Safety Evaluation of Toll Plaza Diverging Area Considering Different Vehicles’ Toll Collection Types

Abstract

Different toll collection types of vehicles and different distribution of tollbooths lead to the toll plaza diverging area becoming a typical vehicle weaving area with frequent crossing behaviors and conflicts on highways. This study aims to identify contributing factors to conflict risks of four RP by developing random parameters ordered logit models with heterogeneity in means and variances. The model can flexibly capture the unobserved heterogeneity of the contributing factors in different vehicle-following patterns. Real-world vehicle trajectory data obtained from the toll plaza diverging area in Nanjing, China, are used for model estimation. The results show that vehicle-following patterns with the same toll collection types have a higher percentage of severe conflict risks. The average acceleration of the following vehicles, lane-marking indicator, the initial lanes and lane changes of vehicles are significantly associated with the collision risk levels. The standard deviation of surrogate safety measures of all vehicles in sub-segments are found to differ significantly between vehicle-following patterns. Furthermore, a series of likelihood ratio tests are adopted to test the spatial dependence in sub-segments of the diverging area. The findings of this study could provide valuable information for safety improvement in toll plazas.

1. Introduction

Toll roads provide a great convenience in the road transportation system. In China, toll collection mainly relies on the toll plazas; they are a crucial component of toll roads. By the end of 2021, the quantity of toll plazas in China had increased to 972 [1]. Since electronic payments have not been fully popularized yet, the majority of toll plazas in China are traditional mainline toll plazas (TMTPs), which consist of both electronic toll collection (ETC) and manual toll collection (MTC) methods.

In the TMTP, drivers using electronic payment can pass through the ETC lanes at low speeds (20 km/h in China), but those paying with cash must stop at MTC lanes to finish the payment. Previous research has shown that there is a considerable speed differential between ETC and MTC vehicles in the diverging area of toll plazas [2]. However, vehicles with different toll collection types are not restricted by lane markings or separation facilities in the diverging area of the TMTP. Vehicles need to complete the diversion and select their matching toll collection lanes within a limited distance. Complex distribution of toll collection lanes and speed difference of ETC and MTC vehicles result in the diverging area becoming a hot spot of frequent crossing behaviors and rapid decelerations [3,4,5,6]. These aggressive driving behaviors caused a large number of conflicts [2,7]. There is an urgent need to explore the conflict mechanism in the diverging area of toll plazas so a corresponding effective policy can be developed.

Modeling the relationship between traffic conflict risk and contributing factors can help us understand how these factors cause conflict risks [8,9,10]. Numerous robust econometric models have been employed to investigate the possible association between conflict risk and influencing factors. Nevertheless, there may be some unobserved and unmeasured heterogeneity variables that make the existing factors less effective in explaining the observed results [11,12]. Especially on complex road nodes like toll plazas, drivers need to process more information simultaneously than on the road sections, such as choosing a matching toll lane with shorter waiting time. Ignoring such heterogeneity and assuming the effect of a contributing factor across all observations as the same will lead to biased and incorrect parameter estimation [13]. The random parameter approach is applied to solve the problem of unobserved heterogeneity. As one of the random parameter approaches, the heterogeneity in the means and variances of the random parameters has been used to explore the mechanism of crash frequency and severity and obtained remarkable results [14,15,16,17].

In previous studies, the safety analysis of toll plazas is mainly based on historical crash data [18,19]. However, due to their insufficient quantity, crash data need to be collected for a long time before they can be used for analysis. Meanwhile, since most traffic conflicts have not further developed to crashes, it is not comprehensive to judge the traffic safety status of a region based only on crash data. With the development of information techniques, a growing number of devices are used to record vehicle traveling information, such as GPS, high-definition cameras, and unmanned aerial vehicles (UAVs). Vehicles’ high-precision trajectory data are extracted in large quantities and are being utilized more frequently to analyze traffic safety. Potential conflicts captured from trajectory data overcome the limitation of insufficient quantity and incomplete assessment of historical crash data. This study analyzes the characteristics of conflicts in the toll plaza diverging area based on the trajectory data collected by UAV.

Although several studies have assessed toll plaza conflict risks based on trajectory data [2,5,7], they do not consider the conflict characteristics and influencing mechanism between vehicles with different toll collection types. Unlike driving on road segments, vehicles are additionally grouped with their toll collection types at toll plaza diverging areas that they need to drive to their matching toll booths. As mentioned above, the frequent crossing behavior between vehicles with different toll collection types is one of the key factors contributing to conflicts in the toll plaza diverging area. Therefore, this study aims to explore the influencing mechanism of vehicle conflict risks in the toll plaza diverging area from the perspective of different vehicle-following patterns with different toll collection types. To achieve this goal, trajectory data of vehicles were collected from a typical toll plaza diversion area in Nanjing, China, and the collision risk between vehicles at any angle was calculated by using the index of extended time-to-collision (ETTC). The conflicts between two vehicles are divided into four categories based on their toll collection types: (1) ETC-ETC, (2) ETC-MTC, (3) MTC-MTC, and (4) MTC-ETC, as shown in Figure 1. Each combination indicates the toll collection type of the leading vehicle followed by the toll collection type of the following vehicle. Considering the ordinal nature of conflict risks and the unobserved heterogeneity of influencing factors, a heterogeneous random parameters ordered logit model is employed to analyze the causes of conflicts among vehicles with different payment methods.

This study focuses on the following directions for the traffic safety analysis in toll plaza diverging areas: (1) comparing the characteristics of conflict risks between vehicles with different toll collection types, and (2) analyzing the differences in the formation mechanism of vehicle conflict risks between vehicles with different toll collection types. The remainder of the paper is organized as follows. Section 2, review studies, is about conflict risk modeling. In Section 3 and Section 4, major methodologies and data description are presented, respectively, followed by likelihood ratio tests in Section 5. The results and discussion are presented in Section 6. The paper concludes in Section 7.

2. Literature Review

2.1. Trajectory Data Used for Conflict Risk Analysis

Conflict data can more comprehensively represent the safety status so that it gradually replaces historical crash data for traffic safety assessment. Compared to historical crash data, traffic conflict data have four advantages [13,20,21]. First, compared with crash data, conflict data can better evaluate traffic safety from a micro perspective. Second, the quality of vehicle trajectory data is typically better than that of historical crash data, because traffic conflict data may contain more details such as various vehicle-following patterns. Third, traffic conflict data are abundant and inexpensive to acquire in comparison to historical crash data. Finally, many crashes may be not reported due to a variety of non-random factors (such as the willingness to report the crashes to police), leading to selection bias in the reported crashes [22]. Arun et al. [23] pointed out that the use of actual crash data may result in the overrepresentation of risky drivers in crash data, whereas conflict-based measures are much less likely to suffer such potential data problems.

High-resolution trajectory data can be used to capture the movement characteristics of vehicles from a microscopic perspective that has been widely used for microscopic conflict-based analysis. For example, Chen et al. [24] investigated the factors affecting the potential conflict risks during the lane change process and analyzed its unobserved heterogeneity based on the NGSIM trajectory dataset. Based on the HighD trajectory dataset, Hu et al. [25] proposed a real-time evaluation method for vehicle conflicts at a lane level. Li et al. [26] evaluated the transition durations of rear-end collision risks based on real trajectory data. Xing et al. [2] used time-varying mixed logit models to explore the traffic conflict risks in toll plaza diverging areas based on the trajectory data collected by UAV. Nevertheless, the contributing factors to vehicle conflict risk in complex road nodes have not been fully explored from the perspective of different vehicle-following patterns.

2.2. Studies Accounting for Heterogeneity

As determined by Mannering et al. [27], heterogeneous impacts may permeate all human, vehicle, traffic, and environmental characteristics that influence the possibility of crashes. While traffic conflicts and crashes share a similar failure mechanism in the driving process, modeling traffic conflicts is still a formidable challenge, especially under complex traffic conditions.

Traffic conflicts data may offer more specific factors associated with conflict risks than conventional traffic crash data. However, there are still a great number of unobserved variables. Ignoring such unobserved heterogeneity may result in misjudgment on the severity of traffic conflicts, mis-specified traffic conflicts models, biased and inefficient model parameter estimations, and ultimately erroneous inferences [21,27,28]. In recent years, therefore, the random parameters approach with heterogeneity in means and variances has dominated many econometric methods [15,17]. For instance, Yan et al. [19] utilized a random parameters logit model with heterogeneity in means and variances to investigate the contributing factors that affect the injury severity involving the driver’s gender. Alnawmasi and Mannering [29] investigated the factors affecting motorcyclist injury severities by employing a random parameters multinomial logit model. Such a method has been confirmed to improve the effect of parameter estimations compared to other random parameter models, but only a few studies have focused on conflict risk of vehicle-following groups conflict and their contributing factors [13,24]. To address the research gaps, the current study utilizes the random parameter ordered logit model with heterogeneity in means and variances to explore conflict risks of different vehicle-following patterns in the toll plaza diverging area, aiming to reveal the interaction mechanism between various ETC and MTC vehicle-following patterns.

3. Methodology

3.1. Surrogate Safety Measure

Among all traffic conflict indicators, the time-to-collision (TTC) has been commonly used to estimate individual vehicle collision risk [30,31,32]. Hayward [33] defined TTC as “the time that remains until a collision between two vehicles would have occurred if the collision course and speed difference are maintained”. It could both identify the safety conditions and assess the severity of vehicle conflicts [34].

