Based on data availability, four patrol routes, two each on Highway 401 and ..... Fu, L., Perchanok, M.S., Moreno, L. F. M., and Shah, Q. A. Effects of Winter ...
8th International Transportation Specialty Conference 8è Conférence internationale spécialisée sur le génie des transports Winnipeg, Manitoba June 9-12, 2010 / 9 au 12 juin 2010
MODELLING THE SEVERITY OF WINTER ROAD COLLISIONS Taimur Usman and Liping Fu Department of Civil & Environmental Engineering, University of Waterloo Waterloo, ON, N2L 3G1, Canada Luis Miranda-Moreno Assistant Professor Department of Civil Engineering & Applied Mechanics McGill University Montreal, Quebec H3A 2K6
Abstract: This paper has presented a multilevel multinomial and ordered nested Logit modeling framework for relating the injury severity of winter road collisions to various influencing factors such as weather, road, vehicle, and driver characteristics. Four Ontario freeway sections, two on Highway 401 and another two on Queen Elizabeth Way (QEW), were selected for this analysis, each representing an actual patrol route covered by a specific maintenance yard. Collisions over a period of three years (2003-2006) were analyzed and multilevel multinomial and ordered Logit models developed for the conditional probability of a collision resulting in one of the pre-defined severity levels. Multilevel multinomial models show better results than binary logistic models. It was found that vehicle safety device, road congestion level, and road surface condition were the major factors contributing to the severity of an accident. When coupled with the collision frequency models, the developed models could be used to estimate the safety effects of alternative maintenance policies, standards and methods. 1. Introduction In countries such as Canada safety is a significant concern in winter seasons. Driving conditions in winter can deteriorate and vary dramatically due to snowfall and ice formation, causing significant reduction in pavement friction and increase in accidents (Andrey et al. 2001). Each year in Canada, approximately 100,000 traffic accidents occur during incremental weather conditions such as rain, snow, freezing rain, high winds, etc. Weather related crashes (property damage and injury) costs are estimated to be in the range of $1 billion per year in Canada (Andrey et al. 2001). Litwin et al (2004) showed that the total accident cost in the province of Ontario, Canada, was $567.1 million ($125.4 million direct and $441.7 million indirect) for the year 1996. A report by Transport Canada (2007) estimated accident related costs in Ontario to be $18 billion for the year 2004. In U.S, injuries, fatalities and property damage from weather related crashes represent an average of $42 billion dollars annually (Qin et al. 2007). These estimates show the huge financial implication of winter road accidents. One of the remedies to mitigate the safety impact of winter weather is effective winter road maintenance, such as ploughing, salting and sanding. Winter road maintenance can improve road safety by maintaining the traction of the road surface. Fu et al (2006) have shown that maintenance operations including antiicing, pre-wet salting with ploughing and sanding have a statistically significant effect on reducing the frequency of accidents. Norrman et al (2000) show that maintenance activities have reduced accident risks for sites that initially were at high risk.
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While essential for keeping road safe, winter maintenance operations also incur significant monetary costs and negative environmental effects. For example, the total cost of winter maintenance in Ontario is about 50% of its total highway maintenance budget (Buchanan et al. 2005). The total winter road maintenance cost is estimated to be $1 billion in Canada, and over $2 billion in U.S (Transport Association of Canada, 2003). These estimates do not include significant indirect costs such as damages on the environment, on the road side infrastructure, and on the vehicles due to salt use (Perchanok et al 1991).A recent study by Environment Canada had concluded that road salts at high levels of concentration pose a risk to plants, animals and aquatic environment (Transport Canada, 2001). A Risk Management Strategy for Road Salts was subsequently developed to provide the measures to manage the risks associated with road salts (RMSRS, 2003). Many researchers have attempted to develop models for various aspects of severity. Andrey et al (2003) has reported based on the data from six cities across Canada that injury accidents increased by 45% due to rain or snow and the effects of snow were more pronounced than rain. In another study by Andrey et al (2003) the relative risk of being involved in a fatal accident, injury accident and property damage (PD) were 1:1.5:2. In a recent study, Andrey, J. (2010) investigated the effects of weather on crash severities in the long run using data from 10 Canadian cities from 1984 to 2002. Severity is classified into four levels, namely, fatal, minimal injury, minor injury, and major injuries. Match pair technique was utilized with control one week apart from the event. It was found that the risk of minimal or minor injury crash increases by 74% due to rain fall and 89% during snowfall. These figures are 46% and 52% for major injury and fatal crash respectively. In an attempt to find the effect of snowfalls on crash