Partial Sufficient Dimension Reduction On The Joint Model Of Recurrent Event And Terminal Event

In the statistical experiment, subjects may experience the outcome of interestmore than once, the event are called recurrent event and the observation of theseoutcomes are called repeated events data or recurrent event data. This type of dataarises in many fields such as medical field, public health, biology, demography andeconomics. To analyze recurrent event data, various models and approaches have beenproposed in the literature. The researches of recurrent events include the number ofevents up to the observation time, the event times and the inter-event times (gaps). Themodels and approaches for the number of events over times were studied by manyauthors. Generally, low dimensional covariates have been assumed in modelingrecurrent event data, but which may be violated in the analysis when the covariates arehigh-dimensional. In recent years, with the development of computers and otherinformation technology, the data and the covariates are more complex from the studiesof biology, medicine, ecology, demography, environment and economics and otherdisciplines. Some useful important information is often hidden behind thehigh-dimensional data. So, recently, the high-dimensional data regression is verypopular in many areas. Due to the "dimension curse" exists, many traditional statisticalmethods have encountered big challenges and unprecedented difficulties. Thus, in ourpaper, we discuss the dimension reduction approaches on the joint model of recurrentevent and terminal event with high-dimensional covariates to solve the dimensionreduction problems in recurrent event model.In our paper, the model we discussed is the joint model of the multiplicativehazard rate function of recurrent event and termination event. The innovation of thispaper is that we use the partial sufficient dimension reduction theory and method onthe joint model. We not only study the dimension reduction problem of the recurrentevent model, but also consider the dimension reduction problem of the terminal eventmodel. The application of the partial sufficient dimension reduction theory and methodon recurrent event model and the dimension reduction problem of terminal eventmodel are worthy to study and not to pay more attention in the literature. In this paper,we mainly focus on two aspects: The first aspect, to solve the dimension reductionproblem of the multiplicative hazard rate function of recurrent event model, first anonparametric estimator is proposed for the baseline function, and the basis of thepartial central subspace and its structural dimension are estimated through the partial sufficient dimension reduction. Based on the estimated structural dimension, the basisof the partial central subspace and the baseline, we can get the estimator of theregression function by using the local linear regression. The second aspect, based onthe frailty factor estimated from the multiplicative hazard rate model of the recurrentevent, we discuss the partial dimension reduction of the multiplicative hazard ratemodel of terminal event. Two methods are proposed in this paper, the first method: weuse the partial dimension reduction to get the estimator of the partial central subspaceand then we use the local partial likelihood function to get the estimators of thebaseline function and the regression function. The second method: the efficientestimation method is used to get the central subspace. For simplicity in calculation, weassume the baseline function is known in the terminal event model.A simulation is performed to conform and assess the theoretical findings, and anexample is also demonstrated on a set of chronic glaucomatous disease data.