Some Consequences of randomization in a Generalization of the Balanced Incomplete Block Design

Abstract

The present paper envisages a generalized situation of the balanced incomplete block design in the sense of allowing for the sampling of sources of experimental material, of blocks within sources, of experimental units within blocks, and of treatments under consideration. A model for an arbitrary observation of a generalized balanced incomplete block design is derived explicitly from the physical way in which the experiment is performed, i.e., from the way in which the sampling and randomization procedures are carried out. The correlational structure of the observations is therefore implicit in the model. The model initially uses no assumptions of additivity of treatments with experimental material. It is shown that expected values of squares of partial observational means, as well as the expected values of products of individual observations, admit simple and easily specifiable expressions in terms of quantities called cap sigmas and denoted by $\Sigma$'s. The expected values of mean squares in the analysis of variance table are then derived. Consequences of the presence of various types of nonadditivity on the usual test of no treatment effects are discussed for fixed, mixed and random situations. For example, when the blocks actually used in the experiment form a random sample from an infinite population of blocks then the presence of interactions of blocks with treatments produces no bias in the comparison of the error and adjusted treatment mean squares. The correlational structure of the observations under the simplifying additivity assumption is examined for the standard balanced incomplete block design. It is shown that the usual forms of estimators of treatment comparisons are appropriate and that the $\Sigma$'s play the roles which the block and plot variances have in the corresponding assumed infinite model. In the presence of nonadditivities of treatments with the experimental material the usual forms of the linear estimators are no longer best.