## 3 Materials and Methods
### Statistical Analysis  
Results from previous studies indicated that the best-fitting relationships between retroreflectivity and age were generally linear and that the linear models were significant despite their low R\({}^{2}\) values [1; 6; 12; 13; 15; 16; 17]. Therefore, linear regression models were conducted in this study. However, possible nonlinearity in the regression model was checked using residual plots, with their remedies attempted through Box-Cox transformations [18] of the response factor.  
The Box-Cox transformation was used to transform the dataset into a more normally distributed one. This transformation procedure is used to modify the distributional shape of the response variable so that the residuals are more normally distributed, which is important for ensuring a stable estimate and valid inference [19; 20], especially with a small sample size. The Box-Cox transformation is a family of power transformation and has the following mathematical form:  
\[Y_{i}^{(\lambda)}=\left\{\begin{array}{l}\frac{Y_{i}^{(\lambda)}-1}{\lambda }\ (\lambda\
eq 0)\\ \log(Y_{i})\ (\lambda=0)\end{array}\right. \tag{1}\]  
where \(Y\) represents the data and \(\lambda\) is the \"power\" to which each data value is raised.  
The Box-Cox plot is a plot of the correlation from the data transformation for given values of \(\lambda\). The Box-Cox plot, shown in Figure 4, is considered as a guide to choosing a value of the parameter of power transformation (\(\lambda\)) that is as close as possible to the one that maximizes the log-likelihood function and is easy to interpret [18]. Figure 4 indicated the optimal value of \(\lambda\) was close to zero for the two datasets, therefore logarithmic regression models were used to transform the data according to Equation (1). The plot of the profile likelihood in Figure 4 is not very flat, indicating that the Box-Cox transformation is likely to be successful with the data [21].  
Linear and logarithmic regression models with multiple predictors were applied to analyze the retroreflectivity, as shown in Figure 5. Firstly, the models were used to find the relationship between retroreflectivity and all the factors in the datasets. A backward stepwise selection [18] was then used to find the best subset selection for the significant factors, as described in Section 4.1. Secondly, logarithmic regression models were used to calculate the performance and deterioration of retroreflective road traffic signs mounted on the roads, as shown in Sections 4.2 and 4.3.  
Figure 4: Profile–likelihood plot for the Box–Cox transformation of the VTI data (**left**) and NMF data (**right**).  
The analysis performed in this work is two-fold. Firstly, all the factors in the datasets (X, Y, age, color, class, GPS position, direction, and \(\beta\)) were included in the analyses and models (Sections 4.1 and 4.2). Secondly, only the factors that can be collected without the need for expensive measuring instruments were used in the models to calculate the retroreflectivity (Section 4.3). The predictors included the age of the road traffic sign, its GPS position, and the direction of the sign, as well as its color and the class of its retroreflective sheeting.  
Linear and logarithmic regression were used to determine equations that produce the minimum distance between the fitted values and all data points, with R\({}^{2}\) used to measure, statistically, how close the data are to the fitted values. R\({}^{2}\) was selected to assess the goodness of fit (model performance) because it had been used in previous studies investigating retroreflectivity deterioration, thereby leaving scope for comparability between studies.  
Because the predictive performance of the model is not the prime concern of this work, we did not conduct any cross-validation or other type of out-of-sample performance assessment.