SHIFTED MULTIPLICATIVE MODELS FOR NONADDITIVE 2-WAY TABLES

This paper presents analysis of a two-way table of data using a 'shifted multiplicative model (SHMM) of the form y(ij) = beta + SIGMA(k=1)t lambda(k)alpha(ik)gamma(jk) + e(ij). The least squares estimates of the multiplicative terms are obtained from singular value decomposition of the matrix Z = [z(ij)] where z(ij) = y(ij) - beta, but the least squares estimate, beta, of the shift parameter depends on estimates of parameters contained in the multiplicative terms. The sum of squares can be minimized " a function only of beta, for which iterative Newton-Raphson and generalized EM algorithms are developed. Expectations of sums of squares owing to sequentially increasing the number, t, of multiplicative terms (presented in an ANOVA format) were obtained by Monte Carlo simulation for the case where errors are i.i.d. N(0, sigma-2) and all lambda(k) = 0.. The analysis is illustrated with several examples from the literature.