The TTC of vehicle $\mathrm{j}$ at time step $\mathrm{t}$ with a leading vehicle $\mathrm{i}$ can be expressed as:

$${\mathrm{T}\mathrm{T}\mathrm{C}}_{\mathrm{j}}\left(\mathrm{t}\right)=\left\{\begin{array}{c}\frac{{\mathrm{X}}_{\mathrm{i}}\left(\mathrm{t}\right)-{\mathrm{X}}_{\mathrm{j}}\left(\mathrm{t}\right)-{\mathrm{L}}_{\mathrm{i}}}{{\mathrm{V}}_{\mathrm{j}}\left(\mathrm{t}\right)-{\mathrm{V}}_{\mathrm{i}}\left(\mathrm{t}\right)},\mathrm{if} {\mathrm{V}}_{\mathrm{j}}\left(\mathrm{t}\right)>{\mathrm{V}}_{\mathrm{i}}\left(\mathrm{t}\right)\\ \infty , \mathrm{if} {\mathrm{V}}_{\mathrm{j}}\left(\mathrm{t}\right)\le {\mathrm{V}}_{\mathrm{i}}\left(\mathrm{t}\right)\end{array}\right.$$

(1)

where $\mathrm{X}\left(\mathrm{t}\right)$ denotes the vehicle position at time $\mathrm{t}$, $\mathrm{V}\left(\mathrm{t}\right)$ is the vehicle speed at time $\mathrm{t}$, and ${\mathrm{L}}_{\mathrm{i}}$ is the length of vehicle $\mathrm{i}$. According to the definition in Equation (1), the larger the TTC is, the more time drivers could use to avoid a crash and thus be safer [35,36].

However, in most studies, the calculation of conventional TTC assumes that the two vehicles are in the same lane [37,38,39]. This assumption would result in errors in the vehicle collision risk calculation when two vehicles approach each other at other angles. To overcome this limitation, the extended TTC (ETTC) is applied for evaluating the vehicles’ collision risk in the diverging area. The ETTC can capture the two-dimensional conflict which is more available for traffic areas without lane markings [10,40]. The ETTC can be expressed as follows:

$${\mathrm{E}\mathrm{T}\mathrm{T}\mathrm{C}}_{\mathrm{j}}=-\frac{{\mathrm{D}}_{\mathrm{i}\mathrm{j}}-0.5{\mathrm{L}}_{\mathrm{i}}-0.5{\mathrm{L}}_{\mathrm{j}}}{\frac{1}{{\mathrm{d}}_{\mathrm{i}\mathrm{j}}}{\left({\mathbf{O}}_{\mathbf{i}}-{\mathbf{O}}_{\mathbf{j}}\right)}^{\mathrm{T}}({\mathbf{V}}_{\mathbf{i}}-{\mathbf{V}}_{\mathbf{j}})}$$

(2)

where ${\mathrm{L}}_{\mathrm{i}}$ and ${\mathrm{L}}_{\mathrm{j}}$ are the length of vehicle $\mathrm{i}$ and $\mathrm{j}$, respectively; ${\mathrm{D}}_{\mathrm{i}\mathrm{j}}$ is the distance between two vehicles’ centroids; ${\mathrm{d}}_{\mathrm{i}\mathrm{j}}$ is the distance between two closest points of vehicles; $\mathbf{O}$ and $\mathbf{V}$ are two-dimensional coordinates and speed vectors of vehicle’s centroid, as shown in Figure 2. More detailed information about ETTC can be referred in our previous studies [7,10].

3.2. The Random Parameter Ordered Logit (RPOL) Model

As mentioned above, the random parameters ordered logit model with heterogeneity in means and variances is widely applied to explore the accident mechanism as it can flexibly capture the unobserved heterogeneity. Thus, the model is employed in this study to investigate factors contributing to conflict risks. The typical form of the utility function is defined as follows:

$${\mathrm{U}}_{\mathrm{m}\mathrm{n}}={\mathsf{\beta }}_{\mathbf{m}}{\mathbf{X}}_{\mathbf{m}\mathbf{n}}+{\mathsf{\epsilon }}_{\mathrm{m}\mathrm{n}}$$

(3)

where ${\mathrm{U}}_{\mathrm{m}\mathrm{n}}$ is the function determining the probability of conflict severity $\mathrm{m}$ in sample $\mathrm{n}$, ${\mathsf{\beta }}_{\mathbf{m}}$ represents a vector of the estimated parameters, ${\mathbf{X}}_{\mathbf{m}\mathbf{n}}$ is the $\mathrm{n} \mathrm{t}\mathrm{h}$ vector of explanatory variables affecting conflict-severity $\mathrm{m}$. ${\mathsf{\epsilon }}_{\mathrm{m}\mathrm{n}}$ is a stochastic error term accounting for unobserved effects and assumed to follow the generalized extreme value distribution yielding a standard multinomial logit model:

$${\mathrm{P}}_{\mathrm{m}\mathrm{n}}=\int \frac{\exp \left({\mathsf{\beta }}_{\mathbf{m}\mathbf{n}}{\mathbf{X}}_{\mathbf{m}\mathbf{n}}\right)}{\sum \exp \left({\mathsf{\beta }}_{\mathbf{m}\mathbf{n}}{\mathbf{X}}_{\mathbf{m}\mathbf{n}}\right)}\mathrm{f}\left(\mathsf{\beta }\right|\mathsf{\phi })\mathrm{d}\mathsf{\beta }$$

(4)

where ${\mathrm{P}}_{\mathrm{m}\mathrm{n}}$ denotes the probability of risk level $\mathrm{m}$ in sample $\mathrm{n}$, $\mathrm{f}\left(\mathsf{\beta }\right|\mathsf{\phi })$ is the density function of vector $\mathsf{\beta }$ and $\mathsf{\phi }$ is the predefined parameter vector of the density function. ${\mathsf{\beta }}_{\mathbf{m}\mathbf{n}}$ is a vector of estimable parameters to capture the heterogeneity in the mean and variance, which is defined as:

$${\mathsf{\beta }}_{\mathbf{m}\mathbf{n}}={\mathsf{\beta }}_{\mathbf{m}}+{\mathrm{v}}_{\mathrm{m}\mathrm{n}}{\mathbf{Z}}_{\mathbf{m}\mathbf{n}}+{\mathsf{\sigma }}_{\mathrm{m}\mathrm{n}}\exp \left({\mathsf{\omega }}_{\mathbf{m}\mathbf{n}}{\mathbf{W}}_{\mathbf{m}\mathbf{n}}\right){\mathrm{v}}_{\mathrm{m}\mathrm{n}}$$

(5)

where ${\mathsf{\beta }}_{\mathbf{m}}$ represents the mean parameters, ${\mathsf{\nu }}_{\mathrm{m}\mathrm{n}}$ is the estimable parameters, ${\mathbf{Z}}_{\mathbf{m}\mathbf{n}}$ is the vector of collision-specific variables with the heterogeneity in the mean, ${\mathbf{W}}_{\mathbf{m}\mathbf{n}}$ is another vector of explanatory variables that explain the standard heterogeneity in the standard deviation ${\mathsf{\sigma }}_{\mathrm{m}\mathrm{n}}$, with the corresponding parameter vector ${\mathsf{\omega }}_{\mathbf{m}\mathbf{n}}$, and ${\mathrm{v}}_{\mathrm{m}\mathrm{n}}$ is an error term.

The potential conflict risk propensity function for $\mathrm{i} \mathrm{t}\mathrm{h}$ vehicle-following pair is determined by Equation (6):

$${\mathrm{Y}}_{\mathrm{m}}^{\mathrm{*}}={\mathsf{\beta }}_{\mathbf{m}}^{\mathbf{\prime }}{\mathbf{X}}_{\mathbf{m}}+{\mathsf{\epsilon }}_{\mathrm{m}}$$

(6)

$$\begin{array}{cc}\hfill {\mathrm{Y}}_{\mathrm{m}}=1,\mathrm{i}\mathrm{f} {\mathrm{Y}}_{\mathrm{m}}^{\mathrm{*}}\le {\mathrm{u}}_{\mathrm{m}0}\hfill & \\ \hfill {\mathrm{Y}}_{\mathrm{m}}=2,\mathrm{i}\mathrm{f} {\mathrm{u}}_{\mathrm{m}0}\le {\mathrm{Y}}_{\mathrm{m}}^{\mathrm{*}}\le {\mathrm{u}}_{\mathrm{m}1}\hfill & \\ \hfill {\mathrm{Y}}_{\mathrm{m}}=3,\mathrm{i}\mathrm{f} {\mathrm{Y}}_{\mathrm{m}}^{\mathrm{*}}\ge {\mathrm{u}}_{\mathrm{m}1}\hfill & \end{array}$$

(7)

where ${\mathrm{Y}}_{\mathrm{m}}=\mathrm{1,2},3$ in Equation (7) denotes the most severe, medium, and lowest collision risk level, respectively. ${\mathrm{u}}_{\mathrm{m}1}$ and ${\mathrm{u}}_{\mathrm{m}0}$ are the estimated thresholds.

The probability of vehicle group $\mathrm{i}$ to be $\mathrm{j} \mathrm{t}\mathrm{h}$ collision risk level is shown in Equation (8):

$$\begin{array}{cc}\hfill \mathrm{P}\left(\mathrm{y}=1\right)=\mathsf{\Phi }(-{\mathsf{\beta }}_{\mathbf{m}}{\mathbf{X}}_{\mathbf{m}})\hfill & \\ \hfill \mathrm{P}\left(\mathrm{y}=2\right)=\mathsf{\Phi }\left({\mathrm{u}}_1-{\mathsf{\beta }}_{\mathbf{m}}{\mathbf{X}}_{\mathbf{m}}\right)-\mathsf{\Phi }\left(-{\mathsf{\beta }}_{\mathbf{m}}{\mathbf{X}}_{\mathbf{m}}\right)\hfill & \\ \hfill \mathrm{P}\left(\mathrm{y}=3\right)=1-\mathsf{\Phi }\left({\mathrm{u}}_1-{\mathsf{\beta }}_{\mathbf{m}}{\mathbf{X}}_{\mathbf{m}}\right)\hfill & \end{array}$$

(8)

where $\mathsf{\Phi }\left(\right)$ denotes the standard normal cumulative distribution function, y = 0 indicates the lowest conflict level, y = 1 indicates the medium conflict risk level, and y = 2 indicates the highest conflict risk level.

