhood functon, whch are the rato of populaton share Q to the sample share H for each level. Actually, the rato of the weghts rather than the value of

9 Ye & Lord 8 weghts themselves make the estmated parameters dfferent, whch maxmze log-lkelhood functon of the WESMLE. The rato of the fve weghts could be easly calculated as below. Snce weght of level s: Q N N weght ( ) (5) H N * (1 rate ( )) [ N * (1 rate ( ))] Where, s the number of level n the populaton, and rate( ) s the unreported rate assumed for level. Then the rato of weght( ) for the fve levels s: : : : : rate1 rate2 rate3 rate4 rate5 If we have the full nformaton about the unreported rates for all fve levels, the above rato wll be the true rato of weghts: : : : : Trate1 Trate2 Trate3 Trate4 Trate5 Where, Trate( ) s the true unreported rate n level. Intutvely, the closer the weghts rato s to the true one, better the estmaton wll be when usng WESMLE (23). In order to llustrate ths dea, a smple example was evaluated. In the smulaton, the true unreported rate was desgned to be 40% n one of the fve levels, but assume that we do not know ths number and our best assumpton for t s 20% or 60%. The total RMSE usng these two unreported rates were calculated for three models, as shown n Table 3. For comparson, the estmaton results based on the MLE method wthout takng account of the true unreported rates were also lsted n ths table. Table 3 showed that the ncorrect unreported rates wth WESMLE ncreased the total RMSE compared to those when the true underreportng nformaton was used. However, t stll provded a better estmaton than those wthout consderng the underreportng n the data (.e., usng the MLE method). Furthermore, the ncorrect unreported rates do not refer to any random numbers used as unreported rates wth the WESMLE. When the assumed unreported rates shft the weghts rato nto another drecton (such as makng the weghts of fve levels n a reverse order as the true ones), t mght gve a larger bas than usng the MLE method alone. Some sense of the unreported rates for each level s defntely needed to get reasonable results usng WESMLE, even f t s not perfect. In addton, the tentatve dea was shown that an unreported rate of 20% had a lower total RMSE than the one equal to 60%. Thus, t supports the hypothess that the closer the weghts rato s to the true value, the better estmaton wll be usng WESMLE. 4. ANALYSIS USING OBSERVED CRASH DATA In Secton 3, we only ncluded one varable whch was assumed to be normally dstrbuted. In addton, all the data were generated separately for the three models. However, crash severty data have a large amount of varaton whch mght lead to dfferent patterns of parameter bas and varablty. Thus, we conducted further analyses usng observed crash data. The prmary data sources ncluded four years ( ) of traffc crash records provded by the Texas Department of Publc Safety (TxDPS) and the Texas Department of

10 Ye & Lord 9 Transportaton (TxDOT) general road nventory. The crash data focused on sngle-vehcle crashes nvolvng fxed objects that occurred on rural two-way hghways (excludng those occurrng at ntersectons). There were a total of 26,175 usable records n the database whch contan nformaton related to weather, roadway, drver, and vehcle condtons as well as the severty of the crash reported at the tme of the crash (same classfcaton as before). In ths dataset, there were 11,844 PDO (45.3%), 5,270 Inj. C (20.1%), 5,807 Inj. B (22.2%), 2,449 Inj. A (9.4%), and 805 Fatal (3.1%) crashes. Frst, usng the full crash dataset, the same three models (MNL, OP and ML) were developed and the model estmaton from the full dataset was consdered as the baselne condton for each model. The estmaton results from the three models were not ncluded here due to the space constrant, whch can be found n the dssertaton (23). Next, the underreported crash datasets were generated by randomly removng some crashes for specfc severty levels from the full dataset accordng to the desgned unreported rates. For smplcty, 30 underreported datasets were replcated based on a desgned unreported rate n crash data (rather than 100 used for the smulated data). By comparng the results wth the baselne condtons, the bas and varance of each parameter were calculated for each model upon whch the total RMSEs were computed as an ndex of underreportng. In addton, the same 30 generated underreported crash datasets for each desgned unreported rate were estmated agan for the three models, usng WESMLE. Smlar to Scenaro 1 descrbed above, two unreported rates 10% and 40% were establshed for each severty level. The unreported crash datasets were appled for three models usng both the MLE and WESMLE methods. The total RMSEs by each unreported rate were shown n Table 4. The OP model was estmated n both ascendng and descendng order to examne whether the order of severty level had effects on the total RMSE when crash data were underreported. Table 4 showed that the results were consstent wth the smulaton output n the prevous secton. For all three models, the larger unreported rates were assocated wth a hgher total RMSE value. Usng WESMLE wth the knowledge of the unreported rates, the total RMSE decreased for all underreportng stuatons for the three models. For the MNL model, when the baselne severty level (fatal or K) was underreported, the total RMSE acheved the largest value than the values attaned when other severty levels had the same unreported rate. For the ML model, though the total RMSE value for the PDO underreportng was slghtly larger than the baselne severty level (fatal) by the same unreported rate, the value of fatal underreportng s much larger than other severty levels (C, B, A). For crash data, as mentoned before, PDO crashes are more lkely to be unreported and fatal crash usually has the hghest reportng rate. Thus, when the MNL and ML models are used to predct the probablty of crash severty level, fatal should be set as the baselne outcome n order to mnmze the bas and varablty. For the OP model, comparng the total RMSE values usng the MLE method from descendng order (KABCO) and ascendng order (OCBAK), lower total RMSE values were obtaned for the underreportng n O, C, and B when the descendng order was used. Snce crash data have more serous underreportng ssue for lower severty crashes, usng descendng order provdes a better approach to reduce the bas and varablty n the estmaton of parameters for the OP model. The analyss descrbed above was based on only one severty level that was underreported. Thus, we further examned the bas and varablty of the estmated parameters when dfferent levels of unreported rates were used. The followng unreported

