category: literaturenote citekey: ayatiinvestigationroletrafficvolume2011 title: Investigation on the role of traffic volume in accidents on urban highways authors: "Ayati, Esmaeel; Abbasi, Ehsan" year: 2011 date: 2011-00-00 2011 doi: 10.1016/j.jsr.2011.03.006 publication: Journal of Safety Research url: "https://doi.org/10.1016/j.jsr.2011.03.006" zotero_key: VEX26R2A zotero_storage: 9Z5XVDJV collections: transpordi_planeerimine folder: 001_artiklid/Road Safety Indicators firstAuthor: "Ayati, Esmaeel"

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Journal of Safety Research

j o u r n a l h om e p a g e : www. e l s ev i e r. c om / l o c a t e / j s r

Investigation on the role of traffic volume in accidents on urban highways

Esmaeel Ayati ⁎, Ehsan Abbasi ¹

Department of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran

article info abstract

Article history: Received 31 March 2010 Received in revised form 4 January 2011 Accepted 1 March 2011 Available online 27 May 2011

Keywords: No injury accidents More severe accidents Urban highways Light non-passenger car vehicles Negative binomial (NB) regression model

Introduction: Roadway safety is a major concern for the general public and public agencies, as roadway crashes claim many lives and cause substantial economic loss each year. In Iran, a large number of vehicles are involved in road accidents each year, which cause many deaths and extensive property damage; such accidents are among the major causes of death and disability in the country. Method: To reduce roadway accidents, the factors that affect the occurrence and severity of accidents should be scrutinized to prevent or reduce their effect. The method that many researchers have adopted to determine the effective parameters surrounding road accidents in recent years is through statistical modeling of accidents. In this article, the role of different kinds of vehicles in traffic flow are investigated separately in terms of the likelihood of crashes on urban highways, and the vehicles are divided into three groups: passenger cars, heavy vehicles, and light nonpassenger car vehicles. Poisson and negative binomial (NB) regression models were applied to model the accidents in this research, which were categorized into two groups: no injury (property damage only) accidents and more severe (injury and fatal) accidents. Results: Ultimately, we conclude that light nonpassenger car vehicles (i.e., taxis and motorcycles) play the largest role in the occurrence of crashes on urban highways for both types of accidents.

1. Introduction

Roadway safety is a major concern for the general public and public agencies. Indeed, roadway crashes claim many lives and cause substantial economic loss each year (Ma, Kockelman, & Damien, 2008). In Iran, a large number of vehicles are involved in road accidents each year, causing significant death, disability, and property damage. Recent investigations in Iran indicate that 25% of casualties due to unnatural deaths are caused by traffic accidents (Attaran, 1996). Based on the results of research done on the cost of traffic accidents in Iran in 2001, 19,727 and 117,600 people have been killed and injured, respectively, in about 450,000 accidents (Legal Medicine Organization of Iran, 2002), which is estimated to have cost more than 3% of the gross national product (Ayati, 2008). Developing sound or reliable statistical models for analyzing motor-vehicle crashes is important in highway safety studies. The models can be used for identifying hazardous sites, predicting motor-vehicle collisions, and developing accident modification factors (Lord & Park, 2008).

In this paper, the role of vehicle type volume in traffic flow in the occurrence of accidents on urban highways is investigated through modeling no injury and more severe accidents by Poisson and NB regression models. The vehicle types are divided into three groups, including: passenger cars; heavy vehicles including truck trailers, trucks, buses, and minibuses; and light non-passenger car vehicles including taxis, pickups, and motorcycles. The purpose of the study was to scrutinize the role of each group of vehicles in the likelihood of accidents. The role of heavy vehicles in accident occurrence is often discussed, but it must be determined whether heavy vehicles result in more accidents or whether passenger cars or light non-passenger car vehicles, like taxis or motorcycles, play a key role in accident occurrence or frequency.

