Logspline Density Estimation under Censoring and Truncation

ABSTRACT. In this paper we consider logspline density estimation for data that may be left-truncated or right-censored. For randomly left-truncated and right-censored data the product-limit estimator is known to be a consistent estimator of the survivor function, having a faster rate of convergence than many density estimators. The product-limit estimator and B-splines are used to construct the logspline density estimate for possibly censored or truncated data. Rates of convergence are established when the log-density function is assumed to be in a Besov space. An algorithm involving a procedure similar to maximum likelihood, stepwise knot addition, and stepwise knot deletion is proposed for the estimation of the density function based upon sample data. Numerical examples are used to show the finite-sample performance of inference based on the logspline density estimation.     

A Poisson-multinomial mixture approach to grouped and right-censored counts

Although count data are often collected in social, psychological, and epidemiological surveys in grouped and right-censored categories, there is a lack of statistical methods simultaneously taking both grouping and right-censoring into account. In this research, we propose a new generalized Poisson-multinomial mixture approach to model grouped and right-censored (GRC) count data. Based on a mixed Poisson-multinomial process for conceptualizing grouped and right-censored count data, we prove that the new maximum-likelihood estimator (MLE-GRC) is consistent and asymptotically normally distributed for both Poisson and zero-inflated Poisson models. The use of the MLE-GRC, implemented in an R function, is illustrated by both statistical simulation and empirical examples. This research provides a tool for epidemiologists to estimate incidence from grouped and right-censored count data and lays a foundation for regression analyses of such data structure.     

Imputation for semiparametric transformation models with biased-sampling data

Widely recognized in many fields including economics, engineering, epidemiology, health sciences, technology and wildlife management, length-biased sampling generates biased and right-censored data but often provide the best information available for statistical inference. Different from traditional right-censored data, length-biased data have unique aspects resulting from their sampling procedures. We exploit these unique aspects and propose a general imputation-based estimation method for analyzing length-biased data under a class of flexible semiparametric transformation models. We present new computational algorithms that can jointly estimate the regression coefficients and the baseline function semiparametrically. The imputation-based method under the transformation model provides an unbiased estimator regardless whether the censoring is independent or not on the covariates. We establish large-sample properties using the empirical processes method. Simulation studies show that under small to moderate sample sizes, the proposed procedure has smaller mean square errors than two existing estimation procedures. Finally, we demonstrate the estimation procedure by a real data example.     

Estimation of a change point in the hazard rate of Lindley model under right censoring

In this article, we consider a single change point model for a sudden change in the hazard rate of Lindley distribution to model right-censored survival data. We derive the quantile function to generate random numbers from the proposed distribution by using the Lambert function. The maximum likelihood estimation method is used to estimate parameters of the change point model. A simulation study is also carried out to analyze the performance of the estimators. To validate our findings, a dataset on bone marrow transplant for patients of acute lymphoblastic leukemia is analyzed using the proposed model and is compared with the existing exponential single change point model.     

Generalized M-estimation for the accelerated failure time model

The accelerated failure time (AFT) model is an important regression tool to study the association between failure time and covariates. In this paper, we propose a robust weighted generalized M (GM) estimation for the AFT model with right-censored data by appropriately using the Kaplan–Meier weights in the GM–type objective function to estimate the regression coefficients and scale parameter simultaneously. This estimation method is computationally simple and can be implemented with existing software. Asymptotic properties including the root-n consistency and asymptotic normality are established for the resulting estimator under suitable conditions. We further show that the method can be readily extended to handle a class of nonlinear AFT models. Simulation results demonstrate satisfactory finite sample performance of the proposed estimator. The practical utility of the method is illustrated by a real data example.     

Consistent estimation of parameters and quantiles of the three-parameter gamma distribution based on Type-II right-censored data

In this paper, we propose a method of estimation of parameters and quantiles of the three-parameter gamma distribution based on Type-II right-censored data. In the proposed method, under mild conditions, the estimates always exist uniquely, and the estimators have consistency over the entire parameter space. Through Monte Carlo simulations, we further show that the proposed method performs well compared with another prominent method of estimation in terms of bias and root mean-squared error in small-sample situations. Finally, two real data sets are used for illustrating the proposed method.     

