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Crash counts are non-negative integer events often analyzed using crash frequency models such as the negative binomial (NB) distribution. However, due to their random and infrequent nature, crash data usually exhibit unique characteristics, such as excess zero observations that the NB distribution cannot adequately model. The negative binomial-Lindley (NBL) and random parameters negative binomial-Lindley (RPNBL) models have been proposed to address this limitation. Despite addressing the issues of excess zero observations, these models may not fully account for unobserved heterogeneity resulting from temporal variations in crash data. In addition, many variables, such as traffic volume, speed, and weather, change with time. Therefore, the analyst often requires disaggregated data to account for their variations. For example, it is recommended to use monthly crash datasets to better account for temporally varying weather variables compared to yearly crash data. Using temporally disaggregated data not only adds the complexity of the temporal variations issue in data but also compounds the issue of excess zero observations. To address these issues, this paper introduces a new variant of the NBL model with coefficients and Lindley parameters that vary by time. The derivations and characteristics of the model are discussed. Then, the model is illustrated using a simulation study. Subsequently, the model is applied to two empirical crash datasets collected on rural principal and minor arterial roads in Texas. These datasets include several time-dependent variables such as monthly traffic volume, standard deviation of speed, and precipitation and exhibit unique characteristics such as excess zero observations. The results of several goodness-of-fit (GOF) measures indicate that using the NBL model with time-dependent parameters enhances the model fit compared to the NB, NBL, and the NB model with time-dependent parameters. Findings derived from crash data collected from both rural minor and principal arterial roads in Texas suggest that the variables denoting the median presence and wider shoulder width are associated with a potential decrease in crash occurrences. Moreover, higher variations in speed and wider road surfaces are linked to a potential increase in crash frequency. Similarly, a higher monthly average daily traffic (Monthly ADT) positively correlates with crash frequency. We also found that it is important to account for temporal variations using time-dependent parameters., Competing Interests: 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., (Copyright © 2024 Elsevier Ltd. All rights reserved.)

Identifying hazardous crash sites (or hotspots) is a crucial step in highway safety management. The Negative Binomial (NB) model is the most common model used in safety analyses and evaluations - including hotspot identification. The NB model, however, is not without limitations. In fact, this model does not perform well when data are highly dispersed, include excess zero observations, or have a long tail. Recently, the Negative Binomial-Lindley (NB-L) model has been proposed as an alternative to the NB. The NB-L model overcomes several limitations related to the NB, such as addressing the issue of excess zero observations in highly dispersed data. However, it is not clear how the NB-L model performs regarding the hotspot identification. In this paper, an innovative Monte Carlo simulation protocol was designed to generate a wide range of simulated data characterized by different means, dispersions, and percentage of zeros. Next, the NB-L model was written as a Full-Bayes hierarchical model and compared with the Full-Bayes NB model for hotspot identification using extensive simulation scenarios. Most previous studies focused on statistical fit, and showed that the NB-L model fits the data better than the NB. In this research, however, we investigated the performance of the NB-L model in identifying the hazardous sites. We showed that there is a trade-off between the NB-L and NB when it comes to hotspot identification. Multiple performance metrics were used for the assessment. Among those, the results show that the NB-L model provides a better specificity in identifying hotspots, while the NB model provides a better sensitivity, especially for highly dispersed data. In other words, while the NB model performs better in identifying hazardous sites, the NB-L model performs better, when budget is limited, by not selecting non-hazardous sites as hazardous., Competing Interests: 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., (Copyright © 2024 Elsevier Ltd. All rights reserved.)

Stay-at-home orders - imposed to prevent the spread of COVID-19 - drastically changed the way highways operate. Despite lower traffic volumes during these times, the rate of fatal and serious injury crashes increased significantly across the United States due to increased speeding on roads with less traffic congestion and lower levels of speed enforcement. This paper uses a mixed effect binomial regression model to investigate the impact of stay-at-home orders on odds of speeding on urban limited access highway segments in Maine and Connecticut. This paper also establishes a link between traffic density and the odds of speeding. For this purpose, hourly speed and volume probe data were collected on limited access highway segments for the U.S. states of Maine and Connecticut to estimate the traffic density. The traffic density was then combined with the roadway geometric characteristics, speed limit, as well as dummy variables denoting the time of the week, time of the day, COVID-19 phases (before, during and after stay-at-home order), and the interactions between them. Density, represented in the model as Level of Service, was found to be associated with the odds of speeding, with better levels of service such as A, or B (low density) resulting in the higher odds that drivers would speed. We also found that narrower shoulder width could result in lower odds of speeding. Furthermore, we found that during the stay-at-home order, the odds of speeding by more than 10, 15, and 20 mph increased respectively by 54%, 71% and 85% in Connecticut, and by 15%, 36%, and 65% in Maine during evening peak hours. Additionally, one year after the onset of the pandemic, during evening peak hours, the odds of speeding greater than 10, 15, and 20 mph were still 35%, 29%, and 19% greater in Connecticut and 35% 35% and 20% greater in Maine compared to before pandemic., Competing Interests: 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., (Copyright © 2023 Elsevier Ltd. All rights reserved.)

