Where the joint effects of two or more factors are not additive, a simple model is proposed for representing the effects. The effects of one factor are assumed to be proportional, rather than equal, at different levels of the other factors. The main effects of the first factor are given as weighted averages of the simple effects at the different levels of the other factors, the weights being the estimated factors of proportionality. The weights are given as the latent vector of a matrix of sums of squares and products corresponding to the largest latent root of the matrix; the sian of squares for the weighted main effect is a multiple of this latent root, and the other latent roots correspond to a partition of the interaction sum of squares. The analysis is closely related to the canonical analysis of a set of variates. Tests of significance of (a) the residual interactions and (b) the adequacy of a proposed set of weights are discussed. For the case where the matrix has only two non-vanishing latent roots, the approach of the joint distribution of the roots to its limiting form is discussed. The joint probability density is expanded as a series of Bessel functions of imaginary argument. Asymptotic formulae for the moments and product-moments of the roots are derived. Exact tests for the adequacy of a proposed set of weights, when there are only two non-vanishing latent roots, are presented. The methods of analysis are illustrated with a numerical example.

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