2. Literature review

Numerous studies have been conducted in recent years to investigate the frequency and severity of accidents on different parts of rural and urban freeways and highways, two-lane roads and arterials, or intersections using various methods of modeling accidents (Anastasopoulos & Mannering, 2009; Chang, 2005; Delen, Shadra, & Bessonov, 2006; Lord & Park, 2008; Lord, Washington, & Ivan, 2007; Ma et al., 2008; Milton, Shankar, & Mannering, 2008; Wang, Quddus, & Ison, 2009). Poisson and NB regression models have been used widely for predicting the occurrence of accidents on highways (Lord, 2006; Lord & Park, 2008; Savolainen & Tarko, 2005). In addition, more advanced regression models such as zero-inflated negative binomials and Conway–Maxwell–Poisson generalized linear models are applied to modeling crashes (Lord, 2006; Lord et al., 2007; Lord & Park, 2008). In previous studies, the average daily traffic (ADT) or annual average daily traffic (AADT) were used as volume-related independent variables for

^⁎ Corresponding author. Tel.: +98 0915 310 7483; fax: +98 0511 8676769. E-mail addresses: esmaeel@ayati.co.uk (E. Ayati), ehsan.abbasi395@gmail.com

⁽E. Abbasi). ¹ Tel.: +98 0915 314 7887.

modeling (Ma et al., 2008; Wang et al., 2009) or, considering the number of lanes on each side, the average daily traffic per lane was applied (Chang, 2005; Lee & Mannering, 2002; Milton et al., 2008). Researchers also applied variables such as the average daily percentage of trucks in the traffic stream or the average daily truck traffic (ADTT) (Anastasopoulos & Mannering, 2009; Milton et al., 2008) to consider the responsibility of heavy vehicles in the frequency and severity of highway accidents.

In their studies on the frequency and severity of highway accidents using a mixed logit model, Milton et al. (2008) concluded that the average daily traffic per lane had an increasing role in the likelihood of no injury accidents on highways (Milton et al., 2008). Chang (2005) concluded, in his investigation on the frequency of freeway accidents with a comparison between NB regression and artificial neural network models using variables such as ADT per lane, the percentage of trucks in the traffic flow and peak hour factor (PHF); Chang (2005) concluded that, given the traffic volume in the peak hour, volume-related variables that led to conflicts between vehicles and increased exposure to potential risks of accidents correlated with an increasing number of vehicles and trucks.

3. Methodology

In this research, the accidents on urban highways in Mashhad were modeled by Poisson and NB regression models, two models for no injury accidents and two for more severe accidents (injury or fatal). The specific contribution of this research is the separation of the total traffic volume into the passenger car volume, the heavy vehicle volume, including truck trailers, trucks, buses, and minibuses, and the light non-passenger car vehicle volume, including taxis, pickups, and motorcycles. The role of volume in the occurrence of accidents with property damage and injury or fatalities was investigated thoroughly to determine what kind of traffic has a significant role in accident occurrence. Thus, the independent variables of both no injury and more severe accident models are as follows:

1. passenger car volume (Veh/hr)
2. heavy vehicle volume (passenger car equivalent)
3. light non-passenger car vehicle volume (passenger car equivalent)

Accident data are usually two-level data; the first and main level is often road segments (i.e., the highway is divided into several parts or segments). The base of segmentation can differ, and this segmentation was based on total traffic volume in this study (i.e., the total traffic volume [passenger car equivalent] is constant for each part of the highway). The second level is the hours of the day, that is, the traffic peak hours are considered as the first sub-level, the day non-peak hours the second, and the night non-peak hours the third. The accident data for Mashhad urban highways were collected by the Transportation and Traffic Organization of Mashhad. After developing the models, the Poisson and NB models for no injury and more severe accidents were compared, and the better model was then validated to prove the efficiency of that model in modeling urban highway accidents.

