category: literaturenote citekey: fadenshorttermsafetyanalysis2025 title: "Short-term safety analysis and interdependencies of mixed-operation freeways with fully separated express lanes: A copula-based poisson lognormal lindley approach" authors: "Faden, Abdulrahman; Abdel-Aty, Mohamed; Wang, Chenzhu; Yang, Samgyu" year: 2025 date: 2025-00-00 2025 doi: 10.1016/j.aap.2025.108050 publication: "Accident Analysis & Prevention" url: "https://doi.org/10.1016/j.aap.2025.108050" zotero_key: Q4G2T6U8 zotero_storage: 8VZB8Z2A collections: transpordi_planeerimine folder: 001_artiklid firstAuthor: "Faden, Abdulrahman"

status: converted

Contents lists available at ScienceDirect

Accident Analysis and Prevention

journal homepage: www.elsevier.com/locate/aap

Short-term safety analysis and interdependencies of mixed-operation freeways with fully separated express lanes: A copula-based poisson lognormal lindley approach

Abdulrahman Faden a,b,* , Mohamed Abdel-Aty ^a , Chenzhu Wang ^a , Samgyu Yang ^a

^a UCF Smart & Safe Transportation Lab, Department of Civil, Environment and Construction Engineering, University of Central Florida, Orlando, FL 32816, United States
^b Department of Traffic and Transportation Engineering, Imam Abdulrahman bin Faisal University, Dammam, Saudi Arabia

ARTICLE INFO

Keywords: Express lanes General-purpose lanes Short-term Freeway Managed Lanes Traffic detectors Copula Poisson Lognormal Lindley Crash frequency

ABSTRACT

The I-4 Ultimate Express Lanes (ELs) are part of a major improvement project on Interstate 4, introducing innovative freeway design that impacts traffic safety and operations. This research examines safety performance and the interdependencies between general-purpose lanes (GPLs) and (ELs). Utilizing crash data from February 2022 to February 2024, short-term crash frequency models are developed at the lane-level, leveraging microscopic traffic detector data and unique geometric design features to estimate annual average weekday crash frequencies and capture temporal safety variations. The study employs Poisson Lognormal Lindley (PLN-L) model to address excessive zeros in crash data, particularly prevalent in ELs, while copula-based framework analyzes the dependency between GPL and EL crash frequencies. Frank copula provided the best fit among five tested copula structures, revealing significant dependency, especially at access points. The analysis incorporates lanelevel traffic characteristics, geometric data (e.g., standard and I-4 Ultimate specific segment types (interaction between ELs and GPLs at access points), their lengths, and ramp lengths), and time-period effects. Significant variables for GPL and EL segments include lane-level traffic exposure and other factors. Key findings indicate that the average lane-occupancy in the rightmost lane significantly impacts EL safety, with higher crash rates at merge segments. For GPLs, I-4 Ultimate segments and their shorter lengths (i.e., Long [5000ft >= (length) > 3000ft]) are associated with GPL-related crashes. These findings contribute to developing safer, more efficient managed lane systems and offer guidance for future freeway design improvements. The study gives policymakers and engineers insights into EL safety, design, and operation.

1. Introduction

Due to demographic growth and expanding vehicle ownership, the utilization of the freeway network has increased dramatically. This expansion has directly contributed to various severe challenges throughout the transportation system, including safety, congestion, delay, reliability, and other concerns. The number of freeway fatalities increased by 11.9 % in 2023 compared to the average from 2017 to 2019 (International Traffic Safety Data and Analysis Group (IRTAD), 2023). Nearly 41,000 people lost their lives in traffic crashes in 2023, making it one of the worst years since 2005 (Crash\*Stats: Early Estimate of Motor Vehicle Traffic Fatalities in, 2023). Traffic safety is a critical priority in the regulation of transportation systems, policy development, and

infrastructure improvements aimed at achieving safer and more efficient travel. Transportation services are inherently dependent on two core requirements: mobility and safety. To enhance these aspects, the adoption of active traffic management (ATM) strategies has become crucial. Accordingly, the Florida Department of Transportation (FDOT) is intensively researching and implementing targeted traffic management approaches to improve the safety and operational efficiency of the roadway network (Florida Department of Transportation (FDOT), 2015; Kadeha et al., 2020). As a result, managed toll lanes (i.e., express lanes) are becoming more widely recognized as an effective approach for reducing traffic congestion and improving safety, especially in the Interstate Highway System in metropolitan areas where there is limited access, policy challenges, and high construction costs (Avelar and

E-mail addresses: Abdulrahman.Faden@ucf.edu, Akfaden@iau.edu.sa (A. Faden), m.aty@ucf.edu (M. Abdel-Aty), chenzhu.wang@ucf.edu (C. Wang).

^* Corresponding author at: UCF Smart & Safe Transportation Lab, Department of Civil, Environment and Construction Engineering, University of Central Florida, Orlando, FL, 32816, United States.

Fitzpatrick, 2018; Thomson et al., 2012). Accordingly, the U.S. has about 500 miles of priced-managed lanes and hundreds more under construction or plan (Fitzpatrick et al., 2016).

Managed toll lanes (MLs), such as express lanes, reversible lanes, and High Occupancy Toll lanes (HOT), have been installed throughout the state, providing safe, efficient, and seamless mobility alternatives. Interstate-4 (I-4) stands out as an important part of the Interstate Highway System, and it plays a more significant role in Florida's transportation network by facilitating efficient commerce, enhancing daily life, and attracting tourists. However, it suffers from excessive congestion and frequent crashes due to economic development and population growth. To address these issues, FDOT initiated the I-4 Ultimate Express toll lanes in February 2022, which is one of MLs strategies. The 1–4 Ultimate project, covering a 21-mile stretch, is part of a transformative infrastructure improvement initiative on Interstate 4 (I-4) in Central Florida. These lanes are fully isolated from the generalpurpose lanes (GPLs) by physical barriers, unlike previous methods using tubular markers, as seen in Fig. 1 (FDOT, 2024).

Evaluating new ELs infrastructure's operational and safety performance is vital for agencies to demonstrate concrete benefits during project assessments, such as for the I-4 Ultimate express lanes. Most studies have focused on environmental factors, mobility impacts, policies, and toll pricing (Abuzwidah et al., 2014; Florida Department of Transportation (FDOT), 2015; Perez et al., 2012). Prior research indicates that ELs improve mobility by enhancing trip time reliability, saving travel time, providing environmental benefits, and reducing delays (Kadeha et al., 2020; Thomson et al., 2012). However, assessing the safety performance of ELs is essential due to the complexity of transportation networks, which may pose additional safety concerns. The literature has rarely explored the safety performance of ELs facilities or developed a crash frequency model, highlighting the need for more attention in this area. Therefore, in this study, a safety analysis has been conducted to explore the crash factors on the I-4 Ultimate Express toll lanes.

Previous studies on managed freeways have explored the safety impacts of adding managed lanes to existing freeways, converting GPLs to managed lanes, and changing high-occupancy vehicle (HOV) lanes to high-occupancy toll (HOT) lanes (Porter et al., 2019). These studies have reported inconsistent results regarding crash rates, with some findings showing increases, decreases, or no change, regardless of the separation type (Fitzpatrick and Avelar, 2016). The literature indicates a strong correlation between safety and key design factors such as cross-section, separation type (buffer or barrier), and access design of the facility (Avelar and Fitzpatrick, 2018; Cai et al., 2018; Saad et al., 2018). Notably, managed lane geometry often varies due to right-of-way limits and road constraints, which can compromise safety. Therefore,

collecting detailed traffic data, using robust statistical models for crash estimation, and identifying significant contributing factors are essential to improving the safety of managed lanes. Accordingly, this study assesses the safety performance of the I-4 Ultimate Express Lanes, incorporating unique geometric design considerations. To the best of the authors' knowledge, no prior research has developed a lane-level crash estimation model for fully separated express lanes while accounting for time variations across peak and off-peak periods. This study aims to bridge that gap by offering a more detailed analysis of traffic patterns and their associated safety impacts.

The managed lane configurations, particularly at access points, also introduce significant interactions between ELs and GPLs. This suggests a potential dependency and correlation between crash frequencies on GPLs and ELs. While traditional univariate methods, such as Poisson Lognormal (PLN) or Negative Binomial (NB) models, treat crash counts as independent and ignore potential correlations, to date, no research has specifically addressed the joint modeling of crash frequencies for GPL and EL segments, leaving a critical gap in understanding the interdependence of crashes between these interconnected lane types.

