Ananlsis Of Binary Longitudinal Data With Empirical Likelihood Method

Binary data appears commonly and exerts a significant application in studying subjects such as biomedical,economic and social science.Longitudinal data is observation information of individuals for finite times,in which the data obtained are correlated while the extent of data correlation is unknown,and the observed data is independent for different individuals.Empirical likelihood(EL)method,comparing with Generalized estimation equations(GEE)proposed by Liang and Zeger,was given by Owen has a lot of significant advantages in analysis of binary longitudinal data.Nonetheless,There is few articles that analyze the properties of covariate of empirical likelihood method under large sample data in a fixed design of regression model for binary longitudinal data.Some only testified the properties of empirical log-likelihood ratio under large sample data,yet they did not give the proof about the maximum of empirical loglikelihood estimator.Other articles presented the assumptions and conditions which are stronger and too hard to be testified in practical application.On the basis above,Firstly there is a brief instruction about development history of the EL method by analyzing its properties and merits under independently identically distributed data,including existence of minimum of the empirical log-likelihood ratio,Consistency and asymptotic normality distribution of the empirical log-likelihood ratio estimation and asymptotic chi-square distribution of the empirical log-likelihood ratio statistic.Secondly,Testify whether the estimation equations are unbiased are given under null hypothesis of the empirical log-likelihood ratio statistic.Later,Importance and practical application of the Logit model are introduced as well as foundational theory of generalized estimation equation.Furthermore,The consistency and asymptotic normality distribution and asymptotic chi-square distribution of empirical log-likelihood estimation are proved under easier being testified lemmas and weaker assumptions,in order that all kinds of constraints of practical complex data is more theoretically reliable in real binary longitudinal data analysis.The results show the asymptotic distributions of empirical log-likelihood ratio test are obtained,and therefore the related result in the literature are improved.Meanwhile,the statistical simulation shows that the parameter fitness of empirical likelihood estimation is more accurate comparing with GEE simulation.