Existência de estimadores de máxima verossimilhança em modelos de regressão logística

Abstract

The logistic regression model is the statistical method frequently used to deal with binary responses and the estimation of their coefficients is usually done using the method of maximum likelihood. But as this method is based on the asymptotic properties of the estimators, it needs sample sizes generally large, so the theory asymptotics results cannot be appropriate or cannot exist, even when we have the use of large samples, but their data are sparse. The logistic data can be classified into three mutually exclusive and exhaustive categories, according to Albert and Anderson (1984): complete separation, quasicomplete separation and overlap. For the first two categories, the maximum likelihood estimators do not exist. This researche has been motivated for two real data sets that are classified on the category of separation quasicomplete and, consequently, there are no maximum likelihood estimators. Then, two proposals from the literature (exact logistic regression and addition of a small constant in data) were discussed and it was presented the new proposal, that consists in taking away randomly the results of any of the non-null cell (with same value from covariable or same response value ) and add it to the null cell. The comparison of the proposals is done by using simulations of Monte Carlo. The criterion used for this comparison was the mean-square error. The best results obtained were based on the addition of a small constant in data and the effectiveness of the new proposal.