2. Previous studies

2.1. Express lanes modeling

Safety analysis has demonstrated significant efficacy in developing statistically estimated models for evaluating and improving roadway safety. Analysis of crash frequency models can assess the safety effects of engineering treatments, helping agencies understand and adjust operational strategies as needed. However, remarkably few published studies focus on the crash estimation of freeways with express lanes (Abuzwidah and Abdel-Aty, 2024; Alluri et al., 2022; Fu et al., 2023; Kitali et al., 2021). In other words, the safety performance of (HOV) lanes has received more attention than the safety performance of (HOT) and ELs. Those few previous studies have explored the safety effects of separation types on managed express lanes (Alluri et al., 2022), factors influencing crash severity in reversible lanes without considering geometry (Fu et al., 2023), and injury risks associated with different separation types, such as concrete barriers (Kitali et al., 2021). However, Abuzwidah and Abdel-Aty (2024) conducted a detailed evaluation of the safety performance of express lanes using crash data from 55 miles of Florida highways (i.e., I-95). Employing cross-sectional and before-after approaches enhanced by empirical Bayesian modeling, the study individually provided a focused analysis of express and general-purpose lanes. The findings revealed improved safety within express lanes, demonstrated by crash modification factors consistently below one across all crash types, while general-purpose lanes experienced a marginal increase in

Fig. 1. A schematic illustration of the I-4 Ultimate Express Managed Lane (FDOT,2023).

crash frequencies, indicating a redistribution of crashes (Abuzwidah and Abdel-Aty, 2024). These results highlight the importance of advancing research to refine express lane safety measures and address the tradeoffs in overall traffic safety. Previous studies have shown that concrete separation types, temporal attributes, and other geometric features have a different impact than pylon, and those features are strongly correlated, whether with ELs in different operation types (i.e., RL, HOV, and HOT) or GPLs. On Interstate-4, the I-4 Ultimate Express functions differently from the conventional approach of employing tubular markers to designate fully separated managed toll lanes. On Interstate-4, the I-4 Ultimate Express functions differently from the conventional approach of using tubular markers to designate fully separated managed toll lanes. Thus, the newly constructed I-4 Ultimate Express may potentially differ in its safety aspects due to its unique features and variable geometry compared to other express lane facilities (FDOT, 2024; Florida Depart ment of Transportation (FDOT), 2015).

Recent research has begun investigating the impact of newly constructed Els separated from GPLs by concrete barriers on freeway safety. A notable study used a crash risk prediction model based on real-time loop detector data to identify traffic characteristics on these freeways (Yang et al., 2024). A key study used a crash risk prediction model with real-time loop detector data to identify traffic characteristics, but a specific crash estimation model for the I-4 Ultimate Express has not yet been developed. Traditional crash frequency estimation models use highly aggregated AADT data, which do not fully capture temporal traffic changes. On the other hand, real-time crash estimation may be unsuitable due to potential instability over short intervals, making it challenging to base long-term decisions on real-time crash risk estimates, as recent studies have demonstrated (Hasan et al., 2023a; Yuan et al., 2021). To address this gap, this study proposes developing a shortterm crash estimation model using high-resolution microscopic traffic detector data. These crash estimation models capture temporal changes in crashes and traffic characteristics, providing more accurate and dynamic crash frequency estimates. This study focuses on the annual average weekday period crash frequency model for GPLs and ELs segments related crashes, considering distinct traffic patterns during different periods (peak/off-peak), which can capture the presence of commuters.

2.2. Copula-Based joint modeling

Numerous studies have extended the Poisson regression model to the negative binomial model to address overdispersion in crash-frequency data. Other approaches include the Poisson-Lognormal model, Hierarchical, hurdle, Zero-Inflated Poisson model, and more (Mannering and Bhat, 2014). Recently, there has been an increased focus on developing time-specific crash estimation models to analyze temporal variations in crash data. However, disaggregating crash counts can lead to the issue of excess zero observations (Dzinyela et al., 2024). Few studies have addressed the predominance of zeros in crash count data by employing a univariate modeling approach that combines Negative Binomial and Lindley distributions (NB-L) (Hasan and Abdel-Aty, 2024; Islam et al., 2023, 2022). A recent study introduced a new mixing distribution model, the Poisson Lognormal Lindley (PLN-L) model, which accounts for overdispersion and the abundance of zeros in crash count data (Tahir et al., 2024). This model aims to improve crash-count estimation by addressing the excessive zeros that arise due to disaggregation levels. Accordingly, the PLN-L was proposed to improve crash-count estimation by taking into account the excessive zeros due to the disaggregation of crash counts.

Most crash analysis studies have traditionally focused on examining the relationship between specific explanatory variables and individual crash outcomes, such as crash frequency, treating these outcomes as independent entities. However, limited attention has been given to jointly modeling crash outcomes to investigate the potential interdependencies or correlations between them (Ahmad et al., 2023; Bhowmik et al., 2021a; Faden et al., 2024; Huang et al., 2023; Nashad et al., 2016; Park et al., 2021). This study aims to estimate crash frequencies for General-Purpose Lanes (GPL) and Express Lanes (EL) separately, recognizing the distinct design of fully separated lanes. Nevertheless, this innovative design introduces significant interactions at access points (Yang et al., 2024) and other managed lane configurations (Cai et al., 2018; Saad et al., 2018), which could influence crash dynamics. Some studies have attempted to integrate GPL and EL segment parameters into a single crash frequency model to estimate crashes occurring on either lane type (Abuzwidah and Abdel-Aty, 2024; Fu et al., 2023, 2022). This suggests the possibility of dependency and correlation between GPL-related and EL-related crash frequencies. While traditional univariate methods, such as Poisson Lognormal (PLN) or Negative Binomial (NB) models, treat crash counts as independent and ignore potential correlations, no existing research has specifically addressed the joint modeling of crash frequencies for GPL and EL segments in this specific innovative design. This leaves a critical gap in understanding the interdependence of crashes between these lane types.

To address the potential correlation between variables associated with General-Purpose Lanes (GPL) and Express Lanes (EL), recent research advocates for employing multivariate approaches (Faden et al., 2024; Hasan et al., 2023b; Mannering et al., 2016; Nashad et al., 2016). These approaches often employ propensity equations based on negative binomial or lognormal models to account for observed factors influencing crash outcomes. Multivariate methods are broadly classified into simulation-based and analytically closed-form approaches. Simulationbased techniques rely on generating repeated error term draws based on distributional assumptions, while closed-form methods use copulabased distributions for direct analytical probability computation. Simulation-based models, including multivariate Poisson regression, lognormal models (Bai and Fan, 2012), the INLA Multivariate Poisson Lognormal model (Faden et al., 2024), and more, have been widely used to capture dependencies across crash-related variables. Nevertheless, very few studies have employed the closed-form approach for developing multivariate crash frequency analysis despite offering a more flexible and direct means of capturing interrelations (Bhowmik et al., 2021a; Nashad et al., 2016; Yasmin et al., 2018). A comparative study examining multivariate crash frequency modeling at the traffic analysis zone (TAZ) level highlights the superiority of copula-based methods over simulation-based approaches in terms of data fit and estimated performance (Bhowmik et al., 2019a). As a result, copula-based closedform approaches have gained popularity in recent years as a strong alternative for crash frequency modeling, as first proposed in the safety literature by Eluru et al. (2010) (Eluru et al., 2010; Mannering and Bhat, 2014). As defined by Bhat and Eluru (2009), a copula is a mathematical function that constructs multivariate distributions by linking random variables with predefined marginal distributions operating within the unit cube (Bhat and Eluru, 2009). Furthermore, to the best of our knowledge, no prior research has specifically addressed the joint analysis of crash frequency with copula-based, specifically in GPL-related crashes and EL-related crashes, leaving a significant gap in understanding the dependence between these critical outcomes.

Therefore, this study introduces a copula-based bivariate Poisson Lognormal Lindley (PLN-L) model to analyze GPLs and ELs crash frequencies, offering clear advantages over the traditional simultaneous models. Unlike previous models that often place restrictions on the nature of this dependency by assuming asymmetric multivariate normal relationship (e.g., Abay et al., 2013; Faden et al., 2024; Hasan et al., 2023), the copula model allows for the empirical comparison of different dependency structures, increasing flexibility and adaptability to realworld data. This flexibility is important in transportation safety, where traffic data often show complex, non-linear, and asymmetric relationships (Anwari et al., 2025). For example, heavy traffic on General Purpose Lanes (GPL) may increase crash likelihood without similarly affecting the Express Lanes (EL). The copula model effectively captures these asymmetric dependencies, enabling transportation engineers to

understand and model the interaction between crashes on different lanes more accurately. Additionally, the copula-based approach provides an analytical formulation, yielding more precise parameter estimates compared to simulation-based methods like MSL or MCMC (Bhowmik et al., 2021b, 2019b). This improves the model's handling of complex, multi-source data and enhances its robustness. Overall, the copula model gives transportation engineers a reliable tool for improving road safety, optimizing resources, and applying cost-effective solutions.