3.1. Poisson regression model

In the Poisson regression model, the i-th observation of the dependent variable yi is modeled as a random Poisson variable with mean λi:

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

The most important property of the Poisson regression model is the equality of the mean and variance of the distribution, which is the main shortcoming of this model and blocks the "overdispersion" of accident data. In the Poisson model, the conditional variance is equal to the conditional mean:

$$V(y_i|\mathbf{x}_i) = E(y_i|\mathbf{x}_i) = \lambda_i = e^{\mathbf{X}_i'\beta}$$ (2)

The log-likelihood of the Poisson regression model is given by

$$L = \sum_{i=1}^{n} \ln \frac{e^{-\lambda_i} \lambda_i^{y_i}}{yi 1} = \sum{i=1}^{n} \left[ -\lambda_i + y_i \ln \lambda_i - \ln y_i \right]$$ (3)

To estimate the regression coefficients by the maximum likelihood method, the derivative of the log-likelihood relative to the vector of coefficients, β, is set equal to zero:

$$\frac{\partial L}{\partial \beta} = \sum_{i=1}^{n} \left( y_i - e^{x_i \beta} \right) \mathbf{x}_i = 0 \tag{4}$$

The estimation of the regression coefficients in the Poisson regression model is not obtained with a direct equation, but the Newton-Raphson iteration procedure is used for estimating the unknown parameters of the model. To perform the estimation process, the corresponding iteration algorithm is analyzed by SAS statistical software to obtain the calculated coefficients. The predicted value of ŷⁱ is the conditional mean or average number of accidents given xi, which is also denoted as λi, that is, the mean of the random variable yi, which follows a Poisson distribution. This value is typically not an integer number, whereas the observed value yi is a count (Anselin, 2002).

3.2. Negative binomial regression model

In the negative binomial (NB) model yi, the i-th observation of the dependent variable has the following probability distribution function:

$$p(y_i) = \frac{\Gamma(y_i + r)}{y_i!\Gamma(r)} \left[ \frac{\mu_i}{\mu_i + r} \right]^{y_i} \left[ \frac{r}{\mu_i + r} \right]^r$$ (5)

The conditional mean of yi, for the vector of observed independent variables, xi, is given by

$$E(y_i|\mathbf{x}_i) = \mu_i = e^{\mathbf{x}_i^i \beta} \tag{6}$$

The NB distribution does not have the disadvantage of the Poisson distribution because there is no equality of the mean and variance of the distribution. The relationship between the mean and variance of an NB distribution is

$$V(Y) = \mu + \frac{1}{r}\mu^2 \tag{7}$$

The variance of the NB distribution is always greater than the mean. Thus, it fits the data with a variance greater than the mean. α= 1/r is the dispersion parameter. For accident data with low dispersion, the Poisson distribution gives plausible results, but, if overdispersion exists in the data, the variance of data will be greater than the mean. In such cases, the NB distribution is preferred. In the NB regression model, the dispersion parameter should be estimated in addition to regression parameters. r must be positive, so r*=lnr is estimated instead, which can take on any value. The log-likelihood function of the NB model for the i-th observation is obtained from the following equation:

$$L{i} = \sum{i=0}^{y{i}-1} \ln(e^{r*} + j) = \ln y{i}! + e^{r} \ln(e^{r}) + y{i} \ln(\mu{i}) - (e^{r} + y{i}) \ln(\mu{i} + e^{r})$$

ð8Þ

To estimate β and r*, as in the Poisson model, the iteration procedure of Newton-Raphson is applied (Agresti, 2002). The related iteration algorithm is written in SAS to obtain the unknown parameters. In the NB regression model, the predicted value of ŷⁱ is the conditional mean or average number of accidents given xi. This is also denoted as μi, that is, the mean of the random variable yi, which follows the NB distribution.

4. Data

The accident data for Mashhad urban highways in the years 2006 and 2007 were used for modeling accidents on urban highways. This information was collected by Traffic and Transportation Organization of Mashhad with the help of GIS and police reports. In this study, the accidents were divided into two groups: accidents with property damage only (no injury) and more severe accidents (injury or fatal). Therefore, accident statistics were collected in two groups: no injury and more severe. Accident data are usually two-level data, the first and main level is often the road segment (i.e., the highway is divided into several parts). In this study, highway segmentation was performed based on total traffic volume. The second level is the hour, that is, traffic peak hours are considered the first sub-level, the day non-peak hours the second, and the night non-peak hours the third. To scrutinize the roles of the different vehicle types in crash occurrence or severity, the traffic was separated into passenger cars, heavy vehicles, and light non-passenger car vehicles. The volume data were not readily available separately from the Traffic and Transportation Organization of Mashhad, which was one of the study limitations. The organization conducts a comprehensive survey once every few years in November and obtains the passing volume of different kinds of vehicles through the main urban roads in terms of daily hours. In November, the traffic conditions in Mashhad are normal, and the models made in this research do not consider the seasonal changes of traffic; this could be another limitation of the research because the statistics of traffic volume in different seasons were not available, and the volume data are related to different hours of the day. According to this statistics, one can perceive the traffic combination and percentage of each vehicle type in traffic and can trust this combination in the future; however, the total (equivalent) traffic volume is updated every year.