Estimation of Hazard, Density and Survivor Functions for Randomly Censored Data

Maximum likelihood estimation and goodness-of-fit techniques are used within a competing risks framework to obtain maximum likelihood estimates of hazard, density, and survivor functions for randomly right-censored variables. Goodness-of- fit techniques are used to fit distributions to the crude lifetimes, which are used to obtain an estimate of the hazard function, which, in turn, is used to construct the survivor and density functions of the net lifetime of the variable of interest. If only one of the crude lifetimes can be adequately characterized by a parametric model, then semi-parametric estimates may be obtained using a maximum likelihood estimate of one crude lifetime and the empirical distribution function of the other. Simulation studies show that the survivor function estimates from crude lifetimes compare favourably with those given by the product-limit estimator when crude lifetimes are chosen correctly. Other advantages are discussed.     

Semiparametric inference for the proportional odds model with time-dependent covariates

This paper studies estimation in the proportional odds model, with time-dependent covariates, based on right-censored data. The estimation procedure is an extension of the Yang and Prentice (J. Amer. Statist. Assoc. 94 (1999) 125) approach to the time-dependent covariate case. The proposed estimators include a class of minimum distance estimators defined through weighted empirical odds function. These estimators are shown to be strongly consistent and asymptotically normal, with variances that can be consistently estimated. It also contains a simulation study making comparison of some of the estimators in the class.     

Sample size determination for clustered right-censored data using copula models

Determination of an adequate sample size is critical to the design of research ventures. For clustered right-censored data, Manatunga and Chen [Sample size estimation for survival outcomes in cluster-randomized studies with small cluster sizes. Biometrics. 2000;56(2):616–621] proposed a sample size calculation based on considering the bivariate marginal distribution as Clayton copula model. In addition to the Clayton copula, other important family of copulas, such as Gumbel and Frank copulas are also well established in multivariate survival analysis. However, sample size calculation based on these assumptions has not been fully investigated yet. To broaden the scope of Manatunga and Chen [Sample size estimation for survival outcomes in cluster-randomized studies with small cluster sizes. Biometrics. 2000;56(2):616–621]'s research and achieve a more flexible sample size calculation for clustered right-censored data, we extended the work by assuming the marginal distribution as bivariate Gumbel and Frank copulas. We evaluate the performance of the proposed method and investigate the impacts of the accrual times, follow-up times and the within-clustered correlation effect of the study. The proposed method is applied to two real-world studies, and the R code is made available to users.     

Estimation of percentile residual life function with left-truncated and right-censored data

This article focuses on the estimation of percentile residual life function with left-truncated and right-censored data. Asymptotic normality and a pointwise confidence interval that does not require estimating the unknown underlying distribution function of the proposed empirical estimator are obtained. Some simulation studies and a real data example are used to illustrate our results.     

Nonparametric estimation in the illness-death model using prevalent data

We study nonparametric estimation of the illness-death model using left-truncated and right-censored data. The general aim is to estimate the multivariate distribution of a progressive multi-state process. Maximum likelihood estimation under censoring suffers from problems of uniqueness and consistency, so instead we review and extend methods that are based on inverse probability weighting. For univariate left-truncated and right-censored data, nonparametric maximum likelihood estimation can be considerably improved when exploiting knowledge on the truncation distribution. We aim to examine the gain in using such knowledge for inverse probability weighting estimators in the illness-death framework. Additionally, we compare the weights that use truncation variables with the weights that integrate them out, showing, by simulation, that the latter performs more stably and efficiently. We apply the methods to intensive care units data collected in a cross-sectional design, and discuss how the estimators can be easily modified to more general multi-state models.     

A Mass Redistribution Algorithm for Right Censored and Left Truncated Time to Event Data

Failure times are often right-censored and left-truncated. In this paper we give a mass redistribution algorithm for right-censored and/or left-truncated failure time data. We show that this algorithm yields the Kaplan-Meier estimator of the survival probability. One application of this algorithm in modeling the subdistribution hazard for competing risks data is studied. We give a product-limit estimator of the cumulative incidence function via modeling the subdistribution hazard. We show by induction that this product-limit estimator is identical to the left-truncated version of Aalen-Johansen (1978) estimator for the cumulative incidence function.     

Nonparametric inference on mean residual life function with length-biased right-censored data

Abstract

In survival or reliability studies, the mean residual life (MRL) function is an important characteristic in understanding the survival or ageing process. In this article, we consider the problem of nonparametric MRL function estimation with length-biased right-censored data. Two nonparametric estimators of the MRL are proposed and their weak convergence is presented. In order to evaluate the performance of these estimators, small Monte Carlo simulations are carried out. Results show that the proposed estimators work well especially when the sample size is small and their calculations are simple. Finally, a real data example is provided.     

