category: literaturenote citekey: mapredictingexpresswaycrashfrequency2017 title: "Predicting expressway crash frequency using a random effect negative binomial model: A case study in China" authors: "Ma, Zhuanglin; Zhang, Honglu; Chien, Steven I-Jy; Wang, Jin; Dong, Chunjiao" year: 2017 date: 2017-00-00 2017 doi: 10.1016/j.aap.2016.10.012 publication: "Accident Analysis & Prevention" url: "https://doi.org/10.1016/j.aap.2016.10.012" zotero_key: KZ7KDZCC zotero_storage: VXZPFJ64 collections: transpordi_planeerimine folder: 001_artiklid/Road Safety Indicators firstAuthor: "Ma, Zhuanglin"

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Accident Analysis and Prevention

jour nal home page: www.elsevier.com/locate/aap

Predicting expressway crash frequency using a random effect negative binomial model: A case study in China

Zhuanglin Ma^a,∗, Honglu Zhang b, Steven I-Jy Chien^a,c, Jin Wang d, Chunjiao Dong^e

^a School of Automobile, Chang'an University, Xi'an, Shaanxi, China
^b Deppon Logistics Corporation, Shanghai, China
^c John A. Reif, Jr. Department of Civil and Environmental Engineering, New Jersey Institute of Technology, Newark, NJ, USA
^d Yunnan Transport Research Institute, Kunming, Yunnan, China
^e Center of Transportation Research, The University of Tennessee, Knoxville, TN, USA

a r t i c l e i n f o

Article history: Received 18 February 2016 Received in revised form 5 October 2016 Accepted 9 October 2016 Available online 17 October 2016

Keywords: Crash frequency Negative binomial model Random effects negative binomial model Goodness-of-fit Prediction

a b s t r a c t

To investigate the relationship between crash frequency and potential influence factors,the accident data for events occurring on a 50 km long expressway in China, including 567 crash records (2006–2008), were collected and analyzed. Both the fixed-length and the homogeneous longitudinal grade methods were applied to divide the study expressway section into segments. A negative binomial (NB) model and a random effect negative binomial (RENB) model were developed to predict crash frequency. The parameters of both models were determined using the maximum likelihood (ML) method, and the mixed stepwise procedure was applied to examine the significance of explanatory variables. Three explanatory variables, including longitudinal grade, road width, and ratio of longitudinal grade and curve radius (RGR), were found as significantly affecting crash frequency. The marginal effects of significant explanatory variables to the crash frequency were analyzed. The model performance was determined by the relative prediction error and the cumulative standardized residual. The results show that the RENB model outperforms the NB model. It was also found that the model performance with the fixed-length segment method is superior to that with the homogeneous longitudinal grade segment method.

1. Introduction

In the advent of rapid economic growth, the length of expressways in China had yielded 104,400 km of roadway by the end of 2013, thus ranking the integrated system in first place worldwide. The widely spread expressway network offers convenient, efficient, and comfortable service to its many road users, albeit it is also associated with significant negative effects, such as increased crash frequency and injury severity. In 2013, there were 8693 crash records occurring on expressways in China, of which 4.38% were reportedtopolice. There were 5843people killedand11,169people injured (Traffic Management Bureau of Ministry of Public Security, 2014) and the fatality and injury rates were 0.67 and 1.28 persons per crash, respectively. These rates are much higher than the national average of 0.30 and 1.08 persons per crash on all roadways. Moreover, the percentage of fatal crashes was 10.34% of the total number of fatal crashes which was much higher than Great Britain's 5%, for example (Department for Transport, 2012). Therefore, improving traffic safety on expressways in China is a critical issue and consequently deserves special attention.

To improve traffic safety, many efforts have focused on modeling the safety issues and factors associated with crash frequency throughout the years. However, only a few studies have explored the impact of road segmentation to the accuracy of crash frequency prediction. Dividing the study-road section into segments based on pre-defined criteria is the first step to consider before developing a crash frequency prediction model. Existing road segmentation methods can be classified into two categories: static and dynamic. The static method divides a road section into segments with a fixed length, and the dynamic method defines segments with variable lengths. Both road segmentation methods have advantages and disadvantages, and there is no consensus on which method performs better in predicting crash frequency.

Several studies examined predicted crash frequency using the static method (Shankar et al., 1995; Cenek et al., 1997; Cafiso et al., 2013). Shankar et al. (1995) investigated crash frequency for rural freeways with an equal length segment method (i.e. 1.6 km). Cenek et al.(1997) predicted crashfrequency for rural roads withthe static

E-mail address: zhuanglinma@chd.edu.cn (Z. Ma).