3. The Objectives

In conclusion, a comprehensive review of the existing literature on ELs as a traffic management and operation strategy reveals several research gaps, particularly in safety performance modeling. Notably, no studies have examined the short-term or traditional crash estimation models for newly constructed ELs on freeway segments. Furthermore, the dependency and correlation between General-Purpose Lanes (GPL) crash frequencies and ELs have not been explored. This study aims to fill key gaps in understanding crash dynamics in express lanes (ELs) and their interactions with GPLs. By focusing on both methodological advancements and practical applications, this study makes the following contributions:

1. Development of Lane-Level Short-Term Crash Frequency Models for GPL and EL Segments: This research proposes a novel framework for crash frequency estimation using high-resolution microscopic traffic data from detectors to estimate crashes at the weekday time-period level. Unlike traditional models that rely on aggregated (AADT) data, this approach captures short-term variations in traffic and crash mechanisms, offering a more accurate model for assessing safety on EL freeway segments at different times of the day.
2. Evaluating Critical Traffic Features Influencing Crash Frequency: The study utilizes microscopic lane-level traffic data to assess how key traffic characteristics—such as volume, speed, lane occupancy, and others—affect crash frequencies. By examining these features at the annual average weekday period (AAWDPT) level, the study accounts for daily traffic fluctuations in lane-specific, providing a more comprehensive and realistic understanding of crash dynamics in managed lane systems, particularly in fully separated ELs.
3. Identifying The Impact of Cross-Weaving Spacing and Geometric Design: Express lanes differ from traditional freeway lanes in design, especially in terms of separation types, access points, and weaving sections. This study evaluates how factors like cross-weaving spacing, barrier separation, functional design differences between GPLs and ELs, and geometric parameters of ELs, such as the length of access points impact crash risk. These insights can help engineers refine ELs designs for improved safety.
4. Joint Modeling of GPL and EL Crash Frequencies Using a Copula-Based Framework: One of this study's novel aspects is applying a copula-based framework to simultaneously model short-term crash frequencies on GPL and EL segments using Poisson Lognormal Lindley distribution. This approach examines the complex dependency and error correlation between crash frequency parameters of GPL and EL segments and provides a more accurate picture of crash dynamics, which is essential for refining safety management on managed lanes.

Through these contributions, this study provides new insights into the safety performance of managed lanes, particularly the unique configuration of the I-4 Ultimate Express lanes and offers practical guidance for improving traffic management and safety strategies on similar freeway systems. This study also provides a deeper understanding of the safety impacts of the I-4 Ultimate Express lanes, highlighting how design and operational features affect crash patterns on newly constructed freeway systems.

4. Data preparation

The study focuses on the Florida I-4 Ultimate Express Section, a 21 mile stretch managed by the FDOT, as shown in Fig. 1. This section employs a managed lanes system with physically separated toll lanes, unlike the previous method of using tubular markers to separate lanes. This new approach aims to optimize traffic flow and enhance safety along Interstate 4. For the modeling process, three types of data were integrated: lane-level traffic detector data, crash data, and geometric information. The traffic detector and geometric information were sourced from FDOT, while the crash data was obtained from Signal Four Analytics (S4A), a statewide web-based crash mapping and analysis system covering the period from February 2022 to February 2024.

One of the key points in this study is classifying road segments by segment types, such as basic and merge, and processing the geometry data to align with the traffic detector information. This alignment was used to gather relevant information about each segment and its geometric characteristics. As a result, each segment was categorized based on its segment type, and the length of each segment was also considered a variable as part of geometry information. Moreover, it was essential to accurately determine the location of each crash, mainly to distinguish between crashes in the GPL and the EL. This distinction was achieved by using the provided crash information, such as descriptions from police reports and the latitude and longitude coordinates of the crash. This precise identification of crash locations ensured a clear understanding of the differences between the GPL and EL.

Before the traffic data was aggregated, the data cleaning process was performed. The data undergoes an outlier handling process to ensure accuracy and reliability. Suspected outliers are identified based on the following criteria and removed: 1) occupancy > 100, 2) speed = 0 and volume > 0, 3) speed > 100 mph, 4) traffic flow > 25 vehicles per 30 sec per lane or 5) volume = 0 with speed > 0 (Abdel-Aty et al., 2004; Abdel-Aty and Pemmanaboina, 2006; Al-Deek et al., 2004). Furthermore, Due to the specification of each detector, the initial aggregation intervals of traffic data could differ. To address these inconsistencies in traffic data intervals, which varied between 30 and 60 s, traffic data is aggregated into 60-second intervals to standardize the analysis. This standardization ensures a uniform time window for analysis, making the data more consistent and easier to compare across different segments and time periods. Moreover, the traffic data, being lane-level, is categorized into rightmost lane, leftmost lane, and through lanes. This categorization helps in understanding the distinct traffic behavior across different lanes. Fig. 2 illustrates a road segment with GPL and EL and shows the directional flow of traffic and lane categorization, including entry and exit points for both lane types. In addition, the traffic data, such as volume, speed, and occupancy, for GPL and EL were further processed by aggregating the lane-level data to generate segment-level data.

The processed traffic data (i.e., the 1 min resolution traffic-detector data) was initially aggregated at an hourly level. Afterward, the data were aggregated into the predefined time periods for every segment: AM Peak (6 AM to 9 AM), PM Peak (3 PM to 6 PM), Daytime (Off Peak) (9 AM to 3 PM), and Nighttime (6 PM to 6 AM). The changes in traffic patterns during time periods are notably different on weekdays, primarily due to the presence of commuters. As a result, this study specifically focused on analyzing data from weekdays, ensuring that the capturing accurately reflects typical weekday traffic conditions. For a more insightful analysis, the traffic data is further aggregated into annual average weekday periods traffic (AAWDPT) by grouping highresolution traffic data at the hourly level into four time periods for each weekday (4 periods × 522 days). These period data were then averaged across all weekdays, resulting in the AAWDPT dataset with four-period observations for each lane-level segment. These time periods are defined in the same way as crash data. Within these time periods, data for each lane is aggregated and analyzed. The aggregated data includes several key metrics, such as volume (AvgVolume), average speed (AvgSpeed), standard deviation of speed (Std.Speed), and average

Fig. 2. The Explanation of the Lane-Level Traffic Variables.

occupancy (AvgOccupancy) within each period. Finally, the aggregated traffic data were integrated with the corresponding crash data and geometry information for each segment. Additionally, each segment was matched with the upstream and downstream segments and the ramp connections. The data captures precisely whether a ramp is present and, if so, whether it connects to the GPL or the EL. For segments with missing data, these were excluded from the analysis. The descriptive statistics of the processed dataset are shown in Table 1.

4.1. The explanation of the segmentation process and geometric variables

There are standard road sections within this stretch, including basic, merge, diverge, and weaving segments, as presented in Fig. 3. Basic segments are simple stretches without significant traffic pattern changes, merge segments involve on-ramps joining the main roadway, diverge segments occur where lanes split into off-ramps, and weaving segments are characterized by mixed traffic flows requiring careful navigation. Besides these standard segments, the study specifically examines areas influenced by general-purpose lanes (GPL) and express lanes (EL), termed "ultimate specific" segments. These unique sections, where ramps connect GPL and EL, lead to mixed traffic flow directions. The ultimate specific segments are further categorized into four types: type 1, type 2, type 3, and type 4, each reflecting different traffic flow and ramp configurations influenced by the express lanes (Yang et al., 2024).

− Type 1 is a segment between on-ramp (merge to GPL) and off-ramp (diverge from GPL), where lane changes of vehicles are most likely to occur frequently.
− Type 2 is the most frequent case, where the same type of ramp exists sequentially. For example, two consecutive on-ramps or two consecutive off-ramps based on GPL. These segments have different traffic behaviors compared to basic segments and have a higher possibility of crashes.
− Type 3 is a segment where different types of ramps (merge, diverge based on GPL) exist, but the sequence differs from Type 1—for example, an on-ramp followed by an off-ramp within the same segment.

− Type 4 is commonly referred to as 'weaving.' Vehicles are simultaneously merging and diverging in these areas, leading to complex traffic patterns.

The I-4 Ultimate specific segments, being unique in their configurations, are further analyzed by length to understand their impact on road safety. Given their distinct characteristics and potential for influencing traffic dynamics, these segments are categorized as Short (length ≤ 1500ft), Medium (3000ft ≥ length > 1500ft), and Long (5000ft ≥ length > 3000ft). This classification allows for a detailed examination of how the extent of these segments influences traffic behavior and crash occurrence, providing valuable insights into their safety implications. Finally, segments are classified into five categories: basic, merge, diverge, weaving, and ultimate specific, as illustrated in Fig. 4. This classification aims to capture the various traffic behaviors and interactions within I-4.