rate, Eisenberg et al (2005) conducted research on relationship between crash and weather based on data obtained in period 1975-2000 for 48 states in the US. They showed that during snowfalls, the number of non fatal injury crashes increased by 24% and property damage crashes increased by 78% but the numbers of fatal crashes decreased by 16%. Hermans et al (2006) analysed the monthly frequency and severity of accident based on monthly data collected from 1974 to 1999 from Belgium using a state space approach. Two severity levels were considered, fatal or major injury and minor injury. Sun light hours and percent of days with precipitation were found to increase both minor injury and major injury or fatal collision risk. Percent of days with freezing temperature was found to be associated with decreased injury and fatal collisions. This analysis was conducted at a very aggregate level (monthly) and therefore values of some predictors might not represent the true conditions at the time of accidents. In a study to quantify the safety benefit of winter road maintenance Usman et al (2010a) have shown the direct link between road surface conditions and the risk of accidents. In another study by Usman et al (2010b), they showed that the timing and types of maintenance treatments were statistically significant factors influencing the risk of accidents. This paper presents the results of a study aimed at developing statistical models for predicting the severity of winter road collisions. The study takes a disaggregate, multilevel approach accounting for the different severity levels of the individual vehicles involved in each collision. This approach has the advantages of making a full use of the information available in collision data while at the same time accounting for possible correlation in severity levels of individual vehicles involved in a given collision. 2. Proposed Methodology There are a large number of factors that influence the severity of collisions under winter conditions (Miaou et al. 2003; Andrew et al. 1998). The major factors can be grouped into three categories, namely,, road driving conditions, vehicle, and driver. Road driving conditions include road geometry, environment, and pavement surface conditions. The later are affected by weather and maintenance operations. Accident severity is commonly modeled in two different approaches. The first approach incorporates severity into the frequency domain by modelling collision frequencies of different severity types directly (Park and Lord, 2007). In the second approach, separate models are developed to relate the conditional probabilities of experiencing individual severity levels for a given collision to various factors (Shankar et
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al. 1996; Dissanayake et al 2002; Yao 2004; Saccomanno et al 1996; Wong et al 2008). In this research we have applied the second approach as detailed in the following section. 2.1. Study Sites Based on data availability, four patrol routes, two each on Highway 401 and Queen Elizabeth Way (QEW) in the Province of Ontario, Canada, were selected as study sites, as shown in Figure 1: • • • •
Hwy 401-R1: Hwy 400 to Morningside Ave (28.0 Km) Hwy 401-R2: Trafalgar Road to Hwy 400 (31.1 Km) QEW-R1: Burloak Drive to Erin mills parkway (17.4 Km) QEW-R2: Erin mills parkway to Eastmall (13.1 Km)
Figure 1- Site Map Showing Location of Selected Patrol Routes These are major inter-urban freeways with multiple lanes in each direction and the AADT ranges from 100,000 to more than 400,000. 2.2. Data sources Data was obtained for accidents and traffic from October 1999 to April 2006, weather data (RCWIS, RWIS and EC) was obtained from October 2003 to April 2006 for the four selected sites. As a results only the last three years of data was used with 2,952 accidents involving 9,144 vehicles. A description of each data source is given below: • •
•
Traffic Volume Data: Traffic volume data was obtained from loop detectors from the Ministry of Transportation of Ontario (MTO), which was then processed and converted into hourly traffic volume data. Traffic Accident Data: The Ontario Provincial Police (OPP) maintains a database of all collisions occurring on Ontario highways. A database including all collision records for the study routes was obtained from MTO. The database includes detailed information on each collision including; accident time, accident location, accident type, impact type, severity level, vehicle information, driver information, and road conditions – surface and geometry, weather conditions, passenger information, etc. Road Condition Weather Information System (RCWIS) data: This data contains information about road surface conditions, maintenance, precipitation type, accumulation, visibility and temperature.
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RCWIS data is collected by MTO maintenance personnel, who patrol the maintenance routes during a storm event 3 ~ 4 times on the average. Road Weather Information System (RWIS) data: This data contains information about temperature, precipitation, visibility, wind speed, road surface conditions, etc., recorded automatically by RWIS stations near the selected maintenance routes Environment Canada (EC) data: Weather data is available from Environment Canada, which is also used to supplement the data from RCWIS. This data contains information about temperature, precipitation, visibility, wind speed etc.