The Traffic and Transportation Organization of Mashhad gives the total traffic volume on different urban roads, particularly highways in peak hours, for future years. According to this total equivalent volume, the passenger car equivalent factors of vehicle types, the percentage of each type in traffic combinations, and the ratio of the total volume in non-peak hours to that in peak hours, the volumes of each vehicle type in peak hours and 2 hours representing the day and night nonpeak hours were obtained. The volume of passenger cars and equivalent volume of heavy vehicles and light non-passenger car vehicles at peak hours and 2 representative hours of non-peak day and night hours were also obtained. These calculations were performed with Excel. The passenger car equivalent factors of vehicle types are given in Table 1 (Transportation and Traffic Organization of Mashhad, 2008).

The total volume (passenger car equivalent) can be obtained with the following equation:

$$V_{te} = e_1 N_1 + \dots + e_n N_n (9)$$

Table 1 The passenger car equivalent factors of vehicle types.

Heavy vehicles	Bicycle and Motorcycle	Intracity bus	Intercity bus	Minibus	Pickup	Taxi	Passenger car
2.5	0.5	5	2.5	2	1	2	1

where Vte is the total equivalent volume and ei and Ni are the equivalent factor and number of vehicles of type i, respectively. The number of each type of vehicle is given by

$$N_i = \frac{Pi V{te}}{e_1 P_1 + \dots + e_n P_n} \tag{10}$$

where Pi is the percentage of vehicles of type i.

Because the number of vehicle types is counted hourly, these data are represented in terms of vehicle per hour, Veh/hr. Eq. (10) was used in Excel to calculate the number of each vehicle type according to the total equivalent volume with the percentage of each vehicle type on any road segment at any hour. The summary statistics of crashes and independent variables of the models in the years 2006 and 2007 for 156 sections of Mashhad urban highways in 936 sub-sections are presented in Table 2.

5. Statistical modeling

After collecting the data, the models were developed for accidents with property damage only (no injury) and more severe (injury or fatal) accidents to identify the factors influencing the accident occurrence of urban highways. Independent variables applied in the models included xij,1, xij,² and xij,3, the volume of passenger cars, heavy vehicles, and light non-passenger car vehicles on the i-th section in the j-th period.

5.1. Poisson regression model

To estimate the unknown parameters of the model, the iteration method of Newton-Raphson is applied. First, a vector of regression parameters is considered as a primary estimate, β̂ 0, and the second parameter estimate is obtained from iteration relation (11) and so on until the difference between two successive estimates becomes less than the tolerance level. The estimate before the last is the desired estimated coefficient. The iteration relation is as follows:

$$\hat{\beta}_{k+1} = \hat{\beta}_k - \hat{H}_k^{-1} \hat{g}_k \tag{11}$$

where ĝ^k is the gradient evaluated at β̂ ^k, the first partial derivative of the log likelihood in the gradient, and Ĥ^k is the Hessian evaluated at β̂ ^k, the second partial derivative of log likelihood, as follows:

$$g = \sum_{i=1}^{n} (y_i - \hat{\lambda}_i) xi, H = -\sum{i=1}^{n} \hat{\lambda}_i x_i' x_i$$ (12)

For evaluating the significance of independent variables, the inverse Hessian obtained at the last iteration will be the asymptotic variance matrix. The variance of the estimates is the diagonal elements and the standard errors their square roots. The t statistic for each parameter is also constructed as the ratio of the parameter estimate over its standard error. If the p-value of a parameter is less than the required level of significance (0.05 or less), the corresponding variable is significant and will stay in the model. Otherwise, it is

Table 2 Summary statistics of accidents and independent variables.

