Strata for randomized experiments

Fisher argued that, for completely randomized designs, block designs and Latin square designs, his proposed randomization justified his proposed analysis of variance in that the latter gives valid estimates of experimental error even under very weak assumptions about the plot effects. In the succeeding 50 years a large number of more complicated experimental structures has been proposed and used. Randomization of these structures is rarely discussed in more than a cursory way, and its relationship with the analysis is mentioned even less frequently. Here I show that Fisher's ideas on validity and randomization may be set in a symmetry framework. Randomization is done by permuting the plot names, and that randomization procedure is used to justify the assumption of a relatively simple model for plot effects. Permutations are chosen at random from a specified permutation group. Except in the simplest cases, determination of the strata for the analysis of variance needs some knowledge of group theory. The relevant results from group theory are presented, together with some straightforward techniques which depend on them. The strata are then found for many large classes of examples, including the majority of experimental structures in current use.