## 6 Study Analysis and Results
### Spearman Correlation Coefficient  
Initially, the Spearman's rank correlation coefficient was calculated between the proposed method's ranking and the ranking using observed crash data. The Spearman rank correlation coefficient is a nonparametric technique that is usually applied to evaluate the degree of linear association between two independent variables [28]. Here, this coefficient  
Figure 2: R output of regression model. is used to measure the degree of association between two lists of hazardous sites ordered based on two ranking criteria. The simple expression for '\(\rho\)', based on the difference between the two ranked variables, is as shown in Equation (2):  
\[\rho=1-\frac{6\sum d_{i}^{2}}{n\;(n^{2}-1)} \tag{2}\]  
where \(\rho\) = Spearman's rank correlation coefficient; \(\text{d}_{i}\) = R(X\({}_{i}\)) - R(Y\({}_{i}\)) is the difference between the two ranks of the subject segment by two compared methods; and n = number of observations. The Spearman rank correlation value varies from \(-1\) to \(+1\). A higher Spearman correlation coefficient indicates a stronger agreement between the ranking by the compared method and the crash history, and 0 means no agreement. Figure 3 shows a scatterplot with site rankings using the proposed method, EB method, and PSI method versus crash history. An examination of Figure 3a, which shows the proposed method versus the crash history rank, reveals that the data points are spread around the diagonal line. The discrepancy between the ranks increases with the increase in rank. The tightness of the data points around the line and the correlation coefficient of 0.686 indicates a moderate correlation between the ranks using the two methods. A perfect correlation is a diagonal line passing through the origin. Figure 3: Rank comparison using different network screening methods versus crash history; (**a**) proposed method versus crash history rank; (**b**) EB method versus crash history rank; and (**c**) PSI method versus crash history rank. The EB method ranking was also compared with the ranking using the observed number of crashes and the results are shown in Figure 3b. In this figure, the proximity of data points to the line passing through the origin (diagonal line) indicates that it is more compact compared to that of the proposed method. This observation is confirmed by a higher correlation coefficient (r = 0.816) between the two rankings. Additionally, Figure 3c shows the comparison of ranking using PSI method versus the ranking using the observed number of crashes. The data points clustered around the diagonal line and the correlation coefficient of 0.879 indicate the very strong association between the ranks. Table 2 presents the [PERSON]'s rank correlation coefficient values for three different methods (proposed method, Empirical Bayes method, and PSI method) based on crash density and crash frequency in different upper tail segments and for the total sample size of 377 segments. In analyzing the crash density for the upper tail segments (20, 40, 60, 80, 100), the proposed method shows moderate to strong positive correlations with crash history, with values ranging from 0.587 to 0.744. The Empirical Bayes method and PSI method consistently exhibit higher correlations, ranging from 0.786 to 0.828 and 0.954 to 0.959, respectively. For the total sample size of 377 segments, the proposed method exhibits a moderate positive correlation (0.683), while the Empirical Bayes method (0.816) and PSI method (0.879) display stronger positive correlations. This shows that, in analyzing crash densities, the Empirical Bayes and PSI methods outperform the proposed method at identifying safety improvement sites. In the case of crash frequency, for the upper tail segments (20, 40, 60, 80, 100), the proposed method shows weak to moderate positive correlations, ranging from 0.502 to 0.703.