7.3 - Split-Split-Plot Design

The idea of split plots can easily be extended to multiple splits. In a 3-factor factorial, for example, it is possible to assign Factor A to whole plots, then Factor B to split-plots within the applications of Factor A, and then split the experimental units used for Factor B into sub-sub-plots to receive the levels of Factor C. The ANOVA follows from the split-plots discussed so far.

For a fixed effect factorial treatment design in a RCBD (with r blocks, a levels of Factor A, b levels of Factor B, and c levels of Factor C) the split-split plot would produce (Hover over the lightbulb   to see where the source or df come from):

Source d.f.
(Whole plots)  
Block r - 1
Factor A a - 1
Whole plot error

(r - 1)(a - 1)

(Sub plots)  
Factor B b - 1
A × B (a - 1)(b - 1)
Sub plot error a(r - 1)(b - 1)
(Sub-sub-plots)  
Factor C c - 1
A × C (a - 1)(c - 1)
B × C (b - 1)(c - 1)
A × B × C (a - 1)(b - 1)(c - 1)
Sub-sub plot error ab(r - 1)(c - 1)
Total (rabc) - 1

The model is specified as we did earlier for the split-plot in an RCBD, retaining only the interactions involving replication where they form denominators for F tests for factor effects. For the model above, we would need to include the block, block × A, and block × A × B terms in the random statement in SAS. In SAS, Block × A × B would automatically include the Block × B effect SS and df.  All other interactions involving replications and factor C would be included in the residual error term. The block × A term is often referred to as ‘Error a’, the Block × A × B term as ‘Error b’, and the residual error as ‘Error c’ because of their roles as the denominator in the F tests.