Estimation For The Survival Function Based On The Partially Interval-censored Data

The censored data arises in a number of applied field, such as biology, engineering, medicine and economics, which include right-censored data, left-censored data and interval-censored data. The right-censored data and the left-censored data in some sense can be regarded as a special interval-censored data. The right-censored data have a lot of research results, such as:Cox and Oakes (1984) [1], Kalbfleisch and Prentice (2002)[2] and so on, these monographs introduce some methods to process right-censored data systematically. But the type II interval censoring is more challenging than right censoring and for such data the methods developed for right censoring do not generally apply. Therefore, it is necessary for us to propose some new methods which can be used to analyze the interval-censored data.By interval censoring, we mean that we often can not obtain the exact failure time, but only know that the failure time of interest lies within an interval. There are many kinds of research methods of interval censoring, such as Turnbull’s algorithm, modified EM algorithm, self-consistency algorithm etc.. Among them, a modification of the iterative procedure as proposed by Turnbull(1967)[3] is widely applied in the nonparametric estimation. In this thesis, we focus on the data that contains both right-censored data and type II interval-censored data. Based on previous research, we introduce the self-consistency algorithm of partial interval-censored data, and prove that this method is nonparametric maximum likelihood estimator. Because of the complex iterative procedure, its high programming problems and complex computation, in this thesis, inspired by Sun(2015)[4] mentioned method, we propose a method that is more simple and direct. We use the imputation method to change interval-censored data approximately to right-censored data. And then estimate the survival function by the Product-Limit estimator(KM estimator). Besides, in this thesis we compared the imputation method with the self-consistency algorithm and then come to a conclusion.The structure of this thesis is as follows:the chapter 1 is introduction, which briefly describes the research background of interval-censored data and the definition of type II interval censored data; in chapter 2, we first briefly introduced the method that is most commonly used in estimating the survival function of interval-censored data and the research background and specific algorithm. Next we introduce the method by using the interpolation method to make interval-censored data approximate to right-censored data and then estimate the survival function by the Product-Limit estimator; the chapter 3 is the model simulation using both algorithms above of II-type interval-censored data and the comparative analysis; In chapter 4, we cite an example by using the data of Time to Cosmetic Deterioration of Breast Cancer Patients; in chapter 5, we point out the inadequacy of our thesis, and make a conclusion and prospect.