Computing maximum likelihood estimates of loglinear models from marginal sums with special attention to loglinear item response theory

In this paper, algorithms are described for obtaining the maximum likelihood estimates of the parameters in log-linear models. Modified versions of the iterative proportional fitting and Newton-Raphson algorithms are described that work on the minimal sufficient statistics rather than on the usual counts in the full contingency table. This is desirable if the contingency table becomes too large to store. Special attention is given to log-linear Item Response Theory (IRT) models that are used for the analysis of educational and psychological test data. To calculate the necessary expected sufficient statistics and other marginal sums of the table, a method is described that avoids summing large numbers of elementary cell frequencies by writing them out in terms of multiplicative model parameters and applying the distributive law of multiplication over summation. These algorithms are used in the computer program LOGIMO, and are illustrated with simulated data for 10,000 cases.