Estimation of general linear-bilinear models for two-way tables

We define the General Linear-Bilinear Model (GLBM) for data arranged as a r x c table as y(ij) = Sigma(k=1)(m) beta(k)x(kij) + Sigma(k=1)(t) lambda(k) alpha(ik)gamma(ik) + e(ij). This includes linear-bilinear models known as Additive Main Effects and Multiplicative Interaction, Rows Regression, Columns Regression, and Shifted Multiplicative models as special cases, but further allows for inclusion of regression on covariates as additional linear terms and for estimation of missing cells. A GLBM is defined as ''balanced'' if least squares estimates of its linear effects are free of the bilinear effects. A closed form least squares solution exists if the GLBM is balanced or if t=c-q-1 (c-q less than or equal to r) and X-k = [x(kij)] is of rank one for all k, where q is the number of linear effects fitted within each (and every) row. In all GLBMs, the least squares estimates of the multiplicative terms are obtained by singular value decomposition of the matrix of deviations y(ij) - Sigma(kj)<(beta)over cap>(k)x(kij), but, if the GLBM is unbalanced, solutions for the <(beta)over cap (k)> depend on the decomposition to be obtained. For such cases, iterative Newton-Raphson and generalized EM algorithms are developed. Closed form solutions for unbalanced GLBMs with t = c-q-1 and all rank(X-k) = 1 can be exploited for finding initial values for iterative solutions for smaller t, as well as for models with some rank(X-k) > 1. An example is presented in which, within each lever of the column factor, there is regression on a covariate and adjustment for incomplete blacking.