Semiparametric Random Censorship Models Research Articles

Semiparametric random censorship models for survival data with long-term survivors

In this article, we study a semiparametric random censorship model for survival data in the presence of long-term survivors. Local likelihood method is employed to estimate the conditional mean regression function of binary random variables. The proposed estimators for the survival function and the cure rate, as well as their asymptotic properties, are investigated based on empirical and U-statistical processes. In particular, the proposed estimator for the cure rate is shown to be superior over the previous estimator considered by Maller and Zhou in the sense of having a smaller asymptotic variance. This semiparametric random censorship model with related estimation methods provide an efficient alternative for survival analysis with long-term survivors.

Semi-parametric survival function estimators deduced from an identifying Volterra type integral equation

Based on an identifying Volterra type integral equation for randomly right censored observations from a lifetime distribution function F , we solve the corresponding estimating equation by an explicit and implicit Euler scheme. While the first approach results in some known estimators, the second one produces new semi-parametric and pre-smoothed Kaplan–Meier estimators which are real distribution functions rather than sub-distribution functions as the former ones are. This property of the new estimators is particular useful if one wants to estimate the expected lifetime restricted to the support of the observation time. Specifically, we focus on estimation under the semi-parametric random censorship model (SRCM), that is, a random censorship model where the conditional expectation of the censoring indicator given the observation belongs to a parametric family. We show that some estimated linear functionals which are based on the new semi-parametric estimator are strong consistent, asymptotically normal, and efficient under SRCM. In a small simulation study, the performance of the new estimator is illustrated under moderate sample sizes. Finally, we apply the new estimator to a well-known real dataset.

Simultaneous confidence bands for Cox regression from semiparametric random censorship.

Cox regression is combined with semiparametric random censorship models to construct simultaneous confidence bands (SCBs) for subject-specific survival curves. Simulation results are presented to compare the performance of the proposed SCBs with the SCBs that are based only on standard Cox. The new SCBs provide correct empirical coverage and are more informative. The proposed SCBs are illustrated with two real examples. An extension to handle missing censoring indicators is also outlined.

Two-sample location–scale estimation from semiparametric random censorship models

When two survival functions belong to a location–scale family of distributions, and the available two-sample data are each right censored, the location and scale parameters can be estimated using a minimum distance criterion combined with Kaplan–Meier quantiles. In this paper, it is shown that using the estimated quantiles from a semiparametric random censorship framework produces improved parameter estimates. The semiparametric framework was originally proposed for the one-sample case (Dikta, 1998), and uses a model for the conditional probability that an observation is uncensored given the observed minimum. The extension to the two-sample setting assumes the availability of good fitting models for the group-specific conditional probabilities. When the models are correctly specified for each group, the new location and scale estimators are shown to be asymptotically as or more efficient than the estimators obtained using the Kaplan–Meier based quantiles. Individual and joint confidence intervals for the parameters are developed. Simulation studies show that the proposed method produces confidence intervals that have correct empirical coverage and that are more informative. The proposed method is illustrated using two real data sets.

Asymptotically efficient estimation under semi-parametric random censorship models

We study the estimation of some linear functionals which are based on an unknown lifetime distribution. The observations are assumed to be generated under the semi-parametric random censorship model (SRCM), that is, a random censorship model where the conditional expectation of the censoring indicator given the observation belongs to a parametric family. Under this setup a semi-parametric estimator of the survival function was introduced by the author. If the parametric model assumption is correct, it is known that the estimated functional which is based on this semi-parametric estimator is asymptotically at least as efficient as the corresponding one which rests on the nonparametric Kaplan–Meier estimator.In this paper we show that the estimated functional which is based on this semi-parametric estimator is asymptotically efficient with respect to the class of all regular estimators under this semi-parametric model.

Inverse censoring weighted median regression

We implement semiparametric random censorship model aided inference for censored median regression models. This is based on the idea that, when the censoring is specified by a common distribution, a semiparametric survival function estimator acts as an improved weight in the so-called inverse censoring weighted estimating function. We show that the proposed method will always produce estimates of the model parameters that are as good as or better than an existing estimator based on the traditional Kaplan–Meier weights. We also provide an illustration of the method through an analysis of a lung cancer data set.

Bootstrap Approximations in Model Checks for Binary Data

Consider a binary regression model in which the conditional expectation of a binary variable given an explanatory variable belongs to a parametric family. To check whether a sequence of independent and identically distributed observations belongs to such a parametric family, we use Kolmogorov–Smirnov and Cramér–von Mises type tests based on a marked empirical process introduced by Stute. We propose and study a new resampling scheme for a bootstrap in this setup to approximate critical values for these tests. We also apply this approach to simulated and real data. In the latter case we check some parametric models that are used to analyze right-censored lifetime data under a semiparametric random censorship model.

The central limit theorem under semiparametric random censorship models

We study integrals for arbitrary Borel-measurable functions with respect to a semiparametric estimator of the distribution function in the random censorship model. Based on a representation of these integrals, which is similar to the one given by Stute for Kaplan–Meier integrals, a central limit theorem is established which generalizes a corresponding result of the Cheng and Lin estimator. It is shown that the semiparametric integral estimator is at least as efficient as the corresponding Kaplan–Meier integral estimator in terms of asymptotic variance if the correct semiparametric model is used. Furthermore, a necessary and sufficient condition for a strict gain in efficiency is stated. Finally, this asymptotic result is confirmed in a small simulation study under moderate sample sizes.

Resampling Methods for Testing a Semiparametric Random Censorship Model

This paper presents a goodness‐of‐fit test for a semiparametric random censorship model proposed by Dikta (1998). The test statistic is derived from a model‐based process which is asymptotically Gaussian. In addition to test consistency, the proposed test can detect local alternatives distinct n‐1/2 from the null hypothesis. Due to the intractability of the asymptotic null distribution of the test statistic, we turn to two resampling approximations. We first use the well‐known bootstrap method to approximate critical values of the test. We then introduce a so‐called random symmetrization method for carrying out the test. Both methods perform very well with a sample of moderate size. A simulation study shows that the latter possesses better empirical powers and sizes for small samples.

The strong law under semiparametric random censorship models

We study integrals for arbitrary Borel-measurable functions with respect to a semiparametric estimator of the distribution function in the random censorship model. A strong law for these integrals is established which generalizes a corresponding result for the Cheng and Lin estimator.