Estimation with right‐censored observations under a semi‐Markov model

The semi‐Markov process often provides a better framework than the classical Markov process for the analysis of events with multiple states. The purpose of this paper is twofold. First, we show that in the presence of right censoring, when the right end‐point of the support of the censoring time is strictly less than the right end‐point of the support of the semi‐Markov kernel, the transition probability of the semi‐Markov process is nonidentifiable, and the estimators proposed in the literature are inconsistent in general. We derive the set of all attainable values for the transition probability based on the censored data, and we propose a nonparametric inference procedure for the transition probability using this set. Second, the conventional approach to constructing confidence bands is not applicable for the semi‐Markov kernel and the sojourn time distribution. We propose new perturbation resampling methods to construct these confidence bands. Different weights and transformations are explored in the construction. We use simulation to examine our proposals and illustrate them with hospitalization data from a recent cancer survivor study. The Canadian Journal of Statistics 41: 237–256; 2013 © 2013 Statistical Society of Canada     

A linear transformation model for multivariate interval‐censored failure time data

This paper discusses multivariate interval‐censored failure time data observed when several correlated survival times of interest exist and only interval censoring is available for each survival time. Such data occur in many fields, for instance, studies of the development of physical symptoms or diseases in several organ systems. A marginal inference approach was used to create a linear transformation model and applied to bivariate interval‐censored data arising from a diabetic retinopathy study and an AIDS study. The results of simulation studies that were conducted to evaluate the performance of the presented approach suggest that it performs well. The Canadian Journal of Statistics 41: 275–290; 2013 © 2013 Statistical Society of Canada     

A generalized Fleming and Harrington's class of tests for interval‐censored data

The class $G^{\rho,\lambda }$ of weighted log‐rank tests proposed by Fleming & Harrington [Fleming & Harrington (1991) Counting Processes and Survival Analysis, Wiley, New York] has been widely used in survival analysis and is nowadays, unquestionably, the established method to compare, nonparametrically, k different survival functions based on right‐censored survival data. This paper extends the $G^{\rho,\lambda }$ class to interval‐censored data. First we introduce a new general class of rank based tests, then we show the analogy to the above proposal of Fleming & Harrington. The asymptotic behaviour of the proposed tests is derived using an observed Fisher information approach and a permutation approach. Aiming to make this family of tests interpretable and useful for practitioners, we explain how to interpret different choices of weights and we apply it to data from a cohort of intravenous drug users at risk for HIV infection. The Canadian Journal of Statistics 40: 501–516; 2012 © 2012 Statistical Society of Canada     

Restricted maximum likelihood estimation of joint mean‐covariance models

The class of joint mean‐covariance models uses the modified Cholesky decomposition of the within subject covariance matrix in order to arrive to an unconstrained, statistically meaningful reparameterisation. The new parameterisation of the covariance matrix has two sets of parameters that separately describe the variances and correlations. Thus, with the mean or regression parameters, these models have three sets of distinct parameters. In order to alleviate the problem of inefficient estimation and downward bias in the variance estimates, inherent in the maximum likelihood estimation procedure, the usual REML estimation procedure adjusts for the degrees of freedom lost due to the estimation of the mean parameters. Because of the parameterisation of the joint mean covariance models, it is possible to adapt the usual REML procedure in order to estimate the variance (correlation) parameters by taking into account the degrees of freedom lost by the estimation of both the mean and correlation (variance) parameters. To this end, here we propose adjustments to the estimation procedures based on the modified and adjusted profile likelihoods. The methods are illustrated by an application to a real data set and simulation studies. The Canadian Journal of Statistics 40: 225–242; 2012 © 2012 Statistical Society of Canada     

Constrained estimation and likelihood intervals for censored data

The authors propose a reduction technique and versions of the EM algorithm and the vertex exchange method to perform constrained nonparametric maximum likelihood estimation of the cumulative distribution function given interval censored data. The constrained vertex exchange method can be used in practice to produce likelihood intervals for the cumulative distribution function. In particular, the authors show how to produce a confidence interval with known asymptotic coverage for the survival function given current status data.     

