Computation of maximum likelihood estimates in association models

In association models for cross-classified data, computation of maximum likelihood estimates (MLE's) is relatively difficult due to-the nonlinear constraints on the parameters. Currently available procedures based on the scoring algorithm for constrained maximum likelihood are relatively unreliable, and other cyclic procedures are relatively slow and do not provide estimated asymptotic standard deviations as by-products of calculations. To facilitate computations, it is noted that in standard association models, removal of constraints results in underidentification of parameters but does not affect the model itself, so that the MLE's of cell probabilities and conditional probabilities are unaffected. Given this observation, maximum likelihood estimation may be accomplished by unconstrained maximization of an objective function with two components: a log-likelihood ratio and a sum of squares representing deviations of parameters from their constraints. The objective function is then maximized by using a modification of the Newton-Raphson algorithm that ensures that successive iterations increase the objective function whenever a local maximum has not been reached. The proposed algorithm is shown to be reliable and relatively rapid. In addition, it is shown that the proposed technique may be used to estimate asymptotic standard deviations of parameter estimates. Use of the algorithm in practice is illustrated through some standard examples of association models.