category: literaturenote citekey: dinurandomparametermodelsaccident2011 title: Random parameter models for accident prediction on two-lane undivided highways in India authors: "Dinu, R.R.; Veeraragavan, A." year: 2011 date: 2011-00-00 2011 doi: 10.1016/j.jsr.2010.11.007 publication: Journal of Safety Research url: "https://doi.org/10.1016/j.jsr.2010.11.007" zotero_key: XUJ6PMQ5 zotero_storage: 66JCVXKR collections: transpordi_planeerimine folder: 001_artiklid/Road Safety Indicators firstAuthor: "Dinu, R.R."

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Journal of Safety Research

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

R.R. Dinu a, ⁎, A. Veeraragavan b,1

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

article info abstract

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

Keywords: Road accident modeling Rural highway Mixed traffic Poisson regression Random parameter models

Introduction: Generalized linear modeling (GLM), with the assumption of Poisson or negative binomial error structure, has been widely employed in road accident modeling. A number of explanatory variables related to traffic, road geometry, and environment that contribute to accident occurrence have been identified and accident prediction models have been proposed. The accident prediction models reported in literature largely employ the fixed parameter modeling approach, where the magnitude of influence of an explanatory variable is considered to be fixed for any observation in the population. Similar models have been proposed for Indian highways too, which include additional variables representing traffic composition. The mixed traffic on Indian highways comes with a lot of variability within, ranging from difference in vehicle types to variability in driver behavior. 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 appropriate for 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, from nearly 200 km of highway segments, is used to calibrate and validate the models. Results: The results of the analysis suggest that the model coefficients for traffic volume, proportion of cars, motorized two-wheelers and trucks in traffic, and driveway density and horizontal and vertical curvatures are randomly distributed across locations. Conclusions: The paper is concluded with a discussion on modeling results and the limitations of the present study.

1. Introduction

A considerable amount of research effort has gone into understanding factors that contribute to road accidents, and statistical models have been developed to predict accident frequencies under given traffic, road geometric, and environmental factors. The generalized linear modeling (GLM) approach proposed by Nelder and Wedderburn (1972), with the assumption of Poisson or negative binomially distributed error, has been the choice in the development of most of these models. Early research in accident modeling (Miaou & Lum, 1993; Miaou, 1994; Maher & Summersgill, 1996; Poch & Mannering, 1996) established the adequacy of this modeling approach and proposed models to estimate expected accident frequencies on highway segments, given the traffic and geometrics of the highway element. The variables such as Average Annual Daily Traffic (AADT), lane and shoulder widths, horizontal curvature, section length, skew angle of intersection, land-use,

¹ Tel.: +91 44 2257 4272.

and so forth were found to be contributing to the variations in accident 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 fixed parameter models, where the model parameters are assumed to be the same for any individual observation in the population. In other words, the magnitude of influence of given explanatory variable on outcome is the same for any individual observation. In the case of accident prediction models, this means that the contribution of given explanatory variable on accidents is the same on any highway segment. Recent research on accident modeling suggests that it would be more appropriate to model accident frequencies with random parameter modeling, which allows the model parameters to vary among individual observations in the population (Anastasopoulos & Mannering, 2009). The argument in favor of this approach is that there could be heterogeneity associated with the process of accident occurrence among highway segments due to variability among vehicle types, driver behavior, and so forth. Hence, the magnitude of influence of explanatory variables on accidents is more likely to be varying across locations than fixed.

^⁎ Corresponding author. Tel.: +91 44 2257 5292.

E-mail addresses: rrdinu@iitm.ac.in (R.R. Dinu), av@iitm.ac.in (A. Veeraragavan).