^∗ Corresponding author.

segment method (i.e. 0.2 km). Cafiso et al. (2013) compared and investigated five segmentation criteria for motorways, and found that the fixed-length segment could lead to the best result.

Some studies predicted crash frequency using the dynamic method (Abdel-Aty and Radwan, 2000; Casfiso et al., 2008; Koorey, 2009; AASHTO, 2010; Borsos et al., 2014). Abdel-Aty and Radwan (2000) divided a 227-km long arterial into 566 segments based on the homogeneity of road geometry and traffic flow characteristics. Later,Casfiso et al.(2008)dividedtwo-lane ruralhighways basedon the homogeneity of road geometry and traffic flow characteristics. Koorey (2009) used the homogeneous segments by aggregating curves and tangents when the radius of curves exceeds a predetermined threshold value. The Highway Safety Manual (AASHTO, 2010) suggested a homogeneous segment method using AADT, number of lanes, lane or median width, shoulder width or type, curvature, driveway density, roadside hazard rating, and clear zone width as potential explanatory variables. Borsos et al. (2014) divided a 919-km long highway into 1357 segments according to homogeneous AADT and road geometry, and the segment lengths ranged from 0.2 km to 6.2 km.

A considerable number of studies have been carried out to establish models representing the relationship between crash frequency and traffic flow as well as geometric and environmental factors, such as the Poisson model, NB model, Poisson-lognormal model, zero-inflated count model, generalized estimating equation model, random effect model, and random parameter count model (Lord and Mannering, 2010). The purpose of these models was to identify the influencing factors that have a significant impact on crash frequency. From a practical perspective, many researchers attempted to analyze the influencing factors of crash frequency, including roadway geometry (e.g., horizontal and vertical alignment, median type, number of lanes, land, shoulder width, etc.), traffic features (e.g., average daily volume, vehicle proportion, etc.), and environmental conditions (e.g., road surface condition, light condition, weather condition, etc.).

The NB model has been widely used to establish the relationship between crash frequency and potential factors on an expressway, owing to overcoming the problem of over-dispersion. Hadi et al. (1995) adopted the Poisson and the NB models to investigate crash frequency for both multi-lane and two-lane roads, focusing on AADT and road environment factors. Shankar et al. (1995) developed an NB model to investigate the frequency of specific accident types on rural freeways. The results showed that weather and geometric conditions significantly affect crash frequency. Later, Shankar (1997) analyzed the impact of roadway geometric designs and environmental conditions on rural crash frequency with an NB model. The results showed that the number of curves and weatherrelated factors (rain, icy, and snow) significantly influenced crash frequency. Abdel-Aty and Radwan (2000) used an NB model to predict crash frequency, and they found that heavy traffic volume and restricted road geometry (e.g., narrow lane/shoulder width, reduced median width, larger number of lanes, etc.) increased the likelihood of crash involvement. Hauer (2004); Hauer et al. (2004) adopted the multiplicative and additive equation forms to investigate crash frequency using a negative binomial distribution.

The NB model has demonstrated itself as a capable approach to deal with over-dispersed crashes, but it has limitations for managing temporal effects, such as the correlation of crashes occurring on the same segment with annual or seasonal changes. Consequently, the standard error of the regression coefficients might be underestimated. Hausman et al. (1984) suggested that using the panel data and considering separate persistent individual effects may overcome the limitations of the NB model. Shankar et al. (1998) thus developed a random effect negative binomial (RENB) model to deal with the temporal issues embedded in the crash data. It was found that the prediction accuracy was significantly improved. Chin and Quddus (2003) investigated the relationship between crash occurrence and potential factors of signalized intersections using an RENB model, which treated the data in a time-series cross-section panel. Yaacob et al.(2010)identified the contributing factors affecting crash frequency using both an NB and a RENB model. They found that the RENB model outperformed the NB model in accommodating data with dispersed temporal and cross-sectional variations. Caliendo et al. (2013) investigated the relationship between severe crashes and potentialfactors for Italian motorway tunnels using the RENB model in order to capture the deterministic temporal effect. Xu et al. (2014) used an NB model and an RENB model to account for the spatial correlation of crash data between intersections along one arterial. They found thattheRENBmodel was the best-fitfor the data at hand. Hosseinpour et al. (2014) assessed the performance of seven models (i.e., Poisson, NB, RENB, hurdle Poisson, hurdle NB, zero-inflated Poisson, and zero-inflated negative binomial models) by predicting the frequency of head-on crashes on a federal road in Malaysia. It was found that the RENB model outperformed the other models subject to goodness-of-fit measures.

