Application Of Firth Penalized Maximum Likelihood Estimation In Logistic Regression For Separation Phenomenon

Objects The phenomenon of separation is observed in the fitting process of a logistic model while there is questionable results or non-existence of the maximum likelihood estimate. The aim is to generalize the firth penalized maximum likelihood estimation to mlutinomial logistic regression.Methods Firth penalized maximum likelihood estimation involves modifying the score function to remove first order bias to reduce the bias of MLE. The text introduce the principles of the method. We analysis three datasets by exact logistic regression and Firth’s penalized maximum Likelihood estimate under SAS system and R and compare the resluts RespectivelyResults Through the resluts of the three datasets,samall sample and separate phenomenon,we can see that, although MLE can turn out the resluts,the estimator of PMLE and LE is smaller than that. When there is separation phenomenon,it produces finite parameter estimates by means of penalized maximum like-lihood estimation and exact logisitic regression.The estimators,standard errors and p value of PMLE is smaller than that of MLE. While The estimators of EL is smaller than that of PMLE,the confidence intervel is wider than that of PMLE.Conclution While there exits the problem of separation, non-existence of the maximum likelihood estimate under special conditions in a sample, exact logistic regression and Firth’s penalized maximum Likelihood estimate can solute it very well. Because of complex calculations, overconditioning and conditional distributions degenarated, The penalized maximum likelihood method can be generally recommended。When sample is small,although MLE can turn out the resluts,it’s resluts is overestimated.