A general recursive procedure for analysis of variance

Abstract

SUMMARY A general recursive least squares procedure for the analysis of experimental designs is described. Any experimental design can be analyzed with a finite sequence of sweeps, in each of which a set of effects for a factor of the model is calculated and subtracted from the vector of observations. The effects are usually either simple means or effective means, which are ordinary means divided by an efficiency factor. The analysis for a particular design and model is characterized by a set of K efficiency factors for each factor of the model, where K is the order of balance of that factor, and by a triangular control matrix of indicators (0 or 1), in which subdiagonal zeros indicate orthogonality between pairs of factors in the model, and diagonal zeros indicate factors that are completely aliased with previous factors. The control matrix determines the minimal sweep sequence for analysis. The procedure may be implemented in an adaptive or 'learning' form, in which the information that characterizes the analysis is determined progressively from preliminary analyses of special dummy variates, each generated from an arbitrarily assigned set of effects for a factor of the model. A simple extension of the procedure produces the multistratum analysis required for stratified designs such as split plots and confounded factorials. Observations commonly arise from designed experiments, the symmetries and pattern of which implicitly affect the analysis and inference from the data, but are not usually explicitly characterized and utilized in general linear model formulations. This paper describes a simple procedure for least squares analysis of experimental designs with respect to a linear factorial model, in which the sequence of operations required is fully determined and controlled by the symmetries and pattern in the design. The analysis process consists of a sequence of sweeps of the data vector, determined by the factors of the model, the sweep being the only form of arithmetic operation required. In a sweep for a factor of the model, a set of effects for that factor is calculated and subtracted