Each segment was split as segment types and the segments length was recalculated based on the defined segment types. It should be noted that all segments that were shorter than 0.04 miles were merged with the neighboring segment (Saha et al., 2016; Srinivasan et al., 2011). This choice was made to better capture the variability and complexity of traffic patterns in urban environments, where changes in traffic conditions can occur more rapidly, such as I-4 Ultimate Express lanes. Additionally, segmentation in this study was conducted using two distinct approaches to address the unique design of the I-4 ultimate express lane. The innovative layout involves two separated sub-freeways, GPL and EL, with significant interaction through access points. To account for these interactions, we divided the segmentation approaches and restructured the data to account for the interaction between the two sub-freeways, as shown in Fig. 4. The first segmentation approach is a general way of categorizing segments based on segment types, as we defined: basic, merge, diverge, weaving, and ultimate specific. This method reflects a traditional classification that is widely applied in similar studies. To be specific, it does not consider the spatial alignment or longitudinal relationships between the GPL and EL. Each lane is divided into their segment types. For instance, as shown in Fig. 4, the GPL consists of 7 segments, while the EL comprises 4 segments. Conversely, the second

Table 1 Descriptive statistics of the input parameters.

| Variables | General Purpose Lanes only (GPLs)
Annual Average Weekday Period Crash Frequency | | | | | Express Lanes only (ELs) | | | | |--------------------------------------------------------------------------|------------------------------------------------------------------------------------|--------------------------------------|--------|----------|---------|-------------------------------------|---------|---------|--| | | | (n = 520 observations, 1745 crashes) | | | | (n = 524 observations, 135 crashes) | | | | | | Min | Max | Mean | SD | Min | Max | Mean | SD | | | Crash Data | | | | | | | | | | | Total Crashes (KABCO) | 0 | 23 | 3.36 | 3.63 | 0 | 3 | 0.26 | 0.53 | | | Traffic Data ~ Annual Avg. Weekday Periods traffic | | | | | | | | | | | Avg Volume (Total) (Veh) | 5421 | 32,591 | 17,917 | 6644.374 | 1285.00 | 15337.00 | 4356.00 | 1737.19 | | | Avg Volume (Rightmost Lane) (Veh) | 220.8 | 9888.8 | 3731.5 | 2101.442 | 0.00 | 5410.00 | 2370.00 | 936.82 | | | Avg Volume (Leftmost Lane) (Veh) | 728.8 | 10527.3 | 4728.8 | 1878.025 | 371.90 | 7246.40 | 1953.90 | 951.03 | | | Avg Volume (Through Lane) (Veh) | 2122 | 27,027 | 9457 | 4888.294 | | | | | | | Avg Speed (Total) (mph) | 30.56 | 88.51 | 64.42 | 7.18 | 56.59 | 83.76 | 69.30 | 4.11 | | | Avg Speed (Rightmost Lane) (mph) | 23.62 | 85.67 | 62.52 | 8.90 | 0.00 | 78.08 | 66.39 | 9.31 | | | Avg Speed (Leftmost Lane) (mph) | 34.58 | 88.77 | 67.74 | 7.81 | 53.52 | 91.64 | 71.26 | 5.32 | | | Avg Speed (Through Lane) (mph) | 29.24 | 90.97 | 63.68 | 8.36 | | | | | | | Std. Speed (Total) | 2.16 | 19.31 | 4.720 | 2.665 | 1.82 | 8.36 | 2.647 | 0.671 | | | Std. Speed (Rightmost Lane) | 1.71 | 18.36 | 4.722 | 2.666 | 0.00 | 5.59 | 2.55 | 0.670 | | | Std. Speed (Leftmost Lane) | 1.47 | 20.59 | 4.23 | 2.73 | 1.53 | 13.36 | 2.70 | 0.94 | | | Std. Speed (Through Lane) | 1.98 | 23.63 | 4.94 | 3.02 | | | | | | | Avg Occupancy (Total) (%) | 1.47 | 17.81 | 5.57 | 2.58 | 0.13 | 9.84 | 2.11 | 1.49 | | | Avg Occupancy (Rightmost Lane) (%) | 0.09 | 23.51 | 4.51 | 3.36 | 0.00 | 7.93 | 2.24 | 1.48 | | | Avg Occupancy (Leftmost Lane) (%) | 0.21 | 21.00 | 5.06 | 2.75 | 0.07 | 9.84 | 1.92 | 1.56 | | | | | | | | | | | | | | Avg Occupancy (Through Lane) (%) | 1.67 | 20.02 | 6.45 | 2.77 | | | | | | | Geometric Design Data | | | | | | | | | | | Segment Length (mile) | 0.041 | 0.737 | 0.265 | 0.129 | 0.045 | 0.551 | 0.271 | 0.094 | | | GPL Ramp Length (mile) | 0.000 | 0.800 | 0.115 | 0.206 | | | | | | | EL Ramp Length (mile) | 0.000 | 0.500 | 0.020 | 0.083 | 0.000 | 0.500 | 0.047 | 0.116 | | | Summary Statistics of Dummies | | | | | | | | | | | Number of GPL Lanes | | | | | | | | | | | Number of lanes (4 or fewer lanes) | 0 | 1 | 0.877 | 0.329 | | | | | | | Number of lanes (5 < and =>6 lanes) | 0 | 1 | 0.123 | 0.329 | | | | | | | Number of EL Lanes | | | | | | | | | | | Number of lanes (=< 1 lanes) | | | | | 0 | 1 | 0.015 | 0.123 | | | Number of lanes (2 lanes) | | | | | 0 | 1 | 0.985 | 0.123 | | | Segment Type | | | | | | | | | | | Weaving | 0 | 1 | 0.131 | 0.337 | | | | | | | Merge | 0 | 1 | 0.146 | 0.354 | 0 | 1 | 0.053 | 0.225 | | | Diverge | 0 | 1 | 0.162 | 0.368 | 0 | 1 | 0.099 | 0.299 | | | Basic | 0 | 1 | 0.562 | 0.497 | 0 | 1 | 0.847 | 0.360 | | | Types of Ultimate Specific Segments in the GPL that are influenced by EL | | | | | | | | | | | Type1 | 0 | 1 | 0.046 | 0.210 | | | | | | | Type2 | 0 | 1 | 0.115 | 0.320 | | | | | | | Type3 | 0 | 1 | 0.054 | 0.226 | | | | | | | Type4 | 0 | 1 | 0.015 | 0.123 | | | | | | | Ultimate Specific Types Segments Length in the GPL | | | | | | | | | | | Short ((length) <= 1500ft) | 0 | 1 | 0.046 | 0.210 | | | | | | | Medium (3000ft >= (length) > 1500ft) | 0 | 1 | 0.069 | 0.254 | | | | | | | Long (5000ft >= (length) > 3000ft) | 0 | 1 | 0.115 | 0.320 | | | | | | | The Access Point to/from GPLs Location on ELs Segments | | | | | | | | | | | Left side of the ELs segment | | | | | 0.046 | 0.209 | 0.046 | 0.209 | | | Right-side of the ELs segment | | | | | 0.107 | 0.309 | 0.107 | 0.309 | | | Periods | | | | | | | | | | | AM Peak (6 am − 9 am) | | | | | | | | | | | | 0 | 1 | 0.250 | 0.433 | 0 | 1 | 0.250 | 0.433 | | | Off Peak (9 AM − 3 PM) | 0 | 1 | 0.250 | 0.433 | 0 | 1 | 0.250 | 0.433 | | | PM Peak (3 PM − 6 PM) | 0 | 1 | 0.250 | 0.433 | 0 | 1 | 0.250 | 0.433 | | | Nighttime (6 PM − 6 AM) | 0 | 1 | 0.250 | 0.433 | 0 | 1 | 0.250 | 0.433 | |

Note: n = sample size (number of observations); min. = minimum; max. = maximum; SD = standard deviation; KABCO = KABCO injury scale (K = killed; ABC = injury categories, decreasing severity; O = no injuries); avg = average; Std. = standard deviation.

approach considers the geographical and longitudinal relationships between the GPL and EL. Segments were redefined to ensure that corresponding sections of the two lanes were aligned geographically and extended equally across both lanes. For instance, as shown in Fig. 4, it prioritizes the geographical and longitudinal consistency of the roadway segments, as depicted by the synchronized segmentation lines across the GPL and EL, with the same number of segments by lanes. In other words, 1st segmentation is independent for GPL and EL, and 2nd is consistent for both GPL and EL.

This alignment facilitates a more direct comparison of traffic and crash characteristics between the two lanes. By accounting for spatial continuity and interaction, this method enhances the accuracy and

reliability of analyses, particularly when investigating interactions and the dependency between the GPL and EL or assessing the impact of geometric design on traffic behavior. By employing these two segmentation methods, we are able to compare the geographical influence of the GPL and EL, providing a better understanding of their spatial interactions and the potential impact of the unique freeway design. Finally, the data processing pipeline for each state is shown in Fig. 5.

