Journal of Safety Research 42 (2011) 39–42

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Journal of Safety Research

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Random parameter models for accident prediction on two-lane undivided highwaysin India

R.R. Dinu a,⁎, A. Veeraragavan b,1

a Doctoral Research Student, Department of Civil Engineering, Indian Institute of Technology Madras, Chennai-600036, Indiab Department of Civil Engineering, Indian Institute of Technology Madras, Chennai-600036, India

⁎ Corresponding author. Tel.: +91 44 2257 5292.E-mail addresses: rrdinu@iitm.ac.in (R.R. Dinu), av@

1 Tel.: +91 44 2257 4272.

0022-4375/$ – see front matter © 2011 National Safetydoi:10.1016/j.jsr.2010.11.007

a b s t r a c t

a r t i c l e i n f o

Article history:

Received 17 January 2010Received in revised form 5 October 2010Accepted 29 November 2010Available online 22 January 2011

Keywords:Road accident modelingRural highwayMixed trafficPoisson regressionRandom parameter models

Introduction: Generalized linear modeling (GLM), with the assumption of Poisson or negative binomial errorstructure, has been widely employed in road accident modeling. A number of explanatory variables related totraffic, road geometry, and environment that contribute to accident occurrence have been identified andaccident prediction models have been proposed. The accident prediction models reported in literature largelyemploy the fixed parameter modeling approach, where the magnitude of influence of an explanatory variableis considered to be fixed for any observation in the population. Similar models have been proposed for Indianhighways too, which include additional variables representing traffic composition. Themixed traffic on Indianhighways comes with a lot of variability within, ranging from difference in vehicle types to variability in driverbehavior. This could result in variability in the effect of explanatory variables on accidents across locations.Random parameter models, which can capture some of such variability, are expected to be more appropriatefor the Indian situation. Method: The present study is an attempt to employ random parameter modeling for

accident prediction on two-lane undivided rural highways in India. Three years of accident history, fromnearly 200 km of highway segments, is used to calibrate and validate the models. Results: The results of theanalysis suggest that the model coefficients for traffic volume, proportion of cars, motorized two-wheelersand trucks in traffic, and driveway density and horizontal and vertical curvatures are randomly distributedacross locations. Conclusions: The paper is concluded with a discussion onmodeling results and the limitationsof the present study.

© 2011 National Safety Council and Elsevier Ltd. All rights reserved.

1. Introduction

A considerable amount of research effort has gone intounderstanding factors that contribute to road accidents, andstatistical models have been developed to predict accident frequen-cies under given traffic, road geometric, and environmental factors.The generalized linear modeling (GLM) approach proposed byNelder and Wedderburn (1972), with the assumption of Poisson ornegative binomially distributed error, has been the choice in thedevelopment of most of these models. Early research in accidentmodeling (Miaou & Lum, 1993; Miaou, 1994; Maher & Summersgill,1996; Poch & Mannering, 1996) established the adequacy of thismodeling approach and proposed models to estimate expectedaccident frequencies on highway segments, given the traffic andgeometrics of the highway element. The variables such as AverageAnnual Daily Traffic (AADT), lane and shoulder widths, horizontalcurvature, section length, skew angle of intersection, land-use,

iitm.ac.in (A. Veeraragavan).

Council and Elsevier Ltd. All rights

and so forth were found to be contributing to the variations inaccident potential of highway elements (Abdel-Aty & Radwan, 2000;Harwood, Council, Hauer, Hughes, & Vogt, 2000; Ivan, Wang, &Bernardo, 2000; Kim, Chung, Song, & Chon, 2005; Cafiso, Graziano,Silvestro, & Cava, 2009).

The models mentioned above come under the category of fixedparameter models, where the model parameters are assumed to bethe same for any individual observation in the population. In otherwords, the magnitude of influence of given explanatory variable onoutcome is the same for any individual observation. In the case ofaccident prediction models, this means that the contribution of givenexplanatory variable on accidents is the same on any highwaysegment. Recent research on accident modeling suggests that it wouldbe more appropriate to model accident frequencies with randomparameter modeling, which allows the model parameters to varyamong individual observations in the population (Anastasopoulos &Mannering, 2009). The argument in favor of this approach is thatthere could be heterogeneity associated with the process of accidentoccurrence among highway segments due to variability amongvehicle types, driver behavior, and so forth. Hence, the magnitude ofinfluence of explanatory variables on accidents is more likely to bevarying across locations than fixed.

reserved.

Motorized two-wheeler, 26%

Non-motorized traffic, 5%

Truck, 31%

Car, 25%

Bus, 13%

Bus, 13%

Car, 24%Truck, 63%

(a)

(b)

Fig. 1. (a) Average traffic composition during day (6 AM – 6 PM). (b) Average trafficcomposition during night (6 PM – 6 AM).

