## 1 Introduction
Based on the same observation, [PERSON] and [PERSON] [7] discussed the use of split-plots in the design of robust products and processes, with the environmental factors placed in WP (since in this case there is no interest in checking their significance). In all of the aforementioned literature, the analysis of the data consisted of the direct presentation of the ANOVA table and/or the normal probability plots of the effects, from which the significant main effects and interactions of the factors were determined. In other words, in most cases, no clarifications on how to calculate contrasts, effects, sum of squares (SS), and parameters for the case of split-plot designs had preceded the presentation of the ANOVA table results. It is worth noticing that this was also the case for introductory textbooks in design and analysis of experiments [1; 2], with the only small exception being a comment in the book by [PERSON], [PERSON], and [PERSON] [2] that the calculation of SS for (balanced) split-plots is the same for any other balanced design. Another (partial) exception was [PERSON] [8], who presented in his paper the formulas for the calculation of SS in replicated split-plots. An apparently unexpected observation related to the studied literature was that there were no examples or discussions on the analysis of unreplicated full factorial designs. The only existing comment on the matter was a comment focused on the fact that in this case either high-order interactions should be assumed to be insignificant and be used as the error for the ANOVA table, or probability plots should be used instead [6]. However, since this type of design looks like the fastest, cheapest, and simplest case of all split-plots, it is very likely to be picked by a non-expert practitioner. One would therefore expect to find instructions on how to work with such a design at the very beginning of textbooks and papers dealing with the proper analysis of split-plots. Instead, the presented examples in the literature usually start with the case of replicated split-plots [1; 2; 3; 4; 6; 8] and may continue with fractional split-plots or other cases [1; 6; 8]. It was only [PERSON] and [PERSON] [3] who pointed out that in unreplicated split-plot designs with only one hard-to-change factor, it is impossible to draw any conclusions on the significance of this particular factor. It is worth mentioning here that [PERSON], [PERSON], and [PERSON] [5] had already published in 1996 a paper on how to analyze unreplicated split-plots with more than one WP factor; however, only the final calculated values were presented. The most reasonable explanation about the existence of all these apparent \"gaps\" in the literature seems to be that, since all the authors were experts in the field, many things seemed self-explanatory to them. A common trait of all the studied literature was that any analysis stopped after finding the significant main effects and interactions, and did not continue with suggesting a model and plotting the residuals. However, if an experimenter wishes to continue with suggesting a model, checking the residuals is a procedure prescribed by renowned books in the field as an integral part of the analysis of factorial experiments [1]. In 2009, [PERSON], [PERSON], and [PERSON] [9] closed this gap when they published their work on measures for checking the adequacy of the proposed model for split-plot experiments, namely R\({}^{2}\), R\({}^{2}\), ARJ, PRediction Error Sums of Squares (PRESS), and R\({}^{2}\)Pred. More specifically, in their work, they proposed two different types of residuals (due to the two types of errors present in split-plots) and briefly explained how to calculate them. They continued with the main focus of their paper, suggesting again two types for each of the measures of adequacy of fit, and then continued by providing solved calculation examples. Identification of the significant factors and interactions affecting a process is usually the first step required before optimization takes place.