General balance, large data sets and extensions to unbalanced treatment structures

Abstract

General balance defines an important class of designs covering virtually all the traditional experimental designs and, in particular, those with several error terms. In these designs the block terms are mutually orthogonal, the treatment terms are also mutually orthogonal, and the contrasts of each treatment term have equal efficiency factors in each of the strata where they are estimated. Generally balanced designs can efficiently be analysed by an algorithm which involves a series of sweep operations for each stratum. Each sweep estimates the effects of a treatment term (by its means divided by the efficiency factor), and subtracts them from the current working vector which then becomes the working vector for the next sweep. Workspace requirements are considerably less than those of multiple-regression style algorithms, so the algorithm has great advantages particularly with large treatment models. With large data sets, however, it may be more difficult to fulfill all the conditions of general balance. One possibility is to retain orthogonal block structure, but allow unbalanced treatment structures. It is shown that the algorithm can successfully be adapted to this situation, but it now requires an iterative sequence of sweeps to fit the treatment model in each stratum.