## 3 Methods and Materials
### Methods
#### 3.2.2 Getis Ord \(Gi^{\star}\)  
For RTC hot spot identification, spatial analysis using local spatial autocorrelation is preferable. Local _Moran's I_ is among the well-known local spatial autocorrelation approach used commonly in motor vehicle crash hot spot analysis [53]. However, the family of Moran indices does not differentiate between hot or cold spots. Getis Ord \(Gi^{\star}\) is, therefore, more appropriate since it distinguishes clusters with high and low feature attribute values among local events. In this study, Getis Ord \(Gi^{\star}\) statistics method was used to identify RTC hot spots. \(Gi^{\star}\) statistics aims to investigate the presence of a spatial pattern for an arbitrary variable \(X\), where a selected event (with a value of \(x_{i}\)) is autonomously connected to the field. As a result, if \(x_{i}\) is analogous to adjacent regions, it can be seen that variable \(X\) has spatial autocorrelation over area \(i\). A simple form of \(Gi^{\star}\) statistics is defined as Equation (2), which is derived by dividing the study zone into \(n\) indefinite extent of regions, each with accurate Cartesian coordinates and a central point \(i\) (\(i\) = 1, 2, 3,... \(n\)) [37]. \[\mathrm{G_{i}^{\star}}=\frac{\sum_{j=1}^{n}w_{ij}x_{i}}{\sum_{j=1}^{n}x_{j}} \tag{2}\]  
where \(Gi^{\star}\) is a statistic that describes the spatial dependency of feature \(i\), \(x_{j}\) is the value of variable \(X\) at feature location \(j\). \(w_{ij}\) is the spatial weight between feature \(i\) and \(j\). The conceptualized-spatial relationship around distance \(d\) (for instance Cartesian distance) is used to calculate \(w_{\bar{y}}\). The result of \(Gi^{\star}\) statistics may vary based on the choice of \(d\). The value of \(d\) is a user-defined threshold. The most straightforward way to think about \(w_{\bar{y}}\) is in binary form, with 1 indicating inclusion and 0 indicating exclusion of the association between \(i\) and \(j\) events. However, in practice, \(w_{\bar{y}}\) can have non-binary values and the total weights (\(W_{i}\)) are expressed in Equation (3). \[W_{i}=\sum_{j=1}^{n}w_{ij} \tag{3}\]Readers are referred to a paper on Getis Ord \(Gi^{\star}\) statistics [54] to learn more about the statistical elements such as \(Gi^{\star}\) expectation, \(Gi^{\star}\) variance and sample mean and variance for the variable \(X\). Finally, as shown in Equation (4), \(Gi^{\star}\) statistics are typically standardized using the sample mean and variance for a normal asymptotical condition. \[Z(G_{i}^{\star})=\frac{\sum_{j=1}^{n}w_{ij}x_{j}-\overline{x}\sum_{j=1}^{n}w_{ ij}^{2}}{s\sqrt{\frac{n\sum_{j=1}^{n}w_{ij}^{2}-\left(\sum_{j=1}^{n}w_{ij} \right)^{2}}{n-1}}} \tag{4}\]  
This standardized \(Gi^{\star}\) statistics is a Z-score, which is associated with the statistical significance of each target region. The null hypothesis for the Getis Ord \(Gi^{\star}\) hot spot analysis tool is that crashes are caused by Complete Spatial Randomness (CSR) or are randomly distributed. The null hypothesis would be rejected or accepted based on the Z-score and \(p\)-values obtained from the Getis Ord \(Gi^{\star}\) analysis tool. The \(p\)-value for the spatial pattern analysis is the probability that the observed crash incidents occurred in some arbitrary manner. When the returned \(p\)-value from the hot spot analysis tool is very small, it is very unlikely that the observed crash incident is to be distributed randomly; thus, the null hypothesis can be rejected. The tails of the standard normal distribution curve have very small \(p\)-values and very high absolute values of Z-scores. When the value of the Z-score is closer to zero it implies that there is a random distribution of spatial events in the region. The maximum absolute values of \(Gi^{\star}\) statistics, on the other hand, represent clusters of low-valued events for negative or high-valued events for positive.