Variable	Mean	S.D.	Min	Max
Dependent variables number of accidents with property loss only number of more severe accidents	7.07 0.97	8.86 1.43	0 0	80 12
Independent variables Passenger car volume Heavy vehicle volume Light non-passenger car vehicle volume	1046 562 551	807 427 419	33 29 42	4593 2425 2225

neglected in the model (Anselin, 2002). Such calculations were proceeding by SAS, estimating the parameters and evaluating their significance for no injury and more severe accidents, which is presented in Table 3. Thus, the evaluated number of no injury accidents on section i in period j is obtained from the following equation:

$$\lambda{ij} = exp(x'{ij}\beta) = exp(0.0003x{ij,1} + 0.0005x{ij,3})$$ (13)

The evaluated number of more severe accidents on section i in period j is also as follows:

$$\lambda{ij} = exp(-0.6134 + 0.00084x{ij,3})$$ (14)

Knowing the evaluated value of λij, the Poisson model developed for estimating the likelihood of yij no injury and more severe accidents is obtained as follows:

$$P(y{ij}) = \frac{e^{-\lambda{ij}} \lambda{ij}^{y{ij}}}{y_{ij}!} \tag{15}$$

5.2. Negative binomial regression model

For estimating the regression coefficients and dispersion parameters, the Newton-Raphson iteration procedure is applied as in the Poisson model. Using r*= 0(r= 1) in the first (Poisson model), the regression parameters β are estimated with the following iteration:

$$\hat{\beta}^{(t+1)} = \hat{\beta}^{(t)} - \left[\hat{H}{\beta}^{(t)}\right]^{-1} \hat{g}{\beta}^{(t)} \tag{16}$$

where g^β = − ∑ i= 1 ∂Li ^∂^β ; ^H^β ⁼ − ∑ i= 1 ∂²Li ∂β∂β′ . Then, r* is estimated iteratively as

$$\hat{r}^{(t+1)} = \hat{r}^{(t)} - \left[\hat{H}_r^{(t)}\right]^{-1} \hat{g}_r^{(t)}$$ (17)

where gr ⁼ −∑ ^∂Li ^∂^r ; Hr ⁼ −∑ ^∂² Li ^∂^r ². Now, r* and the estimated regressors are applied as the initial estimate θ in the following iteration to estimate the updated values of r* and β:

$$\hat{\theta}^{(t+1)} = \hat{\theta}^{(t)} - \left[\hat{H}{\beta r}^{(t)}\right]^{-1} \hat{g}{\beta r}^{(t)} \tag{18}$$

Table 3 Estimation results for Poisson and negative binomial (NB) regression models.

| Variable | No injury accidents | | Injury or fatal
accidents | | | |-------------------------------------------|---------------------|----------------|------------------------------|----------------|--| | | Parameter | t
statistic | Parameter | t
statistic | | | Poisson model | | | | | | | Constant | −0.1611 | −1.77 | −0.6134⁎ | −2.56 | | | Passenger car volume | 0.0003 | 15.78 | −0.00006 | −0.95 | | | Heavy vehicle volume | 0.00002 | 0.63 | −0.00024 | −1.92 | | | Light non-passenger car vehicle
volume | 0.0005 | 12.80 | 0.00084 | 7.31 | | | Negative binomial model | | | | | | | Constant | −0.8112 | −2.91 | −0.9231 | −2.77 | | | Passenger car volume | 0.00042 | 5.29 | −0.00012 | −1.33 | | | Heavy vehicle volume | −0.00001 | −0.10 | −0.00014 | −1.17 | | | Light non-passenger car vehicle
volume | 0.00061 | 4.73 | 0.00102 | 6.33 | | | Dispersion parameter | 1.0874 | 17.04 | 0.7832 | 7.39 | |

^⁎ Highlighted coefficients are significant at 5%.