• •
2.3. Modelling Options Two different approaches were used to aggregate accident data for the analysis. In the first method accident data were aggregated on per accident basis and each collision was classified into different severity classes. Two hierarchical severity classification schemes were used, recognizing the fact that some severity levels are similar to each other and therefore should be considered as a same cluster and then further classified. In the first classification scheme, shown as Option – A in Figure – 2, collisions are first classified into two general categories: fatal and non-fatal. The non-fatal accidents are further classified into ones with major injuries and ones without major injuries. The non-major injury accidents are lastly classified into those with minor and minimal injuries and those with property damage (PD) only. Table – 1 shows the sample accident frequency of each class for the four patrol routes based on this classification scheme. Recognizing the low number of accidents in the fatality category, the second classification scheme, called Option – B, combines fatalities with major injuries. Collisions are classified into two categories: fatalities or major injuries and non fatal or non major injury collisions. In the next level non fatal or non major injury collisions are further divided into minor injury collisions and minimal injury or PD only. Classification scheme for Option B is shown in Figure – 2. Both classifications, Option – A and B are accident based classifications. Table – 2 shows the sample accident frequency of each class for the four patrol routes based on this classification scheme. For both modeling options A and B, ordered Logit model were developed for the hierarchical structure shown in Figure 2. Binary Logit models at individual splits were also developed for both the options however only ordered Logit models will be discussed here. Let Y is the observed severity level and Y* is the unobserved injury severity level and µ1, µ2… µj represents the cut-off points or threshold values for the injury severity levels then Y = 1 if Y* ≤ µ1 Y = 2 if µ1 < Y* ≤ µ2 . . . Y = j if µj-1 < Y* Where j represents the number of injury severity categories. Y* is estimated using Eq. 1 (Liao 1994; Wang 2005). [1] Y = *
K
∑β k =1
k
Xk +ε
Where, βk is model coefficient to be estimated and Xk represents a set of explanatory variables. ε is assumed to be logistic distributed. The probability of a particular injury severity level Y = j can be estimated using Eq. 2 (Liao 1994; Wang 2005) which can be further rewritten as Eq. 3 (Kenneth E. Train 2009).
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[2] P (Y = j ) = P ( µ j −1 < Y < *
[3] P (Y = j ) = P ( µ j −1 < Y < *
K
k =1
K
k =1
µ j ) = F µ j − ∑ β k X k − F µ j −1 − ∑ β k X k
µj) =
µ j − β k xk
µ j −1 − β k xk
e e − µ j − β k xk µ −β x 1+ e 1 + e j −1 k k
TABLE - 1: Accident statistics for Option – A Patrol Fatalities Major Injury Minor Injury 401 – R1 4 26 1415 401 – R2 10 43 1146 QEW – R1 0 6 114 QEW – R2 2 7 179 Total 16 82 2854 TABLE – 2: Accident statistics for Option – B Patrol Fatalities + Major Injury Minor Injury 401 – R1 30 522 401 – R2 53 502 QEW – R1 6 56 QEW – R2 9 70 Total 98 1150
PD only 0 0 0 0 0
Total 1445 1199 120 188 2952
Minimal + PD only 892 644 58 110 1704
Total 1444 1199 120 188 2952
Figure – 2: Data classification scheme for Option – A and B (accident based classification) One of the major limitations of the above mentioned approach is that it does not make use of the full information available in accident records about the individual vehicles involved in each collision. For example, a fatal accident could involve several vehicles with different levels of severity. A third modelling option, called Option – C, is therefore proposed, which considers the individual vehicles involved in each accident at the first level. Each vehicle is then classified into classes in a hierarchical way as Option A. This classification scheme has been shown in Figure – 3. Option – C is vehicle based classification scheme. For this structure multilevel multinomial Logit models were calibrated (Lenguerrand et al., 2006;
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Miranda-Moreno et al., 2009). Multilevel multinomial structure allows modelling directly on the data related to a single vehicle. This approach considers accidents as one level and individual vehicles in a particular accident as another level. Explanatory variables could be added to either of the levels. Model structure is shown in Eq. 4.
severity s ij
= [4] Log r severity ij
β 0 + β1 X ij + U j
Where i and j represents level 1 and 2 respectively. Uj is second level (accident) random effect factor which is assumed normally distributed. Severity with superscript “r” represents the base severity against which other severity levels, denoted by superscript “s”, are compared. In our case Xij represent a set of covariates at level 1 (subscript “i”).