The first motive of this paper is to assess the performance of the NB and the RENB models, and the second motive is to investigate which kind of road segment method is suitable to the straight and flat area expressway. Therefore, this paper adopts both the static and dynamic segment methods for the development of the NB and RENB models to predict crash frequency. To investigate the relationship between crash frequency and potential influence factors, the accident data including 567 crash records (2006 ∼ 2008) occurring on a 50-km long expressway in China were collected and analyzed. Some goodness-of-fit and predicted measures are used to evaluate the performance (i.e., sufficient explanatory and prediction accuracy) of the proposed models. An elasticity analysis was conducted thereafter to determine the marginal effects of the explanatory variables.

2. Data description

A 3-year monitoring period (2006–2008) was carried out on a median-divided expressway that is 62.4 km long, connecting the cities of Guangzhou and Zhuhai in Guangdong Province, China. The starting and ending points of the study expressway section is Tingjiao and Jinding, respectively. Considering the instability of traffic flow in the proximity of the start/end segment, the sections near the start/end points are deleted. Thus, the 50 km section in the middle of this expressway was selected for this study. The horizontal alignment data consists oftangents,transition and circular curves, and the vertical alignment data consists of longitudinal grade and vertical curves, while the cross-section alignment data consists of road widths of four-lane and six-lane segments. There are eight on- ramps and off-ramps on the study section as shown in Fig. 1. There are 567 crash records occurred on this section within the 3-year period.

During the study period, crash records, traffic flow, and roadway geometry data were collected by the Expressway Management Agency.Avariety of details are includedinthe crashrecords,including crash date and time, location, cause, collision type, vehicle type, number of vehicles involved, and injury severity. The road geometric data consists of horizontal alignment(tangent,transition curves, or circular curves), vertical alignment(upgrade or downgrade), and cross-section alignment(number of lanes, lane and shoulder width, vertical curve, and super-elevation). The traffic data consists of daily volume and truck percentage, and the average annual daily traffic (AADT) measure was computed as considering the equivalent vehicles per truck. The studied expressway is located within a subtropical region. The meteorological data showed that the ambient temperature very rarely dropped below freezing. Neither snow

Fig. 1. Location of the study expressway.

nor extreme fog was observed. In addition, the study expressway is located in a rural, straight, and flat area with a design speed of 120 km/h. Therefore, the effect of design speed was not considered.

The length of straight subsections range from 0.077 to 2.629 km, and the radius of horizontal curves ranges from 0.718 to 5.8 km. Furthermore, the longitudinal grade ranges between −3.993% and +3.993% and the length of four-lane (two-lane one-way) and sixlane (three-lane one-way) sub-sections are 13 km and 37 km, respectively.

As mentioned previously,two segment methods were applied in this study. The first method is the static method, or the called fixedlengthmethod, whichdivides the selectedsectionperdirectioninto 100 1-km long segments, such that there are 300 observed samples over a three-year span. The other is the dynamic method, or the called homogeneous longitudinal grade method, which divides the study section into 154 segments, such that there are 462 observed samples. The lengths of these segments vary between 0.4 km and 1.4 km with an average of 0.65 km. Each observed sample is associated with 12 attributes, which are illustrated in Table 1 and defined below.

The variable of "Degree of curvature" is the central angle to the ends of an arc or chord of agreed length. "Curve length ratio" means the percentage of curve length to the segment length, and "Curvature change rate" is the sum of deflection angles divided by the segment length. The variable of "Weighted curvature" is a weighted average of the degree of curvature and curve length ratio on the same segment; and "Grade differences" means the subject segment grade minus the upstream segment grade. For the fixedlength method, the "grade of the segment" is a weighted average. The variable of "Super-elevation" is a binary variable, which is equal to 1 for segments with super-elevation; otherwise, 0 is the value used for segments without super-elevation. "Road width" is the sum of the lane width plus the inside and outside shoulder widths. For the studied expressway, there are 13 km four-lane (two-lane one-way) and 37 km six-lane (three-lane one-way) sections. The width per lane, inside shoulder and outside shoulder are 3.75 m, 0.75 m and 3.00 m respectively. Thus, the road widths for four-lane and six-lane sections are 11.25 m and 15.00 m, respectively. The variable of "RGR" is the ratio of longitudinal grade and curve radius that reflects the interaction between horizontal curve and vertical curve on the same segment. Lastly, "Special segment" is a binary variable, which is equal to 1 for segments with bridge, tunnel, and service area; otherwise, it is equal to 0 for a regular segment.