where

$$\theta = \begin{bmatrix} \beta \ r^* \end{bmatrix}, g{\beta r} = \begin{bmatrix} g{\beta} \ gr \end{bmatrix}, H{\beta r} = \begin{bmatrix} H_{\beta} & -\sum \frac{\partial^2 L_i}{\partial r^* \partial \beta} \ \left(-\sum \frac{\partial^2 L_i}{\partial r^* \partial \beta}\right)' & H_r \end{bmatrix}$$ (19)

The estimated values of r* and β in the iteration before the last, θ̂ are of interest.r= exp(r*) is the inverse dispersion parameter (Agresti, 2002). The significance evaluation of the independent variables of the NB regression model is similar to that of the Poisson model. The results of developing the NB model for no injury and more severe accidents are presented in Table 3. Therefore, the evaluated number of no injury accidents on section i in period j is obtained from following equation:

$$\mu{ij} = exp(x{ij}\beta) = exp(-0.8112 + 0.00042x{ij,1} + 0.00061x{ij,3})$$ (20)

The evaluated number of more severe accidents on section i in period j is also obtained as

$$\mu{ij} = exp(-0.9231 + 0.00102x{ij,3})$$ (21)

Knowing the evaluated value of μij, r= 1/α= 0.92 and r= 1.277 for no injury and more severe accidents, respectively, the NB model for estimating the likelihood of yij no injury and more severe accidents is obtained as

$$p(y{ij}) = \frac{\Gamma(y{ij} + r)}{y{ij}\Gamma(r)} \left[ \frac{\mu{ij}}{\mu{ij} + r} \right]^{y{ij}} \left[ \frac{r}{\mu_{ij} + r} \right]^{r}$$ (22)

6. Goodness-of-fit evaluation and comparison of models

After developing the models and identifying the effective factors in accident occurrence, the evaluation and comparison between models is discussed. To evaluate the goodness-of-fit of the models, the Akaike Information Criterion (AIC) was used. The lower the AIC, the better the model fit, and the model with the least AIC is the best one. The goodness-of-fit of the applied models cannot be evaluated independently, but this fit is evaluated by comparing the value of AIC for each model. For comparing models, first the Poisson and NB models are compared together with respect to the data dispersion. To do this, the significance of the dispersion parameter in the NB model and the likelihood ratio (LR) test were used. The statistic of the LR test is obtained as

$$LR = -2\left(LLp - LL{NB}\right) \tag{23}$$

where LLp and LLNB are the log-likelihood of the Poisson and NB models, respectively. This statistic has a Chi-squared distribution with k−1 degrees of freedom, in which k is the number of regression coefficients. If this statistic is greater than the value of χ0.05 ² (k−1), then the second model is better than the first. The consequence of this test for comparing models is usually equivalent to evaluating the significance of the dispersion parameter. The second stage of comparison is the goodnessof-fit evaluation of models and their fit comparison. The AIC is calculated using the following equation (Liu & Cela, 2008):

$$AIC = -2LL + 2k (24)$$

where LL is log-likelihood and k is the number of parameters. The results of the second stage often approve the conclusion of the first stage. The results of the goodness-of-fit evaluation of models and their comparison are presented in Table 4.

The results for no injury accidents show that the dispersion parameter is significant in the NB model, and the negative value of two times the log likelihood is much greater for the Poisson model than the NB model. The value of the LR statistic is LR=−2(LLp−LLNB)= 3377.3, which is much greater than χ0.05 ² (3) equal to 7.81. Therefore, it is concluded that the NB regression model is better than the Poisson model for no injury accidents. The comparison of the AIC value between models demonstrates that the NB model with lower AIC is better than the Poisson model for no injury accidents.

For more severe accidents, the dispersion parameter is also significant, and the LR statistic is equal to LR=−2(LLp−LLNB)= 148.4, which indicates that the NB model is better than the Poisson model. Because the AIC value for the NB model is less than that of the Poisson model, it is evident that this model is better than the Poisson model for modeling injury or fatal accidents on urban highways. Therefore, the NB regression model is an efficient and worthy model for modeling no injury and more severe accidents on urban highways. This claim can be evaluated by examining the validity of this model in predicting crashes on urban highways.

