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

Abstract

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 C$_i$. When differential treatments are applied to the units, any linear model of treatment effects is expressible by another set of mutually orthogonal idempotent matrices T$_j$. 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 C$_i$, T$_j$ 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.