The random parameter models are estimated with 500 Halton draws as past researchers [19,41] have found that 500 Halton draws could provide stable parameters. The stepwise process is also conducted for the parameter selection. The Akaike Information Criterion (AIC) and McFadden Pseudo R-squared are used to evaluate the performances of the fitted models [42], which are formulated as:

$$\mathrm{A}\mathrm{I}\mathrm{C}=-2\mathrm{L}\mathrm{L}\left({\mathrm{M}}_{\mathrm{F}\mathrm{u}\mathrm{l}\mathrm{l}}\right)+2\mathrm{K}$$

(9)

$${\mathrm{R}}^2=1-\frac{\mathrm{L}\mathrm{L}\left({\mathrm{M}}_{\mathrm{F}\mathrm{u}\mathrm{l}\mathrm{l}}\right)}{\mathrm{L}\mathrm{L}\left({\mathrm{M}}_{\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}}\right)}$$

(10)

where $\mathrm{K}$ is the number of estimated model parameters in the model, ${\mathrm{M}}_{\mathrm{F}\mathrm{u}\mathrm{l}\mathrm{l}}$ is the full model with the constant term and all predicting variables, and ${\mathrm{M}}_{\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}}$ is the intercept model only including the constant term. $\mathrm{L}\mathrm{L}\left({\mathrm{M}}_{\mathrm{F}\mathrm{u}\mathrm{l}\mathrm{l}}\right)$ is the log-likelihood value for ${\mathrm{M}}_{\mathrm{F}\mathrm{u}\mathrm{l}\mathrm{l}}$, and $\mathrm{L}\mathrm{L}\left({\mathrm{M}}_{\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}}\right)$ is the log-likelihood value for ${\mathrm{M}}_{\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}}$. Lower AIC value and larger R² of a model indicate the model has better performance [14].

4. Data Description

4.1. Data Collection

To explore the formation mechanism of conflict risk in the diverging area of the toll plaza, trajectory data were collected at the toll plaza located on G42 freeway in the northeast area of Nanjing, China. G42 is an east–west major corridor from Shanghai to Chengdu, with four main lanes for each direction. This study site focused on the eastbound diverging area of the toll plaza. Figure 3 depicts the layout of the diverging area. The diverging area is approximately 360 m in length, including 60-m main lanes and a 300-m non-marked diverging area. Over 300 m, the width of the diverging area without lane markings is increased from 15 m to 65 m along the driving direction of vehicles. There are 12 toll lanes in total, including 3 ETC lanes on the leftmost side and 9 MTC lanes on the right side. Note that MTC trucks can only use the two rightmost MTC lanes.

As mentioned above, trajectory data contain abundant traffic information, which formed the basis for the microscopic study of conflict risk. In this study, the unmanned aerial vehicle (UAV), which could provide 4K ultra-high quality and 30 frames per second (fps) video, was utilized to record vehicle movements in the diverging area. The data were collected during peak periods on 17 March 2018, a sunny and windless day, to ensure that the UAV can hover steadily and capture clear videos. The UAV flew at an altitude of around 200 m. A total of 1.5 h videos were recorded, and 50 min videos were finally selected for analysis. The videos that did not capture any conflicts between vehicles or failed to record complete vehicle trajectories were removed.

The video data were processed by the video analytics system named Automated Roadway Conflicts Identification System (ARCIS), developed by the University of Central Florida Smart and Safe Transportation (UCF SST) team [43,44]. The system contains video data stabilization, features detection and tracking, coordinate transformation, and error elimination. The ARCIS has higher vehicle detecting and tracking accuracy than a traditional video analysis system. More detailed data processing could be seen in our previous study [10]. The trajectories of 1152 vehicles were obtained after data processing, including 1031 cars, 30 buses, and 42 trucks. The hourly traffic flow rate ranged from 1050 vph to 1740 vph (calculated six times of the 10-min volume).

4.2. Model Formulation

The threshold of TTC usually ranges from 2 s to 4 s in previous studies [9,45]. Considering that the ETTC is an extension of the conventional TTC, the threshold of TTC also applies to ETTC. In this study, 4 s is selected as the threshold for ETTC as it could contain more potentially risky cases.

Dividing the safety status of vehicles into conflict and non-conflict only may ignore the severity of the conflict, which leads to the safety analysis lack of details. Classifying the severity of traffic conflicts between vehicles could assess the state of traffic safety status more accurately. Conflict severity is often divided into multiple categories with an intrinsically ordinal structure, and thus the ordered-response model may be a promising alternative to account for this characteristic [9,41,46].

Therefore, in this study, the conflict risks of different ETC and MTC vehicle-following patterns are divided into three levels: (1) The most severe conflict risk level, indicating a high possibility of collision (ETTC less than 2.5 s). This risk level applies to situations in which a collision will occur if the driver does not take prompt action immediately; (2) The medium risk level, which represents that the two vehicles have the potential for collision but drivers have sufficient reaction time to avoid the collision (ETTC between 2.5–4 s); and (3) The lowest risk level (ETTC larger than 4 s), which represents that the distance between the two vehicles is large enough so that it is nearly impossible to collide. Figure 4 depicts the frequency and proportion of observations for each level of conflict risk. Vehicle-following patterns with the same toll collection types (ETC-ETC, MTC-MTC) have a higher percentage of most severe conflict risk. The MTC-ETC vehicle-following pattern has the highest percentage of slightest conflict risk and the lowest percentage of the most severe and medium risk levels.

To fully explore the differences of conflict risk among different parts of the diverging area, the diverging area is divided into 12 sub-segments. As depicted in Figure 5, the length of each sub-segment is 30 m, numbered from 1 to 12. The first ten sub-segments are in the non-lane marked area, while the last two sub-segments are within the lane marked diverging area.

The candidate variables used in this study are listed in

Table 1. Statistics of candidate variables.

Variable Category	Variable	Variable Description	Mean Value	Standard Deviation
			(All Vehicles)
Location-related variables of the following vehicle $i$	${\mathrm{L}}_{\mathrm{I}\mathrm{N}\mathrm{I}\mathrm{T}}$	The initial lane of the following vehicle, LINIT (1–4) is numbered from left to right of vehicles’ driving direction.	1.909	0.943
	${\mathrm{L}}_{\mathrm{M}}$	An indicator variable for lane-marking features of the sub-segment, 0 if non-lane marked, 1 if lane marked.	0.101	0.301
	${\mathrm{L}}_{\mathrm{w}\mathrm{i}\mathrm{d}\mathrm{t}\mathrm{h}}$	The average width of the sub-segment where the vehicle is located, m.	37.769	15.236
	${\mathrm{N}}_{\mathrm{L}\mathrm{L}\mathrm{C}}$	An indicator variable for the number of lanes vehicle changes to the left, NLLC = 1 for less than two lanes, NLLC = 2 for more than two lanes.	2.031	1.532
	${\mathrm{N}}_{\mathrm{R}\mathrm{L}\mathrm{C}}$	An indicator variable for the number of lanes vehicle changes to the right, NRLC = 1 for less than two lanes, NRLC = 2 for more than two lanes.	2.892	1.673
Speed-related variables of the following vehicle $i$	${\mathrm{V}}_{\mathrm{M}\mathrm{E}\mathrm{A}\mathrm{N}}$	Average speed in whole toll plaza diverging area, m/s2.	17.150	4.275
	${\mathrm{V}\mathrm{A}}_{\mathrm{M}\mathrm{E}\mathrm{A}\mathrm{N}}$	Average acceleration in whole toll plaza diverging area, m/s2.	0.429	0.387
	${\mathrm{V}\mathrm{D}}_{\mathrm{M}\mathrm{E}\mathrm{A}\mathrm{N}}$	Average deceleration in whole toll plaza diverging area, m/s2.	−0.390	0.533
	${\mathrm{V}\mathrm{A}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$	The standard deviation of vehicle acceleration in whole toll plaza diverging area, m/s2.	0.755	0.497
	${\mathrm{V}\mathrm{A}}_{\mathrm{M}\mathrm{A}\mathrm{X}}$	The maximum vehicle acceleration in whole toll plaza diverging area, m/s2.	2.474	1.917
	${\mathrm{T}\mathrm{V}}_{\mathrm{M}\mathrm{E}\mathrm{A}\mathrm{N}}$	The average speed within 3 s.	17.248	4.250
	${\mathrm{T}\mathrm{V}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$	The speed standard deviation of the following vehicle within 3 s, m/s.	0.422	0.370
	${\mathrm{T}\mathrm{A}}_{\mathrm{M}\mathrm{E}\mathrm{A}\mathrm{N}}$	The average acceleration within 3 s.	−0.388	0.541
$\mathrm{Traffic} \mathrm{condition} \mathrm{variables} \mathrm{related} \mathrm{to} \mathrm{the} \mathrm{area} \mathrm{around} \mathrm{the} \mathrm{following} \mathrm{vehicle} i$	${\mathrm{Z}}_{\mathrm{V}\mathrm{O}\mathrm{L}}$	The average traffic volume of the sub-segments, vph.	6389.150	2926.179
	${\mathrm{Z}\mathrm{V}}_{\mathrm{M}\mathrm{E}\mathrm{A}\mathrm{N}}$	The average speed of all vehicles in sub-segments.	17.030	2.676
	${\mathrm{Z}\mathrm{V}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$	The standard deviation of speed of all vehicles in sub-segments.	3.475	1.619
	${\mathrm{Z}\mathrm{D}}_{\mathrm{M}\mathrm{E}\mathrm{A}\mathrm{N}}$	Average deceleration of all vehicles in sub-segments, m/s2.	3.154	0.970
	${\mathrm{Z}\mathrm{N}}_{\mathrm{E}\mathrm{T}\mathrm{T}\mathrm{C}}$	The number of vehicles with ETTC less than 4 s in sub-segments.	1.114	1.109
	${\mathrm{Z}\mathrm{N}\mathrm{P}}_{\mathrm{E}\mathrm{T}\mathrm{T}\mathrm{C}}$	The percentage of vehicles with ETTC less than 4 s in sub-segments.	0.343	0.301
	${\mathrm{Z}\mathrm{E}}_{\mathrm{M}\mathrm{E}\mathrm{A}\mathrm{N}}$	The average ETTC value of all vehicles in sub-segments.	1.556	1.296
	${\mathrm{Z}\mathrm{E}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$	The standard deviation of ETTC of all vehicles in sub-segments.	1.195	0.479
	${\mathrm{Z}}_{\mathrm{T}\mathrm{E}\mathrm{T}}$	The time exposed to collision of the following vehicle, s.	1.195	0.478
	${\mathrm{Z}\mathrm{P}\mathrm{C}\mathrm{T}}_{\mathrm{E}\mathrm{T}\mathrm{C}}$	The percentage of ETC vehicles in sub-segments.	0.441	0.355
	${\mathrm{Z}}_{\mathrm{M}\mathrm{I}\mathrm{X}}$	The mix measure of MTC and ETC vehicles in sub-segments.	0.491	0.468

