# Differences Evaluation of Pavement Roughness Distribution Based on Light Detection and Ranging Data
## 4 Lateral Distribution Evaluation of the Pavement Roughness
### Difference Test for the Lateral Distribution of Pavement Roughness  
To effectively characterize the lateral roughness distribution in a large-scale road network, this study analyzed the variation in the IRI distribution among multiple survey lines and road sections using a dataset covering 26 km of road. Parametric and non-parametric tests were conducted to examine significant differences in the IRI distribution. To ensure data quality, the collected data were divided into 100 m intervals, and only straight pavement sections were included, while turning road sections were excluded. Generally, the roughness distribution of asphalt pavement exhibits a normal or skewed pattern. However, in cases with insufficient data samples, the roughness distribution may deviate from normality according to the law of large numbers. Skewed distributions can affect the validity of the parameter tests, leading to the rejection of null hypotheses. Therefore, the Box-Cox transformation was applied to adjust the data distribution to a normal form and to maintain a constant variance of the dependent variables. The \begin{table}
\begin{tabular}{c c} \hline
**Spacing \(I\) (m)** & **Variance (m\({}^{2}\))** \\ \hline
0.2 & 0.4364 \\
0.4 & 0.7278 \\
0.6 & 1.2621 \\
0.8 & 3.0781 \\
1.0 & 7.7535 \\
1.2 & 4.1404 \\ \hline \end{tabular}
\end{table} 
Table 1: The variance of IRI results under different survey line spacing. Figure 8: IRI results under different survey line spacings. transformation involves converting the original values using a specific function to bring the new data distribution closer to normality. Figure 10a demonstrates the skewed distribution of the pavement roughness prior to the Box-Cox transformation, while Figure 10b shows the normal distribution after the transformation. The formula is shown in Formula (14):  
\[IRI_{after}{}^{(\lambda)}=\frac{IRI_{before}{}^{\lambda}-1}{\lambda} \tag{14}\]  
where \(\lambda\) is related to the distribution of the IRI, which is determined by the maximum likelihood estimation method. Assuming that the transformed IRI follows a normal distribution, \(\lambda\) is the value of \(\lambda\) when the likelihood function value is at its maximum. Upon converting the roughness samples of different survey lines to conform to a normal distribution, variation analysis (ANOVA) can be utilized to investigate significant differences in the roughness distribution. It is important to note that despite multiple survey lines being obtained from the same road section, each profile constitutes an independent sample. The calculation of the IRI involves resampling a road profile, treating each survey line as an independent sample. Table 2 presents the \(p\)-value obtained from the ANOVA test for IRI distribution among different survey line spacings. These results revealed that only when the spacing was  
Figure 10: IRI distribution before and after the Box–Cox transformation. (**a**) Before the Box–Cox transformation. (**b**) After the Box–Cox transformation. Figure 9: Spatial distribution of the pavement roughness. 2.5 m, the \(p\)-value was below the significance level of 0.05, indicating a significant variation in roughness distribution. In other cases, no significant variation was observed. Furthermore, these results revealed a significant difference in the roughness distribution across different lanes, while no significant difference was found among the survey lines within the same lane. Notably, the ANOVA test can only shed light on the absence of significant differences in the overall roughness distribution. It cannot illustrate that the IRI results of different survey lines within one road section do not differ significantly. The ANOVA test presupposes that the sample conforms to a normal distribution. Nevertheless, in the case of small sample sizes, the roughness data may display a stronger skewed trend. Therefore, the Kruskal-Wallis test (KW test) was conducted to analyze the roughness difference among multiple survey lines with small samples.