11 Ye & Lord 10 rates were used: 5%, 20%, 30%, 50%, and 75% for severty KABCO, respectvely. The total RMSE values from the MLE and WESMLE methods wth the knowledge of real unreported rates were lsted n Table 5. As expected, the WESMLE method dramatcally decreased the value of total RMSE compared to the MLE. It ndcates that the WESMLE not only works well when a sngle crash severty s underreported but also when multple severtes have dfferent unreported rates as long as the unreported rates are known (though not always the case for real data). As dscussed n Scenaro 2 above, we also examned the change n total RMSE when we used partal rather than perfect nformaton for the unreported rates. In ths case, nstead of usng 5%, 20%, 30%, 50%, and 75% for severty KABCO for the weght calculaton wth WESMLE, two hypothetcal examples were used. One assumed an unreported rate of 5% n fatal crashes (example 1), whle the other assumed the unreported rate for the PDO was 50% (example 2), wth keepng all other severty levels complete. The results were also shown n Table 5. Ths table llustrated that usng an unreported rate 50% n PDO crashes decreased the total RMSE than that from MLE for all three models, whle, usng an unreported rate 5% n fatal crashes ncreased the total RMSE. After verfyng the rato of the fve severty weghts, the above results were found to be reasonable. The true rato of weghts for KABCO that was used was : : : : : : : : Trate1 Trate2 Trate3 Trate4 Trate5 1 5% 1 20% 1 30% 1 50% 1 75% For the unreported rate of 5% n fatal crashes n example 1, the rato of weghts for KABCO was : : : : : : : : rate1 rate2 rate3 rate4 rate5 1 5% For the unreported rate of 50% n PDO crashes n example 2, the rato of weghts for KABCO was : : : : : : : : rate1 rate2 rate3 rate4 rate % It was obvous that usng unreported rate of 5% n fatal crashes shfted the weghts rato nto an opposte drecton where the weght of lower crash severtes should be larger than the fatal crashes due to the larger unreported rates for the lower crash severty levels. However, the unreported rate 50% n PDO crashes stll followed the same drecton as that of the true weghts rato, n whch the weght of PDO was larger than the other severty levels, though not as accurate. The fndngs here further showed what was found n Scenaro 2: the closer the weghts rato was to the true one, better the estmaton would be wth WESMLE. On the other hand, ncorrectly ncludng the unreported rates n the model estmaton for all three models mght lead to a worse model estmaton wth larger bas and varablty. Thus, t s mportant to formulate the weght of each severty level for a model the same rank as the true one among the fve severty levels. Wthout the full knowledge of the true unreported rates, one conservatve way s to only nclude the unreported rate for the PDO (the largest among all the severty levels) for the weght calculaton, assumng a reasonable unreported rate based on prevous research and as much knowledge as possble about the crash data used for estmatng the crash severty models.