. The variables considered in the model can be categorized into three groups: (1) location-related variables, mainly including the position features of the following vehicle during the whole diverging process at each time step; (2) speed-related variables, the average speed and acceleration of the following vehicle in each sub-segment, and the whole diverging process at each time step; (3) traffic environmental variables, mainly including the conflict risks of vehicles around the following vehicle in the same sub-segment at each time step.

Specifically, in traffic environmental variables, considering the mixed travel of MTC and ETC vehicles is the main feature of the TMTP, the entropy index ${\mathrm{Z}}_{\mathrm{M}\mathrm{I}\mathrm{X}}$ was developed to assess the mixed degree of MTC and ETC vehicles in each sub-segment. Moreover, the length of conflict duration has been proven to be an essential index for analyzing the safety state of a vehicle as the start and end of a collision is a continuous process [26]. Thus, this study applies the time exposed to collision (${\mathrm{T}}_{\mathrm{T}\mathrm{E}\mathrm{C}}$) to the candidate variables.

5. Likelihood Ratio Tests

To examine the differences in conflict severity outcomes between different ETC and MTC vehicle-following patterns and the spatial dependence in sub-segments, four groups of likelihood ratio tests were conducted.

To compare the models developed for four different vehicle-following patterns between MTC and ETC vehicles, the first series of likelihood ratio tests were conducted, which can be expressed as follows [19,29]:

$$\begin{array}{cc}\hfill {\mathsf{\chi }}_1^2=-2[\mathrm{L}\mathrm{L}\left({\mathsf{\beta }}_{\mathrm{A}\mathrm{l}\mathrm{l}}\right)-\mathrm{L}\mathrm{L}\left({\mathsf{\beta }}_{\mathrm{E}\mathrm{T}\mathrm{C}-\mathrm{E}\mathrm{T}\mathrm{C}}\right)-\mathrm{L}\mathrm{L}\left({\mathsf{\beta }}_{\mathrm{E}\mathrm{T}\mathrm{C}-\mathrm{M}\mathrm{T}\mathrm{C}}\right)-\mathrm{L}\mathrm{L}\left({\mathsf{\beta }}_{\mathrm{M}\mathrm{T}\mathrm{C}-\mathrm{E}\mathrm{T}\mathrm{C}}\right)\hfill & \\ \hfill -\mathrm{L}\mathrm{L}\left({\mathsf{\beta }}_{\mathrm{M}\mathrm{T}\mathrm{C}-\mathrm{M}\mathrm{T}\mathrm{C}}\right)]\hfill & \end{array}$$

(11)

where $\mathrm{L}\mathrm{L}\left({\mathsf{\beta }}_{\mathrm{A}\mathrm{l}\mathrm{l}}\right)$ is the log-likelihood at the convergence of the model estimated with all data, $\mathrm{L}\mathrm{L}\left({\mathsf{\beta }}_{\mathrm{E}\mathrm{T}\mathrm{C}-\mathrm{E}\mathrm{T}\mathrm{C}}\right)$, $\mathrm{L}\mathrm{L}\left({\mathsf{\beta }}_{\mathrm{E}\mathrm{T}\mathrm{C}-\mathrm{M}\mathrm{T}\mathrm{C}}\right)$, $\mathrm{L}\mathrm{L}\left({\mathsf{\beta }}_{\mathrm{M}\mathrm{T}\mathrm{C}-\mathrm{E}\mathrm{T}\mathrm{C}}\right),$ and $\mathrm{L}\mathrm{L}\left({\mathsf{\beta }}_{\mathrm{M}\mathrm{T}\mathrm{C}-\mathrm{M}\mathrm{T}\mathrm{C}}\right)$ represent the log-likelihood values at the convergence of the estimated models using different ETC and MTC vehicle-following patterns.

Moreover, to investigate the spatial stability of variables between different sub-segments of the diverging area, three sub-models were developed for three parts of the diverging area (denoted as D₁, D₂, and D₃; D₁ represents the downstream of the diverging area, including sub-segment 1–4; D₂ represents the midstream of the diverging area, including sub-segment 5–8; D₃ represents the upstream of the diverging area, including sub-segment 9–12, as shown in Figure 4). The second series of likelihood ratio tests were utilized [11]:

$${\mathsf{\chi }}_2^2=-2[\mathrm{L}\mathrm{L}\left({\mathsf{\beta }}_{{\mathrm{y}}_1{\mathrm{y}}_2}\right)-\mathrm{L}\mathrm{L}\left({\mathsf{\beta }}_{{\mathrm{y}}_1}\right)]$$

(12)

where $\mathrm{L}\mathrm{L}\left({\mathsf{\beta }}_{{\mathrm{y}}_1{\mathrm{y}}_2}\right)$ is the log-likelihood at convergence of a model containing parameters from y₂ while using data subset y₁, and $\mathrm{L}\mathrm{L}\left({\mathsf{\beta }}_{{\mathrm{y}}_1}\right)$ is the log-likelihood at convergence of the model using data subset y₁.

Table 2. Likelihood ratio test results between different diverging areas (ETC-ETC model).

y1	y2
	D1 (0–120 m)	D2 (120–240 m)	D3 (240–360 m)
D1 (0–120 m)	–	22.918 (8)	27.420 (14)
		[99.65%]	[98.30%]
D2 (120–240 m)	27.428 (9)	–	28.678 (14)
	[99.88%]		[98.85%]
D3 (240–360 m)	23.748 (9)	25.475 (8)	–
	[99.53%]	[99.87%]

Notes: the numbers in parentheses represent the degrees of freedom for the Likelihood Ratio Test.

,

Table 3. Likelihood ratio test results between different diverging areas (ETC-MTC model).

y1	y2
	D1 (0–120 m)	D2 (120–240 m)	D3 (240–360 m)
D1(0–120 m)	–	50.473 (13)	102.1 (14)
		[>99.99%]	[>99.99%]
D2(120–240 m)	25.011 (11)	–	40.862 (14)
	[99.09%]		[99.98%]
D3(240–360 m)	23.345 (11)	38.282 (13)	–
	[98.42%]	[99.97%]

Notes: the numbers in parentheses represent the degrees of freedom for the Likelihood Ratio Test.

,

Table 4. Likelihood ratio test results between different diverging areas (MTC-MTC model).

y1	y2
	D1 (0–120 m)	D2 (120–240 m)	D3 (240–360 m)
D1 (0–120 m)	–	36.378 (14)	30.849 (15)
		[99.90%]	[99.08%]
D2 (120–240 m)	30.608 (13)	–	38.868 (15)
	[99.61%]		[99.93%]
D3 (240–360 m)	37.190 (13)	54.555 (14)	–
	[99.96%]	[>99.99%]

Notes: the numbers in parentheses represent the degrees of freedom for the Likelihood Ratio Test.

and

Table 5. Likelihood ratio test results between different diverging areas (MTC-ETC model).

y1	y2
	D1 (0–120 m)	D2 (120–240 m)	D3 (240–360 m)
D1 (0–120 m)	–	36.236 (13)	41.594 (14)
		[99.95%]	[99.99%]
D2 (120–240 m)	43.114 (12)	–	41.666 (14)
	[>99.99%]		[99.99%]
D3 (240–360 m)	64.023 (12)	38.314 (13)	–
	[>99.99%]	[99.97%]

Notes: the numbers in parentheses represent the degrees of freedom for the Likelihood Ratio Test.

present whether the null hypothesis that the parameters are stable in the three parts of the diverging area can be rejected based on the results. For example, as shown in Table 2, using the converged parameters of the D₂ model (y₂) as the initial values and applying them to the D₁ data (y₁) yields ${\mathsf{\chi }}^2=27.428$ with 9 degrees of freedom, indicating that the null hypothesis that the two parts of the diverging area are the same can be rejected at a 99.88% confidence level. Likewise, using the converged parameters of the D₁ model as the starting values and applying them to the D₂ data gave ${\mathsf{\chi }}^2=22.918$ with 8 degrees of freedom, demonstrating that the null hypothesis that the parts of the diverging area are the same can be rejected at a 99.65% confidence level. Other subsections have similar results that the null hypotheses can be rejected at a high confidence level.