5. Methodology

Crash frequency estimation often relies on Poisson and its extended family of models due to their suitability for count-based, nonnegative

Fig. 3. Illustration of segment types.

integer-valued datasets. Among these, Poisson regression (Qi et al., 2014) and negative binomial models (Mahmoud et al., 2021) that are frequently utilized, particularly to address distributional properties such as overdispersion. As presented in Fig. 6, both crash frequency distribution shows a large right tail and an abundance of zeros, indicating a positive skew in the data. Furthermore, Table 1 shows over-dispersion in the crash data, with the crash frequency standard deviation for both being higher than the mean. Overdispersion is a prevalent phenomenon in such data, caused by various unobserved factors influencing crash occurrence. Therefore, this study proposes the PLN-L model as an advanced statistical model to account for the preponderance of zeros. The proposed statistical models' approach has been separately developed to estimate the short-term crash frequency for GPL and EL segments since both have different freeway traffic operations due to the complete separation from each other. In addition, the Negative Binomial model was developed also as base model.

Crash analysis frequently employs Bayesian methods for a variety of models, such as the NB, Poisson lognormal, and NB-L models (Geedipally et al., 2012; Hasan and Abdel-Aty, 2024; Khodadadi et al., 2022a). Thus, this research employs a Bayesian framework for the PLN-L model, which is more flexible than alternative methods due to its ability to naturally incorporate uncertainty probabilistically (Khodadadi et al., 2022b). By utilizing Markov Chain Monte Carlo (MCMC) simulations, Bayesian methods can effectively sample from the posterior distribution (Lord et al., 2021). It is noteworthy that the authors also explored the use of the NB-L model alongside the proposed approach (Faden et al., 2025). However, the PLN-L models demonstrated superior performance compared to the NB-L models.

The overall freeway design, especially the I-4 Ultimate express lanes, encompasses two sub-freeways: General-Purpose Lanes (GPLs) and Express Lanes (ELs). In light of the substantial interactions between these sub-freeways, particularly at access sites, this study concentrates on the development of a joint model to analyze crashes on both GPLs and ELs. As shown in Table 1, the two dependent variables—GPL crash model and EL crash model—are presumed to be interconnected. Hence, the joint model approach is utilized to explore the intricate relationships between the two dependent variables. By modeling these dependent variables together, the correlation between them can be calculated, which improves the precision of the parameter estimates (Huang et al., 2023; Park et al., 2021). To achieve this objective of our study, we employed a bivariate copula-based approach that combined both independent Poisson-Lognormal Lindley (PLN-L) models by utilizing the traditional maximum likelihood method (Bhowmik et al., 2021a). This approach was utilized to investigate the factors influencing total crash counts (KABCO) in both General-Purpose Lanes (GPL) and Express Lanes (EL). The modeling process in this study can be summarized as follows:

Initially, separate Poisson-Lognormal Lindley (PLN-L) models were estimated for crash frequencies on GPL and EL using the original data structure (1st segmentation approach).
Next, Independent PLN-L models were then developed for GPL and EL crashes using uniformly restructured data (2nd segmentation approach).
Subsequently, A copula-based joint estimation of independent PLN-L models was conducted to capture the dependency between crash frequencies in GPL and EL. Several copula types, including Gaussian, Clayton, T-Copula, Gumbel, and Frank, are attempted in the modeling process.
Finally, the copula-based PLN-L model results were evaluated and compared with those from the independent models to assess performance.

This section provides a detailed methodological framework for the proposed models' methods.

5.1. The Poisson-Lognormal Lindley distribution (PLN − L)

The PLN-L is a statistical model that integrates the distinctive features of the Poisson, the lognormal, and the Lindley distribution to provide a robust framework for modeling count data with heavy tails and overdispersion. Consider i (i = 1,2,3, …) as the index representing different segments, which are the primary units of analysis. Presuming that the crash counts Yi (differentiated by severity levels (i.e., total crashes (KABCO)) follow a Poisson distribution influenced by a random error term ξi, the Poisson lognormal model is described as follows:

$$Yi \sim Poisson(\lambda i | \xi i)$$ (1)

Here, the random error term ξi follows a normal distribution (ξi ~ N (0, σ2 ^μ)), with σ2 μ representing the additional Poisson variation parameter.

Subsequently, combining the Lindley distribution and the Poisson lognormal distribution results in the formation of a novel mixed statistical distribution known as The Poisson-Lognormal Lindley Distribution

Fig. 3. (continued).

(PLN − L). To illustrate the PLN-L distribution within the generalized linear models (GLM) framework, utilize Equation (2) (Tahir et al., 2024).

$$\begin{split} P\left( \, Yi \, = \, yi | \mu i, \, \sigma 2 \, \mu, \, \theta \, \right) \, = \, \int PLN( \, yi \, | \sigma 2 \, \mu, \, \mu i \, \xi i \, \epsilon i \, ) Lindley(\epsilon i | \theta) d\epsilon i; \, \theta \ > \, 0, \, y \, > \, 0 \end{split}$$

In this formulation, PLN (yi ∣ σ² ^μ, μi ξi ∊i) indicates that μi ξi ∊i is the mean of the variable yi, which follows a Poisson lognormal distribution with a multiplicative random error term ξi that adheres to a normal distribution (ξ~N (0, σ² μ)) Here, σ² ^μserves as an additional Poisson variation parameter, and ∊i is distributed according to the Lindley distribution. The log-link functional form of the Poisson lognormal-Lindley model for the expected number of crashes, λi, is given by:

Fig. 4. Illustration and Example of segmentation approach.

$$ln(\lambda i) = ln(\mu i) + \xi i + \epsilon i \tag{3}$$

where μi is defined as exp (β0 ⁺ ^∑^J ^j=1βjXij), with β0 being the intercept and βj the parameter estimate for the j-th covariate Xij.

The mean response function of the PLN-L distribution can be formulated as follows:

$$E(Y_i = y_i) = \left[ \mui \times exp\left(\sigma\mu^2/2\right) \right] \times E(\varepsilon_i)$$ (4)

Here, μⁱ ^× exp( *σ*2 *^μ/*2 ) and E(εi) = ^θ+² θ(θ+1) . Thus, the mean function can be transformed as follows by entering these values in Eq. (5). (Tahir et al., 2024):

$$E(Y_i = y_i) = exp\left(\beta0 + \sum{j=1}^J \betaj X{ij}\right) \times exp\left(\sigma_\mu^2/2\right) \times \frac{\theta + 2}{\theta(\theta + 1)}$$

$$= exp(\beta0 + \sum{j=1}^J \betaj X{ij} + \sigma_\mu^2/2 + \ln(\frac{\theta + 2}{\theta(\theta + 1)}))$$ (5)

The Lindley distribution is reformulated as follows (Zamani and Ismail, 2010):

$$\varepsilon_i \sim \frac{1}{1+\theta} Gamma(2,\theta) + \frac{\theta}{1+\theta} Gamma(1,\theta)$$ (6)

Based on the composition of the mixture component, the previous expression can be expressed as Equation (7).(Geedipally et al., 2012; Shaon et al., 2018).

$$\varepsilon{i} \sim \sum Gamma(1 + z{i}, \theta)Bernoulli(z_{i}|\frac{1}{1 + \theta})$$ (7)

Under the Bayesian framework, the multi-level hierarchical structure of the PLN-L models can be parameterized as follows: (Tahir et al., 2024).

$$\begin{cases} Y{i} \sim Poisson(\lambda{i}|\mu{i}\xi{i}e{i}) \ \mu{i} = exp\left(\beta{0} + \sum{j=1}^{J}\beta{j}X{ij}\right) \ \xi{i} \sim N\left(0,\sigma{\mu}^{2}\right) \end{cases} (8) \ \varepsilon{i} \sim Gamma(\varepsilon{i}|1+z{i},\theta) \ z{i} \sim Bernoulli(z_{i}|\frac{1}{1+\theta}) \end{cases}$$

To understand Equation (8), it is important to recognize that the crash count adheres to a Poisson distribution, influenced by an error term (ξ) from the lognormal distribution and a site-specific fragility component (∊) from the Lindley distribution. These components account for the excess variance in the crash data. The Poisson lognormal-Lindley model's hierarchical structure is suitable for Bayesian implementation. The error term represents a non-informative gamma prior, with the shape parameter (θ) linked to a Bernoulli distribution. To generate Bayesian estimates, specifying appropriate prior distributions is essential. Geedipally et al. (2012) advise applying a beta distribution prior to 1 / (1 + θ) 1/(1 + θ), precisely Beta(n /3, n /2), where n is the total sample size (Geedipally et al., 2012). The regression coefficient (βj) is normally distributed. For a comprehensive understanding of the methodology, refer to Tahir et al. (Tahir et al., 2024).

