## 3 Solved Example
### Assessing the Adequacy of Fit of Model  
As a last step, it was decided to proceed with the assessment of the adequacy of the proposed model. In order to calculate the measures of adequacy of fit of the model, the design had to \"break\" into its two sub-designs (i.e., the WP sub-design and the SP sub-design). Their SS, as well as degrees of freedom of each source, had to be calculated. These calculations for split-plots are exactly the same for CRD. The results for each sub-design are presented in Table 5.  
The values of R\({}^{2}\)WP, R\({}^{2}\)SP, R\({}^{2}\)WP\(\_\)Adj, and R\({}^{2}\)SP\(\_\)Adj were calculated through (5)-(8), respectively, by substituting the values for SS and df from Table 5 as follows:  
\[\text{R}^{2}_{\text{WP}}=\frac{\text{SS}_{\text{Model(WP)}}}{\text{SS}_{\text {Total(WP)}}}=\frac{6.061}{6.135}=0.9879\]  
\[\text{R}^{2}_{\text{SP}}=\frac{\text{SS}_{\text{Model(SP)}}}{\text{SS}_{\text {Total(SP)}}}=\frac{1.948}{4.231}=0.4603\]  
\[\text{R}^{2}_{\text{WP\_Adj}}=1-\frac{\text{SS}_{\text{Residualus(WP)}}/ \text{df}_{\text{SS}_{\text{Residualus(WP)}}}}{\text{SS}_{\text{Total(WP)}}/ \text{df}_{\text{SS}_{\text{Total(WP)}}}}=1-\frac{0.074/4}{6.135/7}=0.9788\]  
\[\text{R}^{2}_{\text{SP\_Adj}}=1-\frac{\text{SS}_{\text{Residualus(SP)}}/ \text{df}_{\text{SS}_{\text{Residualus(SP)}}}}{\text{SS}_{\text{Total(SP)}}/ \text{df}_{\text{SS}_{\text{Total(SP)}}}}=1-\frac{2.284/21}{4.231/24}=0.3832\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{3}{c}{**WP Sub-Design**} & \multicolumn{3}{c}{**SP Sub-Design**} \\ \hline
**Source** & **df** & **SS** & **Source** & **df** & **SS** \\ \hline
**Model** & 3 & 6.061 & **Model** & 3 & 1.948 \\ \hline A & 1 & 0.737 & D & 1 & 0.397 \\ \hline B & 1 & 1.091 & E & 1 & 1.323 \\ \hline C & 1 & 4.233 & ABDE & 1 & 0.228 \\ \hline
**Residual** & & & & **Residual** & \\ (i.e., rest WP sources) & 4 & 0.074 &
\begin{tabular}{c} **(i.e., rest SP sources)** \\ \end{tabular} & 21 & 2.284 \\ \hline
**Total** & 7 & 6.135 & **Total** & 24 & 4.231 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Division of the design into two sub-designs for calculating the SS and degrees of freedom (df) of each source. The large values of \(\rm R^{2}_{WP}\) and \(\rm R^{2}_{WP\_Adj}\) indicate that a large proportion of the variability in the data is explained by the selected WP effects A, B, and C. On the other hand, the small values of \(\rm R^{2}_{SP}\) and \(\rm R^{2}_{SP\_Adj}\) indicate that only a small proportion of the variability in the data is explained by the selected SP factors D and E as well as the interaction ABDE. The assessment of the model adequacy continued with the calculation of \(\rm PRESS_{WP}\) and \(\rm PRESS_{SP}\) through Equations (11) and (12), respectively. For this purpose, the proposed model must \"break\" into its two sub-models, i.e., the WP sub-model and the SP sub-model. Then, the calculation of \(\rm h_{ii}\) for each of the sub-models should follow, using Equation (10). Attention must be paid here that for the case of the WP sub-model, the intercept must also be taken into account. Since this calculation step needs some caution, the detailed calculation  
As one can easily understand, the 1 st column is associated with the intercept, the 2 nd with A, the 3 rd with B, and the 4 th with C. The transpose matrix \(\chi^{\prime}\) 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 should be used, since for this i = 1. Therefore, for i = 1,  
\[\mathrm{x}_{1}= \begin{bmatrix}1&-1&-1&-1\end{bmatrix} \]  
The transpose matrix \(\mathrm{x}_{1}^{\prime}\) can then be calculated easily. After that, the h\({}_{11}\) was calculated from (10) as  
\[\mathrm{h}_{11}=\mathrm{x}_{1}\left(\chi^{\prime}\mathrm{X}\right)^{-1}\mathrm{ x}_{1}^{\prime}=0.125\]  
The same procedure was followed with the rest of the rows, e.g., the 2 nd row in order to calculate h\({}_{22}\), the 3 rd row in order to calculate h\({}_{33}\), etc.