Split Plot Models

Abstract

In an experiment with at least two factors, it is sometimes convenient to apply some of the factors to large experimental units (called whole plots) and then to split the large units into smaller parts to which the remaining factors are applied. The subdivisions of the whole plots are called subplots or split plots. Split plot designs are often used when either (1) the factors applied to whole plots are not of direct interest or (2) some factors require larger experimental units than the other factors. The first case is illustrated with an experiment to evaluate crop yields when using varying levels of a standard herbicide and a new pesticide. If the standard herbicide is not of direct interest, but rather primary interest is in the effects of the pesticide and any possible interaction between the herbicide and the pesticide, then it is appropriate to apply herbicides as whole plot treatments. (It will be seen later that interaction contrasts and comparisons between pesticides are subject to less error than comparisons among herbicides.) The second case that split plot designs are often used for can also be illustrated with this experiment. If the standard herbicide is applied using a tractor, but the new pesticide is applied by crop dusting, then the experimental procedure makes it necessary to use pesticides as whole plot treatments. (Clearly, an airplane requires a larger plot of ground for spraying than does a tractor.) It is of interest to note that a split plot design can be thought of as an (unbal-anced) incomplete block design. In this approach, each whole plot is thought of as a block. Each block contains the treatments that are all combinations of the subplot factor levels with the one combination of whole plot factor levels that was applied. As with other incomplete block designs, a split plot design is necessary when there are not enough blocks available that can accommodate all of the treatment combinations. In split plot designs, this means that there are not enough subplots per whole plot so that all treatment combinations could be applied at the subplot level. If, in addition, there are not enough whole plots so that each treatment combination could be applied to a whole plot, then a split plot design is an attractive option. Mathematically, the key characteristic of a split plot model is the covariance structure. Typically, observations taken on the subplots of any particular whole plot