12 Ye & Lord CONCLUSIONS AND RECOMMENDATIONS Ths paper amed at studyng the effects of underreportng on three commonly used traffc crash severty models. A secondary objectve conssted of quantfyng how the outcomebased samplng method n model estmaton, va WESMLE, can account for specfc underreportng condtons when transportaton safety analysts have full and partal knowledge for dfferent severty unreported rates. A Monte-Carlo approach usng smulated and observed crash data was utlzed for evaluatng the three models. The results of ths study showed that the analyss usng smulated and observed crash data acheved consstent results on the effects of underreportng for the three models wth and wthout accountng for the underreportng for each crash severty level. In order to mnmze the bas and reduce the varablty of the model, fatal crashes should be set as the baselne severty level for the MNL and ML models. For the OP model, the rank of the crash severty should be set from fatal to PDO n a descendng order. It should be ponted out that none of the three models was mmune to ths underreportng ssue. The results also showed that when the actual nformaton about the unreported rates of each severty level was known, the WESMLE method dramatcally mproved the estmaton for all three models compared to the result produced by the MLE whch dd not take nto account the underreportng ssue for crash data. However, for crash data, the unreported rate for each severty level s rarely known wth certanty. When partal or mperfect knowledge about unreported rates are avalable, the WESMLE stll gves better estmaton results than not consderng the underreportng n the data (va the MLE method), though the estmaton s not as robust as when the exact underreportng nformaton s obtanable. In addton, the closer the weghts rato s to the true value, better the estmaton wll be wth the WESMLE. It s the hope that the nformaton provded n ths paper wll be useful for transportaton safety analysts who are nterested n determnng factors that nfluence crash severty. REFERENCE (1) Lord, D., and F. Mannerng. The Statstcal Analyss of Crash-Frequency Data: A Revew and Assessment of Methodologcal Alternatves. Transportaton Research - Part A, Vol. 44, No. 5, 2010, pp (2) Savolanen, P.T., F.L. Mannerng, D. Lord, and M.A. Quddus. The Statstcal Analyss of Hghway Crash-Injury Severtes: A Revew and Assessment of Methodologcal Alternatves. Accdent Analyss & Preventon, n press (DOI: /j.aap ). (3) Hauer, E. and A.S. Hakkert. Extent and Some Implcatons of Incomplete Accdent Reportng. Transportaton Research Record, No.1185, Transportaton Research Board, Natonal Research Councl, Washngton, D.C., 1989, pp.10. (4) Elvk, R. and A.B. Mysen. Incomplete Accdent Reportng Meta-analyss of Studes Made n 13 Countres. Transportaton Research Record, No.1665, Transportaton Research Board, Natonal Research Councl, Washngton, D.C., 1999, pp

13 Ye & Lord 12 (5) Tsu, K.L., F.L. So, N.N Sze, S.C. Wong, and T.F. Leung. Msclassfcaton of Injury Severty among Road Casualtes n Polce Reports. Accdent Analyss & Preventon, Vol.41(1), 2009, pp (6) Hvoslef, H. Under-Reportng of Road Traffc Accdents Recorded by the Polce at the Internatonal Level. Publc Roads Admnstraton, Norway, (7) James, J.L., and K.E. Km. Restrant Use by Chldren Involved n Crashes n Hawa, Transportaton Research Record, No.1560, Transportaton Research Board, Natonal Research Councl, Washngton, D.C., 1996, pp (8) Stutts, J., and Hunter, W. Polce reportng of pedestrans and bcyclsts treated n hosptal emergency rooms. Transportaton Research Record, No.1635, Transportaton Research Board, Natonal Research Councl, Washngton, D.C., 1998, pp (9) Aptel I., L.R. Salm, F. Masson, A. Bourdé, G. Henron, and P. Erny. Road accdent statstcs: dscrepances between polce and hosptal data n a French sland. Accdent Analyss & Preventon, Vol.31, 1999, pp (10) Alsop, J., and J. Langley. Under-reportng of motor-vehcle traffc crash vctms n New- Zealand. Accdent Analyss & Preventon, Vol.33 (3), 2001, pp (11) Cryer, P.C., S. Westrup, A. C. Cook, V. Ashwell, P. Brdger, and C. Clarke. Investgaton of bas after data lnkage of hosptal admsson data to polce road traffc crash reports. Injury Preventon, Vol.7 (3), 2001, pp (12) Dhllon, P.K., A.S. Lghtstone, C. Peek-Asa, and J.F. Kraus. Assessment of hosptal and polce ascertanment of automoble versus chldhood pedestran and bcyclst collsons. Accdent Analyss & Preventon, Vol.33 (4), 2001, pp (13) Rosman, D.L. The Western Australan road njury database ( ): ten years of lnked polce, hosptal and death records of road crashes and njures. Accdent Analyss & Preventon, Vol.33 (1), 2001, pp.888. (14) Amoros, E., J-L Martn, and B. Laumon. Under-reportng of Road Crash Casualtes n France. Accdent Analyss & Preventon, Vol. 38, 2006, pp (15) Hauer E. The Frequency-severty Indetermnacy. Accdent Analyss & Preventon, Vol.38, 2006, pp (16) Blncoe, L., A. Seay, E. Zaloshnja, T..Mller, E. Romano, S.Luchter, and R.Spcer. The Economc Impact of Motor Vehcle Crashes, Publcaton DOT-HS Plans and Polcy,Natonal Hghway Traffc safety Admnstraton, (17) Kumara, S. P., and H. C. Chn. Applcaton of Posson Underreportng Model to Examne Crash Frequences at Sgnalzed Three-Legged Intersectons. Transportaton Research Record, No.1908, Transportaton Research Board, Natonal Research Councl, Washngton, D.C., 2005, pp (18) Ma J.. Bayesan Analyss of Underreportng Posson Regresson Model wth an Applcaton to Traffc Crashes on Two-lane Hghways. Presented at the 88th Annual Meetng of the Transportaton Research Board, Washngton, D.C., 2009.