40 R.R. Dinu, A. Veeraragavan / Journal of Safety Research 42 (2011) 39–42

The aforementioned heterogeneity is more likely on Indian roadswhere there is large variability within and among vehicle types anddriver population. Early research has established the significance ofmixed nature of Indian traffic on accident occurrence, with propor-tions of various types of vehicles contributing to the variability inaccident potential of highway elements (Robert & Veeraragavan,2007; Srinivas, Dinu, & Veeraragavan, 2007). These studies used fixedparameter modeling approach and the appropriateness of randomparameter modeling for the mixed traffic situation has not beeninvestigated so far. The present study aims at modeling accidentfrequencies on Indian two-lane undivided rural highway segments,employing random parameter modeling.

2. Methodology

Number of accidents yi occurring in highway segment i over periodof time t follows a Poisson distribution with probability densityfunction;

P yið Þ = λitð Þyi e− λi tð Þ

yi!ð1Þ

where λi is the expected number of accidents in unit time, or in otherwords, the 'safety' associated with the highway segment. Poissondistributed accident frequencies can be modeled using generalizedlinear modeling approach, with a logarithmic link function as;

λi = exp βxið Þ ð2Þ

where λi is the expected accident frequency on highway segment iover unit period of time, β is the vector of estimable parameters and xiis the vector of explanatory variables. To account for the heterogeneitywith random parameters, the estimable parameters can be written as;

β0

i = βi + νi ð3Þ

where νi is the randomly distributed term. The random parametermodel for Poisson distributed accident frequencies takes the followingform;

λi jν = exp β0xi

� �ð4Þ

where λi|ν is the expected accident frequency on highway segmentover unit period of time, conditional on probability distribution ofrandomparameters ν, β' is the vector of estimable randomparametersand xi is the vector of explanatory variables. This results in log-likelihood function;

LL = ∑ilog∫

νP yi jνð Þg νð Þ∂ν ð5Þ

where g(ν) is the probability density function of the random term.Maximum likelihood estimation of the above function is computa-tionally cumbersome. Simulation assisted maximum likelihoodestimation has been proposed in literature for evaluating likelihoodfunctions that involve such high-dimensional integrals (Train, 2009).Here the integral, which is the probability of a random process, isapproximated from repeated simulation of the random process.

3. Data

The data used in the present study were collected from nearly200 km of highways around the city of Chennai, in the southern Indianstate of Tamil Nadu. The highway segments were divided into 33homogeneous sections, based on traffic volume, carriageway width,and shoulder width. The highway segments were two-lane undivided,with 6.5 m to 7 m wide carriageway and were of length varying from

1 km to 22 km. The accident history for a period of three years, from2001 to 2003, was collected for these highway segments from thepolice records. Nearly 3,000 accidents occurred on the selectedsegments during this period. Traffic surveys carried out along mid-block locations and intersections along the segments showed thatthere is considerable difference in both traffic volume and composi-tion between day (6 a.m.–6 p.m.) and night (6 p.m.– 6 a.m.), withaverage hourly volume of 1,000 vph during the day and 300 vphduring the night. Fig. 1 shows the average composition of day andnight traffic. During the day, significant proportions of motorized two-wheelers and non-motorized vehicles were present in the traffic, tothe tune of an average of 26% and 5% of the traffic, respectively.

The huge differences in traffic volume levels and compositionsuggested modeling of day-time and night-time accidents separately.

Road geometric and environment data were collected for theselected highway segments through detailed inventory surveys,supplemented with GPS tracking. As the highway segments containedmany horizontal (/vertical) curves in it, a weighted averagingprocedure was used to arrive at a single representative index for allthe horizontal (/vertical) curves within. The index for horizontalcurves in a highway segment was calculated as;

CIH = ∑iwiDEGi ð6Þ

where wi is the weight assigned to ith horizontal curve of curvatureDEGi degrees per 100 m. The curvature of horizontal curve wascalculated as;

DEGi =Δi

li× 100 ð7Þ

where Δi is the central angle in degrees and ll is the length in m of ith

horizontal curve. The weight wi was calculated as;

wi =liL

ð8Þ

41R.R. Dinu, A. Veeraragavan / Journal of Safety Research 42 (2011) 39–42

where L is the length of highway segment under consideration. Forvertical curves in a segment, the index was computed as;

CIV = ∑jwjGRADj ð9Þ

where wj is the weight for jth vertical curve with gradient GRADj% per100 m. The gradient of vertical curve was computed as;

GRADj =gjlj

× 100 ð10Þ

where gj is the gradient in % and lj is the length in m of the jth verticalcurve. The weightwj is computed similar to that for horizontal curves,which is shown in Eq. (8).

