## 3 Solved Example
### Design
Due to the small number of samples we had, we decided to proceed with an unreplicated design. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{**Factor**} & \multicolumn{4}{c}{**Level**} \\ \hline  & & \multicolumn{2}{c}{**Min**} & \multicolumn{2}{c}{**Max**} \\ \cline{3-6}
**Name (Units)** & **Symbol** & **Physical** & **Coded** & **Physical** & **Coded** \\  & & **Value** & **Value** & **Value** & **Value** \\ \hline Temperature (\({}^{\circ}\)C) & A & 30 & \(-1\) & 50 & \(1\) \\ Presence of salt & B & L1 & \(-1\) & L2 & \(1\) \\ pH & C & 10 & \(-1\) & 12 & \(1\) \\ A/L (cm\({}^{2}\)/mL) & D & \(1/6\) & \(-1\) & \(1/3\) & \(1\) \\ Time (h) & E & 8 & \(-1\) & 24 & \(1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Factors investigated and their levels. Then we had to decide on the design, and more specifically, assess if a CRD would make sense for our case. Since we had decided to investigate five factors, at two levels each, without replication, we would need \(2^{5}=32\) runs in total. At the same time, work was only allowed during office hours. That meant that if we wanted to run the experiment as a CRD, we would need at least 2 months to complete the experimental part, running only one sample per day, due to the relatively long leaching durations (8-24 h) and the fact that there was only one water bath available. That was already an indication that this experimental structure would be very time-consuming and not efficient at all. The water bath was large enough to accommodate all 16 samples which had to be treated under the same temperature. Therefore, temperature was decided to be a hard-to-change factor, or in other words, a WP factor. At this point, if factor A was the only factor kept as a WP, we would end up facing two problems:  
1. Not being able to evaluate the significance of factor A, since we would have set the temperature at its two different levels only once per level. 2. From a practical point of view, running 16 samples simultaneously would mean that all the different solutions would have to be prepared on the same morning and then the experiments would have to be performed on the same day (so that the solutions do not change with time). Preparation of the solutions was a time-consuming process though. For this reason, it was decided that factors B and C would also be hard-to-change factors. On the contrary, A/L and leaching time were easy-to-change factors; therefore, these factors were set as SP factors. To sum up, we ended up with three WP factors (A, B, and C) and two SP factors (D and E). Therefore, in our design, we had eight WP in total, as well as four SP within each of the WP, as illustrated in Figure 1.  
The order in which the WP was performed was determined randomly. All the SP of a particular WP were run simultaneously in the water bath and the position of each container was randomly selected. The design matrix is given in Table 2, along with the responses for each run. The response variable is the leaching yield (mg/cell) of Mo under the particular experimental conditions. The log transformation of the response was found to give better results; therefore, it was decided to continue the analysis with the log values instead of the original ones. More information on transformations can be found elsewhere (see [PERSON] [1]). Figure 1: Graphical representation of the unreplicated split-plot design with 3 WP factors and 2 SP factors, all at 2 levels each.