14 Ye & Lord 13 (19) Yamamoto, T., J. Hashjb, and V. N. Shankar. Underreportng n Traffc Accdent Data, Bas n Parameters and the Structure of Injury Severty Models. Accdent Analyss & Preventon, Vol.40(4), 2008, pp (20) Cosslett, S.R., 1981a. Effcent estmaton of dscrete-choce methods. In: Mansk, C., McFadden, D. (Eds.), Structural Analyss of Dscrete Choce Data wth Econometrc Applcatons. MIT Press, Cambrdge, MA, pp (21) Cosslett, S.R., 1981b. MLE for choce-based samples. Econometrca, Vol.49, pp (22) Berlare M., and D. McFadden. The estmaton of generalzed extreme value models from choce-based samples. Transportaton Research Part B, Vol.42, 2008, pp (23) Ye, F. Investgatng the Effects of Underreportng of Crash Data on Three Commonly Used Traffc Crash Severty Models. Ph.D. Dssertaton. Zachry Department of Cvl Engneerng, Texas A&M Unversty, College Staton, TX, (24) Xe Y. and C. F. Mansk. The Logt Model and Response-based Samples. Socologcal Methods & Research, Vol.17(3), 1989, pp (25) Khorashad, A. Analyss of Drver Injury Severty Logt Models of Truck Involvement/Truck Causaton. Ph.D. Dssertaton. Unversty of Washngton. UMI Dssertaton Publshng, Seattle, WA, 2003.

15 Ye & Lord 14 LIST OF TABLES AND FIGURES Tables TABLE 1 Total RMSE by Dfferent Unreported Rate usng Smulated Data TABLE 2 Total RMSE for the OP model wth Outcomes n a Descendng Order usng Smulated Data TABLE 3 Total RMSE by Incorrect Unreported Rate usng Smulated Data TABLE 4 Total RMSE by Dfferent Unreported Rates usng Crash Data TABLE 5 Total RMSE by Unreported Rates for Each Severty Level Fgures FIGURE 1 Monte-Carlo Analyss on Underreportng for Smulated Data.

16 Ye & Lord 15 TABLE 1 Total RMSE by Dfferent Unreported Rate usng Smulated Data Outcome n Underreportng Unreported Rate 5% 10% 20% 40% 80% 5% 10% 20% 40% 80% MLE the MNL model WESMLE Level Level Level Level Level the OP model Level Level Level Level Level the ML model Level Level Level Level Level

17 Ye & Lord 16 TABLE 2 Total RMSE for the OP model wth Outcomes n a Descendng Order usng Smulated Data Outcome n Unreported Rate Underreportng 5% 10% 20% 40% 80% MLE Level Level Level Level Level WESMLE Level Level Level Level Level

18 Ye & Lord 17 TABLE 3 Total RMSE by Incorrect Unreported Rate usng Smulated Data Outcome n Underreportng 40% (true) 20% (assumed) 60% (assumed) MLE WESMLE WESMLE WESMLE the MNL model Level Level Level Level Level the OP model Level Level Level Level Level the ML model Level Level Level Level Level

19 Ye & Lord 18 TABLE 4 Total RMSE by Dfferent Unreported Rates usng Crash Data unreported MNL ML OP(KABCO) OP(OCBAK) rate MLE WSMLE MLE WSMLE MLE WSMLE MLE WSMLE O=10% C=10% B=10% A=10% K=10% O=40% C=40% B=40% A=40% K=40%

20 Ye & Lord 19 TABLE 5 Total RMSE by Unreported Rates for Each Severty Level Estmaton method MNL ML OP(K O) OP(O K) MLE real unreported rates WESMLE fatal=5% PDO=50%

21 Ye & Lord 20 Complete Dataset (sample sze=50,000, true parameter values were shown n TABLE 1) baselne r=1 Generate underreportng dataset (Randomly remove data accordng to a desgned unreported rate) & Estmate a model to get ˆ (keep the same varables as r ncluded n the complete dataset) r=r+1 r>100 No Yes Calculate the statstcs for 100 teratons for each parameter: E ˆ ) and Var( ˆ ) ( r r Compute the RMSE of each estmated parameter RMSE Bas Var Bas E( ˆr ) - ) 2 ( baselne Calculate the total RMSE (sum up the RMSE for each parameter) FIGURE 1 Monte-Carlo Analyss on Underreportng for Smulated Data.