New Methods to Reduce Bias and Mean Square Error of Maximum Likelihood Estimators

DESCRIPTION (provided by applicant): Logistic regression is the most frequently used model for binary data and has widespread applicability in the health, behavioral, and physical sciences. Over two thousand research papers were published in 1999 in which logistic regression was in the title of the paper or among the keywords. Maximum likelihood is the nearly universal method for computing estimates of regression coefficients in logistic regression models. These estimates are reliable for problems with large samples and when the proportion of responses is neither too small nor too large. However, it has been known for several years that maximum likelihood estimates can have high bias and mean square error for small, sparse or unbalanced datasets, with the latter referring to a considerable difference between the number of responses and non-responses. Exact logistic regression is a method invented by D. R. Cox that is often useful in such situations. However, exact logistic regression is computationally intensive and is limited in practice in terms of the size of datasets and the number of covariates that it can handle before running out of memory or taking an inordinate amount of computing time. D. Firth has developed a method for reducing bias and mean square error for logistic regression as well as other generalized regression models that is not as computationally demanding. Studies in the literature have shown that the method often improves on maximum likelihood. Firth's method is not available in any commercial software package today. We propose to incorporate Firth's method into LogXact, Cytel's regression package, as well as into PROC LOGXACT, a module that runs seamlessly as a part of the SAS software system. In addition to incorporating Firth's method for logistic regression we intend to develop it to apply to conditional logistic regression, ordered and unordered polytomous regression, Poisson regression and Negative Binomial regression. Firth's method does not perform well over certain ranges of model parameters in moderate sized samples in logistic regression. There are instances when it is worse than maximum likelihood. We have created a novel method that generalizes Firth's method to overcome this shortcoming. We propose to implement this method into LogXact and PROC LOGXACT. Under certain unusual conditions both maximum likelihood and Firth's method produce poor estimates for logistic regression. We have developed a diagnostic measure that identifies this situation and we will incorporate this method as part of our generalization of Firth's method. We will also investigate a Bayesian estimator and the target estimator suggested by Cabrerra and Fernholz that have promise of performing well in this situation.