14.5 - The Strip-Plot Designs

These designs are also called Split-Block Designs. In the case where there are only two factors, Factor A is applied to whole plots like the usual split-plot designs but factor B is also applied to strips which are actually a new set of whole plots orthogonal to the original plots used for factor A. Figure 14.11from the 7th edition of the text is an example of strip-plot design where both of the factors have three levels.

A 1 B 1 A 2 B 1 A 3 B 1 A 3 B 3 A 1 B 3 A 2 B 3 A 3 B 2 A 1 B 2 A 2 B 2 B 1 B 3 B 2 A 3 A 1 A 1 Whole Plots Figure 14.11 (Design and Analysis of Experiments, Douglas C. Montgomery, 7th Edition)

The linear statistical model for this two factor design is:

i=1,2,\ldots,r \\
j=1,2,\ldots,a \\
\end{array}\right. \)

Where, \((\tau \beta)_{ij}\), \((\tau \gamma)_{ik}\) and \(\epsilon_{ijk}\) are the errors used to test Factor A, Factor B and interaction AB, respectively. Furthermore, Table 14.26 shows the analysis of variance assuming A and B to be fixed and blocks or replicates to be random.

Source of Variation Sum of Squares Degree of Freedom Expected Mean Square
Replicates (or Blocks) \(SS_{\text{Replicates}}\) r - 1 \(\sigma_{\epsilon}^2 + ab\sigma_{\tau}^2\)
A \(SS_A\) a - 1 \(\sigma_{\epsilon}^2 + b \sigma_{\gamma \beta}^2 + \dfrac{rb \sum \beta_{j}^2}{a - 1}\)
\(\text{Whole Plot Error}_A\) \(SS_{WP_{A}}\) (r - 1)(a - 1) \(\sigma_{\epsilon} + b \sigma_{\tau \beta}^2\)
B \(SS_B\) b - 1 \(\sigma_{\epsilon}^2 + a \sigma_{\tau \gamma}^2 + \dfrac{ra \sum \gamma_{j}^2}{b - 1}\)
\(\text{Whole Plot Error}_B\) \(SS_{WP_{B}}\) (r - 1)(b - 1) \(\sigma_{\epsilon}^2 + a\sigma_{\tau \gamma}^2\)
AB \(SS_{AB}\) (a - 1)(b - 1) \(\sigma_{\epsilon}^2 + \dfrac{r \sum \sum (\tau \beta)_{jk}^2}{(a - 1)(b - 1)}\)
Subplot Error \(SS_{SP}\) (r - 1)(a - 1)(b - 1) \(\sigma_{\epsilon}^2\)
Total \(SS_T\) rab - 1  
Table 14.26 (Design and Analysis of Experiments, Douglas C. Montgomery, 8th Edition)

It is important to note that the split-block design has three sizes of experimental units where the units for effects of factor A and B are equal to whole plot of each factor and the experimental unit for interaction AB is a subplot which is the intersection of the two whole plots. This results into three different experimental errors which we discussed earlier.