Figure – 3: Data classification scheme for Option – C (vehicle based classification) 3.0 Model Calibration and Results Detail of the variables used is given below: Variables considered Explanation RSI Road surface condition index continuous variable (poor = 0 to good = 1) Patrol Dummy variable representing a site (401-R1 = 1, 401-R2 = 2, QEW-R1 = 3 and QEW-R2 = 4) Safety Device Safety device use (Others = 0, Used = 1, Not used = 2) Light Light condition (Otherwise = 0, Light + Dawn = 1, Dark + Dusk = 2) Day Weekdays = 0, Weekends = 1 Weather Weather condition (Other = 0, Freezing rain = 1, snow = 2) Visibility Visibility in Km Temp Air temperature in oC Precipitation Snow or freezing rain precipitation in mm/hr Traffic Log of the hourly traffic volume Road Alignment Otherwise = 0, Straight = 1, Curve = 2 Models for Option A and B were calibrated using Stata (version 9) 1 and the results are shown in Table – 3 for option – A, and Table – 4 for option – B for the ordered Logit models. Three step-wise models, labeled as Model 1, 2 and 3 were calibrated for each option, differing in the variables included. Model 3 is considered as the final model with all variable included as being significant. 1
www.stata.com
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In general results from Option A and B are consistent in terms of their trend and capability to determine the injury severity level of an accident. However, due to different classification schemes used, the numbers of significant variables are different in both options. More importantly, based on the goodness of fit measures (AIC, BIC and LL), Option – A has a much better overall fit than Option – B and is therefore selected for further interpretation and discussion. Results obtained with Option – A classification shows that severity of an accident is affected by visibility, traffic congestion, light conditions, safety device usage and the specific site. Positive sign indicates that increase in the value of the variable will lead to more severe accidents and vice versa. Increased visibility was associated with occurrences of more severe accidents. Similarly, collisions occurring during congested traffic were less severe in nature than those in less congested traffic. Accidents occurring under good light conditions were less severe in nature compared to those in dark. Moreover use of safety device was found to reduce the severity of an accident. The site specific dummy variable shows that collisions occurring on highway 401 were less severe in nature than those occurring on highway QEW. One reason for this might be the high traffic volume on highway 401 compared to QEW. Similar results were obtained when binary Logit models were calibrated for this classification. Table – 3: Significant factors from option – A Model 1 Variable Coef. Sig. RSI -0.547 0.204 Visibility 0.049 0.008 Air Temperature -0.012 0.414 Precipitation intensity -0.120 0.393 Ln(Traffic) -0.272 0.054 WeekEnds -0.336 0.147 Weekdays 0.000 0.000 Light+Dawn -0.323 0.162 Dark+Dusk 0.000 Road Alignment (straight) -35.112 1.000 Road Alignment (Curve) 0.157 0.627 Road Alignment (Others) 0.000 0.000 Weather (freezing rain) 0.204 0.758 Weather (Snow) 0.692 0.104 Weather (Others) 0.000 0.000 Safety Device (used) -1.229 0.001 Safety Device (Not used) -0.034 0.958 Safety Device (Others) 0.000 401 - R1 -1.186 0.005 401- R2 -0.555 0.185 QEW - R1 -0.460 0.462 QEW- R2 0.000
Model 2 Coef. Sig. -0.484 0.239 0.024 0.130 -0.014 0.300 -0.077 0.492 -0.267 0.045 -0.314 0.160 0.000 0.000 -0.330 0.139 0.000
Model 3 Coef. Sig. -0.537 0.168 0.028 0.050
-1.270 -0.011 0.000 -1.062 -0.395 0.079 0.000
0.000 0.986 0.010 0.332 0.886
-0.304
0.022
-0.355 0.000
0.109
-1.273 -0.033 0.000 -1.094 -0.442 0.028 0.000
0.000 0.957
µ1 µ2
-1.019 0.848
-1.270 0.593
-1.349 0.514
LL AIC BIC
-417.083 880.166 1016.890
-448.265 926.531 1016.369
-449.943 921.886 987.767
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0.007 0.274 0.959
Table – 4: Significant factors from option – B Model 1 Variable Coef. Sig. RSI -0.107 0.532 Visibility 0.004 0.531 Air Temperature 0.000 0.978 Precipitation intensity -0.065 0.113 Ln(Traffic) -0.235 0.000 WeekEnds -0.027 0.755 Weekdays 0.000 Light+Dawn -0.056 0.492 Dark+Dusk 0.000 0.000 Road Alignment (straight) 1.311 0.434 Road Alignment (Curve) -0.237 0.036 Road Alignment (Others) 0.000 Weather (freezing rain) -0.145 0.515 Weather (Snow) 0.058 0.719 Weather (Others) 0.000 Safety Device (used) -0.325 0.382 Safety Device (Not used) -1.099 0.001 Safety Device (Others) 0.000 0.000 401 - R1 -0.352 0.034 401- R2 -0.090 0.599 QEW - R1 0.365 0.121 QEW- R2 0.000
Model 2 Coef. Sig. -0.119 0.465 0.003 0.583
Model 3 Coef. Sig.
