# Determining the Safety Level of State Roads: An Italian Case Study
## 5 Case Study
### Safety Indicator Evaluation  
#### 5.4.1 Empirical-Bayesian Method  
The adoption of predictive models for the assessment of road safety levels aims at mitigating the phenomenon of regression to the mean of the actual crash trend. The selected and calibrated predictive model, as presented in Section 5.3, allows for the determination of the number of predicted crashes (\(N_{pred}\)) according to the specific site conditions (traffic volume, geometric-functional road characteristics). Then, the predicted crashes should be combined with the actual number of observed crashes (\(N_{obs}\)). This is the typical approach of the empirical Bayesian (EB) method, calculating a weighted average between the predicted and observed crashes. The value obtained is usually referred to as the number of expected crashes (\(N_{exp}\)), which represents the best estimate of crash frequency for a given site, combining both the predictive model and reality. The formulation of the EB method is described by Equation (5).  
\[N_{exp}=w\cdot N_{pred}+(1-w)\cdot N_{obs} \tag{5}\]  
The weighting factor (\(w\)) depends on the accuracy of the model, which is derived from the goodness-of-fit of the calibration process. In fact, the weighting factor is a function of the overdispersion parameter (\(k\)) with inverse proportionality, as expressed in Equation (6).  
\[w=\frac{1}{1+\frac{k}{L}\cdot(\underbrace{\sum_{\begin{subarray}{c}\text{all } study\\ \text{years}\end{subarray}}N_{pred}})} \tag{6}\]  
The summary of the EB analysis results obtained for the analyzed road sections is shown in Table 6. It can be observed that the total number of observed crashes is not perfectly equivalent to the total number of predicted crashes since the EB analysis also takes into account '_outliers_' that were excluded during the calibration process. For the total length of the analyzed road section of about 148 km, 254.93 crashes are expected in three years, which corresponds to around 85 crashes per year. This value is based on both the predictive model and the actual crashes that occurred. In addition, the expected crashes can be attributed to each specific homogeneous segment and related to their characteristics. This step is essential to assign to each site the safety performance indicators capable of describing the local safety level. In this way, it is possible to identify and localize potential critical infrastructure conditions.  
Figure 3: Cumulative residuals plot to express the goodness-of-fit of the calibration procedure.  
The overall effect of the EB method is summarized in Figure 4, where the frequency distribution of expected crashes is a combination of the distribution of observed and predicted crashes. The frequency distribution of observed crashes is influenced by the fact that, in reality, many segments (202) are characterized by zero crashes. On the other hand, the frequency distribution of predicted crashes is shifted to the right because the predictive model always leads to a value of crashes greater than one. Thus, since the expected crashes are derived from a weighted average of the observed crashes and predicted crashes, they are characterized by a frequency distribution that lies between those of the previously mentioned entities.