The COVID-19 pandemic caused a significant change in traffic operations and safety. For instance, various U.S. states reported an increase in the rate of fatal and severe injury crashes over this duration. In April and May of 2020, comprehensive stay-at-home orders were issued across the country, including in Maine. These orders resulted in drastic reductions in traffic volume. Additionally, there is anecdotal evidence that speed enforcement had been reduced during pandemic. Drivers responded to these changes by increasing their speed. More importantly, data show that speeding continues to occur, even one year after the onset of the pandemic. This study develops statistical models to quantify the impact of the pandemic on speeding in Maine. We developed models for three rural facility types (i.e., major collectors, minor arterials, and principal arterials) using a mixed effect Binomial regression model and short duration speed and traffic count data collected at continuous count stations in Maine. Our results show that the odds of speeding by more than 15 mph increased by 34% for rural major collectors, 32% for rural minor arterials, and 51% for rural principal arterials (non-Interstates) during the stay-at-home order in April and May of 2020 compared to the same months in 2019. In addition, the odds of speeding by more than 15 mph, in April and May of 2021, one year after the order, were still 27% higher on rural major collectors and 17% higher on rural principal arterials compared to the same months in 2019., Competing Interests: 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., (Copyright © 2022 Elsevier Ltd. All rights reserved.)

Crash data are often highly dispersed; it may also include a large amount of zero observations or have a long tail. The traditional Negative Binomial (NB) model cannot model these data properly. To overcome this issue, the Negative Binomial-Lindley (NB-L) model has been proposed as an alternative to the NB to analyze data with these characteristics. Research studies have shown that the NB-L model provides a superior performance compared to the NB when data include numerous zero observations or have a long tail. In addition, crash data are often collected from sites with different spatial or temporal characteristics. Therefore, it is not unusual to assume that crash data are drawn from multiple subpopulations. Finite mixture models are powerful tools that can be used to account for underlying subpopulations and capture the population heterogeneity. This research documents the derivations and characteristics of the Finite mixture NB-L model (FMNB-L) to analyze data generated from heterogeneous subpopulations with many zero observations and a long tail. We demonstrated the application of the model to identify subpopulations with a simulation study. We then used the FMNB-L model to estimate statistical models for Texas four-lane freeway crashes. These data have unique characteristics; it is highly dispersed, have many locations with very large number of crashes, as well as significant number of locations with zero crash. We used multiple goodness-of-fit metrics to compare the FMNB-L model with the NB, NB-L, and the finite mixture NB models. The FMNB-L identified two subpopulations in datasets. The results show a significantly better fit by the FMNB-L compared to other analyzed models., (Copyright © 2022 Elsevier Ltd. All rights reserved.)

The expected crash frequency is the long-term average crash count for a specific site. It is extensively used to systematically evaluate the crash risk associated with roadway elements. To estimate the expected crashes, the Empirical Bayesian (EB) approach is typically employed. The EB method is a computationally convenient approximation to the Full Bayesian (FB) method, which gained popularity due to its simple interpretation, computational efficiency, and the ability to account for the regression to the mean bias. However, the common EB method used in traffic safety analysis is only applicable when the traditional Negative Binomial (NB) model is used. The NB model, however, is not a suitable choice when data is highly dispersed, skewed, or has a large number of zero observations. The Negative Binomial-Lindley (NB-L) model is a mixture of the NB and Lindley distributions and has shown superior fit compared to the NB model, especially when the dataset is characterized by excess zero observations. Even though several studies have used the NB-L in developing crash prediction models, the application of the NB-L in other safety-related tasks (e.g., hot spot identification) is largely neglected. This study proposed a framework to develop the EB method for the NB-L model and subsequently estimate the expected crash values. A comparison between the EB and FB estimates was performed to validate the approximation framework in general. The results indicated that the proposed EB framework is able to estimate expected crashes with comparable precision to the FB estimate, but with much less computational cost. In addition, a site ranking analysis using the EB estimates was conducted to validate the proposed approximation method in safety studies. However, it should be noted that any other type of safety analysis that requires access to the expected crashes can benefit from the proposed EB method. This study concluded that the proposed EB framework can properly approximate the underlying FB approach and can reasonably be considered as an alternative to the traditional EB formula derived from the NB model. The results of this study can help to extend the application of the advanced predictive models beyond predicting crashes to other safety-related tasks, with no additional computational efforts., (Copyright © 2022 Elsevier Ltd. All rights reserved.)

