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 lightbulbto see where the source or df come from):
|Block||r - 1|
|Factor A||a - 1|
|Whole plot error||
(r - 1)(a - 1)
|Factor B||b - 1|
|A × B||(a - 1)(b - 1)|
|Sub plot error||a(r - 1)(b - 1)|
|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.