5.2. The elasticity

Moreover, the elasticity values are used to measure the relative change in the expected crash frequency with respect to a marginal change in an explanatory variable. Based on the equation (5), the elasticity of Poisson lognormal-Lindley model could be expressed as follows:

$$\textit{Elasticity} = \frac{\partial lnE(Yi)}{\partial X{ij}} #(9)$$

5.3. Copula model

A copula model is used to address the dependence structure between random variables, allowing for greater flexibility in joint distribution modeling. Here, *Y*1 and *Y*2 denotes the crash count for GPLs and ELs, respectively. Assuming the Λ1 and Λ2 being the marginal distribution of *Y*1 and *Y2, respectively. Then, a bivariate distribution Λ1,*2(ν, ε) can be developed as a joint cumulative probability distribution of uniform [0,1] marginal variables *U*1 and *U*2. (Bhat and Eluru, 2009)

$$\Lambda_{1,2}(\nu,\varepsilon) = \Pr\left(\nu_q < \nu, \, \varepsilon_q < \varepsilon\right) = \left[\Lambda_1^{-1}(U_1) < \nu, \Lambda_2^{-1}(U_2) < \varepsilon\right] = \left[U1 < \Lambda{1,2}(\nu), \, U2 < \Lambda{1,2}(\varepsilon)\right]$$ (10)

Assuming the X1, X2,… Xn as the random variables with joint cumulative distribution. Following the research (Sklar, 1973), the joint distribution (of uniform marginal variable) can be generated by a function Cθ:

Fig. 5. Flow chart of Data Processing.

$$\Lambda_{1,2}(\nu, \delta2) = C{\theta}(U1 = \Lambda{1,2}(\nu), U2 = \Lambda{1,2}(\varepsilon))$$ (11)

where C*θ*is a copula function, and θ denotes the dependence parameter defining the link between *Y*1 and *Y*2. In the current study, the dependence parameter θ is parameterized as a function for observed crash occurrence contributors as follows:

$$\theta = fn(\gamma s) \tag{12}$$

where γ is a row vector of unobserved parameters and s denotes a column vector of variables. f denotes the functional form of parameterization. Our study compares five alternative copula forms based on the permissible ranges of dependency parameters. θ = fn(γs) is used for Gaussian, t-Copulas, and Frank copula. Then, for the Clayton copula, we propose θ = exp(γs). For Gumbel copula we use θ = 1 + exp(γs).

5.4. Model performance measure

The Deviance Information Criteria (DIC) and the Watanabe-Akaike Information Criterion (WAIC) were implemented to select the optimal explanatory variables for the PLN-L model with the traditional data structure. DIC is a Bayesian formulation of Akaike's Information Criteria (AIC) that penalizes the model for more parameters. It is defined as:

$$DIC = \overline{D} + \rho_D \tag{13}$$

where PD is the adequate number of parameters and D = E(− 2log(Pr(y|θ)) is represents the expected model deviance under the posterior distribution of the model parameters (collectively denoted as θ). A smaller DIC value indicates a higher estimated power when models are developed using a variety of variable combinations based on the same dataset (Geedipally et al., 2014).

The Watanabe-Akaike Information Criterion (WAIC), proposed by

Fig. 6. Distribution of Crash Frequency for GPLs-related and On-ELs-related Crashes.

Watanabe and Opper (2010), is a robust alternative to the DIC within the Bayesian framework (Watanabe and Opper, 2010). The WAIC leverages the full posterior distribution for its calculations and is designed to be robust against model parameterization alterations, addressing DIC's limitations (Vehtari et al., 2017). Lower values indicate better fit and less overfitting. WAIC is calculated by summing the log pointwise estimated density (lppd) and applying a correction for the effective number of parameters to prevent overfitting. The formulation is as follows (Vehtari et al., 2017):

WAIC = $$-2\left(\sum{i=1}^{n} log\left(1/S\sum{s=1}^{S} p(yi|\theta^{s})\right) - pWAIC\right)$$ (14)

where:

n is the number of observations.
p(yi|θs) is the estimated density for observation yi at the s-th posterior sample θs
S is the number of posterior samples.
pWAIC is the estimated adequate number of parameters, which accounts for model complexity.

To evaluate the optimal model between independent and joint copula-based models, we utilize AIC and BIC (Ahmad et al., 2023; Anwari et al., 2025). The best model is identified as the one achieving the lowest values for AIC and BIC. AIC and BIC are defined as follows:

$$AIC = 2LL + 2K (15)$$

$$BIC = 2LL + ln(N)*K (16)$$

In this context, LL denotes the log-likelihood of the model, in particular, the natural logarithm of the maximal value of the likelihood function. The parameter k represents the number of approximated parameters, while N represents the total number of observations (i.e., the sample size) utilized for model fitting.

6. Results and discussion

6.1. PLN-L model estimation using the original data structure

The study developed PLN-L short-term crash frequency for GPLs and ELs, which are separated by concrete barriers. In here, the sample size n is determined by the total number of observations made across four distinct time periods: AM peak, PM peak, daytime off-peak, and

nighttime off-peak. The sample size is calculated by multiplying the fixed number of segments by the four time periods. The crash frequency models were estimated within a Bayesian framework using WinBUGS, with posterior probabilities of total crashes (KABCO) derived via Markov Chain Monte Carlo (MCMC) simulations. The models' estimated performance was assessed using performance metrics such as DIC, WAIC and Pd. During the model development process, lower DIC and WAIC values indicated a better statistical fit and performance and were crucial in selecting the superior model. Segment length was treated as an offset variable, with the crash frequency defined as the number of annual weekday period crashes per length of the segment. Two PLN-L crash frequency models were developed: one for GPLs and another for ELs. The result has been presented in the Appendix A.2.

The result shows the posterior means and standard deviations of the regression coefficients for the superior model, along with the DIC and WAIC measures. Significant factors, identified using a 90 % confidence interval, had 90 % credible intervals that did not include zero (Hasan and Abdel-Aty, 2024). A positive posterior mean indicated an increase in crash counts with higher variable levels, while a negative posterior mean indicated an increase in crash counts with lower variable levels.

6.2. PLN-L model estimation using uniformly restructured data

6.2.1. Model estimation

As aforementioned, this research aims to comprehensively analyze the factors influencing crash frequency and to investigate the dependency between crash occurrences in General-Purpose Lanes (GPLs) and Express Lanes (ELs). To achieve this, the study evaluates the potential interdependence by employing and comparing five copula structures—Gaussian, Clayton, T-Copula, Gumbel, and Frank—against an independent model after the data was uniformly restructured (2nd segmentation approach). The independent model, representing separate (PLN-L) models for GPL and EL crash frequencies, assumes no dependency between these two sub-freeways. This methodology establishes a robust framework to enhance the safety evaluation of managed freeway lane systems by accurately capturing the intricate interactions between GPL and EL crash dynamics. The analysis incorporates key performance metrics, including the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC), to ensure rigorous model evaluation and comparison. These metrics are computed for all candidate models, including the independent PLN-L models. The most appropriate copula structure is selected based on the lowest AIC and BIC values, ensuring a reliable determination of the dependency between crash frequencies (Ahmad et al., 2023; Nashad et al., 2016).

In Table 2, the coefficient estimation results of the independent GPL and EL models are presented, including only those variables that achieved statistical significance at the 90 % confidence level, while Table 3 illustrates a comprehensive overview of the copula model development outcomes, focusing on dependence structure analysis (i.e., the copula dependence parameters (θ)) and model goodness-of-fit metrics. The analysis reveals a broadly consistent pattern among the independent and five copula models in terms of the variables' influence on the outcome variables. Nevertheless, certain variables exhibit variations in their effects and outputs. This uniformity highlights the strength of the copula framework in capturing the complex interdependencies between the two response variables analyzed in this study (Huang et al., 2023). In the following subsections, the authors discuss the results related to the development of independent PLN-L models and the estimated regression parameters, as well as a summary of the five copula model outcomes.

6.2.2. Discussion on independent model estimation

According to the estimated posterior parameters of the independent PLN-L crash estimation model for EL, as shown in Table 2, the natural logarithm of the annual average weekday period volume in the rightmost is significant at the 90 % credible interval and correlates positively with crash frequency. This suggests that congestion along the corridor—particularly in the rightmost lane—could elevate the likelihood of the side-swipe or rear-end crashes on GPLs segments. The model reveals that higher average lane occupancy in the rightmost lane significantly negatively affects crash frequency in ELs. It suggests that more uniform traffic flow and reduced speed variability can lead to fewer crashes. One possible explanation for this is that drivers in the rightmost lane tend to be more cautious and either avoid the faster left lane or prepare to exit. Interaction variables were analyzed to better understand crash mechanisms on ELs. The results show that higher average lane occupancy in merge segments is associated with an increased crash frequency compared to diverge and basic segments. In contrast, the average occupancy in the rightmost lane itself shows a negative correlation with crash frequency. This finding is likely due to the numerous lane changes that occur within a limited time and space in these segments. Moreover, the model highlights an interrelation between GPLs and ELs operations at access points, even though they are physically separated (Fu et al., 2023). Traffic and crash data were aggregated into four weekday periods to study traffic patterns. PM peak poses a higher risk potentially due to

Table 3 Performance Measures and Copula Parameter of Different Copula Models.

