Random Parameters Negative Binomial Research Articles

Comparative Evaluation of Crash Hotspot Identification Methods: Empirical Bayes vs. Potential for Safety Improvement Using Variants of Negative Binomial Models

The empirical Bayes (EB) method is widely acclaimed for crash hotspot identification (HSID), which integrates crash prediction model estimates and observed crash frequency to compute the expected crash frequency of a site. The traditional negative binomial (NB) models, often used to estimate crash predictive models, typically struggle with accounting for the unobserved heterogeneity in crash data. Complex extensions of the NB models are applied to overcome these shortcomings. These techniques also present new challenges, for instance, applying the EB procedures, especially for out-of-sample data. This study applies a random parameter negative binomial (RPNB) model within the EB framework for HSID using out-of-sample data, comparing its performance with a varying dispersion parameter NB model (VDPNB). The research also evaluates the potential for safety improvement (PSI) scores for both models and compares them with EB estimates using three generalised criteria: high crashes consistency test (HCCT), common sites consistency test (CSCT), and absolute rank differences test (ARDT). The results yield dual insights. Firstly, the study highlights associations between crash covariates and frequency, emphasising the significance of roadway geometric design characteristics (e.g., lane width, number of lanes, and parking type) and traffic volume. Some variables also influenced overdispersion parameters in the VDPNB model. In the RPNB model, annual average daily traffic (AADT) and lane width emerged as random parameters. Secondly, the HSID performance assessment revealed the superiority of the EB method over PSI. Notably, the RPNB model, compared to the VDPNB, demonstrates superior performance in EB estimates for HSID with out-of-sample data. This research recommends adopting the EB method with RPNB models for robust HSID.

Effects of design consistency on run-off-road crashes: An application of a Random Parameters Negative Binomial Lindley model

Run-off-road crashes are one of the most common crash types, especially in rural roadway environments contributing significantly to fatalities and severe injuries. These crashes are complex and multi-dimensional events, and factors like road geometry, driver behaviour, traffic characteristics and roadside features contribute to their occurrence, separately or interactively. Sudden changes in road geometry, in particular, can influence driver behaviour, and therefore, in developing a micro-level crash risk model for run-off-road crashes, one of the challenges is incorporating the effects of driver behaviour (disaggregated information) that may arise from the variations in road geometry (aggregated information). This study aims to examine the interaction between road geometry and driver behaviour through a set of measures for design consistency on two-lane rural roads. Multiple data sources, including crash data for 2014–18, traffic data, probe speed data and roadway geometric data, for twenty-three highways in Queensland, Australia, have been fused for this study. Seventeen types of design consistency measures with regard to alignment consistency, operating speed consistency and driving dynamics are tested. A run-off-road crash risk model is estimated by employing the Random Parameters Negative Binomial Lindley regression framework, which accounts for excess zeros in the crash counts and captures the effects of unobserved heterogeneity in the parameter estimates. Results indicate that the geometric design consistency capturing the interaction between driver behaviour and operational factors better predicts run-off-road crashes along rural highways. In addition, roadside attributes like clear zone width, infrastructures, terrain, and roadway remoteness also contribute to run-off-road crashes. The findings of the study provide a comprehensive understanding of the influence of variations in roadway geometry on driver behaviour and run-off-road crashes along rural highways.

Investigating factors affecting rural crash frequency at marginal areas around cities in Iran: a random parameter modeling approach

ABSTRACT A significant portion of rural crashes in Iran occur in the marginal areas around cities. Exclusive crash frequency models should be developed to identify the factors affecting the occurrence of crashes in these areas. In this study, the rural crash data of highways leading to the city of Isfahan were employed to develop a negative binomial (NB) model as a constant-parameter model, along with random effects negative binomial (RENB) and random parameters negative binomial (RPNB) models as the models based on the random parameter approach. The modeling results indicated the superiority of the RPNB model over the two other models. According to the findings of this study, the traffic volume of roads and the proportion of local and non-local vehicles, the proportion of longitudinal distance-limit violation, the way of controlling access points, and the access to rest areas affected the crash frequency in the marginal areas around cities.

