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.

The linear statistical model for this two factor design is:

\(y_{ijk}=\mu+\tau_i+\beta_j+(\tau\beta)_{ij}+\gamma_k+(\tau\gamma)_{ik}+(\beta\gamma)_{jk}+\varepsilon_{ijk}

\left\{\begin{array}{c}

i=1,2,\ldots,r \\

j=1,2,\ldots,a \\

k=1,2,\ldots,b

\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.