7. Validating the NB regression model

To validate the NB regression model for modeling accidents, this model was used to predict the number and likelihood of no injury and more severe accidents on other urban highways to examine the validity of the model by comparing the observed and predicted numbers of accidents on each section of highway over a specified period. The statistics for the accidents and traffic flow on the highway were provided as before. The Pearson residuals test was used for validation, which is described as follows (Agresti, 2002):

$$S = \sum{i=1}^{n} \sum{j=1}^{3} s{ij} = \sum{i=1}^{n} \sum{j=1}^{3} \left( \frac{y{ij} - M(y{ij})}{\sqrt{V(y{ij})}} \right)^{2} = \sum{i=1}^{n} \sum{j=1}^{3} \frac{(y{ij} - \mu{ij})^{2}}{\mu{ij} + \alpha \mu{ij}^{2}}$$ (25)

where yij is the observed and μij the predicted number of accidents on i-th section of the highway in the j-th period of the day, α is the dispersion parameter and n is the number of sections of the highway. S follows a Chi-squared distribution with N-k degrees of freedom, where N is the number of observations and k is the number of regression coefficients. The closer together the observed and predicted numbers of accidents are, the more valid the model is. The critical value of S is equal to χ0.05 ² (N−k), and a value of S less than it implies the validity of the model. Referring to the development of the NB model, the predicted numbers of no injury and more severe accidents on the i-th section of the highway in the j-th period of the

Table 4 Results of goodness of fit evaluation and comparison between the Poisson and NB models.

	Poisson	Negative binomial
No injury accidents
alpha		1.0874 (17.04)
-2LL	9054.8	5377.6
AIC	9062.8	5387.6
Injury or fatal accidents
alpha		0.7832 (7.39)
-2LL	2485.3	2336.9
AIC	2493.3	2346.9

day were obtained from Eqs. (20) and (21), respectively. The highway was divided into 40 sections and three periods of the day, leading to N= 120. For accidents with property loss only, S was equal to 41.32, which is less than χ0.05 ² (120−3)= 143.21 and implies that the model is valid. However, for more severe accidents, S was equal to 39.72, which is less than χ0.05 ² (120−2)= 144.62 and implies the validity of NB regression model for modeling more severe accidents.

8. Conclusions

The results of this research indicate that the volume of passenger cars and light non-passenger car vehicles plays a significant role in the likelihood of no injury accidents, but the volume of heavy vehicles does not have much effect on the likelihood of no injury accidents on urban highways. The volume of light non-passenger car vehicles increases the likelihood of more severe accidents, but the volume of passenger cars and heavy vehicles do not have much influence on the likelihood of more severe accidents. Therefore, it is concluded that light non-passenger car vehicles, including taxis and motorcycles, play a considerable part in the occurrence of accidents on urban highways, including both no injury and more severe accidents, but heavy vehicles do not. These conclusions will be helpful for future projects involving the management of traffic on urban highways. In these projects, planning to control the role of taxis and motorcycles in accident occurrence should be adopted. This could include the separation of these vehicles types from normal flow on urban highways. The results also indicate that the negative binomial regression model is a valid and efficient model for modeling and predicting the accidents on urban highways. The study had some limitations, two of which are related to the research data, as mentioned in Section 4: the volume data for each kind of vehicle were not readily available, and the models did not consider the seasonal changes of traffic because the statistics for traffic volume in different seasons were not available. The other limitation of the research is related to the models used for modeling accidents. Poisson and NB regression models have their own limitations.

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Ehsan Abbasi is a M.Sc. He received his M.D. in the Road and Transportation Engineering from the Ferdowsi University of Mashhad, Iran. His areas of expertise include safety of traffic and modeling accidents. His recent work has been on modeling accidents on urban highways in terms of traffic flow and road geometry characteristics, using Poisson, NB and ZI regression models.

Dr. Esmaeel Ayati is a Full Professor in the Department of Engineering at Ferdowsi University of Mashhad, Iran. He received his M.D. in Road and Civil Engineering from Tehran University and his Ph.D. in Road and Transportation Engineering from Aston University. His area of expertise is safety of traffic and his researches on evaluating the cost of traffic accidents is unique in Iran and his book titled "The Cost of Traffic Accidents in Iran" is the only valid reference in this area in Iran.

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