Computing Maximum Likelihood Estimates in Log-Linear Models

We develop computational strategies for extended maximum likelihood estimation, as defined in Rinaldo (2006), for general classes of log-linear models of widespread use, under Poisson and product-multinomial sampling schemes. We derive numerically efficient procedures for generating and manipulating design matrices and we propose various algorithms for computing the extended maximum likelihood estimates of the expectations of the cell counts. These algorithms allow to identify the set of estimable cell means for any given observable table and can be used for modifying traditional goodness-of-fit tests to accommodate for a nonexistent MLE. We describe and take advantage of the connections between extended maximum likelihood estimation in hierarchical log-linear models and graphical models.