## 4 Methodology  
In this research, the following two hypothesis were tested:  
1. [label=()]
2. \(\mathrm{H_{0}}\) = There is no difference in the expected crashes estimated by utilizing FRA and field-validated data (\(\alpha\) = 5%).  
To test the first hypothesis, we utilized the original FRA HRGCs inventory data and the field-validated inventory data (both merged with the FRA HRGCs accident/incident database) for the 560 HRGCs to estimate expected crashes by utilizing the FRA's 2020 Accident Prediction (AP) model (Equations (1)-(4)). The two sets of expected crashes (crash predictions) were then statistically evaluated for differences at \(\alpha\) = 5% (a 5% chance of incorrectly rejecting the null hypothesis). This step aimed to assess how erroneous or missing values in the HRGCs inventory data affect the estimation of expected crashes at HRGCs when the crash prediction models currently employed by the federal transportation agencies are utilized.
3. \(\mathrm{H_{0}}\) = There is no difference in the estimated parameters' coefficients of new crash frequency prediction models from the two datasets (FRA Vs field-validated) (\(\alpha\) = 5%).  
For this hypothesis, we developed two new crash prediction models using both FRA and field-validated HRGCs inventory data. The objective was to determine whether the coefficient parameters in these models exhibited statistically significant differences. A base model was first estimated by utilizing the original FRA inventory data, and then another comparison model was estimated using the field-validated inventory data. For both models, the dependent variable was the past 5-year HRGCs crash counts (2017-2021). Data exploration showed that the variance of the crash counts was greater than its mean value, indicating overdispersion in the data. During the 5-year period, most HRGCs had zero crashes, with only a few crossings reporting 28 crashes. This condition warranted the estimation of a two-part Zero-inflated Negative Binomial Model (ZINB). The ZINB model provides a better goodness of fit when compared with the Poisson and the Negative Binomial (NB) Model [20; 21; 22; 23; 24; 25; 26; 27]. It also provides parametric expressions for counts of predicted crashes and expected crash count variance [24].  
A general ZINB model is written as [23]:  
\[E[\mathrm{a}]=\mu(1-\mathrm{p}) \tag{5}\]  
\[V[\mathrm{a}]=\mu(1-\mathrm{p})(1+\mu(\mathrm{p}+\alpha)) \tag{6}\]  
where \(E[\mathrm{a}]\) is the predicted value of crash count, \(V[\mathrm{a}]\) is the variance of the predicted estimate of crash count, \(\mu\) is the mean of the Negative Binomial process, \(\mathrm{p}\) is the likelihood of the entity being in the \"always 0\" case in the finite mixture model, and \(\alpha\) is the over dispersion parameter of the Negative Binomial model (equal to \(1/\theta\), theta is an estimated parameter for the model).  
Table 3 presents the candidate independent variables chosen for model estimation. Before finalizing the model's specification, a check of the multi-collinearity between candidate variables was performed, and those variables were chosen with a Variance Inflation Factor (VIF) \(<\) 10 and Tolerance (TOL) \(>\) 1 [2]. According to past research, the continuous variables (e.g., AADT) were natural Log-transformed to reduce the range of values within the variables [2]. PSCL package in R computing (R-project.org) was used for model estimation (accessed on 1 January 2020) [25]. Also, the model performance criteria were estimated, such as Akaike information criterion (AIC) and Bayesian information criterion (BIC) [24; 25; 26].