The analysis of randomized experiments with orthogonal block structure II Treatment structure and the general analysis of variance

Any orthogonal block structure for a set of experimental units has been previously shown to be expressible by a set of mutually orthogonal, idempotent matrices Ci. When differential treatments are applied to the units, any linear model of treatment effects is expressible by another set of mutually orthogonal idempotent matrices Tj. The analysis of any experiment having any set of treatments applied in any pattern whatever to units with an orthogonal block structure, is expressible in terms of the matrices Ci, Tj and the design matrix N, which lists the treatments applied to each unit. A unit-treatment additivity assumption and a valid randomization are essential to the validity of the analysis. The relevant estimation equations are developed for this general situation, and the idea of balance is given a generalized definition, which is illustrated by several examples. An outline is sketched for a general computer program to deal with the analysis of all experiments with an orthogonal block structure and linear treatment model.