## 2 Theory
### Analysis of Unreplicated Split-Plots with All the Factors at Two Levels Each  
For the unreplicated split-plots with all the factors at two levels each, the formulas used for calculating contrasts, (SS), effects, and parameters are exactly the same with the ones for the case of CRD (more details about these sizes in CRD can be found elsewhere, see [1]). Thus, for m number of design factors,  
\[\text{Contrast}_{\text{ABC...M}}=(\text{a}\pm 1)(\text{b}\pm 1)(\text{c}\pm 1) \ldots(\text{m}\pm 1) \tag{1}\]  
(the sign of 1 is negative for the factors included in the contrast and positive for the ones not included)  
\[\text{SS}_{\text{ABC...M}}=\frac{1}{2^{\text{k}}}(\text{Contrast}_{\text{ ABC...M}})^{2} \tag{2}\]  
\[\text{Effect}_{\text{ABC...M}}=\frac{2}{2^{\text{k}}}\text{Contrast}_{\text{ ABC...M}} \tag{3}\]  
\[\text{Parameter}_{\text{ABC...M}}=\frac{1}{2}\text{Effect}_{\text{ABC...M}} \tag{4}\]  
Then, normal (or half-normal) probability plots are constructed for the WP and SP effects separately and, through them, the significant main effects and interactions are detected (i.e., the ones which fall away from the straight line). The different levels in randomization of split-plots (one randomization between WP and another randomization for the SP within each WP) is the reason why two separate plots are needed for WP and SP [5]. It should be stressed here that all the main effects and interactions between the WP factors belong to the WP, so they should be plotted in the (half) normal probability plot of the WP. All the main effects and any interaction which includes SP factors belong to the SP; even the interactions between SP and WP factors. Therefore, these effects are plotted in the SP (half) normal probability plot [5]. For example, if factors A and B are WP factors and factors C and D are SP factors, the interaction AB belongs to the WP, while the interactions, e.g., AC, CD, and ABCD, are part of the SP.  
After the significant factors are detected, a model is proposed. Model adequacy checking through examination of the residuals should always follow [1]. In split-plot designs, there are two types of residuals: the WP and the SP residuals. Each of the WP residuals is the average of the ordinary residuals e (ordinary residual = response--fitted value from the proposed model) of the runs belonging to the particular WP [9]. Therefore, all the runs within a WP have the same WP residual. The SP residuals are calculated by subtracting the WP residual from the respective ordinary residual for each run [9].  
Investigation of the residuals usually includes checking (i) their normal probability plots for violations of the normality assumption and (ii) the plots of residuals vs. the run order and/or vs. the fitted values and/or vs. any other factor that may have affected the variance. In case of point (ii), the plots should look structureless and with a relatively constant variance per plot [1]. It is important to highlight here that for the case of plots of residuals vs. fitted values in split-plots, it is the SP residuals which should be plotted against the fitted values. The WP residuals should be plotted against the average fitted value per WP. Finally, in split-plot designs, it is also recommended to plot the WP residuals against the SP residuals, to test for their general level of randomness [9].