# Comprehensive Evaluation of a Sparse Dataset, Assessment and Selection of Competing Models
## 1 Introduction  
Sometimes drivers of vehicles leave the travel lane and encroach onto natural or artificial subjects on the roadside, causing a crash. These types of crashes account for a significant proportion of high-severity crashes [1]. The installation of traffic barriers would be warranted if the severity of hitting these barriers would be less severe compared with hitting other roadside fixed objects. However even if this would be the case, the severity of colliding with these barriers would still persist. Previous studies mostly focused on analyzing the severity of traffic barrier crashes by analyzing traditional traffic barrier crash data, without aggregating those crashes across different barriers. In addition, previous studies mostly have ignored the frequency of those crashes in addition to the impacts of different crash severity types, e.g., fatality versus injury crashes. However, the crash frequency or crashseverity alone could not account for the severity of crashes, especially if the objective of the assessment is to optimize the geometric characteristics of roadside barriers. One way to account for both crash frequency and crash severity is to consider property damage only (PDO) crashes as one single group and all other crashes against this PDO group based on their levels of severity, which would be weighted and converted into PDO crash equivalents. This measurement is known as the equivalent property damage only (EPDO) benchmark. One of the biggest challenges of analyzing crash frequency along with crash severity is the distributional aspect of a model which cannot be modeled with distributions such as Poisson or gamma distributions. Despite a lack of studies employing EPDO, many studies have explored other aspects of traffic safety by considering only the frequency or severity of crashes. Nevertheless, it should be noted that the majority of those datasets which modeled crash counts were impacted by some forms of over- or under-dispersion resulting from the sparse nature of their crash counts [2]. Some observations have an excess number of zeroes along with many small property damage only (PDO) crashes, which result in an overall mean that is close to zero. These types of distributions could not be modeled by using a traditional Poisson distribution, which considers the mean to be equal to the variance. However, this sparsity could be handled by using various methods incorporating various shape parameters, such as a negative binomial distribution. Wyoming has one of the highest fatality rates in the United States [3]. This is a result of many factors such as adverse weather conditions and mountainous areas with challenging roadways. Due to the presence of a large number of curves and mountainous areas, there is a significant number of traffic barriers in Wyoming. These barriers help in avoiding the severity of hitting other roadside fixed objects, but they still account for a number of severe crashes in the state. The Wyoming Department of Transportation (WYDOT) funded a project to measure various aspects of more than 1 million linear feet of traffic barriers in the state. The objective of that project was to measure various aspects of the barriers' geometric characteristics such as their heights and shoulder widths to better manage and enhance them. One of the main reasons for collecting the barrier geometric characteristics was that many of the barriers in the state are not within the recommended dimensions of barrier designs, and they should be enhanced. Having well-designed barriers is important because barrier crash severity would be further exacerbated if they are not well designed, resulting in possible override or underride barrier crashes. Unfortunately, some of the barriers in the state are shorter or taller than the recommended heights, and some of those barriers have not been able to prevent any crashes.