Evaluating the correlation between risky riding behaviour and self-reported crashes in India: Minimizing unobserved heterogeneity

Riding Behaviour is found to be the main cause of Powered Two-Wheeler (PTW) crashes in more than90% of the crash events. The high percentage of PTW crashes resulting in fatalities has sought a serious need for research to examine risky riding Behaviour. A widely used instrument for measuring the self-reported riding Behaviour of PTW riders is the Motorcycle Rider Behaviour Questionnaire (MRBQ). In this study, exploratory factor analysis of the MRBQ revealed a four-factor solution viz., traffic errors, control errors, speed violations, and stunts. Despite the popularity of MRBQ, it is capable of covering only a small fraction of the large number of elements that affect the riding Behaviour. Many other elements remain overlooked in the analysis, resulting in unobserved heterogeneity. Therefore, the present study uses a random parameter negative binomial (RPNB) model to minimize the effect of unobserved heterogeneity. It was inferred from the RPNB model that variables like gender, control error, and speed violation have a randomly distributed regression coefficient. Further, it is found that traffic errors are the most significant predictor of crash risk. Additionally, results depict that male riders are positively associated with crashes, and they are more likely to involve in crashes as compared to female riders. The finding of this paper will be valuable for policymakers and decision-makers to improve the rider training program, licensing system, and design road safety campaigns.

Non-Stationary Modeling of Microlevel Road-Curve Crash Frequency with Geographically Weighted Regression

Vehicle crashes on roads are caused by many factors. However, the influence of these factors is not necessarily homogenous across locations, which is a challenge for non-stationary modeling approaches. To address this problem, this paper adopts two types of methods allowing parameters to fluctuate among observations, that is, the random parameter approach and the geographically weighted regression (GWR) approach. With road curvature, curve length, pavement friction, and traffic volume as independent variables, vehicle crash frequencies are modeled by two non-spatial methods, including the negative binomial (NB) model and random parameter negative binomial (RPNB), as well as three spatial methods (GWR approach). These models are calibrated in microlevel using a dataset of 9415 horizontal curve segments with a total length of 1545 kilometers for a period of three years (2016–2018) over the State of Indiana. The results revealed that the GWR approach can capture spatial heterogeneity and therefore significantly outperforms the conventional non-spatial approach. Based on the Akaike Information Criterion (AICc), geographically weighted negative binomial regression (GWNBR) was proved to be a superior approach for statewide microlevel crash analysis.

Assessing the explanatory and predictive performance of a random parameters count model with heterogeneity in means and variances

Random parameters model has been demonstrated to be an effective method to account for unobserved heterogeneity that commonly exists in highway crash data. However, the predefined single distribution for each random parameter may limit how the unobserved heterogeneity is captured. A more flexible approach is to develop a random parameters model with heterogeneity in means and variances by allowing the mean and variance of potential each random parameter to be an estimable function of explanatory variables. This burgeoning technique for modelling unobserved heterogeneity has been increasingly applied to various safety evaluation scenarios recently. However, the predictive performance of this emerging method, which determines the practicability of the model for a specific circumstance, has never been investigated as far as our knowledge. In addition, the explanatory power by including heterogeneous means and variances of random parameters need to be further investigated to confirm the potential merits of this method in crash data analysis.In this paper, a random parameters negative binomial with heterogeneity in means and variances (RPNBHMV) model, a standard random parameters negative binomial (RPNB) model and a traditional fixed parameters negative binomial (NB) model were estimated using the same dataset. The explanatory and predictive performance of the three models were thoroughly evaluated and compared. Results showed that: 1) the RPNB model fitted the data significantly better than the NB model, and the RPNBHMV model further improved the statistical fit of the RPNB model but the improvement was slight; 2) more insights into interactions of safety factors were inferred from the RPNBHMV model, which demonstrates the explanatory benefit of this model; 3) the RPNBHMV and RPNB models had both advantages (e.g., produced overall better prediction accuracy) and disadvantages (e.g., provided reduced prediction accuracy across the range of explanatory variables) when applied to in-sample observations (i.e., observations used to estimate the model); 4) the RPNBHMV and RPNB models might be less precise than the NB model when applied to out-of-sample observations. These findings indicate that the RPNBHMV model offers more insights and may be used for explanatory safety analysis for sites where reliable data can be collected. However, the simple NB model is more reliable – at least with the dataset used in this study - than its random parameters model counterparts for other sites where the data are unavailable or unreliable, which is a common safety evaluation scenario in practice.