6. Results and Discussions

A total of 24 variables are considered. All dummy variables are transformed to binary indicator variables, and numbers are appended to the variable names to distinguish the indicator variables (e.g., ${\mathrm{N}}_{\mathrm{L}\mathrm{L}\mathrm{C}}1$, ${\mathrm{N}}_{\mathrm{R}\mathrm{L}\mathrm{C}}1$). Prior to modeling, the correlations between the variables are analyzed, and those with high correlations are removed. In addition, Z-score is used to standardize continuous data in order to eliminate the effects of dimension and make them comparable with other variables.

The results of the best-fitted random parameters ordered logit (RPOL) models and RPOL models with heterogeneity in means (RPOLMH) are shown in

Table 6. Parameter estimation results for all vehicle-following patterns.

Variables	RPOL		RPOLMH
	Coefficient	t-Stat	Coefficient	t-Stat
Nonrandom parameters
Threshold parameters for probabilities	3.345 ***	67.06	3.478 ***	66.63
Constant	−4.448 ***	−19.48	−4.583 ***	−15.87
${\mathrm{L}}_{\mathrm{I}\mathrm{N}\mathrm{I}\mathrm{T}}2$	−0.285 ***	−3.83	−0.606 ***	−3.29
${\mathrm{L}}_{\mathrm{I}\mathrm{N}\mathrm{I}\mathrm{T}}3$	−0.709 ***	−9.34	−0.365 **	−2.00
${\mathrm{L}}_{\mathrm{I}\mathrm{N}\mathrm{I}\mathrm{T}}4$	−0.813 ***	−9.4	−0.255 ***	−2.411
${\mathrm{Z}\mathrm{P}\mathrm{C}\mathrm{T}}_{\mathrm{E}\mathrm{T}\mathrm{C}}$	−0.178 ***	−7.14	−0.176 ***	−3.14
${\mathrm{L}}_{\mathrm{M}}$	−0.176 ***	−7.08	−0.448 ***	−6.80
${\mathrm{V}\mathrm{A}}_{\mathrm{M}\mathrm{A}\mathrm{X}}$	−0.144 **	−6.24	−0.246 ***	−4.54
${\mathrm{T}\mathrm{V}}_{\mathrm{M}\mathrm{E}\mathrm{A}\mathrm{N}}$	0.176 ***	6.35	2.146 ***	6.05
${\mathrm{T}\mathrm{A}}_{\mathrm{M}\mathrm{E}\mathrm{A}\mathrm{N}}$	−0.396 ***	−14.64	1.488 ***	46.53
${\mathrm{Z}\mathrm{N}\mathrm{P}}_{\mathrm{E}\mathrm{T}\mathrm{T}\mathrm{C}}$	0.545 ***	21.20	0.5543 ***	21.15
${\mathrm{N}}_{\mathrm{L}\mathrm{L}\mathrm{C}}1$	0.352 **	4.70	0..315 ***	4.12
${\mathrm{V}\mathrm{A}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$	0.146 ***	6.05	2.146 ***	6.05
Random parameters
${\mathrm{Z}}_{\mathrm{V}\mathrm{O}\mathrm{L}}$	−0.446 ***	−14.61	−0.476 ***	−5.03
$\mathrm{Standard} \mathrm{deviation} \mathrm{for} \mathrm{the} \mathrm{random} \mathrm{parameter} ({\mathrm{Z}}_{\mathrm{V}\mathrm{O}\mathrm{L}}$)	0.906 ***	17.18	0.968 ***	59.12
${\mathrm{Z}}_{\mathrm{M}\mathrm{I}\mathrm{X}}$	−0.286 ***	−4.12	−0.674 ***	−6.00
$\mathrm{Standard} \mathrm{deviation} \mathrm{for} \mathrm{the} \mathrm{random} \mathrm{parameter} ({\mathrm{Z}}_{\mathrm{M}\mathrm{I}\mathrm{X}}$)	0.903 ***	46.43	0.368 ***	21.49
Heterogeneity in the means of random parameters
${\mathrm{Z}}_{\mathrm{V}\mathrm{O}\mathrm{L}}$:			−0.156 *	−1.86
${\mathrm{L}}_{\mathrm{I}\mathrm{N}\mathrm{I}\mathrm{T}}3$
${\mathrm{Z}}_{\mathrm{V}\mathrm{O}\mathrm{L}}$:			−0.272 **	−2.93
${\mathrm{L}}_{\mathrm{I}\mathrm{N}\mathrm{I}\mathrm{T}}4$
${\mathrm{Z}}_{\mathrm{V}\mathrm{O}\mathrm{L}}$:			0.106 ***	3.83
${\mathrm{L}}_{\mathrm{M}}$
${\mathrm{Z}}_{\mathrm{M}\mathrm{I}\mathrm{X}}$			0.214 ***	2.66
${\mathrm{L}}_{\mathrm{I}\mathrm{N}\mathrm{I}\mathrm{T}}2$
${\mathrm{Z}}_{\mathrm{M}\mathrm{I}\mathrm{X}}$			0.193 ***	3.68
${\mathrm{Z}\mathrm{P}\mathrm{C}\mathrm{T}}_{\mathrm{E}\mathrm{T}\mathrm{C}}$
${\mathrm{Z}}_{\mathrm{M}\mathrm{I}\mathrm{X}}$			0.059 ***	2.29
${\mathrm{L}}_{\mathrm{M}}$
${\mathrm{Z}}_{\mathrm{M}\mathrm{I}\mathrm{X}}$			0.048 *	1.88
${\mathrm{V}\mathrm{A}}_{\mathrm{M}\mathrm{A}\mathrm{X}}$
Number of parameters	17			24
Number of observations	11,306			11,306
Log-likelihood with constants only	−11,214.896			−11,214.896
Log-likelihood at convergence	−9828.941			−9786.598
Akaike information criterion (AIC)	19,697.9		19,635.2

Notes: *, **, and *** denote the statistical significance at 0.1, 0.05, and 0.01 levels, respectively.

,

Table 7. Parameter estimation results for ETC-ETC vehicle-following pattern.

Variables	RPOL		RPOLMH
	Coefficient	t-Stat	Coefficient	t-Stat
Nonrandom parameters
Threshold parameters for probabilities	3.381 ***	29.69	4.366 ***	28.18
Constant	−5.576 ***	−10.37	−5.408 ***	−8.160
${\mathrm{L}}_{\mathrm{I}\mathrm{N}\mathrm{I}\mathrm{T}}2$	0.657 ***	4.97	0.817 ***	5.520
${\mathrm{L}}_{\mathrm{I}\mathrm{N}\mathrm{I}\mathrm{T}}3$	1.600 ***	10.36	2.019 ***	11.460
${\mathrm{L}}_{\mathrm{I}\mathrm{N}\mathrm{I}\mathrm{T}}4$	1.943 ***	7.67	2.339 ***	8.320
${\mathrm{Z}}_{\mathrm{M}\mathrm{I}\mathrm{X}}$	0.211 ***	3.19	−0.5679 *	−0.260
${\mathrm{L}}_{\mathrm{M}}$	−0.268 ***	−4.82	−0.294 **	−2.220
${\mathrm{V}\mathrm{A}}_{\mathrm{M}\mathrm{A}\mathrm{X}}$	−0.122 **	−2.34	−0.164 ***	−2.830
${\mathrm{L}\mathrm{T}\mathrm{V}}_{\mathrm{M}\mathrm{E}\mathrm{A}\mathrm{N}}$	1.707 ***	22.03	2.122 ***	22.440
${\mathrm{T}\mathrm{A}}_{\mathrm{M}\mathrm{E}\mathrm{A}\mathrm{N}}$	−0.572 ***	−9.36	−0.747 ***	−10.720
${\mathrm{Z}\mathrm{N}\mathrm{P}}_{\mathrm{E}\mathrm{T}\mathrm{T}\mathrm{C}}$	0.789 ***	13.18	0.332 *	1.910
${\mathrm{N}}_{\mathrm{L}\mathrm{L}\mathrm{C}}1$	0.319 **	2.37	0.417 ***	2.780
Random parameters
${\mathrm{T}\mathrm{V}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$	0.184 ***	2.69	0.404 ***	2.910
Standard deviation for the random parameter (${\mathrm{T}\mathrm{V}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$)	0.674 ***	17.18	1.013 ***	20.310
${\mathrm{Z}}_{\mathrm{V}\mathrm{O}\mathrm{L}}$	−0.286 ***	−4.12	−1.290 ***	−9.340
Standard deviation for the random parameter (${\mathrm{Z}}_{\mathrm{V}\mathrm{O}\mathrm{L}}$)	0.903 ***	26.85	1.381 ***	28.010
${\mathrm{Z}\mathrm{E}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$	0.262 ***	4.62	0.382 ***	3.80
Standard deviation for the random parameter (${\mathrm{Z}\mathrm{E}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$)	0.975 ***	23.14	1.086 ***	22.490
Heterogeneity in the means of random parameters
${\mathrm{T}\mathrm{V}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$:			0.895 *	1.420
${\mathrm{Z}}_{\mathrm{M}\mathrm{I}\mathrm{X}}$
${\mathrm{T}\mathrm{V}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$:			−0.239 **	−2.110
${\mathrm{L}}_{\mathrm{M}}$
${\mathrm{T}\mathrm{V}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$:			−0.149 **	−2.540
${\mathrm{Z}\mathrm{N}\mathrm{P}}_{\mathrm{E}\mathrm{T}\mathrm{T}\mathrm{C}}$
${\mathrm{Z}}_{\mathrm{V}\mathrm{O}\mathrm{L}}$:			0.937 *	1.270
${\mathrm{Z}}_{\mathrm{M}\mathrm{I}\mathrm{X}}$
${\mathrm{Z}}_{\mathrm{V}\mathrm{O}\mathrm{L}}$:			0.613 ***	8.270
${\mathrm{Z}\mathrm{N}\mathrm{P}}_{\mathrm{E}\mathrm{T}\mathrm{T}\mathrm{C}}$
${\mathrm{Z}\mathrm{E}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$:			0.107 *	1.800
${\mathrm{L}}_{\mathrm{M}}$
${\mathrm{Z}\mathrm{E}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$:			−0.144 **	−2.410
${\mathrm{Z}\mathrm{N}\mathrm{P}}_{\mathrm{E}\mathrm{T}\mathrm{T}\mathrm{C}}$
Number of parameters	18			25
Number of observations	2466			2466
Log-likelihood with constants only	−3000.274			−3000.274
Log-likelihood at convergence	−2308.335			−2294.59157
Akaike information criterion (AIC)	4652.7			4639.2
McFadden Pseudo R-squared	0.230			0.235

Notes: *, **, and *** denote the statistical significance at 0.1, 0.05, and 0.01 levels, respectively.