3. Methodology

3.1. Model description

The crash occurrence process at a roadway segment may be viewed as a Bernoulli trial and each crash has unequal probabilities of independent events. A Bernoullitrial has two possible outcomes: success (i.e., a crash) and failure (i.e., no crash). The number of trials with success in a certain time period follows a binomial distribution. If the number of trials is large, the binomial distribution may be approximated with a Poisson distribution. Then, the probability function of segment i in a time period can be written as

$$f(y_i) = \frac{\exp(-\lambda_i) \cdot \lambda_i^{y_i}}{y_i \cdot 1} \tag{1}$$

Table 1 List of explanatory variables.

Explanatory variable	Fixed length				Homogeneous longitudinal grade
	Maximum	Minimum	Mean	Standard Error	Maximum	Minimum	Mean	Standard Error
Crash frequency	11	0	1.89	1.96	9	0	1.23	1.40
Degree of curvature (m−1)	1.75	0.00	0.49	0.38	1.75	0.00	0.45	0.36
Curve length ratio (%)	1.00	0.00	0.75	0.35	1.00	0.00	0.76	0.36
Curvature change rate (◦/km)	49.15	0.00	14.94	9.94	49.15	0.00	13.54	10.10
Weighted curvature (%/m)	1.53	0.00	0.41	0.33	1.75	0.00	0.39	0.33
Longitudinal grade (%)	2.32	−2.32	1.17	0.72	3.99	−3.80	1.17	1.58
Grade differences (%)	6.69	−5.70	−0.02	2.34	6.69	−5.70	−0.14	2.85
Super-elevation	1	0	0.38	0.49	1	0	0.35	0.48
Road width (m)	15.00	11.25	14.03	1.65	15.00	11.25	14.12	1.59
RGR (%/km)	2.00	0.00	0.40	0.39	2.00	0.00	0.29	0.35
Special segment	1	0	0.07	0.46	1	0	0.58	0.49
AADT (103 passenger cars per day)	6.3	4.7	5.3	0.04	6.3	4.7	5.3	0.05

where yⁱ and ⁱ respectively represent the numbers of actual and expected crashes in a time period on segment i.

The expected crashes can be expressed as

$$\lambda_i = \exp(\mathbf{\beta} \mathbf{X}_i) \tag{2}$$

where Xⁱ is a vector of explanatory variables for segment i, and is a vector of estimated parameters associated with Xi.

The nature of a Poisson distribution is that the mean and variance of crashes occurring on a roadway segment are equal. In reality, however, they are not equal. Thus, an error term is introduced to deal with this situation. Thus,

$$\lambda_i = \exp(\mathbf{\beta} \mathbf{X}_i + \varepsilon_i) \tag{3}$$

where εi is the error term and exp(εi) is a gamma-distributed error term with mean 1 and variance 1/ϕ2, ϕ is the inverse of the overdispersion parameter.

With such a modification, the relationship between mean and variance can be expressed as

$$Var(y_i) = E(y_i)[1 + E(y_i)/\varphi] = \lambda_i(1 + \lambda_i/\varphi)$$ (4)

where E(yi) and Var(yi) respectively represent the mean and the variance of yi.If 1/ϕ is greater than zero (i.e. ϕ is small), the crash data is either over-dispersed or under-dispersed. However, if 1/ϕ is equal to zero (i.e. ϕ is infinity), a negative binomial distribution is a Poisson distribution. Thus,the probability function of segmenti in a time period for a negative binomial distribution can be formulated as

$$f(y_i) = \frac{\Gamma(y_i + \varphi)}{y_i! \Gamma(\varphi)} \left(\frac{\varphi}{\lambda_i + \varphi}\right)^{\varphi} \left(\frac{\lambda_i}{\lambda_i + \varphi}\right)^{y_i}$$ (5)

where (•) is a gamma function.

For an NB model, the number of crashes on segment i for year t is assumed as independent; however, in the real-world they are likely correlated over time owing to the location-specific effects, such as horizontal and vertical alignment, the number of lanes, and shoulder width. To this end, the RENB model seems superior to the NB model because the RENB model is capable of managing random effects with temporal correlation.

Therefore, the expected crash frequency and the associated explanatory variables in the RENB model will also vary temporally and spatially. Thus,

$$\lambda{it} = \exp\left(\beta \mathbf{X}{it} + \varepsilon_{it} + \nu_i\right) \tag{6}$$

where it is the expected number of crashes for segment i in year t, Xit is a vector of explanatory variables for segment i in year t, εit is the error term for segment i in year t, and vⁱ is the unknown intercept for segment i, which follows a probability distribution over time.

