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 subplots within the applications of Factor A, and then split the experimental units used for Factor B into sub-subplots to receive the levels of Factor C.
For a fixed effect factorial treatment design in an RCBD (with blocks for the levels of Factor A, levels of Factor B, and levels of Factor C), the split-split-plot would produce the following table. (Hover over the error rows to see where the source or df comes from):
| Source | d.f. |
|---|---|
| (Whole plots) | |
| Block | r - 1 |
| Factor A | a - 1 |
| Whole plot error |
(r - 1)(a - 1) |
| (Subplots) | |
| Factor B | b - 1 |
| A × B | (a - 1)(b - 1) |
| Subplot error | a(r - 1)(b - 1) |
| (Sub-subplots) | |
| 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-subplot error | ab(r - 1)(c - 1) |
| Total | (rabc) - 1 |
The model is specified as we did earlier for the split-plot in 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’ (‘Whole plot error’ in the table), the Block × A × B term as ‘Error B’ (‘Subplot error’ in the table), and the residual error as ‘Error C’ (‘Sub-subplot error’ in the table) because of their roles as the denominator in the F-tests.