,

Table 8. Parameter estimation results for ETC-MTC vehicle-following pattern.

Variables	RPOL		RPOLMH
	Coefficient	t-Stat	Coefficient	t-Stat
Nonrandom parameters
Threshold parameters for probabilities	3.565 ***	27.37	3.869 ***	26.88
Constant	−7.383 ***	−11.32	−7.475 ***	−10.41
${\mathrm{L}}_{\mathrm{I}\mathrm{N}\mathrm{I}\mathrm{T}}2$	−1.140 ***	−8.85	−1.141 ***	−8.46
${\mathrm{Z}}_{\mathrm{M}\mathrm{I}\mathrm{X}}$	−0.130 *	−1.81	−0.420	−2.2
${\mathrm{L}}_{\mathrm{M}}$	0.075 *	2.59	−0.395 **	−2.53
${\mathrm{V}\mathrm{A}}_{\mathrm{M}\mathrm{A}\mathrm{X}}$	0.184 ***	3.29	0.244 ***	4.17
${\mathrm{T}\mathrm{V}}_{\mathrm{M}\mathrm{E}\mathrm{A}\mathrm{N}}$	1.903 ***	23.28	2.077 **	23.57
${\mathrm{T}\mathrm{A}}_{\mathrm{M}\mathrm{E}\mathrm{A}\mathrm{N}}$	−0.447 ***	−5.85	−0.453 ***	−5.67
${\mathrm{Z}\mathrm{N}\mathrm{P}}_{\mathrm{E}\mathrm{T}\mathrm{T}\mathrm{C}}$	0.694 ***	10.81	0.224	1.27
${\mathrm{Z}}_{\mathrm{T}\mathrm{E}\mathrm{T}}$	−0.224 ***	−3.5	−0.270 ***	−4.05
${\mathrm{N}}_{\mathrm{R}\mathrm{L}\mathrm{C}}1$	0.983 ***	8.8	1.095 ***	9.37
Random parameters
${\mathrm{Z}}_{\mathrm{V}\mathrm{O}\mathrm{L}}$	−0.799 ***	−10.33	−1.746 ***	−9.57
Standard deviation for the random parameter (${\mathrm{Z}}_{\mathrm{V}\mathrm{O}\mathrm{L}}$)	0.641 ***	21.28	0.744 ***	22.33
${\mathrm{Z}\mathrm{E}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$	0.192 ***	3.07	0.0402 ***	3.13
Standard deviation for the random parameter (${\mathrm{Z}\mathrm{E}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$)	0.577 ***	14.45	0.520 ***	12.95
${\mathrm{T}\mathrm{V}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$	−0.518 *	−72	0.990 *	6.60
Standard deviation for the random parameter (${\mathrm{T}\mathrm{V}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$)	0.815 ***	17.24	0.977 ***	18.68
Heterogeneity in the means of random parameters
${\mathrm{Z}}_{\mathrm{V}\mathrm{O}\mathrm{L}}$:			0.151 ***	0.052
${\mathrm{L}}_{\mathrm{M}}$
${\mathrm{Z}}_{\mathrm{V}\mathrm{O}\mathrm{L}}$:			0.197 **	0.094
${\mathrm{Z}}_{\mathrm{M}\mathrm{I}\mathrm{X}}$
${\mathrm{T}\mathrm{V}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$:			0.380 ***	0.076
${\mathrm{Z}\mathrm{N}\mathrm{P}}_{\mathrm{E}\mathrm{T}\mathrm{T}\mathrm{C}}$
${\mathrm{Z}\mathrm{E}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$:			0.186 **	0.072
${\mathrm{Z}}_{\mathrm{M}\mathrm{I}\mathrm{X}}$
${\mathrm{Z}\mathrm{E}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$:			−0.416 ***	0.066
${\mathrm{Z}\mathrm{N}\mathrm{P}}_{\mathrm{E}\mathrm{T}\mathrm{T}\mathrm{C}}$
${\mathrm{Z}}_{\mathrm{V}\mathrm{O}\mathrm{L}}$:			0.019 *	0.106
${\mathrm{L}}_{\mathrm{M}}$
${\mathrm{Z}}_{\mathrm{V}\mathrm{O}\mathrm{L}}$:			−0.405 ***	0.074
${\mathrm{Z}}_{\mathrm{M}\mathrm{I}\mathrm{X}}$
${\mathrm{Z}}_{\mathrm{V}\mathrm{O}\mathrm{L}}$:			0.272 ***	0.065
${\mathrm{Z}\mathrm{N}\mathrm{P}}_{\mathrm{E}\mathrm{T}\mathrm{T}\mathrm{C}}$
Number of parameters	17			25
Number of observations	2561			2561
Log-likelihood with constants only	−2402.155			−2402.155
Log-likelihood at convergence	−1776.738			−1751.157
Akaike information criterion (AIC)	3587.5			3552.3
McFadden Pseudo R-squared	0.260			0.271

Notes: *, **, and *** denote the statistical significance at 0.1, 0.05, and 0.01 levels, respectively.

,

Table 9. Parameter estimation results for MTC-MTC vehicle-following pattern.

Variables	RPOL		RPOLMH
	Coefficient	t-Stat	Coefficient	t-Stat
Nonrandom parameters
Threshold parameters for probabilities	3.613 ***	40.07	3.597 ***	40.1
Constant	−5.793 ***	−14.85	−6.211 ***	−12.86
${\mathrm{L}}_{\mathrm{I}\mathrm{N}\mathrm{I}\mathrm{T}}2$	−0.167 *	1.82	0.211 **	2.28
${\mathrm{L}}_{\mathrm{M}}$	−0.249 ***	−5.55	−0.558 ***	−4.85
${\mathrm{V}\mathrm{A}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$	−0.147 ***	−3.5	−0.180	−1.60
${\mathrm{Z}\mathrm{D}}_{\mathrm{M}\mathrm{E}\mathrm{A}\mathrm{N}}$	0.179 ***	4.51	0.176 ***	4.40
${\mathrm{Z}}_{\mathrm{M}\mathrm{I}\mathrm{X}}$	0.58 *	11.4	0.242 *	1.82
${\mathrm{T}\mathrm{V}}_{\mathrm{M}\mathrm{E}\mathrm{A}\mathrm{N}}$	1.624 ***	29.14	1.656 ***	29.34
${\mathrm{T}\mathrm{A}}_{\mathrm{M}\mathrm{E}\mathrm{A}\mathrm{N}}$	−0.395 ***	−7.84	−0.331 ***	−6.39
${\mathrm{Z}\mathrm{N}\mathrm{P}}_{\mathrm{E}\mathrm{T}\mathrm{T}\mathrm{C}}$	0.420 ***	9.47	0.264 **	2.44
${\mathrm{Z}\mathrm{E}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$	0.385 ***	8.41	0.359 ***	7.74
${\mathrm{N}}_{\mathrm{R}\mathrm{L}\mathrm{C}}1$	0.266 ***	3.45	0.239 ***	3.09
Random parameters
${\mathrm{Z}}_{\mathrm{V}\mathrm{O}\mathrm{L}}$	−0.672 ***	−12.08	−1.100 ***	−9.81
Standard deviation for the random parameter (${\mathrm{Z}}_{\mathrm{V}\mathrm{O}\mathrm{L}}$)	0.920 ***	34.14	0.916 ***	34.10
${\mathrm{T}\mathrm{V}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$	0.331 ***	6.4	0.954 ***	8.58
Standard deviation for the random parameter (${\mathrm{T}\mathrm{V}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$)	0.597 ***	20.02	0.526 ***	18.26
Heterogeneity in the means of random parameters
${\mathrm{Z}}_{\mathrm{V}\mathrm{O}\mathrm{L}}$:			0.157 ***	3.47
${\mathrm{L}}_{\mathrm{M}}$
${\mathrm{Z}}_{\mathrm{V}\mathrm{O}\mathrm{L}}$:			0.177 ***	3.76
${\mathrm{V}\mathrm{A}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$
${\mathrm{Z}}_{\mathrm{V}\mathrm{O}\mathrm{L}}$:			−0.090 *	−1.69
${\mathrm{Z}}_{\mathrm{M}\mathrm{I}\mathrm{X}}$
${\mathrm{Z}}_{\mathrm{V}\mathrm{O}\mathrm{L}}$:			0.155 ***	3.31
${\mathrm{Z}\mathrm{N}\mathrm{P}}_{\mathrm{E}\mathrm{T}\mathrm{T}\mathrm{C}}$
${\mathrm{T}\mathrm{V}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$:			−0.241 ***	−7.02
${\mathrm{V}\mathrm{A}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$
${\mathrm{T}\mathrm{V}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$:			−0.912 **	−2.24
Number of parameters	17		22
Number of observations	3817		3817
Log-likelihood with constants only	−4259.636		−4259.636
Log-likelihood at convergence	−3400.384		−3229.269
Akaike information criterion (AIC)	6834.8		6502.5
McFadden Pseudo R-squared	0.202		0.242

Notes: *, **, and *** denote the statistical significance at 0.1, 0.05, and 0.01 levels, respectively.

and

Table 10. Parameter estimation results for MTC-ETC vehicle-following pattern.

