# Split-plot designs in design of experiments

## What is a split-plot design?

A split-plot design is a designed experiment that includes at least one hard-to-change factor that is difficult to completely randomize because of time or cost constraints. In a split-plot experiment, levels of the hard-to-change factor are held constant for several experimental runs, which are collectively treated as a whole plot. The easy-to-change factors are varied during these runs, each combination of which is considered a sub-plot within the whole plot. You should randomize the order in which you run both the whole plots and the sub-plots within whole plots.

## Example of a split-plot design

A large-scale bakery is designing a new brownie recipe. They are experimenting with two levels of chocolate and sugar using two different baking temperatures. However, to save time they decide to bake more than one tray of brownies at the same time instead of baking each tray individually. The brownie example includes 2 whole plots replicated twice (total of 4 whole plots). Each whole plot contains 4 sub-plots. The whole plot is all the trays of brownies being baked at the temperature. The sub-plots are each individual tray of brownies.

 Tray 1 (Chocolate 1, Sugar 1) Tray 2 (Chocolate 1, Sugar 2) Tray 3 (Chocolate 2, Sugar 1) Tray 4 (Chocolate 2, Sugar 2)
 Tray 1 (Chocolate 1, Sugar 1) Tray 2 (Chocolate 1, Sugar 2) Tray 3 (Chocolate 2, Sugar 1) Tray 4 (Chocolate 2, Sugar 2)
 Tray 1 (Chocolate 1, Sugar 1) Tray 2 (Chocolate 1, Sugar 2) Tray 3 (Chocolate 2, Sugar 1) Tray 4 (Chocolate 2, Sugar 2)
 Tray 1 (Chocolate 1, Sugar 1) Tray 2 (Chocolate 1, Sugar 2) Tray 3 (Chocolate 2, Sugar 1) Tray 4 (Chocolate 2, Sugar 2)

Split-plot designs were originally used in agriculture where the whole plots referred to a large area of land and the sub-plots were smaller areas within each whole plot.

## Error terms for split-plot designs

There is no single error term for testing all factor effects in a split-plot design. If the levels of factor A form the sub-plots, then the mean square for Block * A will be the error term for testing factor A. There are two schools of thought for what should be the error term to use for testing B and A * B. If you enter the term Block * B, the expected mean squares show that the mean square for Block * B is the correct term for testing factor B and that the remaining error (which is Block * A * B) will be used for testing A * B. However, it is often assumed that the Block * B and Block * A * B interactions do not exist and these are then lumped together into error. You might also pool the two terms if the mean square for Block * B is small relative to Block * A * B. If you don't pool, enter Block A Block * A B Block * B A * B in Model and what is denoted as Error is really Block * A * B. If you do pool terms, enter Block A Block * A B A * B in Model and what is denoted as Error is the set of pooled terms. In both cases enter Block as a random factor.

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