A Correlated Random Parameters Model with Heterogeneity in Means to Account for Unobserved Heterogeneity in Crash Frequency Analysis

Unobserved heterogeneity induced by omitted variables is a major challenge in developing reliable road safety models. In recent years, the random parameters negative binomial (RPNB) model has been used frequently in crash frequency analysis to account for unobserved heterogeneity. However, the majority of past studies of the RPNB model assumed that there was no correlation between different sources of unobserved heterogeneity, which is not always true given the complex interactions of safety factors. Compared with the RPNB model, a more flexible random parameters model that is the correlated random parameters negative binomial with heterogeneity in means (CRPNBHM) model was proposed in this study. Results indicate that the CRPNBHM model could not only capture the otherwise unobserved heterogeneity, but also track the underlying correlation among different sources of unobserved heterogeneity, thus outperforming the RPNB model. In addition, new insights into the interactions of safety factors (e.g., the joint safety effects of heavy trucks and pavement rutting depth) were obtained from the CRPNBHM model and these are expected to be beneficial in developing effective safety countermeasures. Results from this study demonstrated the CRPNBHM model to be a good alternative for crash frequency analysis, particularly when unobserved heterogeneity was detected.

Public acceptance of and heterogeneity in behavioral beach trip responses to offshore wind farm development in Catalonia (Spain)

A combined travel cost – contingent behaviour survey of residents and tourists in Catalonia is conducted on-site to examine the effects on beach recreational demand of developing an offshore wind farm (OWF) project. The survey considers four potential OWF scenarios with different degrees of visual impact. We allow for heterogeneity in trip preferences among individuals and control for on-site sampling through the use of a random parameters negative binomial (RPNB) model and a Multivariate Poisson log-normal (MPLN) model, respectively. The welfare measures derived from the RPNB model relate to the current beach users only, whereas those from the MPLN model refer to the general population of residents and tourists in Catalonia. The results show the importance of the specific place of location of the OWF project and how the installation of wind turbines would significantly decrease the demand for trips, depending on their degree of visual impacts, leading to a substantial welfare loss. However, the results also show that the project mainly would cause a displacement of trips to other beaches within Catalonia rather than outside Catalonia and that the welfare per trip measures generated by the RPNB and MPLN models substantially differ. Policy implications of these findings are discussed.

Applying a Correlated Random Parameters Negative Binomial Lindley Model to Examine Crash Frequency Along Highway Tunnels in China

Highway tunnels have a higher risk of crashing than open roads, which require a systematic approach to tunnel safety. However, previous research had the following problems: 1) Studies have largely focused on open roads, with very little research on tunnels. 2) The collected crash contributing factors involve narrow ranges, with very little tunnel crash data including both tunnel design features, traffic conditions and pavement conditions. 3) None of the studies considered both excess zero observations and unobserved heterogeneity with its interactions. To address these issues, this paper first established an appropriate tunnel dataset containing 3 to 5 years of crash data from several highways in China and the influence factors of tunnel design features, traffic conditions and pavement conditions. A correlated random parameters negative binomial Lindley (CRPNB-L) model that considers both excess zero observations and unobserved heterogeneity with its interaction effects was then proposed. Compared to the uncorrelated random parameters negative binomial Lindley (URPNB-L) model, fixed parameters negative binomial Lindley (FPNB-L) model and fixed parameters negative binomial (FPNB) model, the CRPNB-L model solves the deviation that arises from excess zero observations by introducing the Lindley distribution and considers the unobserved heterogeneity with its interactions by introducing correlated random parameters. In the comparisons, the CRPNB-L model achieves the best effects in the goodness-of-fit. Furthermore, the estimated results of the CRPNB-L model showed that segment length, traffic volume, proportion of class 5 vehicle (heavy trucks and trailers), tunnel entrance and exit segments, and steep uphill and downhill segments were associated with higher crash frequency, while curvature, tunnel length, pavement damage condition index (PCI) and skid resistance index (SRI) were associated with lower crash frequency. In addition, the random variables of the curvature, the steep downgrade indicator, the proportion of class 5 vehicle and SRI were identified and their intercorrelations were analyzed.

Evaluating the predictive power of an SPF for two-lane rural roads with random parameters on out-of-sample observations