Variables	RPOL		RPOLMH
	Coefficient	t-Stat	Coefficient	t-Stat
Nonrandom parameters
Threshold parameters for probabilities	9.666 ***	24.17	12.017 ***	21.79
Constant	−15.119 ***	−16.04	−18.890 ***	−15.86
${\mathrm{L}}_{\mathrm{I}\mathrm{N}\mathrm{I}\mathrm{T}}2$	0.730 ***	3.35	0.820 ***	3.35
${\mathrm{L}}_{\mathrm{I}\mathrm{N}\mathrm{I}\mathrm{T}}3$	0.231 *	8.9	0.541 *	1.84
${\mathrm{L}}_{\mathrm{I}\mathrm{N}\mathrm{I}\mathrm{T}}4$	1.382 ***	3.54	1.564 ***	3.56
${\mathrm{Z}}_{\mathrm{M}\mathrm{I}\mathrm{X}}$	−0.682 ***	−6.45	−0.814 ***	−4.55
${\mathrm{L}}_{\mathrm{M}}$	−0.319 ***	−4.36	−1.367 ***	−5.54
${\mathrm{V}\mathrm{A}}_{\mathrm{M}\mathrm{A}\mathrm{X}}$	−0.380 ***	−5.11	−0.425 ***	−5.05
${\mathrm{T}\mathrm{V}}_{\mathrm{M}\mathrm{E}\mathrm{A}\mathrm{N}}$	3.653 ***	21.47	4.587 ***	20.13
${\mathrm{T}\mathrm{A}}_{\mathrm{M}\mathrm{E}\mathrm{A}\mathrm{N}}$	−0.501 ***	−6.06	−0.561 ***	−5.99
${\mathrm{Z}\mathrm{N}\mathrm{P}}_{\mathrm{E}\mathrm{T}\mathrm{T}\mathrm{C}}$	1.176 ***	10.24	1.356 ***	10.24
${\mathrm{Z}\mathrm{E}}_{\mathrm{M}\mathrm{E}\mathrm{A}\mathrm{N}}$	0.592 ***	5.42	0.812 ***	6.5
${\mathrm{N}}_{\mathrm{L}\mathrm{L}\mathrm{C}}1$	0.944 ***	4.48	1.083 ***	4.56
${\mathrm{N}}_{\mathrm{R}\mathrm{L}\mathrm{C}}1$	0.756 ***	3.64	0.695 ***	2.98
Random parameters
${\mathrm{Z}}_{\mathrm{V}\mathrm{O}\mathrm{L}}$	−1.531 ***	−12.79	−2.236 ***	−14.28
Standard deviation for the random parameter (${\mathrm{Z}}_{\mathrm{V}\mathrm{O}\mathrm{L}}$)	2.337 ***	23.37	3.014 ***	21.36
${\mathrm{Z}\mathrm{E}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$	0.335 ***	3.88	−0.228	−1.31
Standard deviation for the random parameter (${\mathrm{Z}\mathrm{E}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$)	2.090 ***	21.03	2.122 ***	19.06
${\mathrm{T}\mathrm{V}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$	0.454 ***	4.97	1.077 ***	7.1
Standard deviation for the random parameter (${\mathrm{T}\mathrm{V}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$)	0.925 ***	15.04	1.541 ***	17.2
Heterogeneity in the means of random parameters
${\mathrm{Z}}_{\mathrm{V}\mathrm{O}\mathrm{L}}$			0.730 ***	0.094
${\mathrm{L}}_{\mathrm{M}}$
${\mathrm{Z}\mathrm{E}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$:			0.560 ***	0.102
${\mathrm{Z}}_{\mathrm{M}\mathrm{I}\mathrm{X}}$
${\mathrm{Z}\mathrm{E}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$:			−0.157 *	0.087
${\mathrm{L}}_{\mathrm{M}}$
${\mathrm{T}\mathrm{V}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$:			−0.348 ***	0.087
${\mathrm{Z}}_{\mathrm{M}\mathrm{I}\mathrm{X}}$
${\mathrm{T}\mathrm{V}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$:			−0.614 ***	0.136
${\mathrm{L}}_{\mathrm{M}}$
Number of parameters	20			25
Number of observations	2462			2462
Log-likelihood with constants only	−2597.626			−2597.626
Log-likelihood at convergence	−1971.890			−1959.918
Akaike information criterion (AIC)	3983.780			3969.836
McFadden Pseudo R-squared	0.241			0.246

Notes: *, and *** denote the statistical significance at 0.1 and 0.01 levels, respectively.

, as the variance heterogeneity is not statistically significant. These results consist of the estimation of parameter, t-statistics, and goodness-of-fit statistics (simulated log-likelihood at convergence, restricted log-likelihood, McFadden’s pseudo R², and AIC). On the basis of the RPOLMH models, likelihood ratio tests show that the null hypothesis of the performance of four models (Table 7, Table 8, Table 9 and Table 10) are the same with that of the combined model (Table 6) which is rejected with over 99.9% confidence, indicating the separation of different vehicle-following patterns is reasonable.

For four vehicle-following patterns, the AIC values of the RPOL models are larger than those of RPOLMH, which demonstrates the RPOLMH models have better performance. The values of McFadden’s pseudo R² of four RPOLMH models also indicate the same result. The number of significant variables is almost the same in models for different vehicle-following patterns (Table 7, Table 8, Table 9 and Table 10). Specifically, nine variables, including the average traffic volume of the sub-segment (${\mathrm{Z}}_{\mathrm{V}\mathrm{O}\mathrm{L}}$), overall acceleration standard deviation (${\mathrm{V}\mathrm{A}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$), lane-marking indicator (${\mathrm{L}}_{\mathrm{M}}$), the average acceleration within 3 s (${\mathrm{T}\mathrm{A}}_{\mathrm{M}\mathrm{E}\mathrm{A}\mathrm{N}}$), the average speed within 3 s (${\mathrm{T}\mathrm{V}}_{\mathrm{M}\mathrm{E}\mathrm{A}\mathrm{N}}$), the speed standard deviation of the following vehicle within 3 s (${\mathrm{T}\mathrm{V}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$), the standard deviation of ETTC of all vehicles in sub-segment (${\mathrm{Z}\mathrm{E}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$), the mix measure of MTC and ETC vehicles (${\mathrm{Z}}_{\mathrm{M}\mathrm{I}\mathrm{X}}$), the percentage of vehicles with ETTC less than 4 s in sub-segment (${\mathrm{Z}\mathrm{N}\mathrm{P}}_{\mathrm{E}\mathrm{T}\mathrm{T}\mathrm{C}}$) are significantly associated with risk status in the toll plaza diverging area. Moreover, three variables are identified to be random parameters with heterogeneity in means (${\mathrm{Z}}_{\mathrm{V}\mathrm{O}\mathrm{L}}$, ${\mathrm{Z}\mathrm{E}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$, and ${\mathrm{T}\mathrm{V}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$).

6.1. Traffic Condition Related Characteristics

According to the results of four RPOLMH models presented in Table 7, Table 8, Table 9 and Table 10, the coefficients of the proportion of risky vehicles around the vehicles (${\mathrm{Z}\mathrm{N}\mathrm{P}}_{\mathrm{E}\mathrm{T}\mathrm{T}\mathrm{C}}$) are positive, indicating that an increase in ${\mathrm{Z}\mathrm{N}\mathrm{P}}_{\mathrm{E}\mathrm{T}\mathrm{T}\mathrm{C}}$ has significant positive effects on the conflict risk for all vehicle-following patterns. One possible reason might be that when approaching the toll collection lanes, vehicles usually have an increase in lane-changing behaviors and acceleration and deceleration due to the toll booth restriction and speed limitation [47].

According to a significant body of studies, the collision risk level increases as traffic volume rises. In our investigation, the coefficient of the average traffic volume of the diverging area (${\mathrm{Z}}_{\mathrm{V}\mathrm{O}\mathrm{L}}$) is found to be a random parameter with heterogeneity in means in all vehicle-following patterns, indicating that, in some cases, the collision risk level increases with low traffic volume. Such a result is reasonable since drivers are more likely to compensate for the increased traffic volume with more cautious behavior in the toll plaza diverging area. Similarly, another possible explanation for this outcome is that vehicles must wait in line to pass the toll lane when the traffic volume is high, thereby reducing the conflict risk.

