Parameter Redundancy and the Existence of Maximum Likelihood Estimates in Log-linear Models.

In fitting log-linear models to contingency table data, the presence of zero cell entries can have an adverse effect on the estimability of parameters, due to parameter redundancy. We describe a general approach for determining whether a given log-linear model is parameter redundant for a pattern of observed zeros in the table. We derive the estimable parameters or functions of parameters and show how to reduce the unidentifiable model to an identifiable one. Parameter redundant models have a flat ridge in their likelihood function. Orthogonality of this ridge to some vectors in the parameter space may impose additional parameter constraints on the model. These constraints can lead to obtaining unique maximum likelihood estimates for parameters that otherwise would not have been estimable. In contrast to other frameworks, the proposed approach informs on those constraints, elucidating the model is actually being fitted.