-0.065 -0.232
0.055 0.000
-0.070 -0.252
0.017 0.000
-0.062 0.000 0.001 -0.239 0.000
0.441 0.000 0.999 0.034
-0.006 -0.248 0.000
0.996 0.027
-0.332 -1.104 0.000 -0.350 -0.092 0.377 0.000
0.371 0.000 0.000 0.034 0.590 0.108
-0.318 -1.094 0.000 -0.357 -0.108 0.394 0.000
0.392 0.001 0.000 0.030 0.523 0.093
µ1 µ2
-3.795 -0.671
-3.461 -0.340
-3.522 -0.403
LL AIC BIC
-2303.545 4651.089 4782.852
-2305.875 4641.750 4731.589
-2306.813 4635.626 4701.507
For Option C, models were calibrated using MLwin 2 software for all the data pooled as a single set. A dummy variable representing the specific patrol routes was also added to cover for site related unobserved heterogeneities. PD only accidents were used as the base case for the analysis. Results for all the sites are consistent in terms of their intuitiveness and are presented in Table 5.
2
Rasbash, J., Charlton, C., Browne, W.J., Healy, M. and Cameron, B. (2005) MLwiN Version 2.02. Centre for Multilevel Modeling, University of Bristol.
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Table 5: Results for option – C
Variable Constant WeekEnds Weekdays Dark+Dusk Light+Dawn RSI Safety Device (used) Safety Device (Not used) Safety Device (Others) Visibility Ln(Traffic) 401 - R1 401- R2 QEW - R1 QEW- R2
Log(Minor injury/PD) Coef. Sig. 2.833 0.000 -0.145 0.002 0.000 0.010 0.824 0.000 -0.289 0.002 0.455 0.000 1.172 0.000 0.000 -0.004 0.182 -0.292 0.000 -0.401 0.000 -0.299 0.002 0.130 0.363 0.000
Log(Major injury/PD) Coef. Sig. 2.287 0.098 0.352 0.089 0.000 0.396 0.069 0.000 -0.528 0.204 -1.236 0.000 2.140 0.000 0.000 0.009 0.520 -0.420 0.001 -1.812 0.000 -1.134 0.001 -0.163 0.736 0.000
Log(Fatal injury/PD) Coef. Sig. 3.529 0.094 -0.812 0.005 0.000 -0.878 0.004 0.000 -3.805 0.000 -0.127 0.768 1.907 0.005 0.000 0.243 0.000 -0.805 0.000 -3.165 0.000 -1.548 0.000 -11.061 0.798 0.000
The following observations could be made about the results: •
Fatal collisions are less likely to occur on weekends than weekdays. This result is consistent with the finding from the model of Option A. This might be attributed to the differences in traffic composition and trip purposes. Weekend traffic involves more family recreational trips and drivers are more likely to drive more carefully with less urgency.
•
Accidents occurring in poor light conditions will increase the probability of major severity as compared to PD only whereas the chances of fatalities are reduced.
•
The worse the road surface conditions, the less likely severe road collisions result. This is likely due to drivers’ behavior adjustment to the road driving conditions. Drivers tend to be more attentive and careful when the road is slippery. Furthermore, most drivers would adjust their speed according the prevailing road conditions.
•
As expected, the use of safety device reduced severe accidents. The effect of safety device is more pronounced in case of major injury accidents.
•
Better visibility was found to be associated with higher number of fatal accidents. This could again be attributed to drivers’ speed adjustment under poor visibility.
•
The probability of having severe collisions decreased as traffic volume increases. This result could be attributed to speed adjustment. Under congested conditions, drivers tend to reduce their speed and become more attentive.