The aforementioned heterogeneity is more likely on Indian roads where there is large variability within and among vehicle types and driver population. Early research has established the significance of mixed nature of Indian traffic on accident occurrence, with proportions of various types of vehicles contributing to the variability in accident potential of highway elements (Robert & Veeraragavan, 2007; Srinivas, Dinu, & Veeraragavan, 2007). These studies used fixed parameter modeling approach and the appropriateness of random parameter modeling for the mixed traffic situation has not been investigated so far. The present study aims at modeling accident frequencies on Indian two-lane undivided rural highway segments, employing random parameter modeling.

2. Methodology

Number of accidents yi occurring in highway segment i over period of time t follows a Poisson distribution with probability density function;

$$P(y_i) = \frac{(\lambda_i t)^{y_i} e^{-(\lambda_i t)}}{y_i!} \tag{1}$$

where λⁱ is the expected number of accidents in unit time, or in other words, the 'safety' associated with the highway segment. Poisson distributed accident frequencies can be modeled using generalized linear modeling approach, with a logarithmic link function as;

$$\lambda_i = \exp(\beta x_i) \tag{2}$$

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

$$\beta_i = \beta_i + \nu_i \tag{3}$$

where νⁱ is the randomly distributed term. The random parameter model for Poisson distributed accident frequencies takes the following form;

$$\lambda_i | \nu = \exp(\beta' x_i) \tag{4}$$

where λi|ν is the expected accident frequency on highway segment over unit period of time, conditional on probability distribution of random parameters ν, β' is the vector of estimable random parameters and xi is the vector of explanatory variables. This results in loglikelihood function;

$$LL = \sum{i} \log \int{\nu} P(y_i | \nu) g(\nu) \partial \nu$$ (5)

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

3. Data

The data used in the present study were collected from nearly 200 km of highways around the city of Chennai, in the southern Indian state of Tamil Nadu. The highway segments were divided into 33 homogeneous 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, from 2001 to 2003, was collected for these highway segments from the police records. Nearly 3,000 accidents occurred on the selected segments during this period. Traffic surveys carried out along midblock locations and intersections along the segments showed that there is considerable difference in both traffic volume and composition between day (6 a.m.–6 p.m.) and night (6 p.m.– 6 a.m.), with average hourly volume of 1,000 vph during the day and 300 vph during the night. Fig. 1 shows the average composition of day and night traffic. During the day, significant proportions of motorized twowheelers and non-motorized vehicles were present in the traffic, to the tune of an average of 26% and 5% of the traffic, respectively.

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

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

$$CI{H} = \sum{i} w{i} DEG{i} \tag{6}$$

where wi is the weight assigned to i th horizontal curve of curvature DEGi degrees per 100 m. The curvature of horizontal curve was calculated as;

$$DEG_i = \frac{\Delta_i}{l_i} \times 100 \tag{7}$$

where Δⁱ is the central angle in degrees and ll is the length in m of i th horizontal curve. The weight wi was calculated as;

$$w_i = \frac{l_i}{L} \tag{8}$$

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

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

$$CI_V = \sum_i w_j GRAD_j$$ (9)

where wj is the weight for j th vertical curve with gradient GRADj% per 100 m. The gradient of vertical curve was computed as;

$$GRAD_j = \frac{g_j}{l_i} \times 100 \tag{10}$$

where gj is the gradient in % and lj is the length in m of the j th vertical curve. The weight wj 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 a given highway segment in a year. Separate models were developed for day time and night time accidents. The traffic variables considered in the modeling were logarithm of average hourly volume and proportion of the different vehicle types in traffic. Length of the highway segment was considered to account for vehicle-kilometers traveled. The road environment variable included in the modeling was driveway density per kilometer, which is the number of access roads joining the main roadway in one kilometer length. Among road geometric variables, the candidate variables were width of shoulder, weighted index for horizontal curvature CIH and weighted index for vertical curvature CIV along the segment.