A study of R2 measure under the accelerated failure time models

For right-censored data, the accelerated failure time (AFT) model is an alternative to the commonly used proportional hazards regression model. It is a linear model for the (log-transformed) outcome of interest, and is particularly useful for censored outcomes that are not time-to-event, such as laboratory measurements. We provide a general and easily computable definition of the R² measure of explained variation under the AFT model for right-censored data. We study its behavior under different censoring scenarios and under different error distributions; in particular, we also study its robustness when the parametric error distribution is misspecified. Based on Monte Carlo investigation results, we recommend the log-normal distribution as a robust error distribution to be used in practice for the parametric AFT model, when the R² measure is of interest. We apply our methodology to an alcohol consumption during pregnancy data set from Ukraine.     

Analysis of spatial frailty models by a weighted estimating equation

Spatially correlated survival data are frequently observed in ecological and epidemiological studies. An assumption in the clustered survival models is inter-cluster independence, which may not be adequate to model the dependence in spatial settings. For survival data, the likelihood function based on a spatial frailty may be complicated. In this paper, we develop a weighted estimating equation for spatially right-censored data. Some large sample properties for the estimate are developed. We also conduct simulations to compare estimation performance with other methods. A data set from a study of forest decline in Wisconsin is used to illustrate the proposed method.     

Local composite partial likelihood estimation for length-biased and right-censored data

Length-biased data, which are often encountered in engineering, economics and epidemiology studies, are generally subject to right censoring caused by the research ending or the follow-up loss. The structure of length-biased data is distinct from conventional survival data, since the independent censoring assumption is often violated due to the biased sampling. In this paper, a proportional hazard model with varying coefficients is considered for the length-biased and right-censored data. A local composite likelihood procedure is put forward for the estimation of unknown coefficient functions in the model, and large sample properties of the proposed estimators are also obtained. Additionally, an extensive simulation studies are conducted to assess the finite sample performance of the proposed method and a data set from the Academy Awards is analyzed.     

Absolute error criteria for bandwidth selection in density estimation from censored data

In Kernel density estimation, a criticism of bandwidth selection techniques which minimize squared error expressions is that they perform poorly when estimating tails of probability density functions. Techniques minimizing absolute error expressions are thought to result in more uniform performance and be potentially superior. An asympotic mean absolute error expression for nonparametric kernel density estimators from right-censored data is developed here. This expression is used to obtain local and global bandwidths that are optimal in the sense that they minimize asymptotic mean absolute error and integrated asymptotic mean absolute error, respectively. These estimators are illustrated fro eight data sets from known distributions. Computer simulation results are discussed, comparing the estimation methods with squared-error-based bandwidth selection for right-censored data.     

A note on Bayesian nonparametric regression function estimation

In this note the problem of nonparametric regression function estimation in a random design regression model with Gaussian errors is considered from the Bayesian perspective. It is assumed that the regression function belongs to a class of functions with a known degree of smoothness. A prior distribution on the given class can be induced by a prior on the coefficients in a series expansion of the regression function through an orthonormal system. The rate of convergence of the resulting posterior distribution is employed to provide a measure of the accuracy of the Bayesian estimation procedure defined by the posterior expected regression function. We show that the Bayes’ estimator achieves the optimal minimax rate of convergence under mean integrated squared error over the involved class of regression functions, thus being comparable to other popular frequentist regression estimators.     

Nonparametric estimation of mean residual quantile function under right censoring

In this paper, we develop non-parametric estimation of the mean residual quantile function based on right-censored data. Two non-parametric estimators, one based on the empirical quantile function and the other using the kernel smoothing method, are proposed. Asymptotic properties of the estimators are discussed. Monte Carlo simulation studies are conducted to compare the two estimators. The method is illustrated with the aid of two real data sets.     

The performance of standard and hybrid em algorithms for ml estimates of the normal mixture model with censoring

Many studies have been made of the performance of standard algorithms used to estimate the parameters of a mixture density, where data arise from two or more underlying populations. While these studies examine uncensored data, many mixture processes are right-censored. Therefore, this paper addresses the accuracy and efficiency of standard and hybrid algorithms under different degrees of right-censored data. While a common belief is that the EM algorithm is slow and inaccurate, we find that the EM generally exhibits excellent efficiency and accuracy. While extreme right censoring causes the EM to frequently fail to converge, a hybrid-EM algorithm is found to be superior at all levels of right-censoring.s