Negative binomial (NB) regression is among the most common statistical modeling methods used to model crash frequencies due to its simple functional form and ability to handle over-dispersion commonly found in crash data. However, a drawback of this approach is that regression parameters are assumed to be the same across observations, which could contribute to biased parameter estimates. To alleviate this concern, the random parameters negative binomial (RPNB) model was recently proposed, which allows regression parameters to differ across observations following some known distribution. The resulting coefficients should be less biased, and thus the RPNB approach is believed to provide a more accurate relationship between independent variables and expected crash frequency. However, the prediction accuracy of the RPNB model relative to the standard NB model has not been thoroughly evaluated, particularly with respect to out-of-sample observations for which unique random parameters cannot be estimated. In this paper, the predictive power of the RPNB and NB models are examined using two-lane rural highway data from three engineering Districts in Pennsylvania. Multiple evaluation metrics are applied—root-mean-square error (RMSE) and mean absolute error (MAE), coefficients from calibration functions and cumulative residual (CURE) plots—to assess each model type. The results show that the RPNB model outperforms the NB model when applied to within sample observations (i.e., those used to estimate the model) by making use of the observation-specific coefficients. However, the predictive power of the RPNB model appears to be similar to or slightly less precise than the traditional NB model when applied to out-of-sample observations. Since the RPNB model is estimated using a simulation-based approach, sensitivity tests were also performed to see how the parameter estimates change with the number of Halton draws used to perform the simulation. For the sample sizes used in this paper, the estimates were fairly insensitive when more than 50 Halton draws were used. The findings suggest that the RPNB model is more reliable when applied to the same set of sites that were used to estimate the model but might not be as robust as the traditional NB model when applied to other sites.

Investigating factors of crash frequency with random effects and random parameters models: New insights from Chinese freeway study

In response to the rapid economic growth in China, its freeway system has become the longest in the world and likely will continue to expand. Unfortunately, the safety issues on freeways in China have grown as well and are of great concern to Chinese transportation authorities and drivers. While many proven safety countermeasures developed and implemented by other countries are available for reference, they may be not fully transferrable to China due to the differences in driving cultures and conditions. As a result, an investigation of China’s unique safety factors and effective relevant countermeasures are urgently needed.The study presented in this paper thoroughly investigated the factors contributing to freeway crashes in China based on detailed crash data, traffic characteristics, freeway geometry, pavement conditions, and weather conditions. To properly account for the over-dispersion of data and unobserved heterogeneity, a random effects negative binomial (RENB) model and a random parameters negative binomial (RPNB) model were applied, along with a negative binomial (NB) model. The analysis revealed a large number of crash frequency factors, including several interesting and important factors rarely studied in the past, such as the safety effects of climbing lanes. Moreover, the RENB and RPNB models were found to considerably outperform the NB model; however, although the RPNB exhibited better goodness-of-fit than the RENB model, the difference was rather small. The findings of this study shed more light on the factors influencing freeway crashes in China. The results will be useful to highway designers and engineers for creating, building, and operating safe freeways as well as to safety management departments for developing effective safety countermeasures. The study presented in this paper also provides additional guidance for choosing relevant methods to analyze safety and to identify safety factors.

Applying a random parameters Negative Binomial Lindley model to examine multi-vehicle crashes along rural mountainous highways in Malaysia

Road safety in rural mountainous areas is a major concern as mountainous highways represent a complex road traffic environment due to complex topology and extreme weather conditions and are associated with more severe crashes compared to crashes along roads in flatter areas. The use of crash modelling to identify crash contributing factors along rural mountainous highways suffers from limitations in data availability, particularly in developing countries like Malaysia, and related challenges due to the presence of excess zero observations. To address these challenges, the objective of this study was to develop a safety performance function for multi-vehicle crashes along rural mountainous highways in Malaysia. To overcome the data limitations, an in-depth field survey, in addition to utilization of secondary data sources, was carried out to collect relevant information including roadway geometric factors, traffic characteristics, real-time weather conditions, cross-sectional elements, roadside features, and spatial characteristics. To address heterogeneity resulting from excess zeros, three specialized modelling techniques for excess zeros including Random Parameters Negative Binomial (RPNB), Random Parameters Negative Binomial – Lindley (RPNB-L) and Random Parameters Negative Binomial – Generalized Exponential (RPNB-GE) were employed. Results showed that the RPNB-L model outperformed the other two models in terms of prediction ability and model fit. It was found that heavy rainfall at the time of crash and the presence of minor junctions along mountainous highways increase the likelihood of multi-vehicle crashes, while the presence of horizontal curves along a steep gradient, the presence of a passing lane and presence of road delineation decrease the likelihood of multi-vehicle crashes. Findings of this study have significant implications for road safety along rural mountainous highways, particularly in the context of developing countries.

