## 3 Solved Example
### Analysis
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Contrast** & **SS** & **Effect** & **I Effect\(\dagger\)** & **Parameter** & **Plot** & **Significance** \\ \hline A & 4.855 & 0.737 & 0.303 & 0.303 & 0.152 & WP & Yes \\ B & 5.909 & 1.091 & 0.369 & 0.369 & 0.185 & WP & Yes \\ C & 11.639 & 4.233 & 0.727 & 0.727 & 0.364 & WP & Yes \\ D & \(-\)3.563 & 0.397 & \(-\)0.223 & 0.223 & \(-\)0.111 & SP & Yes \\ E & 6.507 & 1.323 & 0.407 & 0.407 & 0.203 & SP & Yes \\ AB & 0.823 & 0.021 & 0.051 & 0.051 & 0.026 & WP & \\ AC & 0.209 & 0.001 & 0.013 & 0.013 & 0.007 & WP & \\ BC & \(-\)1.277 & 0.051 & \(-\)0.080 & 0.080 & \(-\)0.040 & WP & \\ AD & \(-\)0.717 & 0.016 & \(-\)0.045 & 0.045 & \(-\)0.022 & SP & \\ BD & 1.153 & 0.042 & 0.072 & 0.072 & 0.036 & SP & \\ CD & 0.303 & 0.003 & 0.019 & 0.019 & 0.009 & SP & \\ AE & \(-\)0.091 & 0.000 & \(-\)0.006 & 0.006 & \(-\)0.003 & SP & \\ BE & 1.327 & 0.055 & 0.083 & 0.083 & 0.041 & SP & \\ CE & \(-\)1.963 & 0.120 & \(-\)0.123 & 0.123 & \(-\)0.061 & SP & \\ DE & \(-\)0.721 & 0.016 & \(-\)0.045 & 0.045 & \(-\)0.023 & SP & \\ ABC & \(-\)0.155 & 0.001 & \(-\)0.010 & 0.010 & \(-\)0.005 & WP & \\ ABD & \(-\)1.817 & 0.103 & \(-\)0.114 & 0.114 & \(-\)0.057 & SP & \\ ACD & \(-\)0.987 & 0.030 & \(-\)0.062 & 0.062 & \(-\)0.031 & SP & \\ BCD & 2.615 & 0.214 & 0.163 & 0.163 & 0.082 & SP & \\ ABE & 0.529 & 0.009 & 0.033 & 0.033 & 0.017 & SP & \\ ACE & 0.447 & 0.006 & 0.028 & 0.028 & 0.014 & SP & \\ BCE & 3.261 & 0.332 & 0.204 & 0.204 & 0.102 & SP & \\ ADE & 0.709 & 0.016 & 0.044 & 0.044 & 0.022 & SP & \\ BDE & \(-\)1.457 & 0.066 & \(-\)0.091 & 0.091 & \(-\)0.046 & SP & \\ CDE & \(-\)0.015 & 0.000 & \(-\)0.001 & 0.001 & 0.000 & SP & \\ ABCD & 3.165 & 0.313 & 0.198 & 0.198 & 0.099 & SP & \\ ABCE & \(-\)2.009 & 0.126 & \(-\)0.126 & 0.126 & \(-\)0.063 & SP & \\ ABDE & \(-\)2.699 & 0.228 & \(-\)0.169 & 0.169 & \(-\)0.084 & SP & Yes \\ ACDE & \(-\)0.857 & 0.023 & \(-\)0.054 & 0.054 & \(-\)0.027 & SP & \\ BCDE & 2.901 & 0.263 & 0.181 & 0.181 & 0.091 & SP & \\ ABCDE & 3.107 & 0.302 & 0.194 & 0.194 & 0.097 & SP & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Contrasts, SS, effects, parameters, and significant factors for Mo leaching. that the run named (1) belonged to the 2 nd WP. Therefore, its WP residual was the average of all ordinary residuals e of the runs belonging to the 2 nd WP (i.e., runs (1), d, e, and de):  
\[\text{WP resid}_{(1)}=\text{WP resid}_{\text{d}}=\text{WP resid}_{ \text{e}}=\text{WP resid}_{\text{de}}\] \[=\frac{\epsilon_{(1)}+\epsilon_{\text{d}}+\epsilon_{\text{e}}+ \epsilon_{\text{d}}}{4}=\frac{-0.124-0.316+0.377+0.056}{4}=-0.002\]  
The calculation of the SP residuals followed, in the way indicated in the Theory section: each SP residual was equal to the ordinary residual of the run minus the WP residual of the same run. For instance, for the run named (1), the SP residual was calculated as follows:  
\[\text{SP resid}_{(1)}=\text{e}_{(1)}-\text{WP resid}_{(1)}=-0.124-(-0.002)=-0.122\]  
Next, the construction of the normal probability plots for the WP and SP residuals followed, as shown in Figures 3a and 3b, respectively. Since all the residuals fell approximately on the respective straight line, no serious violations of the normality assumption were observed. Checking the residual plots vs. fitted values, run order and any other factor which could have affected the variance must follow. Therefore, the two residuals, WP and SP, were plotted first against the run order, as shown in Figures 4a and 4b, respectively. Both  
Figure 2: Normal probability plots for the (**a**) WP effects and (**b**) SP effects. The straight lines were drawn manually for the WP effects and automatically by MATLAB for the SP effects. residual plots looked structureless with the residuals distributed around zero and no big differences were observed in the variances; therefore, the plots were considered satisfactory.