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.