Model type	AIC	BIC	Copula parameter θ
Clayton	2338.914	2421.834	0.542
t-Copula	2336.890	2419.809	0.373
Gaussian	2338.783	2421.702	0.342
Gumbel	2338.865	2421.785	1.268
Frank	2335.783	2418.702	2.377

due to increased congestion.

As illustrated in Table 2, the GPLs model offers valuable insights into the crash characteristics of I-4 Ultimate Express GPLs and demonstrates the effectiveness of crash estimation modeling. The estimated parameters of the independent GPL model reveal that the natural logarithm of traffic volume in both the rightmost and through lanes is significant and positively correlated with crash frequency. This trend aligns with most freeway crash frequency models, identifying traffic congestion as a critical factor contributing to crash occurrence (Abdel-Aty et al., 2023; Faden et al., 2024; Fu et al., 2022). In addition, the standard deviation of speed in the rightmost and leftmost lanes was identified as a significant factor positively associated with crash frequency. It suggests that higher speed variability could be a key factor in rear-end crashes (Wang et al., 2024). In other words, when there are large variations in speed, it means that drivers are likely to make frequent changes in their speed, which increases the likelihood of crashes. This behavior is likely due to increased traffic flow instability in these lanes, often caused by vehicles frequently entering or exiting ramps or EL access points. These findings are consistent with prior research studies (Hasan et al., 2023a; Rim et al., 2023; Wang et al., 2017). Among the key traffic-related factors, the average occupancy for the GPL through lanes was significant at the 10 % level. This variable was positively correlated with the frequency of GPL-related crashes. The finding suggests that higher occupancy levels on the through lane are associated with an increased likelihood of crashes, potentially due to the presence of stop-and-go traffic conditions in these segments.

The results highlight the significant influence of geometric design on GPL crash frequency, specifically, the type of freeway segment (i.e., weaving and diverge segment), ultimate specific segment types influenced by the interaction between GPLs and ELs (i.e., Type3), and the

Table 2 The Independent Copula Model Estimation Result for Total Crashes (KABCO).

Period		Annual Weekday Period Crash Frequency
	PLN-L
		GPLs (N = 832)				ELs (N = 832)
Variables	Mean	SD	90 % CI		Elasticity	Mean	SD	90 % CI		Elasticity
			Lower Limit	Upper Limit				Lower Limit	Upper Limit
(Intercept)	− 5.496	0.629	− 6.530	− 4.461		− 5.568	1.982	− 8.827	− 2.308
Ln (Volume for Through Lane)	0.279	0.057	0.185	0.373	1.1934
Ln (Volume for Rightmost Lane)	0.285	0.046	0.210	0.361	1.1913	0.523	0.268	0.083	0.963	2.5439
Std. Dev. Of Speed for Rightmost Lane	0.047	0.010	0.030	0.064	0.1217
Std. Dev. Of Speed for Leftmost Lane	0.049	0.011	0.031	0.067	0.0851
Avg Occupancy for Through Lane	0.067	0.013	0.046	0.088	0.2206
Avg Occupancy for Rightmost Lane						− 0.280	0.093	− 0.432	− 0.127	− 4.3907
Avg Occupancy for Rightmost Lane * Merge						0.287	0.116	0.096	0.478	0.2097
Segment Type (Ref. Others)
Weaving Segment	0.326	0.065	0.219	0.433	0.0317
Diverge Segment	0.297	0.070	0.181	0.412	0.0211
Types of Specific Segments in the GPL (Ref. Others)
Type3	0.374	0.117	0.181	0.567	0.0117
Specific Segment Types Length in the GPL (Ref. Others)
Long (5000ft >= (length) > 3000ft)	− 0.470	0.105	− 0.642	− 0.297	− 0.0286
Periods (Ref. Others)
PM Peak	0.202	0.067	0.093	0.312	0.0275	0.531	0.252	0.117	0.946	0.9468
Nighttime	0.663	0.077	0.536	0.790	0.0865
AIC	3236.778					776.228
BIC	3293.464					799.847

Note: n = sample size (number of observations), Volume = Annual average weekday periods of number of vehicles; CI = Confidence Interval; SD = standard deviation.

levels of ultimate specific segment length (i.e., Long, which exceeds 3000 feet between ramps). The ramp-related segments, such as weaving and diverge segments, exhibit higher crash likelihood compared to basic and merge segments, as indicated by the PLN-L crash model. In addition to standard segment types, the study considers the abovementioned ultimate specific segment types influenced by the interaction between GPLs and ELs. The analysis shows that type 3 experiences more crashes than other types. This is likely because drivers tend to accelerate or decelerate within these segments due to numerous lane changes occurring in a limited time and space. Three levels of ultimate specific segment length were considered: Long, Medium, and Short. The Long length of these specific segment types was negatively correlated with GPL-related crashes. This finding indicates that shorter segments have a higher potential for crashes due to frequent lane changes occurring within limited space. The findings suggest that longer lengths (i.e., exceeding 3000 feet) between ramps could be an effective countermeasure to prevent crashes, consistent with prior studies (Wang et al., 2022).

Period dummy variables were incorporated into this analysis. A positive coefficient for both the PM peak and nighttime periods indicates that crashes are more likely to occur during these times compared to other periods. The higher crashes during nighttime can be attributed to lower traffic volumes and high speed with reduced visibility and fatigue (Kitali et al., 2021), while the PM peak poses a higher risk potentially also due to increased congestion or lane-changing behavior.

In addition, the elasticity comparison between GPLs and ELs models indicates that crash frequency in ELs is more sensitive to changes in key traffic and environmental factors. Variables including traffic volume for Rightmost Lane and PM peak exhibit significantly larger elasticity values in the EL model compared to the GPL model, suggesting a stronger influence on crash occurrence. Notably, the Ln (Volume for Rightmost Lane) shows an elasticity of 2.5439 in ELs compared to 1.1913 in GPLs, indicating that an increase in volume has a much greater impact on EL crash frequency. Additionally, the interaction effect between rightmost lane occupancy and merging is only significant in the EL model, highlighting the role of congestion and lane-changing behavior in EL crashes. Furthermore, the influence of peak-hour periods is more pronounced in ELs, reinforcing the notion that operational factors such as speed variability and driver behavior contribute more significantly to crash risk in ELs than in GPLs. These findings suggest that express lane crashes are more susceptible to traffic dynamics and congestion effects, necessitating targeted safety interventions and operational management strategies to effectively mitigate risk.

6.3. Model comparison

The estimated models' performance measures (e.g., AIC) for each

response in each model was evaluated, and this helped us understand how well our models estimate crashes. Hence, the AIC and BIC were calculated to evaluate the estimation models' accuracy and compare the developed crash frequency models using uniformly restructured data. The better estimation accuracy is the one with the lower values. This comparison between NB, and proposed method (PLN-L) provided a better understanding of the boundaries of each model according to weekday periods traffic aggregation dataset structure. Fig. 7 represents the model comparison parameters between the negative binomial and the PLN-L. By comparing the two mentioned model selection criteria, it was observed that the Poisson log-normal Lindley model performed better than the other model. In terms of the AIC and BIC values, the PLN-L model outperformed the negative binomial (NB) model since PLN-L accounts for overdispersion and the abundance of zeros in crash count data (Tahir et al., 2024).

6.4. The copula dependence parameters (θ)

The estimated copula dependence parameters (θ) and goodness-of-fit metrics, including AIC and BIC, for each copula are summarized in Table 3. The results indicate that copula-based models provide a significantly better fit to the data compared to independent models, as evidenced by their superior AIC and BIC metrics. Among the models, the Frank copula model outperforms all other copulas, achieving the lowest AIC (2335.783) and BIC (2418.702) (Tamakloe et al., 2020). The Frank copula stands out due to its flexibility, as it accommodates both positive and negative dependence. A positive θ value indicates a positive correlation among shared variables influencing the two dimensions of interest (i.e., General Purpose Lanes (GPL) and Express Lanes (EL)), while a negative θ value denotes negative dependency. According to its theoretical properties, the Frank copula exhibits independence between variables when θ = 0 (Bhat and Eluru, 2009). Furthermore, this copula is particularly adept at capturing strong central dependence, although it performs relatively less in modeling tail dependencies (Eluru et al., 2010).

