# Module 12

# Split-Plot Design

*Split-plot design* is also a partially nested design and it is very similar to a *repeated measure design* with respect to the analysis, EXCEPT FOR the design originated from the Agriculture/Biology field of study. In many situations, *split-plot design* makes a lot of sense rather the most efficient *completely randomized design*. For example, an agronomist wants to test the effectiveness of irrigation methods and fertilizer types on the yield of a crop. As it is difficult to apply irrigation (evenly) in a small plot, she decided to use one irrigation method (factor *A*) in one of the big plots (known as *whole plot*) and another method in another big plot as in Figure 13. As the irrigation method factor is applied to whole plot, this factor *A* (irrigation method) is known as the *whole plot factor*. Then she divides each big plot (or whole plot) into four small plots known as *sub-plots* or *split-plots* as in Figure 14. Four different types of fertilizer are randomly applied to each of these *split-plots* as in Figure 15. Therefore, the fertilizer (factor *B*) is known as the *split-plot factor*. Ideally, in a completely randomized design, any of the eight split plots would have an equal chance of getting any treatment combination as in Figure 16. However, running experiment like this would be very expensive and does not make much practical sense. Therefore, in contrast to the theoretically most efficient completely randomized design, *split-plot design* is a preferred way of running an experiment like this.

# Analysis Model

As the whole plot is nested within the whole plot factors, the model can be written as Equation 7. It can be noted that this is the exact same model as in the repeated measure in Equation 6. In a repeated measure design, the subjects are nested in the between-subject factors. Therefore, the whole plot is comparable to the subject in the repeated measure design. Moreover, in the randomized block design, the whole plot is comparable to the blocks; in fact, it is a block.

# Split-Plot Design

# Example

An agronomist wants to see the effectiveness of four types of fertilizer with two irrigation method in crop production. She run the experiment as a split plot design. The experimental layout is provided in Figure 17. She (randomly) selected three sites where she can run the experiments. She divides each site into two big plots (whole plots) to apply one of the irrigation methods to each. Then she divides each whole plot into four sections (split-plot) to apply four types of fertilizer. Crop production was measured as a response variable in the experiment. Data is provided in Table 5.

# Analysis

Analysis outputs are provided in Figure 18 and Figure 19. Comparing both output tables, it can be seen that the whole plot factor Irrigation uses its own error term, the whole plot error which is *Sites(Irrigation)*. Therefore, the ANOVA table in Figure 18 can be rearranged as whole plot and split plot parts, similar to the between-subject and within-subject parts in the repeated measure design. The rearranged outputs are provided in Figure 20. The post-hoc analyses are provided for the significant variables in Figure 21.

# Results Explained/ Contextual Conclusion

Irrigation method 2 produced significantly more yield.

Both fertilizer 3 and 4 produced significantly more yield.

Fertilizer 1 was observed to be the worst, while fertilizer 2 was in the middle with respect to the yield.