This article discusses estimation of the cure rate by means of the bounded cumulative hazard (BCH) model using interval censored data. The parametric and nonparametric estimation methods within the framework of the EM algorithm were employed for cure rate estimation and their results compared. The Turnbull estimator was used in the nonparametric estimation while in parametric method both the exponential and Weibull distributions were considered. We show via simulation that the nonparametric method is a viable alternative to the parametric one when the censoring rate is rapidly increasing.

Nonparametric versus Parametric Estimation of the Cure Fraction Using Interval Censored Data

Estimating cure-rate is a popular research topic in the study of immunity of individuals against terminal diseases such as cancers and HIV. In the literature, mainly estimation based on parametric models have been studied so far, with a few exceptions. In this paper, we use the Poisson smoothing idea of Chaubey and Sen (1996) to propose two novel non-parametric estimators under Case-1 interval censoring, which improve upon previously proposed ones (Sen and Tan, 2008). We show that our estimators are consistent. In addition, their asymptotic normality is studied, and we apply our estimator to estimate cure-rate on data collected for lung tumor in mice (Finkelstein
and Wolfe, 1985). Finally, the smoothing parameter for optimum estimation has been determined by using Jackknife method.

Cure rate estimation under case-1 interval censoring via smoothing

Over the last 25 years, extreme value theory has become a substantial part of statistical analysis. Often, extremes extract from data essential information that can hardly be retrieved from central tendency measures. We first deal with the main ingredients of classical extreme value theory, indicating its potentials and generalizations.


Keywords:

extreme value theory;
extreme value distributions;
generalized Pareto distributions;
quantiles;
Bayesian analysis;
multivariate extremes

Risk Analysis, Extremes in

In survival studies, it is often of interest to study cure rates. Sometimes the event of interest (such as death or the occurrence of a disease) may not be experienced by individuals under study, and the cure rate is the probability of the latter eventuality. Furthermore, survival times (i.e., the time to the event of interest) may be subject to random censoring due to dropping out or late entry of individuals during the study period. Interestingly, random censoring facilitates the estimation of the cure rate.

In this research, we study cure rates for multivariate survival times under multivariate random censoring. Specifically, three topics have been studied. In the first topic of this thesis, a new non-parametric multivariate cure rate estimator, based on a multivariate Kaplan-Meier estimator, is proposed. The asymptotic normality and an estimator of asymptotic variance of this estimator are obtained. In the second topic, a non-parametric cure rate estimator in the presence of covariates is constructed via kernel smoothing. The asymptotic normality of this estimator is obtained, and the optimal choice of the bandwidth via cross-validation is discussed. In the third topic of this thesis, we develop a test for the presence of immunes, i.e., we test if the cure rate is zero against the alternative that it is positive,
under univariate random censoring. The limiting distribution of the test statistic under the null hypothesis is obtained using extreme-value theory. Theoretical results are supported by simulation studies.

Inference on Cure Rate Under Multivariate Random Censoring

Cure-rate estimation under Case-1 interval censoring

We consider nonparametric estimation of cure-rate based on mixture model under Case-1 interval censoring. We show that the nonparametric maximum-likelihood estimator (NPMLE) of cure-rate is non-unique as well as inconsistent, and propose two estimators based on the NPMLE of the distribution function under this censoring model. We present a cross-validation method for choosing a `cut-off' point needed for the estimators. The limiting distributions of the latter are obtained using extreme-value theory. Graphical illustration of the procedures based on simulated data are provided.

Fang Tan

Papers

Cure-rate estimation under Case-1 interval censoring

Cure-rate estimation under Case-1 interval censoring