# Comprehensive Evaluation of a Sparse Dataset, Assessment and Selection of Competing Models
## 2 Method
Another issue with the zero-inflated model for modeling crash data is that this model assumes the extra zeros come from two states: a true-zero state, where the roadway locations are inherently safe meaning that no crash would occur on those roadway locations at any time, and non-zero state where only no crashes occurred in the observation periods [16]. It should be noted that it is unrealistic to consider any road segment as safe at all times, which makes the zero-inflated model unsuitable for crash counts with an excess number of zeros. On the other hand, the hurdle model addresses the assumption of the zero-inflated model by assuming that roadway segments with zeros are only safe over the study period and not at all times [9]. The hurdle model, as a substitute for the zero-inflated model, is a model that could be used to account for excess numbers of zeros and over-dispersion. There are two layers for these models including a truncated count and values greater than zero, which could be accounted for by Poisson, geometric or negative binomial distributions. On the other hand, a hurdle component models zero count by using binary logistic regression to evaluate a change from zero crash to crash areas. For the hurdle model, a probability of mass function (PMF) could be written as [7]:  
\[P_{r}(Y=y)=\left\{ \begin{array}{ll}1-\pi_{i},&y=1,2,3,\ldots\\ \pi_{i},&y=0\end{array} \right. \tag{5}\]  
where \(\pi\) is the probability that the response \(Y=0\) and \(1-\pi\) is the probability that \(Y>0\). For the Poisson process as a starting point of count data, the above formula for zero-truncated Poisson has the PMF as follows:  
\[P_{r}(Y=y)=\left\{ \begin{array}{ll}\dfrac{\lambda_{i}y}{(e^{\lambda}-1)y!}& y=1,2,3,\ldots\\ 0&y=0\end{array} \right. \tag{6}\]  
where \(y_{i}\) is an expected number of 0, 1, 2,..., \(y\) and \(\lambda_{i}\) is the predicted EPDO for every barrier. Thus, from Equations (1) and (2), the unconditional probability mass function for \(Y\) is:  
\[P_{r}(Y=y)=\left\{ \begin{array}{ll}\pi_{i}&y=0\\ 1-\pi_{i}\dfrac{\lambda_{i}y}{(e^{\lambda}-1)y!}&y=1,2,3,\ldots\end{array} \right. \tag{7}\]  
If \(\pi_{i}\) is modeled by log-log and \(\lambda_{i}\) is modeled by log link, with some algebra for the hurdle model and Poisson count, we would have:  
\[\pi_{i}=e^{-e^{x_{i}}\;\beta_{i}} \tag{8}\]  
and  
\[\lambda_{i}=e^{x_{i}\beta_{2}} \tag{9}\]  
and after some algebra the log likelihood could be written as:  
\[lnL=ln[L_{1}(\beta_{1})]+ln[L_{2}(\beta_{2})] \tag{10}\]  
It can be seen from the Equation (10) that the log likelihood of zero and not zero could be written as the sum of the two. In other words, this model estimates the probability of two models: one for zero values and one for non-zero values. In the results section, the hurdle and zero-inflated models are two-layer models. The first layer for the hurdle model is a logistic model governing observations with zero and greater than zero values. The second part is a zero-truncated count model governing the outcomes with positive counts. This model is flexible not only to account for excess zeros but for under- or over-dispersion distribution based on the defined distribution for the second layer. Figure 1 is presented to highlight the sparse nature of barrier EPDO crashes. As can be seen from this figure, EPDO is highly skewed to the left with most observations being less than 50, EPDO is mainly 1, and there are few large observations. The description of this response in Table 1 provides more information. In this study, all the discussed methodologies were implemented to highlight the impact of considering inappropriate distributions. Although it has been discussed that using the wrong distribution for data analysis would result in biased and erroneous results, this study delves into the possible erroneous outcomes by estimating all the coefficients for different distributions and evaluating them based on various goodness-of-fit measures.