## 2 Theory
### Measures for Checking the Adequacy of Fit of Model  
The computing of measures of adequacy of fit for split-plot models also has some differences compared to the CRD case. [PERSON], [PERSON], and [PERSON] [9] proposed the computation of two R\({}^{2}\), R\({}^{2}\)-Adjusted, Prediction Error Sums of Squares (PRESS), and R\({}^{2}\)-Prediction statistics; one of each for the WP and one of each for the SP sub-model. These measures of adequacy reveal whether the correct WP and SP effects have been included in the proposed model and describe the predictive performance of each group of effects. It becomes clear from the previous description that, in order to calculate the measures of adequacy in split-plots, the proposed design and model have to \"break\" into two sub-designs and two sub-models: the WP sub-design and sub-model and the SP sub-design and sub-model. One can easily understand that in the WP sub-design, only the WP main effects and interactions will be included, while in the SP sub-design, only the SP main effects and interactions will be included. Similar is the case for the sub-models. However, what is not always clear is that the grand average (or intercept) is also a part of the WP sub-model and must be taken into account for some of the calculations, as explained later. After creating the WP and SP sub-models, [PERSON], [PERSON], and [PERSON] [9] suggest to calculate the measures of adequacy of fit of the model as follows:  
\[\mathrm{R}^{2}_{\mathrm{WP}}=\frac{\mathrm{SS}_{\mathrm{Model}(\mathrm{WP})}} {\mathrm{SS}_{\mathrm{Total}(\mathrm{WP})}} \tag{5}\]  
\[\mathrm{R}^{2}_{\mathrm{SP}}=\frac{\mathrm{SS}_{\mathrm{Model}(\mathrm{SP})}} {\mathrm{SS}_{\mathrm{Total}(\mathrm{SP})}} \tag{6}\]  
\[\mathrm{R}^{2}_{\mathrm{WP\_Adj}}=1-\frac{\mathrm{SS}_{\mathrm{Residuals(WP)} }/\mathrm{df}_{\mathrm{SS}_{\mathrm{Residuals(WP)}}}}{\mathrm{SS}_{\mathrm{ Total}(\mathrm{WP})}/\mathrm{df}_{\mathrm{SS}_{\mathrm{Total}(\mathrm{WP})}}} \tag{7}\]  
\[\mathrm{R}^{2}_{\mathrm{SP\_Adj}}=1-\frac{\mathrm{SS}_{\mathrm{Residuals(SP)} }/\mathrm{df}_{\mathrm{SS}_{\mathrm{Residuals(SP)}}}}{\mathrm{SS}_{\mathrm{ Total}(\mathrm{SP})}/\mathrm{df}_{\mathrm{SS}_{\mathrm{Total}(\mathrm{SP})}}} \tag{8}\]  
where df is the degrees of freedom of the respective source. According to [PERSON] [1], the R\({}^{2}\) statistics is loosely interpreted as the proportion of the variability in the data \"explained\" by the respective suggested model, while the R\({}^{2}\)\({}_{\mathrm{Adj}}\) statistics is a variation in the R\({}^{2}\), in which the number of factors in the model is reflected. The latter statistics becomes particularly useful as the number of design factors increases and the practitioner wishes to evaluate the impact of increasing or decreasing the number of terms included in the model. High values of R\({}^{2}\) and R\({}^{2}\)\({}_{\mathrm{Adj}}\) are desirable. Regarding PRESS, an easy and fast way to calculate it for factorial designs in general is  
\[\mathrm{PRESS}=\sum_{i=1}^{n}\biggl{(}\frac{\mathrm{e}_{i}}{1-\mathrm{h}_{ii}} \biggr{)}^{2} \tag{9}\]where the term \(\frac{\mathrm{e_{i}}}{1-\mathrm{h_{ii}}}\) is equal to the ith PRESS residual \(\mathrm{e_{(i)}}\), as explained by [PERSON] [1]. Moreover, \(\mathrm{h_{ii}}\) are the diagonal elements of the hat matrix H of the model and they can be calculated through the formula  
\[\mathrm{h_{ii}}=\mathrm{x_{i}}\big{(}\mathrm{X^{\prime}}\mathrm{X}\big{)}^{-1} \mathrm{x_{i}^{\prime}} \tag{10}\]  
As previously mentioned, two PRESS statistics should also be computed for split-plots, one for WP and one for SP, using the respective sub-models [9]. Therefore,  
\[\mathrm{PRESS_{WP}}=\sum_{i=1}^{n}\Bigg{(}\frac{\mathrm{e_{i(WP)}}}{1-\mathrm{ h_{ii(WP)}}}\Bigg{)}^{2} \tag{11}\]  
\[\mathrm{PRESS_{SP}}=\sum_{i=1}^{n}\Bigg{(}\frac{\mathrm{e_{i(SP)}}}{1-\mathrm{ h_{ii(SP)}}}\Bigg{)}^{2} \tag{12}\]  
where \(\mathrm{e_{i(WP)}}\) and \(\mathrm{e_{i(SP)}}\) are the WP residuals and SP residuals, respectively.