To illustrate the variation of segment effect over time, the associated dispersion parameters are not supposed to be constant. In addition, ϕi/(1 +ϕi) is assumed to follow a beta distribution in the probability density function of the RENB model. Thus,

$$f(y{it}|\mathbf{X}{it}) = \frac{\Gamma(a+b)\Gamma(a+\sum{T}\lambda{it})\Gamma(b+\sum{T}\lambda{it})}{\Gamma(a)\Gamma(b)\Gamma(a+b+\sum{T}\lambda{it}+\sum{T}y{it})} \prod{T} \frac{\Gamma(\lambda{it}+y{it})}{\Gamma(\lambda{it})\Gamma(y_{it}+1)}$$ (7)

where yit is the number of crashes on segmenti in year t; a and b are distributional parameters for ϕi. Note that the standard maximum likelihood method used here is to estimate parameters a, b, and ˇ.

3.2. Model evaluation

To evaluate the performance of the developed model (i.e., sufficient explanatory and prediction accuracy), the log-likelihood ratio and the adjusted log-likelihood ratio are employed as the goodness-of-fit measures, while the relative prediction error and the cumulative standardized residual are employed as the prediction performance measures.

The log-likelihood ratio is a suitable goodness-of-fit measure for the model developed based on random and discrete data, which is formulated as

$$\rho^2 = 1 - \frac{l\left(\beta\right)}{l\left(0\right)} \tag{8}$$

where l(ˇ) is the log-likelihood at converge of the "unrestricted" model, and l(0) is the log-likelihood at converge of the "restricted" model which includes the constant term only.

When new variables are introduced to the restricted model, the log-likelihood ratio of the developed model will increase. To cope with this disadvantage, the adjusted log-likelihood ratio formulated as Eq. (9) is employed to incorporate correction for the number of covariates (Ben-Akiva and Lerman, 1985). Thus,

$$\bar{\rho}^2 = 1 - \frac{l\left(\beta\right) - m}{l\left(0\right)} \tag{9}$$

where m is the number of covariates.

The relative prediction error R^e can be calculated as:

$$R{e} = |\frac{\sum{i=1}^{I} \sum{t=1}^{T} (y{it} - \hat{y}{it})}{\sum{i=1}^{I} \sum{t=1}^{T} y{it}}| \times 100\%$$ (10)

where yˆit is the predicted number of crashes of the ith segment in year t, I is the number of segments, and T is the number of years.

The cumulative residual analysis (Lord and Persaud, 2000; Hauer, 2004; Cafiso et al., 2010) was conducted to examine the

Table 2 Results of the NB and RENB models with fixed-length segments.

| Explanatory variable | NB model | | RENB model | | | |-----------------------|-------------|---------|-------------|---------|--| | | coefficient | p-value | coefficient | p-value | | | Intercept | −3.201 | <0.001 | −2.805 | 0.005 | | | Curve length ratio | 1.217 | 0.014 | – | – | | | Curvature change rate | −0.041 | 0.031 | – | – | | | Longitudinal grade | −0.441 | <0.001 | −0.455 | <0.001 | | | RGR | 0.439 | 0.004 | 0.481 | 0.005 | | | Road width | 0.245 | <0.001 | 0.212 | <0.001 | | | AADT | – | – | 3.96 | 0.002 | | | 1/ϕ | 0.249 | | – | | | | a | – | | 28.861 | | | | b | – | | 6.394 | | | | 2 | 0.057 | | 0.099 | | | | ¯ 2 | 0.047 | | 0.092 | | | | Re (%) | 2.89 | | 1.81 | | | | CSR | 296.31 | | 285.60 | | |

adequacy of the developed model. The residual for each segment is equal to the difference between the actual and predicted crash frequencies. The standardized residual for the ith segment in year t denoted as SRit is

$$SR{it} = \frac{y{it} - \hat{y}{it}}{\sqrt{\hat{y}{it} + y_{it}^2/\varphi}} \tag{11}$$

Finally,the cumulative standardized residual denoted as CSR for the total segment can be derived as Eq. (12):

$$CSR = \sum{i=1}^{I} \sum{t=1}^{T} SR_{it}$$ (12)

3.3. Elasticity analysis

To investigate the insights into the implications of the prediction results, an elasticity analysis was conducted to determine the marginal effects of the explanatory variables to the predicted crash frequency. As formulated in previous studies (Abdel-Aty and Radwan, 2000; Washington et al., 2003), the elasticity of the predicted crash frequency to the explanatory variables can be estimated via Eq. (13):

$$E{x{itj}}^{\lambdai} = \frac{\partial \lambda{it}}{\partial x{itj}} \frac{x{itj}}{\lambda_{it}}$$ (13)

where Eⁱ xitj is the elasticity coefficient ofthe jthexplanatory variable, and xitj is the value of the j th explanatory variable for segment i in year t.

