## 3 Results
### Linear Relationships in Sewage Sludge Samples  
The multiple linear equations obtained for the target elements in the sewage sludge samples are:  
\[[\text{As(pred)}]=2.297+0.0427[\text{Pb(s)}], \tag{10}\]  
\[[\text{As(pred)}]=-5.06+0.970[\text{As(s)}], \tag{11}\]  
\[[\text{Cd(pred)}]=-0.089+0.00146[\text{Zn(s)}]+0.00321[\text{Pb(s)}], \tag{12}\]  
\[[\text{Cr(pred)}]=0.747[\text{Cr(s)}], \tag{13}\]  
\[[\text{Cr(pred)}]=-54.2+0.0299[\text{Cl(m)}]+0.0345[\text{Ti(s)}], \tag{14}\]  
\[[\text{Hg(pred)}]=804+7.85[\text{Pb(s)}]-46.1[\text{K(s)}], \tag{15}\]  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{6}{c}{_p-Values_} \\ \cline{2-7}
**Equation** & _p_**(B)** & _p_**(C)** & _p_**(D)** & **r** & **Mean 1** & _N(n) 3_ ** \\ \hline (1) & 0.000 & 0.006 & & 0.967 & 4.70 & 14(1) \\ (2) & 0.000 & 0.030 & 0.029 & 0.980 & 4.99 & 13(13) \\ (3) & 0.000 & 0.014 & & 0.989 & 0.503 & 14(1) \\ (4) & 0.000 & 0.009 & 0.031 &\[[\text{Hg}(\text{pred})]=235+6.75[\text{Pb}(\text{s})]-44.0[\text{K}(\text{s})]+45.3[ \text{Al}(\text{m})], \tag{16}\]  
\[[\text{Ni}(\text{pred})]=-34.7+0.406[\text{Ti}(\text{s})]-0.468~{}[\text{Fe}( \text{m})], \tag{17}\]  
\[[\text{Ni}(\text{pred})]=-26.9+4.81[\text{Al}(\text{m})], \tag{18}\]  
\[[\text{Se}(\text{pred})]=658+52.2[\text{P}(\text{m})]-5.79~{}[\text{Fe}(\text {m})], \tag{19}\]  
\[[\text{Se}(\text{pred})]=567+55.9[\text{P}(\text{m})]-1.91[\text{Ba}(\text{s} )], \tag{20}\]  
Their statistics are shown in Table 4. The correlation coefficients of the aqua-regia-soluble elemental concentrations, X(ae), and the pXRF readings are shown as supplementary material (Table S2). The Pb(s) readings showed a high correlation with As(ae) and Hg(ae), and thus it was used as the first factor for the calculation of the linear equations of these elements (Equation (10), for As and Equations (15) and (16) for Hg). Successive improvements in the Hg equations were obtained when including the K(s) and Al(m) readings. Zn(s) showed the highest correlation coefficient with Cd(ae) and it was used for modeling Cd (Equation (12)). An additional Pb(s) term was also used in the Equation (12). The readings Cl(m) correlated with Cr(ae) in all 11 sewage sludge samples (Table S2) and was used in Equation (14) with a second significant Ti(s) coefficient. Ti(s) was also correlated with Ni(ae) (Equation (17)) but the Al(m) coefficient was better in nine sewage sludge samples and Equation (18) was also obtained for Ni. The Fe(m) readings further improved Equation (17). The P(m) readings showed a correlation with Se(ae) and Equations (19) and (20) used P(m). The second coefficients in these equations were Fe(m) and Ba(s). In a certain number of samples, significant and high correlations were observed for the As(s) and Cr(s) readings with the corresponding values of As(ae) and Cr(ae) such that direct relationships were fitted between these parameters (Equations (11) and (13)) when the corresponding pXRF readings were above the LOD. All the element coefficients included in the fitting equations were statistically significant (\(p\)-values \(<0.05\)). In general, the equations for sludge cannot be considered quantitatively acceptable for calibration (\(r<0.98\)) [28]. The studentized deleted residuals for several sewage sludge equations are shown in Figure 2. No outliers were observed in these plots except one Cr (no. 15) and one Ni data. Furthermore, the absence of a pattern in the distribution of residuals in the corresponding plots in Figure 2 indicates that there was likely no significant bias affecting the equations.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & \multicolumn{4}{c}{\(p\)**-Values**} \\ \cline{2-6}
**Equation** & \(p\)**(B)** & \(p\)**(C)** & \(p\)**(D)** & \(r\)** & **Mean 1** & **N(\(n\)) 2** \\ \hline (10) & 0.000 & & & 0.927 & 7.85 & 11(5) \\ (11) & 0.001 & & & 0.969 & 9.31 & 6(6) \\ (12) &  
Figure 2: Studentized deleted residuals for the indicated regression equations for compost (blue dots) and 4 for sewage sludge (red dots).