## 4 Panel Data Analysis-Based Surface Texture Prediction Model
### Influence of Panel Data Time Scale on Surface Texture Prediction Model  
The above analysis has led to the random-effects model as the most suitable form of the surface texture model with respect to the indicator of sensor measured texture depth. In the following, the surface texture prediction models using one year, two years, three years, and four years sensor measured texture depth data are developed to clarify the influence of different time scales of sensor measured texture depth data on the regression result, the short-term prediction accuracy, and the long-term prediction robustness of the developed model. Table 8 shows the prediction formulas of random-effects model based on different time scales of sensor measured texture depth data. It is seen from Table 8 that when different time scales of SMTD data are used to develop the prediction model, the sign of the coefficient in front of each independent variable does not change except for monthly rainfall. The reason may be the high variation of the monthly rainfall measured in the same month of different years. The coefficient in front of each independent variable varies slightly from the one year panel data-based prediction model to the two years panel data-based model, and similarly, from the three years panel data-based prediction model to the four years panel data-based model. Overall, the coefficients in front of these independent variables change very little when different time scales of SMTD data are used to develop the prediction model. It reveals that the SMTD data time scale has minor effect on the significance level of these independent variables. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Test** & **Statistic** & **df** & **\(p\) Value** & **Result** \\ \hline F test & 18.508 & 18,658 & 0.000 & Rejected \\ \hline LM test & 2782.975 & 171 & 0.000 & Rejected \\ \hline Hausman test & 0.000 & 7 & 1.000 & Accepted \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Examination results of panel data analysis-based surface texture model. A comparison of the variable significance level of random-effects panel data analysis model using different time scales of SMTD data is shown in Table 9. It is seen from Table 9 that except for the two variables of monthly average temperature (MAT) and monthly rainfall (MR), the obtained \(p\) values of other variables are less than 0.1, revealing that the significance levels of CSAL, MAH, CA, FAC, and FAF are basically the same and they have significant effects on asphalt pavement surface texture. With the extension of the time scale of SMTD data, the \(p\) value of MAT increases, indicating that the significance level of MAT gradually decreases; the \(p\) value of MR decreases, indicating that the significance level level of MR gradually increases. Based on 15 years skid number panel data of 255 sections, Ref. [29] developed a fixed-effects model of skid number (SN) considering the annual average precipitation (AAP), the annual average freezing index (AAFI), the annual average humidity (AAH), the annual average daily truck traffic (AADTT), and the service age (AGE). It was concluded that the influences of these five factors on skid number were significant. In our study, the influences of the cumulative standard axle loads (CSAL) and the monthly average humidity (MAH) on surface texture are significant, while the influence of monthly rainfall (MR) on surface texture varies with the time scale of the input panel data. However, monthly data are collected and analyzed in this study, whereas in Ref [29] annual data were analyzed. The influencing factor of freezing index is not considered in this study. Previous studies [27; 28] have developed the panel data analysis-based model for the assessment of skid number (SN) using the variables such as annual average temperature and annual rainfall. In this study, the monthly SMTD data are used to generate the panel dataset. The advantage is that a surface texture prediction model with a shorter testing cycle can be provided.