Owing to the different attributes of continuous and discrete explanatory variables, the elasticity computation equation for continuous and discrete explanatory variables can be expressed as Eqs. (14) and (15), respectively.

$$E{x{itj}}^{\lambda_i} = \betaj \cdot x{itj} \tag{14}$$

$$E{x{itj}}^{\lambda_i} = \frac{\exp(\beta_j) - 1}{\exp(\beta_j)} \tag{15}$$

where ˇj is the coefficient of the jth explanatory variable.

For simplification, the elasticity coefficient of the jth explanatory variable may be donated as E^j and Eqs. (14) and (15) can be expressed as Eqs. (16) and (17), respectively.

$$E_j = \beta_j \cdot \bar{x}_j \tag{16}$$

$$E_j = \frac{\exp(\beta_j) - 1}{\exp(\beta_j)} \tag{17}$$

where x¯^j is the average of the jth explanatory variable.

Table 3 Elasticity analysis (RENB) with fixed-length segments.

Explanatory variable	Mean	Coefficient	Elasticity coefficient
Longitudinal grade	1.17	−0.155	−0.514
Road width	14.03	0.212	2.992
RGR	0.40	0.481	0.195
AADT	5.27	3.96	2.087

4. Results and discussion

4.1. Fixed-length segmentation

To compare the performance of the NB and the RENB models on the basis of fixed-length road segmentation, the annual crash frequency of the selected expressway was predicted using the ML method. A mixed stepwise procedure was then used to examine the significance of explanatory variables under the hypothesis of a 95% confidence interval; and the goodness-of-fit as well as the prediction measures are calculated. The estimated results of the two models are shown in Table 2.

As illustrated in Table 2, five explanatory variables (i.e., curve length ratio, curvature change rate, longitudinal grade, RGR, and road width) significantly affectthe crash frequency in the NB model on the basis of fixed-length road segmentation. However, only four explanatory variables (i.e., longitudinal grade, RGR, road width, and AADT) are suggested to remain in the RENB model. Besides, the over-dispersion parameter ofthe NB model is 0.249, and 1/(1 +˛i)is assumed to follow a beta distribution with parameters a (= 28.861) and b (= 6.394) in the RENB model. For the NB model, the loglikelihood ratio (2) and the adjusted log-likelihood ratio (¯ 2) were 0.083 and 0.077, respectively. For the RENB model, ² and ¯ ² were 0.087 and 0.081, respectively. The values of ² and ¯ ² are relatively low because there are many variables (e.g. human factors, weather, lighting, traffic signs) that are not measured. From the value of ² and ¯ 2, the RENB model outperforms the NB model because of greater log-likelihood and adjusted likelihood ratios.

The comparison between predicted crash frequencies of the NB and RENB models on the basis of fixed length segmentation is shown in Fig. 2. It was found that the predicted crash frequency from the RENB model is closer to the actual crash frequency. The relative prediction error of the NB model is 2.89%, larger than that of the RENB model (1.81%). By the same token, one can find that the cumulative standardized residual of the NB models, 296.31, which is greater than 285.60 from the RENB model, also indicates that the RENB model outperforms the NB model in terms of prediction accuracy.

Table 4 Results of the NB and RENB models with homogeneous longitudinal grade segments.

| Explanatory variable | NB model | | RENB model | | | |----------------------|-------------|---------|-------------|---------|--| | | coefficient | p-value | coefficient | p-value | | | Intercept | −3.297 | <0.001 | −2.232 | 0.033 | | | Curve length ratio | 0.948 | <0.001 | – | – | | | Weighted curvature | −1.241 | 0.019 | – | – | | | Degree of curvature | 1.042 | 0.020 | 0.531 | 0.003 | | | Longitudinal grade | −0.326 | <0.001 | −0.207 | 0.004 | | | Road width | 0.207 | <0.001 | 0.125 | 0.007 | | | AADT | – | – | 5.289 | <0.001 | | | 1/ϕ | 0.036 | | – | | | | a | – | | 34.375 | | | | b | – | | 3.694 | | | | 2 | 0.052 | | 0.067 | | | | ¯ 2 | 0.059 | | 0.071 | | | | Re (%) | 2.34 | | 1.96 | | | | CSR | 351.41 | | 338.26 | | |

Fig. 2. Comparison of actual and predicted crash frequency with the fixed length segment method.

In Table 3, the elasticity analysis results of the RENB model on the basis of fixed-length segmentation are illustrated. Longitudinal grade is highly correlated with crash frequency (P < 0.001). The negative value of the associated coefficient indicates that upgrade segments appear to experience fewer crashes than downgrade segments. This resultis consistent with the findings of previous studies (Montella et al., 2008; Pardillo and Llemas, 2003). The longitudinal grade elasticity is −0.514, indicating that a decrease of 10% in longitudinal grade increases 5.14% in crash frequency.

