## 2 Methodology
### Statistical Analysis of Survey Responses
Before using this data for the proposed model for identifying the risk factors, the Chi-square independence test established the level of association (weak, moderate, or strong) between the design functionality, visibility adequacy, and driver satisfaction level of the street-lighting system. An example of the null and alternative hypotheses for illumination brightness adequacy is provided in the following:  
**H1. **_The null hypothesis: illumination brightness adequacy is a perfectly independent factor and does not affect the driver's visual performance._  
**H2. **_The alternative hypothesis: illumination brightness adequacy is a dependent factor and somehow affects the driver's visual performance._  
The Chi-square method is based on expected frequencies at which the null hypothesis holds true. The expected frequencies for all the factors against the drivers' opinions were calculated using the following relationship [37]:  
\[e_{ij}=\frac{o_{i}\times o_{j}}{N} \tag{2}\]  
where \(e_{ij}\) represents the expected frequency, \(o_{i}\) represents the marginal column frequencies, \(o_{j}\) represents the marginal row frequencies, and \(N\) is the total number of responses. As the \(o_{i}\) and \(o_{j}\) differ, the residuals were estimated as:  
\[r_{ij}=o_{ij}-e_{ij} \tag{3}\]  
A larger \(r_{ij}\) value (absolute) denotes a large difference between the observed responses and the null hypothesis. The Chi-square \(\big{(}\chi^{2}\big{)}\) test statistic was estimated by adding all the residuals as:  
\[\chi^{2}=\sum\frac{\big{(}o_{ij}-e_{ij}\big{)}^{2}}{e_{ij}} \tag{4}\]  
Next, for a given \(\chi^{2}\) and degree of freedom, the factors' independence in the population (\(p\)-value) was estimated as:  
\[d_{f}=(i-1)\times(j-1) \tag{5}\]  
where \(i\) and \(j\) are the number of rows and columns in the contingency table. The estimated Chi-squared \(\big{(}\chi^{2}\big{)}\) compared with the critical Chi-squared \(\big{(}\chi^{2}_{C}\big{)}\), obtained from the \(\chi^{2}\) distribution at \(p\) < 0.05, established the acceptance and rejection of the null hypothesis of independence. The critical values were 3.84, 5.99, 7.82, 9.49, 11.07, 12.59, and 26.296, with related \(d_{f}\), estimated from Equation (5), of 1, 2, 3, 4, 5, 6, and 16. The \(\chi^{2}\) higher than the \(\chi^{2}_{c}\) rejected the null hypothesis. As the \(\chi^{2}\) test's performance depends on a sufficiently large sample size, the estimated significance does not clearly state the degree of the effect. The effect size (ES), estimating the strength of the association between the sub-criteria (given in Table 1) and the drivers' opinion about the overall performance of the USLS (Question 10 in Table 1), was estimated using the results of \(\chi^{2}\) test for each sub-criterion using [PERSON]'s \(V\). [PERSON]'s \(V\) was estimated by:  
\[V=\sqrt{\frac{\chi^{2}}{n.d_{f}}} \tag{6}\]  
where \(n\) denotes the total number of responses. As per the interpretation of ES presented by [PERSON] [33], ES \(\leq\) 0.2 represents a small association, 0.2 \(<\) ES \(\leq\) 0.6 a medium association, and ES \(>\) 0.6 a large association between the parameters.