Simulation assisted maximum likelihood estimation approach was employed for parameter estimation. To account for possible heterogeneity in parameters, each parameter was initially assumed to be normally distributed. After trials with varying number of draws, 500 random draws of each parameter was found to result in stable estimate. The correctness of the assumption of heterogeneity in parameters was tested with the statistical significance of the estimate of standard deviation of parameter distribution. T-test was conducted at 95% level of significance and the parameter was treated as random when the standard deviation of parameter distribution was found significant. The parameters with statistically insignificant variance were treated as fixed parameters.

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

$$\rho^2 = 1 - \frac{LL(\beta)}{LL(0)} \tag{11}$$

where LL(β) is the log-likelihood at convergence of the full model and LL(0) is that of the constant-only model. A ρ² value closer to one suggests that the model explains most of the variability in the dependent variable. Out of the three years data, the data pertaining to the first two years is used as the calibration data set and the data for the third year is kept aside as the validation data set. The model parameters were estimated with the calibration data set, employing the simulation based maximum likelihood estimation procedure. The models were then validated by computing the X² statistic using the following relationship;

$$X^{2} = -2[LL(\beta){c + \nu} - LL(\beta){c} - LL(\beta)_{\nu}]$$

$$(12)$$

where LL(β)c, LL(β)^c and LL(β)c⁺ ^v are the log-likelihood values at convergence for the full model, computed over the calibration data set, the validation data set, and both the data sets combined, respectively. The X² statistic follows a distribution with degrees of freedom equal to the number of parameters in the model under consideration. The statistic was tested against χ² distribution at 95% level of significance. Only those variables that were found significant at 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 the explanatory variables, especially the ones related to traffic

Table 1 Estimation results for model parameters.

Variable	Estimate (t-statistic in parenthesis)
Dependent variable	Day-time accidents per year	Night-time accidents per year
Constant	-6.47 (-7.43)	1.39 (2.35)
Logarithm of hourly traffic volume, vph	1.41 (11.10)	0.60 (5.56)
Standard deviation of parameter distribution	0.02 (5.07)	0.03 (6.17)
Length of highway segment, km	0.16 (28.60)	0.13 (23.46)
Proportion of buses in traffic, %		2.75 (2.77)
Proportion of cars in traffic, %	-8.43 (-13.54)	-11.18 (-11.13)
Standard deviation of parameter distribution	1.95 (17.91)	2.85 (17.79)
Proportion of motorized two-wheelers in traffic, %	1.78 (3.97)
Standard deviation of parameter distribution	1.26 (14.24)
Proportion of trucks in traffic, %		-3.96 (-7.56)
Standard deviation of parameter distribution		0.28 (4.76)
Driveway density, per km	0.05 (9.21)	0.03 (4.64)
Standard deviation of parameter distribution		0.04 (10.71)
Width of shoulder, m	-0.21 (-4.51)
Weighted index for horizontal curvature, CIH	0.06 (3.16)
Standard deviation of parameter distribution	0.02 (2.88)
Weighted index for vertical curvature, CIV	0.19 (2.47)	0.21 (2.36)
Model summary
LL(β)	-219.49	-206.97
LL(0)	-527.53	-434.80
ρ2	0.58	0.52
Model Validation
LL(β)c	-219.49	-206.97
LL(β)v	-104.73	-102.33
LL(β)c+ v	-326.68	-310.45
2 χcritical	18.31	15.51
X2	2 4.92 (bχcritical )	2 2.3 (bχcritical )

composition, have standard deviation significantly different from zero, when modeled as random. This suggests that there is heterogeneity associated with the variables, which are not implicitly accounted for in the data. The result of the random parameter modeling is a good reason to believe that the different explanatory variables do not influence accident occurrence by the same amount on all highway segments. As hypothesized earlier, this randomness could be due to the difference among vehicle types in terms of maneuverability, performance, and so forth, as well as the difference in driver behavior and other differences in road condition, weather, and so forth, which are not measured and included in the model.