Road width is associated with high crash frequency (P < 0.001). The study section consists of four-lane and six-lane subsections. Thepositive coefficient value indicates thatthe crashrisk onthe sixlane section was higher than that on the four-lane section. It seems that under the same traffic volume, drivers have a greater degree of freedomformaneuvering or overtaking ona roadsection withmore lanes, which results in more flow interruptions between adjacent lanes. This result is also consistent with the findings discussed in previous studies (Persaud and Dzbik, 1993; Potts et al., 2007). The road width elasticity is 2.992, which indicates that a 10% increase in road width leads to a 29.92% increase in crash frequency.

RGR is the ratio of longitudinal grade to curve radius and is also highly correlated with crash frequency (P = 0.005). The positive coefficient value indicates that if the increase of RGR results in increasing crashes. It seems thatthe segment with a combination of steep grade and sharp curve could lead to a complicated road environment that is complex to drivers, which increases the probability of crashes. This result is consistent with the findings discussed in previous studies (Ma et al., 2012). The elasticity for RGR is 0.195, which indicates that an increase of 10% in RGR leads to a 1.95% increase in crash frequency.

AADT is also correlated with crash frequency (P = 0.002). The positive value of the associated coefficient indicates that the number of crashes appears to increase with the increase of AADT. This is also consistent with the findings discussed in previous studies (Martin, 2002; Qin et al., 2004). The AADT elasticity of the study expressway is 2.003, whichmeans that a 10%increase inAADT leads to a nearly 20% increase in crash frequency if all other parameters hold.

4.2. Homogeneous longitudinal grade segmentation

To compare the performance of the NB and the RENB models on the basis of data processed based on homogeneous longitudinal grade segmentation, the crash frequency was predicted using the ML method. A mixed stepwise procedure was used to examine the significance of explanatory variables under the hypothesis of a 95% confidence interval, and the goodness-of-fit as the prediction measures is calculated. The estimated results using the two models are shown in Table 4.

As illustrated in Table 4, five explanatory variables (i.e., curve length ratio, weighted curvature, degree of curve, longitudinal grade, and road width) significantly affect crash frequency pre-

Fig. 3. Comparison of actual and predicted crash frequency with the homogeneous longitudinal grade segment method.

Table 5 Elasticity analysis (RENB) with homogeneous longitudinal grade segments.

Independent variable	Mean	Coefficient	Elasticity coefficient
Degree of curvature Longitudinal grade	0.45 1.17	0.531	0.240
Road width	14.03	−0.207 0.125	−0.234 1.772
AADT	5.27	5.289	2.787

dicted in the NB model. On the other hand, in the RENB model, four explanatory variables (i.e., degree of curve, grade difference, road width, and AADT) were found to significantly affect crash frequency. Furthermore, the dispersion parameter of the NB model is 0.360, and 1/(1 +˛i) is assumed to follow a beta distribution with parameter a (=34.375) and b (=3.694) in the RENB model. For the NB model, the log-likelihood ratio index (2) and the adjusted log-likelihood ratio index (¯ 2) were 0.052 and 0.059, respectively. While for the RENB model, ² and ¯ ² were 0.067 and 0.071, respectively. The value of ² and ¯ ² is relative low because there are many variables (e.g. human factors, weather, lighting, traffic signs) which are not measured. From the value of ² and ¯ 2, the RENB model outperforms the NB model because of greater log-likelihood and adjusted likelihood ratios.

The comparison of the actual and predicted crash frequencies by the NB and RENB models is shown in Fig. 3. The predicted crash frequency by the RENB model seems closer to the actual frequency. The relative prediction error of the NB model is 2.34% larger than that of the RENB model (1.96%). By the same token, one can find that the cumulative standardized residual of the NB model, which is 351.41, is greater than that (338.26) from the RENB model. In this study, apparently the RENB model outperforms the NB model.

The results of the elasticity analysis of the RENB model on the homogeneous longitudinal grade segment are shown in Table 5. The degree of curvature is correlated with crash frequency (P = 0.003). The positive value ofthe associated coefficientindicates that the higher the degree of curve, the higher the probability of a crash occurring, which were discussed in previous studies (Abdel-Aty and Radwan, 2000; Ma and Kockelman, 2006). The presumed reason is that a small curve radius may resultin a higher probability of driving error. The elasticity for road width is 0.240, which means that an increase of 10% in degree of curvature leads to an increase of 2.4% in crash frequency.

Longitudinal grade is a significant factor affecting crash frequency (P = 0.004). The negative coefficient indicates that a downhill segment increases crash frequency. This is consistent with findings discussed in previous studies (Montella et al., 2008; Pardillo and Llemas, 2003). The elasticity for longitudinal grade is −0.234, which means that an increase of 10% in longitudinal grade leads to a decrease of 2.34% in crash frequency.