6.2. Driving Behavior Related Characteristics

In terms of the variables related to initial lane and vehicle lane changes, interesting results have been found from the model estimation. Only ${\mathrm{L}}_{\mathrm{I}\mathrm{N}\mathrm{I}\mathrm{T}}2$ have significant effects on conflict risk level in ETC-MTC (Table 8) and MTC-MTC models (Table 9). However, the results are significantly different when the following is an ETC vehicle (Table 7 and Table 10). In ETC-ETC and MTC-ETC models, all effects of variables associated with the initial lane (${\mathrm{L}}_{\mathrm{I}\mathrm{N}\mathrm{I}\mathrm{T}}2$, ${\mathrm{L}}_{\mathrm{I}\mathrm{N}\mathrm{I}\mathrm{T}}3$, ${\mathrm{L}}_{\mathrm{I}\mathrm{N}\mathrm{I}\mathrm{T}}4$) are significant. When the initial lanes of the following ETC vehicles are ${\mathrm{L}}_{\mathrm{I}\mathrm{N}\mathrm{I}\mathrm{T}}2$, ${\mathrm{L}}_{\mathrm{I}\mathrm{N}\mathrm{I}\mathrm{T}}3$, or ${\mathrm{L}}_{\mathrm{I}\mathrm{N}\mathrm{I}\mathrm{T}}4$, the conflict risk increases because ETC vehicles from initial lane 2 may intersect with vehicles coming from inside the diverging area. This implies that different initial lanes chosen by vehicles can result in differences in four vehicle-following patterns.

The average acceleration of the following vehicle within 3 s (${\mathrm{T}\mathrm{A}}_{\mathrm{M}\mathrm{E}\mathrm{A}\mathrm{N}}$) is found to be a fixed parameter with significantly negative effect on the conflict risk level, indicating that traffic safety status improves as the average acceleration of the following vehicles increases. It is likely because the following vehicle would take an acceleration maneuver only when the driver perceives no collision risk around, and thus the collision risk level between the leading vehicle and following vehicle would decrease. Furthermore, the maximum acceleration in the whole diverging area of the following vehicle (${\mathrm{V}\mathrm{A}}_{\mathrm{M}\mathrm{A}\mathrm{X}}$) also significantly increases the collision risk level of the ETC-MTC (Table 8) pattern, while decreasing the collision risk level of other patterns.

6.3. Random Parameters with Heterogeneity in Means

Except for the MTC-MTC pattern, the speed standard deviation of the following vehicle within 3 s (${\mathrm{T}\mathrm{V}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$) and the standard deviation of ETTC of all vehicles in sub-segment (${\mathrm{Z}\mathrm{E}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$) have random effects on collision risk level. According to Table 7, Table 8, Table 9 and Table 10, both the means and standard deviations of the two variables are statistically significant, indicating that the effects of the factors vary across different vehicle-following patterns.

Specifically, ${\mathrm{T}\mathrm{V}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$ is identified as statistically significant random parameters in all RPOL models with heterogeneity in means for all vehicle-following patterns, indicating that the effect of ${\mathrm{T}\mathrm{V}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$ has considerable variations across all observations. Overall, the coefficients of ${\mathrm{T}\mathrm{V}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$ are positive, indicating that larger speed changes will result in higher levels of conflict risk. Regarding the standard deviation of ETTC of vehicles in sub-segment (${\mathrm{Z}\mathrm{E}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$), it produced significant random parameters among models for ETC-ETC, MTC-ETC, and ETC-MTC patterns. The effects of ${\mathrm{Z}\mathrm{E}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$ differ greatly across three patterns. To some extent, the larger standard deviation of ETTC reflects the fluctuation of traffic flow near the following vehicle, which may increase the conflict risk level. Nevertheless, a random parameter is not observed in the RPOLMH model for the MTC-MTC vehicle-following pattern. The possible reason is that MTC vehicles are required to stop before toll collection lanes and wait in queues, so the conflict risk of vehicles in the MTC-MTC pattern is quite different from that of other patterns.

Theoretically, vehicles choosing the toll lanes corresponding to their initial lanes directly do not need to change lanes, resulting in fewer conflicts. Interestingly, changing lanes to the left (${\mathrm{N}}_{\mathrm{L}\mathrm{L}\mathrm{C}}$1) has significant effects on collision risk level in the RPOLMH models for ETC-ETC and MTC-ETC patterns. It indicates that the lane change operation directly affects the collision risk level. If ETC vehicles change to the inner side of the main-line road in advance, there will be less interweaving between ETC and MTC vehicles in the diverging area. The coefficients of variable changing lanes to the right (${\mathrm{N}}_{\mathrm{R}\mathrm{L}\mathrm{C}}1$) are significant in the models for ETC-MTC, MTC-ETC, and MTC-MTC patterns. According to the layout of ETC and MTC lanes, MTC vehicles need to operate more lane changes to pass through the diverging area. Therefore, more vehicle-to-vehicle interactions may increase the conflict risk when MTC vehicles are involved in the vehicle-following groups.

Another important issue that cannot be overlooked is that some random parameters are not sufficient in measuring conflict risk alone and should be considered together with the other variables. In Table 7, Table 8, Table 9 and Table 10, the means of the random parameter for ${\mathrm{Z}\mathrm{E}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$ and ${\mathrm{T}\mathrm{V}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$ are associated with lane-marking indicator (${\mathrm{L}}_{\mathrm{M}}$) and the percentage of vehicles with ETTC less than 4 s in sub-segment (${\mathrm{Z}\mathrm{N}\mathrm{P}}_{\mathrm{E}\mathrm{T}\mathrm{T}\mathrm{C}}$). It is noteworthy that ${\mathrm{L}}_{\mathrm{M}}$ can be regarded as a primary contributor to collision risk level. For example, in the model for the ETC-ETC pattern in Table 7, when ${\mathrm{T}\mathrm{V}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$ is involved with ${\mathrm{L}}_{\mathrm{M}}$, the mean of the random parameter is 0.165 (0.404 − 0.239 = 0.165). Such a result indicates that the effects of ${\mathrm{T}\mathrm{V}}_{\mathrm{S}\mathrm{T}\mathrm{D}}$ on the conflict risk level will greatly differ in sub-segments with and without lane markings to a large extent. One possible reason is that the psychology and behaviors of drivers change with the absence of land markings. Apart from the lane-marking indicator, ${\mathrm{Z}\mathrm{N}\mathrm{P}}_{\mathrm{E}\mathrm{T}\mathrm{T}\mathrm{C}}$ are also found to have an effect on the mean of the random parameter for TV_STD in the ETC-ETC pattern.

7. Conclusions

This study explored the conflict risk in the diverging area of a toll plaza for different ETC and MTC vehicle-following patterns. Four patterns were compared according to the combination of vehicle toll collection types: ETC-ETC, ETC-MTC, MTC-MTC, and MTC-ETC. The extended time to collision (ETTC) was used to measure the conflict risk for each vehicle-following pattern. Using random parameters ordered logit models with heterogeneity in means, the effects of traffic condition characteristics, driving behavior characteristics, and roadway characteristics were evaluated using vehicle trajectory data obtained from a toll plaza diverging area in Nanjing, China. The major conclusions are concluded as follows:

(1)

Vehicle-following patterns with the same toll collection types (ETC-ETC, MTC-MTC) have a higher percentage of severe conflict risk. This indicates that although the mixture of vehicles with different toll collection types leads to an increase in vehicle conflicts in the toll plaza diverging area, it does not mean that the severity of the conflict risk would increase. Therefore, in the toll plaza diverging area, attention should be paid to conflicts not only between vehicles with different toll collection types, but also between vehicles with the same toll collection type.

(2)

The random parameters ordered logit models with heterogeneity in means outperform the conventional random parameters ordered logit models and produce more insightful results. Different models of different vehicle-following patterns have different random parameters, indicating that heterogeneity exists in different combinations. The effects of the speed standard deviation of the following vehicle within 3 s (TV_STD), the standard deviation of ETTC of all vehicles in sub-segment (ZE_STD), and the average traffic volume of the sub-segment (Z_VOL) on the conflict risks considerably vary across vehicle-following patterns. In addition, lane-marking indicator (L_M), the number of vehicles with ETTC less than 4 s in sub-segment (ZNP_ETTC), and the mix measure of MTC and ETC vehicles in sub-segment (Z_MIX) are associated with the means and the variances of these random parameters. More flexible management measures, such as variable speed limits (VSL) and dynamic message signs (DMS) should be implemented instead of the fixed speed limits and signs to further enhance the safety at toll plazas.

(3)

Interesting results are observed in the coefficients of driving behavior-related characteristics. In ETC-ETC and MTC-ETC models, all effects of variables related to the initial lane (${\mathrm{L}}_{\mathrm{I}\mathrm{N}\mathrm{I}\mathrm{T}}2$, ${\mathrm{L}}_{\mathrm{I}\mathrm{N}\mathrm{I}\mathrm{T}}3$, ${\mathrm{L}}_{\mathrm{I}\mathrm{N}\mathrm{I}\mathrm{T}}4$) are significant. When the initial lanes of the following ETC vehicles are ${\mathrm{L}}_{\mathrm{I}\mathrm{N}\mathrm{I}\mathrm{T}}2$, ${\mathrm{L}}_{\mathrm{I}\mathrm{N}\mathrm{I}\mathrm{T}}3$, or ${\mathrm{L}}_{\mathrm{I}\mathrm{N}\mathrm{I}\mathrm{T}}4$, the conflict risk increases. Thus, guiding ETC vehicles to change lanes in advance to the inner side of the main road before they are entering the diverging area could effectively reduce the conflict risks in the diversion area.

(4)

The utilization of random parameters ordered logit models with heterogeneity in means and variances can more comprehensively and accurately reveal the mechanisms of different factors affecting the conflict risks of vehicles with different toll collection types. The findings of this study are significant for assessing the traffic safety situation at toll plazas and formulating corresponding conflict mitigation measures. Nonetheless, this study could be improved in a few ways. First, the scope of this study is restricted to a single toll plaza in Nanjing, China. Additionally, the weather is limited to sunny days with clear vision. More scenarios such as rainy days or foggy days should be considered. Additionally, how the characteristics of drivers, such as age, gender, and interacting with other contributing factors which may have significant effects on conflict risks should be further explored. The authors suggest further research on these topics.