# Estimating Tetrachloroethene Sorption Coefficients Based on Soil Properties in Organic-Poor Soils
## 3 Results and Discussion
### Univariate and Multivariate Linear Regression  
Firstly, the relationship between \(K_{\text{d}}\) and individual soil properties was assessed using univariate linear regression, with prediction models summarized in Table 6. Each model was based on 34 observations, except for SSA, which had 21. Significant soil properties such as SSA, OC, clay, silt and sand content were further investigated. The correlation between \(K_{\text{d}}\) and SSA showed the highest [PERSON]'s correlation (0.679, with linear regression indicating statistically significant relationships between \(K_{\text{d}}\) and both SSA and OC. SSA accounted for 51.1% of the variability in \(K_{\text{d}}\), while OC contributed 37.8% from Table 6. These results emphasize OC as a crucial factor in PCE sorption, even in soils with OC levels below 1%. SSA is strongly correlated with OC and inversely related to particle size. Despite achieving the best fit through univariate regression, the models exhibited low precision. Although clay and silt content are generally considered important for organic substance sorption, the regression analysis revealed that silt was not statistically significant, displaying the lowest coefficient of determination alongside the highest RMSE and MAPE values. The second approach utilized multivariate analysis based on the AHC, Kruskal-Wallis, and LDA results. The models were developed by combining significant soil parameters: \(f_{\text{OC}},f_{\text{Clay}},f_{\text{Silt}},f_{\text{Sand}}\), and SSA. Multiple regression consistently achieved higher \(R^{2}\) than univariate linear regression. As with the PCA analysis, multivariate models were applied to two separate datasets. Table 7 summarizes the models for each soil class from Figure 3B (sandy (1), clay particles (2), and silt particles (3)) without including SSA, while models incorporating SSA are presented in Table S5. Multicollinearity, especially among \(f_{\text{Sand}}\)  
Figure 4: Correlation of loadings and score plot with confidence ellipses: (**A**) the loading plot represents the correlation between the soil properties in the classes and the discriminant functions, (**B**) the score plot displays soil samples, and their position indicates their score on the discriminant functions. The confidence ellipses (95%) visualize the separation between groups and the variability within each group. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & **OC (\%)** & **Clay (\%)** & **Silt (\%)** & **Sand (\%)** & & & \(K_{\text{d}}\) (L kg\({}^{-1}\)) & & \\ \hline
**Class** & **Mean** & **Mean** & **Mean** & **Mean** & **Mean** & **Std. Dev.** & **Variance** & **Min** & **Median** & **Max** \\ \hline
**1** & 0.17 & 14.22 & 22.87 & 62.92 & 0.69 & 0.21 & 0.05 & 0.38 & 0.65 & 1.15 \\
**2** & 0.34 & 41.78 & 39.99 & 18.21 & 1.10 & 0.53 & 0.28 & 0.40 & 0.95 & 2.27 \\
**3** & 0.41 & 10.61 & 61.81 & 27.58 & 1.01 & 0.36 & 0.13 & 0.66 & 0.94 & 1.53 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Mean values of significant soil properties and descriptive statistics of \(K_{\text{d}}\) by class. \(f_{\rm Silt}\), and \(f_{\rm Clay}\), was detected using tolerance (<3.0 \(\times\) 10\({}^{-6}\)) and variance inflation factor (VIF) metrics (>3.4 \(\times\) 10\({}^{5}\)). As a result, silt was excluded from the multivariate models to mitigate multicollinearity. The multivariate models for each class using \(f_{\rm OC}\). \(f_{\rm Clay}\) and \(f_{\rm Sand}\) provided better fits than single-variable regression. These models explained 63.6%, 69.8% and 98.5% of the variance in \(K_{\rm d}\) for Classes 1, 2, and 3, respectively. The highest \(R^{2}\) of 98.5% represents excellent fit but should be interpreted cautiously due to the small sample size in that class. Predictions deviated on average by 0.147, 0.329, and 0.07 units from measured \(K_{\rm d}\) values, demonstrating high accuracy with low RMSE and MAPE values. Figure 5 compares predicted and measured \(K_{\rm d}\), providing insights into model errors for each soil class.