4. Analysis

The dependent variable modeled was the accident frequency on agiven highway segment in a year. Separatemodels were developed forday time and night time accidents. The traffic variables considered inthe modeling were logarithm of average hourly volume andproportion of the different vehicle types in traffic. Length of thehighway segment was considered to account for vehicle-kilometerstraveled. The road environment variable included in themodelingwasdriveway density per kilometer, which is the number of access roadsjoining the main roadway in one kilometer length. Among roadgeometric variables, the candidate variables were width of shoulder,weighted index for horizontal curvature CIH and weighted index forvertical curvature CIV along the segment.

Simulation assisted maximum likelihood estimation approach wasemployed for parameter estimation. To account for possible hetero-geneity in parameters, each parameter was initially assumed to benormally distributed. After trials with varying number of draws, 500random draws of each parameter was found to result in stableestimate. The correctness of the assumption of heterogeneity inparameters was tested with the statistical significance of the estimateof standard deviation of parameter distribution. T-test was conducted

Table 1Estimation results for model parameters.

Variable Estima

Dependent variable Day-ti

Constant -6.4Logarithm of hourly traffic volume, vph 1.4Standard deviation of parameter distribution 0.0Length of highway segment, km 0.1Proportion of buses in traffic, %Proportion of cars in traffic, % -8.4Standard deviation of parameter distribution 1.9Proportion of motorized two-wheelers in traffic, % 1.7Standard deviation of parameter distribution 1.2Proportion of trucks in traffic, %Standard deviation of parameter distributionDriveway density, per km 0.0Standard deviation of parameter distributionWidth of shoulder, m -0.2Weighted index for horizontal curvature, CIH 0.0Standard deviation of parameter distribution 0.0Weighted index for vertical curvature, CIV 0.1Model summaryLL(β) -219.4LL(0) -527.5ρ2 0.5Model ValidationLL(β)c -219.4LL(β)v -104.7LL(β)c+ v -326.6χcritical2 18.3

X2 4.9

at 95% level of significance and the parameter was treated as randomwhen the standard deviation of parameter distribution was foundsignificant. The parameters with statistically insignificant variancewere treated as fixed parameters.

The goodness-of-fit (GOF) of the model was tested with log-likelihood ratioρ2, computed as;

ρ2 = 1− LL βð ÞLL 0ð Þ ð11Þ

where LL(β) is the log-likelihood at convergence of the full model andLL(0) is that of the constant-only model. A ρ2 value closer to onesuggests that the model explains most of the variability in thedependent variable. Out of the three years data, the data pertaining tothe first two years is used as the calibration data set and the data forthe third year is kept aside as the validation data set. The modelparameters were estimated with the calibration data set, employingthe simulation based maximum likelihood estimation procedure. Themodels were then validated by computing the X2 statistic using thefollowing relationship;

X2 = −2 LL βð Þc + v−LL βð Þc−LL βð Þv� � ð12Þ

where LL(β)c, LL(β)c and LL(β)c+ v are the log-likelihood values atconvergence for the full model, computed over the calibration dataset, the validation data set, and both the data sets combined,respectively. The X2 statistic follows a distribution with degrees offreedom equal to the number of parameters in the model underconsideration. The statistic was tested against χ2 distribution at 95%level of significance. Only those variables that were found significantat 0.05 level in the two-tailed t-test were included in the final models.Table 1 gives the summary of the final models.

5. Discussion of results

The results of model parameter estimation show that many of theexplanatory variables, especially the ones related to traffic

te (t-statistic in parenthesis)

me accidents per year Night-time accidents per year

7 (-7.43) 1.39 (2.35)1 (11.10) 0.60 (5.56)2 (5.07) 0.03 (6.17)6 (28.60) 0.13 (23.46)

2.75 (2.77)3 (-13.54) -11.18 (-11.13)5 (17.91) 2.85 (17.79)8 (3.97)6 (14.24)

-3.96 (-7.56)0.28 (4.76)

5 (9.21) 0.03 (4.64)0.04 (10.71)

1 (-4.51)6 (3.16)2 (2.88)9 (2.47) 0.21 (2.36)

9 -206.973 -434.808 0.52

9 -206.973 -102.338 -310.451 15.512 (bχcritical

2 ) 2.3 (bχcritical2 )

42 R.R. Dinu, A. Veeraragavan / Journal of Safety Research 42 (2011) 39–42

composition, have standard deviation significantly different fromzero, when modeled as random. This suggests that there isheterogeneity associated with the variables, which are not implicitlyaccounted for in the data. The result of the random parametermodeling is a good reason to believe that the different explanatoryvariables do not influence accident occurrence by the same amount onall highway segments. As hypothesized earlier, this randomness couldbe due to the difference among vehicle types in terms of maneuver-ability, performance, and so forth, as well as the difference in driverbehavior and other differences in road condition, weather, and soforth, which are not measured and included in the model.

