PARAMETER REDUNDANCY AND THE EXISTENCE OF MAXIMUM LIKELIHOOD ESTIMATES IN LOG-LINEAR MODELS

Log-linear models are typically fitted to contingency table data to describe and identify the relationships between categorical variables. However, these data may include observed zero cell entries, which can have an adverse effect on the estimability of the parameters, owing to parameter redundancy. We describe a general approach to determining whether a given log-linear model is parameter-redundant for a pattern of observed zeros in the table, prior to fitting the model to the data. We derive the estimable parameters or the functions of the parameters, and explain how to reduce the unidentifiable model to an identifiable model. Parameter-redundant models have a flat ridge in their likelihood function. We explain when this ridge imposes additional parameter constraints on the model, which can lead to 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 being fitted.