4. Conclusions and Future Research This paper has presented three alternative models for explaining the variation of and predicting the severity level of winter road collisions. Data from four maintenance routes were used to calibrate and compare the proposed models. It was found that vehicle safety device, road congestion level, and road surface condition were the major factors contributing to the severity of an accident. When coupled with the collision frequency models, these models could be used to estimate the safety effects of alternative maintenance policies, standards and methods.
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It should be noted that these results are preliminary and need to be further validated using data from different classes of highways. 5. Acknowledgements This research was supported by MTO in part through the Highway Infrastructure and Innovations Funding Program (HIIFP). The first author also wishes to thank Higher Education Commission, Pakistan for their financial support. The authors wish to acknowledge in particular the assistance of Zoe Lam and Max Perchanok from MTO. 6. References Andrew V. and J. Bared (1998). “Accident Models for Two-Lane Rural Segments and Intersections”. TRR 1635, Paper No. 98-0294 Andrey, J. (2010). Long-term trends in weather-related crash risks. Journal of Transport Geography 18 (2010) 247–258. Andrey, J., and Christopher Knapper (2003). Weather and Transportation in Canada. Department of Geography, University of Waterloo, publication series No. 55. Andrey, J. Brian Mills, Jeff Suggett, and Mike Leahy (2003). “Weather-Related Road Accident Risks in Mid Sized Canadian Cities”. Journal of Natural hazards. Volume 28, Numbers 2-3 / March, 2003 Andrey, J. Brian Mills, and Jessica Vandermolen (2001). Weather Information and Road Safety. Institute for Catastrophic Loss Reduction, Toronto, Ontario, Canada. Buchanan, F., and Gwartz, S. E (2005). Road Weather Information Systems at the Ministry of Transportation, Ontario. Presented at 2005 Annual Conference of the Transportation Association of Canada Calgary, Alberta. Dissanayake, Sunanda and John Lu (2002). Analysis of Severity of Young Driver Crashes Sequential Binary Logistic Regression Modeling. Transportation Research Record: Journal of the Transportation Research Board, No. 1784, Transportation Research Board of the National Academies, Washington, D.C., 2002, pp. 108–114. Paper No. 02-2302. Eisenberg, Daniel; Kenneth E Warner. “Effects of Snowfalls on Motor Vehicle Collisions, Injuries, and Fatalities”. American Journal of Public Health; Jan 2005; 95, 1; ABI/INFORM Global pg. 120. Fu, L., Perchanok, M.S., Moreno, L. F. M., and Shah, Q. A. Effects of Winter Weather and Maintenance Treatments on Highway Safety. Paper No. 06 – 0728. TRB 2006 Annual Meeting CD-ROM. Hermans, E., Geert Wets, and Filip Van Den Bossche (2006). Frequency and Severity of Belgian Road Traffic Accidents Studied by State-Space Methods. Journal of Transportation and Statistics. Volume 9 Number 1, 2006. Kenneth E. Train (2009). Discrete Choice Methods with Simulation, second edition. Cambridge University Press. Liao, T. F. (1994). Interpreting probability models: Logit, probit, and other generalized linear models. Sage University Paper series on Quantitative Applications in the Social Sciences, series No. 07-101. Thousand Oaks, CA: Sage. Lenguerrand, E., J.L. Martin, B. Laumon (2006). Modelling the hierarchical structure of road crash data— Application to severity analysis. Accident Analysis and Prevention 38 (2006) 43–53. Litwin, N. and Turrittin, T. “Spine and Brain Injuries from Vehicle Crashes: The Human and Economic Cost”. Transport 2000 Ontario. Report 04-01. January, 2004. Miranda-Moreno, Luis F., Liping Fu, Satish Ukkusuri, and Dominique Lord (2009). How to incorporate accident severity and vehicle occupancy into the hotspot identification process? 88th Annual Meeting of the Transportation Research Board, 2009. Paper No. 09 -2824 Miaou, Shaw-pin; Joon Jin Song, and Bani K. Mallick (2003). “Roadway Traffic Crash Mapping: A SpaceTime Modeling Approach”. Journal of transportation and Statistics. Volume 6 Number 1, Pp 33 – 57. Norrman, J., Eriksson, M., and Lindqvist, S (2000). Relationships between road slipperiness, traffic accident risk and winter road maintenance activity. Climate Research Vol 15: 185–193, September 05. Park, E. S., Dominique Lord (2007). Multivariate Poisson–Lognormal Models for Jointly Modeling Crash Frequency by Severity. Transportation Research Record: Journal of the Transportation Research Board,No. 2019, Washington D.C., 2007, pp. 1–6.
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