## 3 Solved Example
### Assessing the Adequacy of Fit of Model
#### 3.3.2 SP Sub-Model  
The SP sub-model consisted of all significant SP main effects and interactions. Therefore, the design matrix X of the SP sub-model was  
\[\left[\begin{array}{rrr}-1&-1&1\\ -1&-1&1\\ -1&-1&-1\\ -1&-1&-1\\ -1&-1&-1\\ -1&-1&1\\  
The 1 st column was associated with D, the 2 nd with E, and the 3 rd with ABDE. The transpose matrix \(\mathsf{X}^{\prime}\) can now be calculated easily.  
In order to calculate the first diagonal element of the hat matrix h\({}_{11}\), the first row of the X matrix was used, since for this i = 1. Therefore, for i = 1:  
\[\mathrm{x_{1}}=\begin{bmatrix}-1&-1&1\end{bmatrix}\]  
The transpose matrix \(\mathsf{x}^{\prime}_{1}\) can then be calculated easily.  
The h\({}_{11}\) had to be calculated next through Equation (10) as  
\[\mathrm{h_{11}}=\mathrm{x_{1}}\left(\mathsf{X}^{\prime}\mathsf{X}\right)^{-1} \mathsf{x}^{\prime}_{1}=0.094\]  
The same procedure was followed for the rest of the rows. The results are presented in Table 7.  
Before calculating PRESSSP, the PRESSSP residuals had to be calculated. To illustrate their calculation, let us consider the 2 nd PRESSSP residual. This is the one for which i = 2, i.e., the one associated with the 2 nd row of the SP sub-design matrix. This residual was calculated as  
\[\mathrm{2 nd\ PRESSSP residual}=\frac{\mathrm{e_{2(SP)}}}{1-\mathrm{h_{22(SP)}}}= \frac{0.035}{1-0.094}=0.039\]  
The values of all PRESSSP residuals are given in Table 7. Now PRESSSP could easily be calculated by substituting the values of PRESSSP residuals into (12), and it was found  
\begin{table}
\begin{tabular}  
to be equal to 2.508. This relatively high value of PRESSSP indicated that the performance of the SP sub-model in predicting the responses in new and future experiments was relatively poor.  
The last measure of adequacy left to calculate for the SP sub-model was R2 SP Prediction. It could easily be calculated now through Equation (14):  
\[\mathrm{R}_{\mathrm{SP\_Pred}}^{2}=1-\frac{\mathrm{PRESS}_{\mathrm{SP}}}{ \mathrm{SS}_{\mathrm{Total}(\mathrm{SP})}}=1-\frac{2.508}{4.231}=0.407\]  
The low value of R2 SPPred confirmed the poor predictive performance of the SP sub-model.  
In total, the values of the calculated measures of adequacy of the WP and SP sub-models showed that the suggested WP sub-model explained a high proportion of the variability of the data and had a good predictive capability, while the opposite was true for the suggested SP sub-model. One of the reasons for the poor performance of the SP sub-model could be the inclusion of non-significant terms in it. However, it is worth mentioning that when the removal of these terms was tested, distortions in the residuals' normal probability plots were caused; therefore, it was decided to keep the proposed model unchanged. Another reason for the poor performance of the SP sub-model could be that a linear model is not suitable if some significant curvature exists. In that case, extra experiments should be performed and the design and analysis will have to change, in order to enable the inclusion of, e.g., quadratic terms in the model. Significant cell-to-cell differences can also be suspected. It is always likely as well that some mistakes happened during the performance of the experiments. Further investigation of the variables which could have caused the low predictability of the SP sub-model would be advisable, in case the model is planned to be used for predicting the leaching efficiency of Mo in the future.