# Comprehensive Evaluation of a Sparse Dataset, Assessment and Selection of Competing Models
## 5 Results
### Model Comparisons
It is of interest not only to find the best-fit model but to make a comparison across parameters' estimations with various distributions. From Table 2, three GLM models and two zero-augmented models were considered. It should be noted that both zero-augmented models used logit link function for their first layers. It can be seen from Poisson and quasi-Poisson that the estimates are almost identical. However, while almost all the included variables are important at 0.05 \(p\)-value significance for the Poisson model, many estimations are significant not for quasi-Poisson models. This is due to the fact that the scale parameter is fixed as 1 for Poisson while this value is computed for quasi-Poisson. Not accounting for over-dispersion, even if that model results in very similar output coefficient estimates, it would result in unreasonable inference/interpretation. Moving to GLM with the family of negative binomial, it can be seen that while there are some similarities between negative binomial and Poisson, and also quasi-Poisson models. It can be observed that while there are similar signs for various driver action variables between negative binomial results and the other GLM models, there are significant differences for barriers and roadway characteristic predictors, shoulder width, barrier height, and the correspondent interaction terms. While quasi-Poisson is not associated with a fitted likelihood, the negative binomial improved the fit dramatically compared to the Poisson distribution (AIC\({}_{\text{NB}}\) = 6955 versus AIC\({}_{\text{Poisson}}\) = 29,630). In summary, even considering a different family of GLM results in complete variations across some predictors in terms of signs and magnitudes. For zero-augmented models, two components need to be clarified for the models. One component for values greater than zero and another component for zero observations, which are often defined by the binary logistic regression model. For the non-zero distribution, logit link function was considered for both the zero-inflated and hurdle models. The Poisson distribution is considered for the non-zero hurdle model as well, which will be evaluated in the next section. For the most part, the estimates for both zero-augmented models are very close with similar signs for all significant predictors in the non-zero layer. The inverted signs of the zero-inflated model are expected. As for the hurdle model, the zero hurdle component describes the likelihood of observing a crash while the zero-inflated component predicts the probability of observing a zero count. In summary, both zero-augmented models have two layers: the first layers controls whether a vehicle hits a barrier or not and the second component controls how many times and with what crash severity level a vehicle would hit a barrier. In summary, a significant improvement was observed by implementing the GLM with negative binomial distribution compared with the GLM Poisson model. These results were expected as Poisson distribution does not account for over-dispersion. Although a slight improvement was observed from the negative binomial to the zero-inflated model, the improvement was greater when compared to the hurdle model.