Comparison of modelling methods accounting for temporal correlation in crash counts

ABSTRACTTemporal correlation in crash counts is an important issue that road safety researchers face when developing crash prediction models. Several modelling methods have been proposed to date to address this issue. Random effects negative binomial (RENB) model, random parameters negative binomial (RPNB) model, multilevel negative binomial (MLNB) model, negative multinomial (NM) model, and generalized estimating equation (GEE) model are the most frequently used ones. Although those models have been applied successfully, it is not clear to researchers what their differences are and how to make a choice among them. This study attempts to answer these questions through describing the main features of the five models, showing their connections, illustrating their applications to predict the crash frequency on basic freeway segments, and discussing their differences that can guide the choice making. The application results show that the RENB, MLNB, and RPNB models are superior to others in terms of predictability (for the current data set), capability, usability, and interpretability. Three fundamental differences, including the ways to handle temporal correlation, whether to account for other heterogeneities, and the interpretation of estimated coefficients, provide some guidance on the model choice.

Developing a Random Parameters Negative Binomial-Lindley Model to analyze highly over-dispersed crash count data

The existence of preponderant zero crash sites and/or sites with large crash counts can present challenges during the statistical analysis of crash count data. Additionally, unobserved heterogeneity in crash data due to the absence of important variables could negatively impact the estimated model parameters. The traditional negative binomial (NB) model with fixed parameters might not adequately handle highly over-dispersed data or unobserved heterogeneity. Many research efforts that have involved the negative binomial–Lindley (NB-L) model or the random parameters negative binomial (RPNB) model, for example, have attempted to improve the inference of estimated coefficients by explicitly accounting for extra variation in crash data. The NB-L is a mixed modeling approach which provides flexibility to account for additional dispersion in data. The RP modeling approach accommodates the effect of unobserved variables by allowing the model parameters to vary from one observation to another. The following study proposes a combination of these models – the random parameters NB-L (RPNB-L) generalized linear model (GLM) – to account for underlying heterogeneity and address excess over-dispersion. The results show that the RPNB-L model not only provides a superior goodness-of-fit (GOF) with the sample data, but also offers a better understanding about the effects of potential contributing factors. The paper uses the Bayesian framework to provide a strategy for eliminating the potential for poor mixing in the Markov Chain Monte Carlo (MCMC) chains during the estimation of the RPNB-L model.

The effect of variations in spatial units on unobserved heterogeneity in macroscopic crash models

Macroscopic safety models establish a relationship between crashes and the contributing factors in a defined spatial unit. Negative binomial (NB) and Bayesian negative binomial models with conditional autoregressive prior (CAR) are techniques widely used to establish this relationship. However, these models do not account for unobserved heterogeneity and their output is global and fixed irrespective of the spatial unit of the analysis. There is a timely need to understand how variations in spatial units affect unobserved heterogeneity. This study uses two advanced modeling techniques, the random parameter negative binomial (RPNB) and the semi-parametric geographically weighted Poisson regression (S-GWPR), to investigate whether explanatory variables found to be significant and random in one spatial aggregation will remain significant and random when another spatial aggregation is used. The key finding is that variations in spatial units do have an impact on unobserved heterogeneity. We also found that variations in spatial units have a greater impact on unobserved heterogeneity in the RPNB models compared to the S-GWPR models. We found that the S-GWPR model performs better than the RPNB model with the lowest value of mean absolute deviation (MAD) and Akaiki information criterion (AIC) but the two modeling techniques produce similar results in terms of the sign of the coefficients across the selected spatial units of analysis. Overall, the study provides a methodological basis for assessing the impact of spatial units on unobserved heterogeneity.

Spatiotemporal and random parameter panel data models of traffic crash fatalities in Vietnam

This paper investigates factors associated with traffic crash fatalities in 63 provinces of Vietnam during the period from 2012 to 2014. Random effect negative binomial (RENB) and random parameter negative binomial (RPNB) panel data models are adopted to consider spatial heterogeneity across provinces. In addition, a spatiotemporal model with conditional autoregressive priors (ST-CAR) is utilised to account for spatiotemporal autocorrelation in the data. The statistical comparison indicates the ST-CAR model outperforms the RENB and RPNB models. Estimation results provide several significant findings. For example, traffic crash fatalities tend to be higher in provinces with greater numbers of level crossings. Passenger distance travelled and road lengths are also positively associated with fatalities. However, hospital densities are negatively associated with fatalities. The safety impact of the national highway 1A, the main transport corridor of the country, is also highlighted.

