**NOTE!**It is worth mentioning the fact that the notation used in this section (especially the use of Greek rather than Latin letters for the random error terms in the linear models) is not our preference. But, we decided to keep the text book’s notation to avoid any possible confusion.

There exist some situations in multifactor factorial experiments where the experimenter may not be able to randomize the runs completely. Three good examples of split-plot designs can be found in the article: "How to Recognize a Split-Plot Experiment" by Scott M. Kowalski and Kevin J. Potcner, *Quality Progress*, November 2003.

Another good example of such a case is in the textbook in Section 14.4. The example is about a paper manufacturer who wants to analyze the effect of three pulp preparation methods and four cooking temperatures on the tensile strength of the paper. The experimenter wants to perform three replicates of this experiment on three different days each consisting of 12 runs (3 × 4). The important issue here is the fact that making the pulp by any of the methods is cumbersome. This method is a “hard to change” factor. It would be economical to randomly select any of the preparation methods, make the blend and divide it into four samples and cook each of them with one of the four cooking temperatures. Then the second method is used to prepare the pulp and so on. As we can see, in order to achieve this economy in the process, there is a restriction on the randomization of the experimental runs.

In this example, each replicate or block is divided into three parts called **whole plots** (Each preparation method is assigned to a whole plot). Next, each whole plot is divided into four samples which are **split-plots** and one temperature level is assigned to each of these split-plots. It is important to note that since the whole-plot __treatment__ in the split-plot design is confounded with __whole plots__ and the split-plot treatment is not confounded, if possible, it is better to assign the factor we are most interested in to split plots.

##
Analysis of Split-Plot designs
Section* *

In the statistical analysis of split-plot designs, we must take into account the presence of two different sizes of experimental units used to test the effect of whole plot treatment and split-plot treatment. Factor *A* effects are estimated using the whole plots and factor *B* and the *A*B* interaction effects are estimated using the split plots. Since the size of the whole plot and split plots are different, they have different precisions. Generally, there exist two main approaches to analyze the split-plot designs and their derivatives.

- First approach uses the Expected Mean Squares of the terms in the model to build the test statistics and is the one discussed by the book. The major disadvantage to this approach is the fact that it does not consider the randomization restrictions which may exist in any experiment.
- Second approach which might be of more interest to statisticians and the one which considers any restriction in randomization of the runs is considered as the traditional approach to the analysis of split-plot designs.

Both of the approaches will be discussed but there will be more emphasis on the second approach, as it is more widely accepted for analysis of split-plot designs. It should be noted that the results from the two approaches may not be much different.

The linear statistical model given in the text for the split-plot design is:

\(y_{ijk}=\mu+\tau_i+\beta_j+(\tau\beta)_{ij}+\gamma_k+(\tau\gamma)_{ik}+(\beta\gamma)_{jk}+(\tau\beta\gamma)_{ijk}+\varepsilon_{ijk} \left\{\begin{array}{c} i=1,2,\ldots,r \\ j=1,2,\ldots,a \\ k=1,2,\ldots,b \end{array}\right. \)

Where, \(\tau_i\) , \(\beta_j\) and \((\tau \beta)_{ij}\) represent the whole plot and \(\gamma_k\), \((\tau \gamma)_{ik}\), \((\beta \gamma)_{jk}\) and \((\tau \beta \gamma)_{ijk}\) represent the split-plot. Here \(\tau_i\), \(\beta_j\) and \(\gamma_k\) are block effect, factor *A* effect and factor *B* effect, respectively. The sums of squares for the factors are computed as in the three-way analysis of variance without replication.

To analyze the treatment effects we first follow the approach discussed in the book. Table 14.17 shows the expected mean squares used to construct test statistics for the case where replicates or blocks are random and whole plot treatments and split-plot treatments are fixed factors.

Factor | rRi | aFj | bFk | lRh | Expected Mean Square | ||
---|---|---|---|---|---|---|---|

Whole plot | \(\tau_i\) | 1 | a | b | 1 | \(\sigma^2 + ab\sigma_{\tau}^2\) | |

\(\beta_j\) | r | 0 | b | 1 | \(\sigma^2 + b \sigma_{\tau \beta}^2 + \dfrac{rb \sum \beta_{j}^2}{a-1}\) | ||

\( (\tau \beta)_{ij}\) | 1 | 0 | b | 1 | \(\sigma^2 + b \sigma_{\tau \beta}^2\) | ||

Subplot | \(\gamma_k\) | r | a | 0 | 1 | \(\sigma^2 + \sigma_{\tau \gamma}^2\ + \dfrac{ra \sum \gamma_{k}^2}{(b-1)})\) | |

