# Comprehensive Evaluation of a Sparse Dataset, Assessment and Selection of Competing Models
## 2 Method  
In this study, barriers with crashes and risky barriers which did not experience any crashes were included in this study as crashes are random in nature. Not experiencing a crash does not mean that abarrier is not risky. This is especially true in a state like Wyoming with very low traffic. Although many of these barriers pose a high risk due to poor design, they have not experienced any crashes and there is a great chance that with an increase in traffic they would experience crashes. When these barriers experience crashes, the situation would be more severe if they are not within the recommended heights. Different predictors were considered in the analysis to account for exposure resulting from differences across various barriers. For instance, variables such as barrier length and traffic were incorporated in the analysis. Some of the issues with the crash data, especially EPDO, are over-dispersion or under-dispersion. Over-dispersion occurs when sample variance is higher the mean, while the reverse is true for under dispersion. These challenges could be a source of errors by specifying a model with the wrong distribution, which results in potential errors and biased coefficient estimates. When the data is over-dispersed, some changes in the model parameters could be applied to account for that over dispersion, e.g., application of a negative binomial model [12]. In general, the count model can be modeled by the generalized linear model (GLM). For Poisson, the variance would be set as identical to the mean and the dispersion parameter is fixed at \(\varphi=1\). In other words, for the Poisson model, the variance is restricted to the mean as shown below:  
\[Var[Y]=\mu \tag{1}\]  
Poisson model has a linear form as:  
\[log(\lambda_{i})=\beta_{0}+\beta_{1}x_{i1}+\beta_{2}x_{i2}+\ldots+\beta_{m}x_ {im} \tag{2}\]  
where \(\beta\)s are the estimated regression coefficients, and xs are different explanatory variables. Quasi-Poisson model is another method of GLM that can handle over-dispersion. For this model, compared with a standard Poisson, \(\varphi\) would be unrestricted on the left. In other words, \(\varphi\), instead of being fixed, would be calculated from the data. Negative binomial is a type of Poisson model, Poisson-gamma mixture model, that can be used to relax the constraint of equality between the mean and variance. The negative binomial can be written as follows:  
\[log(\lambda_{i})=\beta_{0}+\beta_{1}x_{i1}+\beta_{2}x_{i2}+\ldots+\beta_{m}x_ {im}+\varepsilon_{i} \tag{3}\]  
where the Poisson parameter \(\lambda_{i}\) follows a gamma probability distribution, \(\text{Exp}(\varepsilon_{i})\) is a gamma-distributed error term, and other parameters were defined earlier. As discussed, a negative binomial can be used as gamma mixture of Poisson distribution. This model has \(\varphi=1\) where variance differs with the Equation (4):  
\[V(\mu)=\mu+\frac{\mu^{2}}{\theta} \tag{4}\]  
where \(\theta\) is a shape parameter, and \(\mu\) is the mean. In addition to over-dispersion, many count data exhibit an excess number of zero observations, which might not be addressed by GLM models [13]. The zero-inflated model, for example, is a method that can account for over-dispersion and the presence of excess zeros. This model allows zeros to come from both at-risk (sample zeros), and not-at-risk populations (structural zeros), while the hurdle model only allows zeros to come from at-risk populations [14]. For those models, sample zeros would be calculated by a binary logistic model, while structural zeros would be estimated by the count part of the model, e.g., negative binomial or Poisson model. By design, if the interest is to design a model that incorporates only sample zeros and at-risk samples, then the hurdle model is preferred. Although it has been discussed that zero-inflated is preferred in many count datasets due to its improved statistical fit, it has been argued that the inherent dual state process underlining the development of this model is inconsistent with crash data [15].