The estimated copula parameter θ is 2.377 for the Frank copula, indicating a moderate positive dependency between crash counts in GPL and EL. This suggests that segments with low-to-moderate crash counts in GPL are moderately likely to exhibit similar crash patterns in EL. This indicates that interventions aimed at mitigating crash risks on one lane could potentially benefit the other, emphasizing the need for holistic traffic safety strategies that simultaneously address both lane types. It is worth pointing out that the Frank Copula performs exceptionally well in capturing central dependency, making it highly suitable for typical crash scenarios where low-to-moderate crash counts in GPL show a moderate correlation with those in EL (Bhat and Eluru, 2009; Eluru et al., 2010).

Fig. 7. Performance measure results in different models.

7. Conclusions

The I-4 Ultimate Express Lanes introduce an unconventional freeway design on Interstate 4, impacting traffic safety and operations. Most prior studies on ELs have focused on operational efficiency, while remarkably, few studies have focused on the crash estimation model for freeways with express lanes. The I-4 Ultimate Express lanes on Interstate-4 differ significantly from the conventional design of managed toll lanes, which typically use tubular markers to separate lanes. Instead, the I-4 Ultimate Express features unique configurations and variable geometry, potentially influencing its safety aspects compared to other express lane facilities. With the availability of microscopic traffic data, real-time crash prediction models have been applied to evaluate the safety impacts of Ultimate Express lanes, particularly on generalpurpose lanes (GPLs), marking a notable advancement (Yang et al., 2024). However, this study generally contributes to the freeway safety literature by providing methodological enhancement and empirical insights. First, this study is the first to develop short-term safety models for ELs, considering microscopic traffic from detectors and geometric design impact to estimate crash occurrence and identify the contributed factors. Second, to our knowledge, this is the first study to apply a copula framework to jointly model crashes related to general-purpose lanes (GPLs) and express lanes (ELs), highlighting the interdependencies between these lane types. Short-term estimated models are expected to produce more accurate and dynamic crash frequency estimates, bridging the gap between traditional and real-time crash estimation methods.

The independent Poisson Lognormal Lindley (PLN-L) models have been developed to estimate I-4 Ultimate Express Lanes (ELs) crash frequency, effectively addressing the high number of zero-crash instances and skewed data distributions. For the ELs crash frequency model, the result shows that the average volume in the rightmost lanes significantly impacts the traffic safety of ELs segments. Additionally, average lane occupancy in the rightmost lane has a significant negative effect on ELs crash frequency. However, higher lane occupancy in merge segments—where vehicles transition from general-purpose lanes (GPLs) to ELs—correlates with increased crash frequencies. On GPLs, greater speed variability (standard deviation of speed) in the rightmost and leftmost lanes leads to unstable traffic conditions and higher crash frequencies, which means drivers are likely to make frequent changes in their speed. Furthermore, the influence areas on GPLs (i.e., diverge, weaving, and type 3 of ultimate specific segment type) experience more crashes compared to other types. Regarding the length of ultimate specific segment types, when connecting to GPL and EL, the long length (5000ft >= (length) > 3000ft) has most likely fewer crashes compared to other short and medium lengths. Hence, transportation agencies may develop specific criteria for designing the spacing distance to prevent traffic conflicts on those ultimate specific freeway segment types. Despite the physical separation, the models reveal an interaction between General-Purpose Lanes (GPLs) and Express Lanes (ELs) at the access points. Therefore, this study employs a copula-based bivariate PLN-L model to simultaneously examine the factors affecting crash frequencies on both GPLs and ELs, while accounting for their potential interdependence. Model estimation was performed using five distinct copula structures; the Frank copula was determined to be the best fit according to goodness-of-fit results, which show that the copula-based model outperforms models assuming independence, providing significant evidence of the interaction of GPL and EL crashes. These findings contribute to developing safer, more efficient managed lane systems and offer guidance for future freeway design improvements.

Despite its contributions, given more data availability, this research could be extended to evaluate the safety effectiveness of ELs for different crash severities (e.g., fatal, injury, and property-damage-only crashes) and types (e.g., rear-end, side-swipe, and single-vehicle crashes) which could provide a more comprehensive understanding of EL performance. Also, given more length of ELs, this research could refine its accuracy by using o.1 mile as minimum of segment length. The findings of this study hold significant implications for policymakers and transportation engineers. This research can help future managed lane systems designers achieve their goals of optimum safety and operational efficiency by providing a more detailed picture of the effects of ELs on traffic safety, and the study, which uses copula-based models, gives us a better idea of how GPLs and ELs relate to each other Consequently, the study's significance lies in providing a foundation for future safety evaluation research on newly constructed express lanes.

Author contributions

The authors confirm their contribution to the paper as follows: study conception and design: A. Faden, M. Abdel-Aty, and S. Yang; data collection: A. Faden, M. Abdel-Aty, and S. Yang; analysis and interpretation of results: A. Faden, M. Abdel-Aty and C. Wang; draft manuscript preparation: A. Faden, M. Abdel-Aty, C. Wang and Yang. All authors reviewed the results and approved the final version of the manuscript.

CRediT authorship contribution statement

Abdulrahman Faden: Writing – review & editing, Writing – original draft, Visualization, Project administration, Methodology, Formal analysis, Data curation, Conceptualization. Mohamed Abdel-Aty: Writing – review & editing, Project administration, Conceptualization. Chenzhu Wang: Writing – review & editing, Methodology. Samgyu Yang: Writing – review & editing, Visualization, Data curation, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The crash and traffic detector data used in this study were provided by the Florida Department of Transportation (FDOT).

Appendix

As shown in the Appendix A.1, the detectors are spaced quite closely together, with a minimum spacing of only 0.1 mile in several cases. This dense detector coverage ensures that each segment will either have its own dedicated detector or be adjacent to one, facilitating a more accurate assignment of microscopic traffic data. To be specific, where a segment doesn't have a direct detector, data from the nearest adjacent detector can still be reliably used, as the spacing between detectors is small enough to ensure the data reflects the traffic conditions in that segment accurately. Furthermore, in cases where two or more adjacent detectors exist, the average of the traffic data from these detectors is allocated to the segment, maintaining the accuracy of the assignment.

Appendix A1. Information of the selected study routes' detectors

Lane	Direction	Number of Detectors	Minimum Spacing (Mile)	Maximum Spacing (Mile)	Average Spacing (Mile)
Express Lane	East	66	0.1	0.8	0.302
	West	66	0.1	0.7	0.313
General Purpose Lane	East	52	0.1	0.6	0.380
	West	53	0.2	0.8	0.392

Appendix A2. Model estimation result for Total crashes (KABCO)

| Period | | Annual Average Weekday Period Crash Frequency | | | | | | | | | |--------------------------------------------------------|----------|-----------------------------------------------|-------------|-------------|---------|-------|-------------------------------|-------------|--|--| | | | PLN-L
GPLs
(n = 520 observations) | | | | | | | | | | | | | | | | | ELs
(n = 524 observations) | | | | | Variables | Mean | SD | 90 % CI | Upper Limit | Mean | SD | 90 % CI | | | | | | | | Lower Limit | | | | Lower Limit | Upper Limit | | | | (Intercept) | − 0.078 | 0.581 | − 1.098 | 0.854 | − 2.974 | 2.130 | − 6.316 | 0.197 | | | | Ln (Volume for Total) | | | | | 0.415 | 0.267 | 0.018 | 0.831 | | | | Ln (Volume for Rightmost Lane) | 0.203 | 0.070 | 0.090 | 0.324 | | | | | | | | Std. Dev. of Speed for Total | 0.124 | 0.018 | 0.095 | 0.154 | | | | | | | | Avg Occupancy for Rightmost Lane | | | | | − 0.316 | 0.102 | − 0.488 | − 0.146 | | | | Avg Occupancy for Rightmost Lane * Merge | | | | | 0.336 | 0.133 | 0.114 | 0.552 | | | | Segment Type (Ref. Others) | | | | | | | | | | | | Weaving Segment | 0.256 | 0.140 | 0.030 | 0.488 | | | | | | | | Diverge Segment | 0.353 | 0.138 | 0.132 | 0.582 | 0.656 | 0.340 | 0.121 | 1.190 | | | | Types of Specific Segments in the GPL (Ref. Others) | | | | | | | | | | | | Type3 | 0.702 | 0.236 | 0.321 | 1.096 | | | | | | | | Specific Segment Types Length in the GPL (Ref. Others) | | | | | | | | | | | | Long (5000ft >= (length) > 3000ft) | − 0.619 | 0.178 | − 0.921 | − 0.327 | | | | | | | | Periods (Ref. Others) | | | | | | | | | | | | PM Peak | | | | | 0.535 | 0.259 | 0.116 | 0.950 | | | | Nighttime | 0.432 | 0.112 | 0.250 | 0.616 | | | | | | | | DIC | 2049.030 | | | | 621.882 | | | | | | | WAIC | 1985.912 | | | | 633.176 | | | | | | | Pd | 302.901 | | | | 61.676 | | | | | |