7.2 - Split-Plot in CRD

 See Textbook Section 27.6

Split plot designs came out of agricultural field experiments and our text uses an example of an agricultural experiment to illustrate the principles of split plot design.

Two factors are of interest, Irrigation (Factor A at 2 levels) and Fertilizer (Factor B at 2 levels) and they are crossed to form a factorial treatment design. The fertilizer treatment can be easily applied at a small scale, but the irrigation treatment is problematic. Irrigating one plot may influence neighboring plots, and furthermore, the irrigation equipment is most efficiently used in a large area. As a result, the investigators want to apply the irrigation to a large ‘whole plot’ and then split the whole plot into 2 smaller subplots in which they can apply the fertilizer treatment levels.

In the first step, the levels of the irrigation treatment are applied to four experimental (fields) to end up with 2 replications:

Field 1 Field 2 Field 3 Field 4
A2 A1 A1 A2

Following that, the fields are split into two subplots and a level of Factor B is randomly applied to sub-plots within each application of the Irrigation treatment:

Field 1 Field 2 Field 3 Field 4
A2 B2 A1 B1 A1 B2 A2 B1
A2 B1 A1 B2 A1 B1 A2 B2

This design can now be viewed as 2 replications of the CRD with the split-plot sequence of randomization of the factors. The consequence of this design is seen in the error terms used in constructing F tests for the two factors.

Source DF Expected Mean Square Error Term
(Whole Plots)      
A 1 Var(Residual) + 2 Var(field(a)) +  Q(A, A*B) MS(field(a))
field(A) 2 Var(Residual) + 2 Var(field(A))  MS(Residual)
B 1 Var(Residual) + Q(B, A*B) MS(Residual)
A*B 1 Var(Residual) + Q(A*B) MS(Residual)
Residual 2 Var(Residual)  

The important thing to see here is that the test for the treatment applied to whole plots requires a different error term than the other factors. Older software, such as SAS proc ANOVA or proc GLM do not perform this analysis correctly. Proc mixed and Minitab’s GLM procedures do handle this situation properly.

  Using SAS

In SAS, the code would be:

proc mixed data=example_8_2 method=type3;
class factora factorb field;
model resp=factora factorb factora*factorb;
random field(factora);

The random effect we need to serve as the denominator for the F test for Factor A as a split plot in a CRD is the Experimental unit (Whole plot factor), and in this case that is the field (Factor A).

  Using Minitab

In Minitab the field(FactorA) term would need to be constructed in the Random / Nesting options box under the ANOVA > General Linear Model > Fit the General Linear Model.