In Chapter 1, we examined the idea that more than one random-effect term may be necessary in regression analysis, but this idea arises also in the analysis of designed experiments. In the simplest experimental designs, the fully randomized design leading to the one-way anova and the randomized complete block design leading to the two-way anova, it is sufficient to recognize a single source of random variation, the residual term. In designs with several treatment factors, a single random-effect term is still sufficient. But in designs with more elaborate block structures, it is necessary to recognize the block effects as random, either explicitly by the specification of a mixed model, or implicitly by using an anova protocol specific to the design in question. Such designs are known as *incomplete block designs*, because the blocks do not contain a complete set of treatments. One of the simplest designs of this type is the split plot design.

The theory of the split plot design was worked out in the 1930s in the context of field experiments on crops, and such experiments provide a simple context in which to illustrate it. (The literature from this period is not very accessible, but some account of it is given by Cochran and Cox (1957, Chapter 7, pp. 293–316).) If two *treatment factors*, such as the choice of crop variety and the ...

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