Abstract
When a logistic regression model is used under a small sample size and a high or a low event occurrence probability, it is important to confirm the existence of the complete or the quasi-complete separation. If the complete or the quasi-complete separation exists, a maximum likelihood estimator cannot be obtained. However, some statistical softwares such as SAS, S-PLUS or R execute an iteration method to obtain a maximum likelihood estimate. Commercial softwares present a result of the iteration with a warning message regarding the existence of the complete or the quasi-complete separation, or failing in convergence of the iteration. However, glm function implemented in R presents the result of the iteration with regard to the maximum likelihood estimate in spite of failing in convergence of the iteration. In this case, a standard error for the regression parameter estimate is very large. We show that it is possible to confirm the existence of the complete or the quasi-complete separation from the standard error for the regression parameter estimate. Firth (1993) suggested a method to eliminate a bias of the maximum likelihood estimator. As a result, Firth's method can estimate the regression parameter under the complete or the quasi-complete separation and it is possible to use Wald test using the standard error for the regression parameter estimate derived from Firth's method. However, Wald test using both the maximum likelihood method and Firth's method is very conservative under the small sample size and the high (or the low) event occurrence probability. The aim of this paper is to suggest a test for the regression parameter using the bootstrap method instead of Wald test under the small sample size and the high event occurrence probability that tends to near the complete or the quasi-complete separation. Under a null hypothesis, the probability of the type I error in the proposed method is compared with that in Wald test. We show that the proposed method for the slope parameter improves the type I error and assures the prescribed α significance level under the small sample size and the high event occurrence probability.