Safety analysts usually use post-modeling methods, such as the Goodness-of-Fit statistics or the Likelihood Ratio Test, to decide between two or more competitive distributions or models. Such metrics require all competitive distributions to be fitted to the data before any comparisons can be accomplished. Given the continuous growth in introducing new statistical distributions, choosing the best one using such post-modeling methods is not a trivial task, in addition to all theoretical or numerical issues the analyst may face during the analysis. Furthermore, and most importantly, these measures or tests do not provide any intuitions into why a specific distribution (or model) is preferred over another (Goodness-of-Logic). This paper ponders into these issues by proposing a methodology to design heuristics for Model Selection based on the characteristics of data, in terms of descriptive summary statistics, before fitting the models. The proposed methodology employs two analytic tools: (1) Monte-Carlo Simulations and (2) Machine Learning Classifiers, to design easy heuristics to predict the label of the 'most-likely-true' distribution for analyzing data. The proposed methodology was applied to investigate when the recently introduced Negative Binomial Lindley (NB-L) distribution is preferred over the Negative Binomial (NB) distribution. Heuristics were designed to select the 'most-likely-true' distribution between these two distributions, given a set of prescribed summary statistics of data. The proposed heuristics were successfully compared against classical tests for several real or observed datasets. Not only they are easy to use and do not need any post-modeling inputs, but also, using these heuristics, the analyst can attain useful information about why the NB-L is preferred over the NB - or vice versa- when modeling data., (Copyright © 2017 Elsevier Ltd. All rights reserved.)

Severity distribution functions (SDFs) are used in highway safety to estimate the severity of crashes and conduct different types of safety evaluations and analyses. Developing a new SDF is a difficult task and demands significant time and resources. To simplify the process, the Highway Safety Manual (HSM) has started to document SDF models for different types of facilities. As such, SDF models have recently been introduced for freeway and ramps in HSM addendum. However, since these functions or models are fitted and validated using data from a few selected number of states, they are required to be calibrated to the local conditions when applied to a new jurisdiction. The HSM provides a methodology to calibrate the models through a scalar calibration factor. However, the proposed methodology to calibrate SDFs was never validated through research. Furthermore, there are no concrete guidelines to select a reliable sample size. Using extensive simulation, this paper documents an analysis that examined the bias between the 'true' and 'estimated' calibration factors. It was indicated that as the value of the true calibration factor deviates further away from '1', more bias is observed between the 'true' and 'estimated' calibration factors. In addition, simulation studies were performed to determine the calibration sample size for various conditions. It was found that, as the average of the coefficient of variation (CV) of the 'KAB' and 'C' crashes increases, the analyst needs to collect a larger sample size to calibrate SDF models. Taking this observation into account, sample-size guidelines are proposed based on the average CV of crash severities that are used for the calibration process., (Copyright © 2016 Elsevier Ltd. All rights reserved.)

The Highway Safety Manual (HSM) prediction models are fitted and validated based on crash data collected from a selected number of states in the United States. Therefore, for a jurisdiction to be able to fully benefit from applying these models, it is necessary to calibrate or recalibrate them to local conditions. The first edition of the HSM recommends calibrating the models using a one-size-fits-all sample-size of 30-50 locations with total of at least 100 crashes per year. However, the HSM recommendation is not fully supported by documented studies. The objectives of this paper are consequently: (1) to examine the required sample size based on the characteristics of the data that will be used for the calibration or recalibration process; and, (2) propose revised guidelines. The objectives were accomplished using simulation runs for different scenarios that characterized the sample mean and variance of the data. The simulation results indicate that as the ratio of the standard deviation to the mean (i.e., coefficient of variation) of the crash data increases, a larger sample-size is warranted to fulfill certain levels of accuracy. Taking this observation into account, sample-size guidelines were prepared based on the coefficient of variation of the crash data that are needed for the calibration process. The guidelines were then successfully applied to the two observed datasets. The proposed guidelines can be used for all facility types and both for segment and intersection prediction models., (Copyright © 2016 Elsevier Ltd. All rights reserved.)

Crash data can often be characterized by over-dispersion, heavy (long) tail and many observations with the value zero. Over the last few years, a small number of researchers have started developing and applying novel and innovative multi-parameter models to analyze such data. These multi-parameter models have been proposed for overcoming the limitations of the traditional negative binomial (NB) model, which cannot handle this kind of data efficiently. The research documented in this paper continues the work related to multi-parameter models. The objective of this paper is to document the development and application of a flexible NB generalized linear model with randomly distributed mixed effects characterized by the Dirichlet process (NB-DP) to model crash data. The objective of the study was accomplished using two datasets. The new model was compared to the NB and the recently introduced model based on the mixture of the NB and Lindley (NB-L) distributions. Overall, the research study shows that the NB-DP model offers a better performance than the NB model once data are over-dispersed and have a heavy tail. The NB-DP performed better than the NB-L when the dataset has a heavy tail, but a smaller percentage of zeros. However, both models performed similarly when the dataset contained a large amount of zeros. In addition to a greater flexibility, the NB-DP provides a clustering by-product that allows the safety analyst to better understand the characteristics of the data, such as the identification of outliers and sources of dispersion., (Copyright © 2016 Elsevier Ltd. All rights reserved.)