Macroscopic modeling of pedestrian and bicycle crashes: A cross-comparison of estimation methods

The paper presents a cross-comparison of different estimation methods to model pedestrian and bicycle crashes. The study contributes to macro level safety studies by providing further methodological and empirical evidence on the various factors that influence the frequency of pedestrian and bicycle crashes at the planning level. Random parameter negative binomial (RPNB) models are estimated to explore the effects of various planning factors associated with total, serious injury and minor injury crashes while accounting for unobserved heterogeneity. Results of the RPNB models were compared with the results of a non-spatial negative binomial (NB) model and a Poisson-Gamma-CAR model. Key findings are, (1) the RPNB model performed best with the lowest mean absolute deviation, mean squared predicted error and Akaiki information criterion measures and (2) signs of estimated parameters are consistent if these variables are significant in models with the same response variables. We found that vehicle kilometers traveled (VKT), population, percentage of commuters cycling or walking to work, and percentage of households without motor vehicles have a significant and positive correlation with the number of pedestrian and bicycle crashes. Mixed land use is also found to have a positive association with the number of pedestrian and bicycle crashes. Results have planning and policy implications aimed at encouraging the use of sustainable modes of transportation while ensuring the safety of pedestrians and cyclist.

A Development of Traffic Accident Model by Random Parameter : Focus on Capital Area and Busan 4-legs Signalized Intersections

This study intends to build a traffic accident predictive model considering road geometrics, traffic and enviromental characteristics and identify the relationship of 4-legs intersection accidents in Seoul and Busan metropolitan area. The RPNB(Random Parameter Negative Binomial) model shows improvement over the fixed NB(Negative Binomial) and out of 53 variables, 10 variables (main road number of lane, main road vehicle traffic volume(left), minor road vehicle traffic volume(right), main road drive restriction, minor road sight distance, minor road median strip, minor road speed limit, minor road speed restriction) showed to have significant variables affecting traffic accident occurrences in 4-legs signilized intersections. Also, among 10 significant variables, 2 variables(minor road sight distance, minor road speed restriction) found to be random parameters.

RPNB 모형을 이용한 고속도로 인터체인지 구간에서의 교통사고모형 - 인터체인지 형태별/지역별로

PURPOSES : The objective was to develop the advanced method which could not explain each observation's specific characteristic in the present negative binomial method that results in under-estimation of the standard error(t-value inflation) and affects the confidence of whole derived results. METHODS : This study dealt with traffic accidents occurring within interchange segment on highway main line with RPNB(Random Parameter Negative Binomial) method that enables to take account of heterogeneity. RESULTS : As a result, AADT and lighting installation type on the road were revealed to have random parameter and in terms of other geometric variables, all were derived as fixed parameter(same effect on every segment). Also, marginal effects were adapted to analyze the relative effects on traffic accidents. CONCLUSIONS : This study proves that RPNB method which considers each observation's specific characteristics is better fitted to the accident data with geometrics. Thus, it is recommended that RPNB model or other methods which could consider the heterogeneity needs to be adapted in accident analysis.

Random parameter models of interstate crash frequencies by severity, number of vehicles involved, collision and location type

A nine-year (1999–2007) continuous panel of crash histories on interstates in Washington State, USA, was used to estimate random parameter negative binomial (RPNB) models for various aggregations of crashes.A total of 21 different models were assessed in terms of four ways to aggregate crashes, by: (a) severity, (b) number of vehicles involved, (c) crash type, and by (d) location characteristics. The models within these aggregations include specifications for all severities (property damage only, possible injury, evident injury, disabling injury, and fatality), number of vehicles involved (one-vehicle to five-or-more-vehicle), crash type (sideswipe, same direction, overturn, head-on, fixed object, rear-end, and other), and location types (urban interchange, rural interchange, urban non-interchange, rural non-interchange). A total of 1153 directional road segments comprising of the seven Washington State interstates were analyzed, yielding statistical models of crash frequency based on 10,377 observations. These results suggest that in general there was a significant improvement in log-likelihood when using RPNB compared to a fixed parameter negative binomial baseline model. Heterogeneity effects are most noticeable for lighting type, road curvature, and traffic volume (ADT). Median lighting or right-side lighting are linked to increased crash frequencies in many models for more than half of the road segments compared to both-sides lighting. Both-sides lighting thereby appears to generally lead to a safety improvement. Traffic volume has a random parameter but the effect is always toward increasing crash frequencies as expected. However that the effect is random shows that the effect of traffic volume on crash frequency is complex and varies by road segment. The number of lanes has a random parameter effect only in the interchange type models. The results show that road segment-specific insights into crash frequency occurrence can lead to improved design policy and project prioritization.