Road width (four-lane vs. six-lane) also significantly affects crash frequency (P = 0.007). The positive value of the associated coefficient indicates that the crash risk on the six-lane section was higher than that on the four-lane road. A wider road usually constitutes more lanes that allow greater degrees of spatial freedom for overtaking, which results in more conflicts between adjacent lanes of traffic. This is consistent with discussions in previous studies (Persaud and Dzbik, 1993; Potts et al., 2007). The elasticity for road width is 1.772, which means that an increase of 10% in road width leads to an increase of 17.72% in crash frequency.

AADT is a significant factor influencing crash frequency (P < 0.001). The positive value ofthe associated coefficientindicates that the number of crashes appears to increase as AADT increases. This is consistent with the findings in previous studies (Martin, 2002; Qin et al., 2004). The elasticity for annual traffic volume is 2.787, which means that an increase of 10% in AADT leads to 27.87% increase in crash frequency.

4.3. Comparative analysis

To compare the performance of the developed RENB models with fixed-length and homogeneous longitudinal grade segment methods,the relative prediction error and cumulative standardized residual were calculated. The prediction errors of fixed-length segment and homogeneous longitudinal grade segment are 1.81% and 1.96%, respectively, while for the cumulative standardized residuals, the results are 285.60 and 338.26, respectively.

It was found that both the relative prediction error and cumulative standardized residual with the fixed-length segments are less than those with the homogeneous longitudinal grade segments. To this end, the RENB model developed with the data generated from fixed-length segments is recommended. Since the study expressway is located in a straight and flat area, the fixed-length segment method seems fine under this condition.

5. Conclusions

The collected crash data (2006–2008) of a 50 km long expressway section in China are employed to develop predictionmodels for crash frequency. The fixed-length and homogeneous longitudinal grade segment methods are applied to generate data for developing the NB and RENB models.

Owing to the consideration of the temporal-spatial correlation of crash data, the RENB model is superior to the NB model based not only on the fixed-length segment, but also on the homogeneous longitudinal grade method. Three explanatory variables, including longitudinal grade, road width, and AADT, have significant effects on crash frequency for the RENB model based on both the fixed-length and homogeneous longitudinal grade method. While longitudinal grade may have a decreasing impact on crash frequency, road width and AADT both have an increasing impact on crash frequency. From the elasticity analysis perspective, wider road width and higher AADT will lead to higher crash frequency. Finally, the fixed-length segment method is superior to the homogeneous longitudinal grade segment method in terms of less relative prediction error and cumulative standardized residual. Since the studied expressway is located in a straight and flat area, the fixed-length method is recommended. Assessing the performance ofthe fixed-length and the homogeneous longitudinal grade segment methods for predicting crash frequency on mountainous expressways would be an interesting extension of this study.

While the prediction accuracy by the RENB model was found to be statistically satisfied and outperformed the NB model, there are still some unanswered questions. Although explanatory variables were identified and found to be significantly affecting crash frequency, the cause of crashes remains unknown. In addition, one explanatory variable, AADT, may be replaced by hourly volume as a factor to help explain crash phenomena that is related to the time of day. Unfortunately, accurate hourly volumes of segments are difficult to obtain. In addition, inclement weather, lighting, and traffic signage may also influence crash frequency, and are thus suggested to be included as variables in the crash records.

The range of segment lengths is between 0.4 km and 1.4 km, but some research suggests that examining too short of an expressway segment accentuates the chances that a feature of the road in one segment may have triggered a crash officially located in another segment. In addition, if the segment length is long enough, more than one horizontal curve and vertical curves are aggregated in a segment, which give us more explanatory variables to be selected, such as the number of horizontal curves or vertical curves, the longest and shortest horizontal length in one segment, etc. It can help to examine more of the possible relationships between potential influence factors and crash frequency. Therefore, the factor of suitable length segment should be more deeply investigated in the future.

For the RENB model, it is worth noting that the vector of estimable parameters is a two-dimension span covering all segments over a period of three years. Due to limited data attributes in each crash record, some factors that might affect crash frequency may possibly be neglected as potential explanatory variables. To cope withthisproblem, a randomparametermodelmaybe adopted (Venkataraman et al., 2011, 2013; Mannering et al., 2016). Developing an enhanced methodology to investigate and deal with the randomness issues withinthe crashrecords would be animmediate extension of this study.

Acknowledgements

This paper was supported by the research projects sponsored by the Natural Science Foundation of China (No. 51208052), and the State Scholarship Fund of the China Scholarship Council (No. 201406565054).