The Analysis of Randomized Experiments with Orthogonal Block Structure. I. Block Structure and the Null Analysis of Variance

J. A. Nelder

Abstract

Nearly all the experimental designs so far proposed assume that the experimental units are or can be grouped together in blocks in ways that can be described in terms of nested and cross-classifications. When every nesting classification employed has equal numbers of subunits nested in each unit, then the experimental units are said to have a simple block structure. Every simple block structure has a complete randomization theory. Any permutation of the labelling of the units is permissible which preserves the block structure, and this defines a valid randomization procedure. In a null experiment all units receive the same treatment, and the population of all possible vectors of yields generated by the randomization procedure gives the null randomization distribution. The covariance matrix of this distribution, the null analysis of variance, and the expectations of the various mean squares in it are all derivable from the initial description of the block structure. All simple block structures can be characterized by a set of mutually orthogonal idempotent matrices C$_i$. Such sets of matrices may also exist for non-simple block structures (e.g. those having unequal numbers of plots in a block), and the existence of such a set defines an orthogonal block structure. Non-simple orthogonal block structures do not have a complete randomization theory and inferences from them require further assumptions.