\( (\tau \gamma)_{ik}\) | 1 | a | 0 | 1 | \(\sigma^2 + \sigma_{\tau \gamma}^2\) | ||

\( (\beta \ gamma)_{jk}\) | r | 0 | 0 | 1 | \(\sigma^2 + \sigma_{\tau \beta \gamma}^2 + \dfrac{r \sum \sum (\beta \gamma)_{jk}^2}{(a - 1)(b - 1)}\) | ||

\( (\tau \ beta \gamma)_{ijk}\) | 1 | 0 | 0 | 1 | \(\sigma^2 + \sigma_{\tau \beta \gamma}^2\) | ||

\(\epsilon_{ijk}\) | 1 | 1 | 1 | 1 | \(\sigma^2\) (not estimable) | ||

Table 14.17 (Design and Analysis of Experiments, Douglas C. Montgomery, 8th Edition) |

The analysis of variance for the tensile strength is shown in Table 14.16.

Source of Variation | Sum of Squares | Degrees of Freedom | Mean Square | \(F_0\) | P-Value | ||
---|---|---|---|---|---|---|---|

Replicates (or Blocks) | 77.55 | 2 | 38.78 | ||||

Preparation method (A) | 128.39 | 2 | 64.20 | 7.08 | 0.05 | ||

\(\text{Whole Plot Error (replicates (or Blocks)} \times A\) | 36.28 | 4 | 9.07 | ||||

Temperature (B) | 434.08 | 3 | 144.69 | 41.94 | <0.01 | ||

\(\text{Replicates (or Blocks)} \times B\) | 20.67 | 6 | 3.45 | ||||

AB | 75.17 | 6 | 12.53 | 2.96 | 0.05 | ||

\(\text{Subplot Error (replicates (or Blocks)} \times AB\) | 50.83 | 12 | 4.24 | ||||

Total | 822.97 | 35 | |||||

Table 14.18 (Design and Analysis of Experiments, Douglas C. Montgomery, 8th Edition) |

As mentioned earlier analysis of split-plot designs using the second approach is based mainly on the randomization restrictions. Here, the whole plot section of the analysis of variance could be considered as a Randomized Complete Block Design or RCBD with Method as our single factor (If we didn’t have the blocks, it could be considered as a Complete Randomized Design or CRD). Remember how we dealt with these designs (Step back to Chapter 4). The error term which we used to construct our test statistic (The sum of the square of which was achieved by subtraction) is just the interaction between our single factor and the Blocks. (If you recall, we mentioned that any interaction between the Blocks and the treatment factor is considered part of the experimental error). Similarly, in the split-plot section of the analysis of variance, all the interactions which include the Block term are pooled to form the error term of the split-plot section. If we ignore method, we would have an RCBD where the blocks are the individual preparations. However, there is a systematic effect due to the method, which is taken out of the Block effect. Similarly, the block by temperature has a systematic effect due to method*temperature, so a SS for this effect is removed from the block*temperature interaction. So, one way to think of the SP Error is that it is Block*Temp+Block*Method*Temp with 2*3+2*2*3=18 d.f.

The Mean Square error terms derived in this fashion are then be used to build the *F* test statistics of each section of the ANOVA table, respectively. Below, we have implemented this second approach for data. To do so, we have first produced the ANOVA table using the GLM command in Minitab, assuming a full factorial design. Next, we have pooled the sum of squares and their respective degrees of freedom to create the SP Error term as described.

DF | SS | MS | F | P-value | |
---|---|---|---|---|---|

Blocks | 2 | 77.556 | 38.78 | 4.28 | 0.1014 |

Method | 2 | 128.389 | 64.19 | 7.08 | 0.0485 |

WP Error (Blocks*Methods) | 4 | 36.278 | 9.07 | ||

Temp | 3 | 434.083 | 144.69 | 36.43 | 0.0000 |

Method*Temp | 6 | 75.167 | 12.53 | 3.15 | 0.0272 |

SP Error | 18 | 71.5 | 3.97 | ||

35 | 822.973 |

As you can see, there is a little difference between the output of analysis of variance performed in this manner and the one using the Expected Mean Squares because we have pooled Block*Temp and Blocks*Method*Temp to form the subplot error.

##
Advantages and Disadvantages of Split-Plot Experiments
Section* *

In summary, when one of the treatment factors needs more replication or experimental units (material) than another or when it is hard to change the level of one of the factors, these design become important. The primary disadvantage of these designs is the loss in precision in the whole plot treatment comparison and the statistical complexity.