Hourly traffic volume, length of highway segment, proportion of cars and motorized two-wheelers in traffic, driveway density, width of shoulder, and horizontal and vertical curvatures were found to be significantly influencing day-time accident frequencies. While increase in proportion of cars and width of shoulder were found to decrease the accident frequencies, increase in the other variables resulted in an increase in accidents. In case of night-time accidents, hourly traffic volume, length of highway segment, proportion of buses, cars, and trucks, driveway density, and vertical curvature were found to be significant. Here the proportion of cars and trucks in traffic were found to cause a decrease in accidents, while all other variables had a positive coefficient, showing an increase in accident frequencies.

In the model for day-time accidents, the parameters for logarithm of hourly volume, proportions of cars and motorized two-wheelers, and weighted index for horizontal curvature were found to be randomly distributed with standard deviation significantly different from zero. In the model for night-time accidents, the parameters for logarithm of hourly volume, proportion of cars and trucks, and driveway density were found to be randomly distributed. The randomness associated with the explanatory variables related to traffic volume and composition could be attributed to the variability among and within the different vehicle types in terms of dimensions and dynamics. The randomness of parameters of road geometrics and access could be due to variability in driver behavior. The significance of standard deviations of these variables in the models show that the random parameter modeling approach is promising in Indian context of mixed traffic as well.

As the random parameters were assumed to be normally distributed, their range of influence on accident frequencies can be inferred from the estimates of mean and standard deviation. Thus the parameter for logarithm of hourly volume can be expected to vary between 1.37 and 1.48 in the day-time accidents model and between 0.54 and 0.66 in the night-time accidents model for 95% of the highway segments. The range of variability of other random parameters can also be estimated in a similar manner. This would help safety engineers in evaluating more realistic ranges of safety benefits that can be expected from a corrective measure.

6. Conclusions

This study is an attempt to employ random parameter modeling to develop accident prediction models for Indian two-lane undivided rural highways that operate under mixed traffic conditions. The heterogeneity in the process of accident occurrence across locations due to a number of factors (e.g., variability among vehicle types, variability in driver behavior) is expected to be accounted for in this modeling approach. Accident prediction models were developed for day-time and night-time accidents over a three year accident history of nearly 200 km of two-lane undivided highway segments in India. The explanatory variables considered for modeling included hourly traffic 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.

The model coefficients were assumed to be normally distributed and a simulation based maximum likelihood method was used for parameter estimation. The results of the analysis showed that the model parameters for hourly volume, proportion of cars, two-wheelers and trucks in traffic, driveway density, and horizontal and vertical curvature had statistically significant standard deviation, suggesting that the parameters are randomly distributed. As the models proposed in the study accommodate for the unaccounted heterogeneity, they are expected to help safety engineers in computing more realistic ranges of safety benefits of remedial measures.

Although this study employs random parameter modeling successfully to model accidents under mixed traffic conditions, it has the following limitations:

The comparison of random parameter models with fixed parameter counterparts has not been done in the study. Such a comparison would be necessary to assess whether the improvement in modeling with this approach would really result in significant differences when applied to a real world situation, considering the computational complexity involved.
The random parameters in the present study were assumed to be normally distributed. It is possible that the variables are indeed random, but the probability distribution is other than normal. Evaluating suitability of different possible probability distributions or having a methodology to identify the distribution is necessary here.

References

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Anastasopoulos, P. C., & Mannering, F. L. (2009). A note on modeling vehicle accident frequencies with random-parameters count models. Accident Analysis and Prevention, 41(1), 153−159.

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

Dr. Veeraragavan holds a Master's Degree in Transportation Engineering from Indian Institute of Technology Madras and a Ph.D degree in Civil Engineering from Bangalore University, India. He was employed at Bangalore University, Bangalore, India during 1985-2004, and since 2004 he has been employed as Professor of Civil Engineering at the Indian Institute of Technology Madras, Chennai, India. His research interests include road safety under mixed traffic flow, traffic management, and road asset management.

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