8.3: 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 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, levels of Factor A, levels of Factor B, and levels of Factor C), the split-split-plot would produce the following table:

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.

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