Average Values of Mean Squares in Factorials

Abstract

1. Summary. The assumptions appropriate to the application of analysis of variance to specific examples, and the effects of these assumptions on the resulting interpretations, are today a matter of very active discussion. Formulas for average values of mean squares play a central role in this problem, as do assumptionis about interactions. This paper presents formulas for crossed (and, incidentally, for nested and for non-interacting completely randomized) classifications, based on a model of sufficient generality and flexibility that the necessary assumptions concern only the selection of the levels of the factors and not the behavior of what is being experimented upon. (This means, in particular, that the average response is an arbitrary function of the factors.) These formulas are not very complex, and specialize to the classical results for crossed and nested classifications, when appropriate restrictions are made. Complete randomization is only discussed for the elementary case of no interactions with experimental units and randomized blocks are not discussed. In discussion and proof, we give most space to the two-way classification with replication, basing our direct proof more closely on the proof independently obtained by Cornfield [17], than on the earlier proof by Tukey [201. We also treat the three-way classification in detail. Results for the general factorial are also stated and proved. The relation of this paper to other recent work, published and unpublished, is discussed in Section 4 (average values of mean squares) and in Section 11 (various types of linear models). INITIAL DISCUSSION 2. Introduction. During the last years of the last decade it was relatively easy to believe that the analysis of variance was well understood. Eisenhart's summary article of 1947 [5], when combined with the work of Pitman [13] and Welch [15] on the randomization approach (work published in 1937-1938, which ever since has been far too much neglected), seemed to provide a simple, easily understandable account of the foundations. But as the years have passed, both statisticians and users of analysis of variance have gradually become aware of a number of areas in which we needed to deepen our understanding. One of these is the relation of formulas for average values of mean squares to assumptions. These are of central importance, since the choice of an error term as a basis for either