A modified score function estimator for multinomial logistic regression in small samples

Logistic regression modelling of mixed binary and continuous covariates is common in practice, but conventional estimation methods may not be feasible or appropriate for small samples. It is well known that the usual maximum likelihood estimates (MLEs) of (he log-odds-ratio parameters are biased in finite samples, and there is a non-zero probability that an MLE is infinite, i.e., does not exist, In this paper, we extend the approach proposed by Firth (Biometrika 80 (1993) 27) for bias reduction of MLEs in exponential family models to the multinomial logistic regression model, and consider general regression covariate types. The method is based on a suitable modification of the score function that removes first order bias. We apply the method in the analysis of two datasets: one is a study of disease prognosis and the other is a disease prevention trial. In a series of simulation studies in small samples, the modified-score estimates for binomial and trinomial logistic regressions had mean bias closer to zero and smaller mean squared error than other approaches, The modified-score estimates have properties that make them attractive for routine application in logistic regressions of binary and continuous covariates, including the advantage that they can be obtained in samples in which the MLEs are infinite. (C) 2002 Elsevier Science B.V. All rights reserved.