Analysis Of Multivariate Recurrent Event Data With Nonignorable Missing Covariates And Informative Censoring

Multivariable recurrent events usually occur in epidemiological studies,reliability experiments and longitudinal data studies in which each subject may experience multiple recurrence events.Duo to timely prevent relapse or disease progression events,analysis of recurrent events is more important compared with the acquaintance of first occurrence time and hospitalization information.The most important feature of recurrence event data is ordered and correlated,which can be regarded as multivariable ordered survival data.To analyze the multivariate recurrent event data,some inference procedures have been developed.The research is more brittle or marginal distribution model based on fragility model.The covariates can be observed with time dependent and time independent two types,along with the information censored.Huang,Qin and Wang(2010)study the problem of recurrence events with time-dependent covariates and information censoring,they propose a pairwise pseudolikelihood approach^[1,2]and Generalized Estimating Equations approach^[3,4]for estimating coefficients of time-dependent and time-independent covariates,respectively.Meanwhile,the estimation of intensity basis function can be obtained by modifying the truncation product-limit estimator^[5,6]and using the inverse probability weighting technique.Finally,the large sample properties of the estimation are proved^[7].In the follow-up work,Zhao,based on the borrow-strength estimation process proposed by Huang&Wang,Liu and Liu(2012)extend the problem to multivariate recurrent event data that means the subject may experience multiple recurrence events and propose the estimation equation of the frailty variables^[8].But the above methods or models are based on covariates and can be observed.the covariates we study may be missing for some reasons in actual situation,or even can not be ignored.In this paper,we follow their previous models and assume that there exists an nonignorable missing which satisfies the exponential tilting in time-independent covariates,in which the tilt parameters and non-parametric functions are unknown.Based on the idea of two-step moment estimation proposed by Shao and Wang(2016),we introduce instrumental variables into covariates to construct estimation equations and profile the nonparametric component using a kernel-type estimator and then estimate the tilting parameter based on the profiled estimating equations and the generalized method of moments^[9].Due to the missing data of time-independent covariates,we obtain the parameter estimation with inverse probability weighting method to improve the original generalized estimation equation based on the above estimated probability.Finally,we validate our theoretical results by simulation.At the end of the paper,some problems that may exist in this method and the problems that will be further studied in the future are given.