Hourly traffic volume, length of highway segment, proportion ofcars and motorized two-wheelers in traffic, driveway density, widthof shoulder, and horizontal and vertical curvatures were found to besignificantly influencing day-time accident frequencies. While in-crease in proportion of cars and width of shoulder were found todecrease the accident frequencies, increase in the other variablesresulted in an increase in accidents. In case of night-time accidents,hourly traffic volume, length of highway segment, proportion ofbuses, cars, and trucks, driveway density, and vertical curvature werefound to be significant. Here the proportion of cars and trucks in trafficwere found to cause a decrease in accidents, while all other variableshad a positive coefficient, showing an increase in accident frequencies.

In the model for day-time accidents, the parameters for logarithmof hourly volume, proportions of cars and motorized two-wheelers,and weighted index for horizontal curvature were found to berandomly distributed with standard deviation significantly differentfrom zero. In the model for night-time accidents, the parameters forlogarithm of hourly volume, proportion of cars and trucks, anddriveway density were found to be randomly distributed. Therandomness associated with the explanatory variables related totraffic volume and composition could be attributed to the variabilityamong and within the different vehicle types in terms of dimensionsand dynamics. The randomness of parameters of road geometrics andaccess could be due to variability in driver behavior. The significanceof standard deviations of these variables in the models show that therandom parameter modeling approach is promising in Indian contextof mixed traffic as well.

As the random parameters were assumed to be normallydistributed, their range of influence on accident frequencies can beinferred from the estimates of mean and standard deviation. Thus theparameter for logarithm of hourly volume can be expected to varybetween 1.37 and 1.48 in the day-time accidents model and between0.54 and 0.66 in the night-time accidents model for 95% of thehighway segments. The range of variability of other randomparameters can also be estimated in a similar manner. This wouldhelp safety engineers in evaluating more realistic ranges of safetybenefits that can be expected from a corrective measure.

6. Conclusions

This study is an attempt to employ random parameter modeling todevelop accident prediction models for Indian two-lane undividedrural highways that operate under mixed traffic conditions. Theheterogeneity in the process of accident occurrence across locationsdue to a number of factors (e.g., variability among vehicle types,variability in driver behavior) is expected to be accounted for in thismodeling approach. Accident prediction models were developed forday-time and night-time accidents over a three year accident historyof nearly 200 km of two-lane undivided highway segments in India.The explanatory variables considered for modeling included hourlytraffic volume, length of highway segment, proportion of buses, cars,motorized two-wheelers and trucks in the traffic, driveway density,shoulder width, and horizontal and vertical curvatures.

Themodel coefficientswere assumed to be normally distributed anda simulation based maximum likelihood method was used for

parameter estimation. The results of the analysis showed that themodel parameters for hourly volume, proportion of cars, two-wheelersand trucks in traffic, driveway density, and horizontal and verticalcurvature had statistically significant standard deviation, suggestingthat the parameters are randomly distributed. As the models proposedin the study accommodate for the unaccounted heterogeneity, they areexpected to help safety engineers in computingmore realistic ranges ofsafety benefits of remedial measures.

Although this study employs random parameter modeling suc-cessfully to model accidents under mixed traffic conditions, it has thefollowing limitations:

• The comparison of random parameter models with fixed parametercounterparts has not been done in the study. Such a comparisonwould be necessary to assess whether the improvement inmodeling with this approach would really result in significantdifferences when applied to a real world situation, considering thecomputational complexity involved.

• The random parameters in the present study were assumed to benormally distributed. It is possible that the variables are indeedrandom, but the probability distribution is other than normal.Evaluating suitability of different possible probability distributionsor having amethodology to identify the distribution is necessary here.

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Mr. Dinu holds a Master's Degree in Transportation Engineering from University ofKerala. He is pursuing his doctoral research in the Department of Civil Engineering,Indian Institute of Technology Madras. His research interests include road safetystudies under mixed traffic flow and road accident modeling.

Dr. Veeraragavan holds a Master's Degree in Transportation Engineering from IndianInstitute of Technology Madras and a Ph.D degree in Civil Engineering from BangaloreUniversity, India. He was employed at Bangalore University, Bangalore, India during1985-2004, and since 2004 he has been employed as Professor of Civil Engineering atthe Indian Institute of Technology Madras, Chennai, India. His research interestsinclude road safety under mixed traffic flow, traffic management, and road assetmanagement.