Efficient estimation for the proportional hazards model with competing risks and current status data

The proportional hazards model is the most commonly used model in regression analysis of failure time data and has been discussed by many authors under various situations (Kalbfleisch & Prentice, 2002. The Statistical Analysis of Failure Time Data, Wiley, New York). This paper considers the fitting of the model to current status data when there exist competing risks, which often occurs in, for example, medical studies. The maximum likelihood estimates of the unknown parameters are derived and their consistency and convergence rate are established. Also we show that the estimates of regression coefficients are efficient and have asymptotically normal distributions. Simulation studies are conducted to assess the finite sample properties of the estimates and an illustrative example is provided. The Canadian Journal of Statistics © 2009 Statistical Society of Canada     

Constrained nonparametric maximum likelihood estimation of stochastically ordered survivor functions

This paper considers estimators of survivor functions subject to a stochastic ordering constraint based on right censored data. We present the constrained nonparametric maximum likelihood estimator (C‐NPMLE) of the survivor functions in one‐and two‐sample settings where the survivor distributions could be discrete or continuous and discuss the non‐uniqueness of the estimators. We also present a computationally efficient algorithm to obtain the C‐NPMLE. To address the possibility of non‐uniqueness of the C‐NPMLE of $S_1(t)$ when $S_1(t)\le S_2(t)$ , we consider the maximum C‐NPMLE (MC‐NPMLE) of $S_1(t)$ . In the one‐sample case with arbitrary upper bound survivor function $S_2(t)$ , we present a novel and efficient algorithm for finding the MC‐NPMLE of $S_1(t)$ . Dykstra ( 1982 ) also considered constrained nonparametric maximum likelihood estimation for such problems, however, as we show, Dykstra's method has an error and does not always give the C‐NPMLE. We corrected this error and simulation shows improvement in efficiency compared to Dykstra's estimator. Confidence intervals based on bootstrap methods are proposed and consistency of the estimators is proved. Data from a study on larynx cancer are analysed to illustrate the method. The Canadian Journal of Statistics 40: 22–39; 2012 © 2012 Statistical Society of Canada     

Local likelihood density estimation for interval censored data

The authors propose a class of procedures for local likelihood estimation from data that are either interval‐censored or that have been aggregated into bins. One such procedure relies on an algorithm that generalizes existing self‐consistency algorithms by introducing kernel smoothing at each step of the iteration. The entire class of procedures yields estimates that are obtained as solutions of fixed point equations. By discretizing and applying numerical integration, the authors use fixed point theory to study convergence of algorithms for the class. Rapid convergence is effected by the implementation of a local EM algorithm as a global Newton iteration. The latter requires an explicit solution of the local likelihood equations which can be found by using the symbolic Newton‐Raphson algorithm, if necessary.     

Semiparametric median residual life model and inference

For randomly censored data, the authors propose a general class of semiparametric median residual life models. They incorporate covariates in a generalized linear form while leaving the baseline median residual life function completely unspecified. Despite the non‐identifiability of the survival function for a given median residual life function, a simple and natural procedure is proposed to estimate the regression parameters and the baseline median residual life function. The authors derive the asymptotic properties for the estimators, and demonstrate the numerical performance of the proposed method through simulation studies. The median residual life model can be easily generalized to model other quantiles, and the estimation method can also be applied to the mean residual life model. The Canadian Journal of Statistics 38: 665–679; 2010 © 2010 Statistical Society of Canada     

Least squares estimation of varying‐coefficient hazard regression with application to breast cancer dose‐intensity data

To enhance modeling flexibility, the authors propose a nonparametric hazard regression model, for which the ordinary and weighted least squares estimation and inference procedures are studied. The proposed model does not assume any parametric specifications on the covariate effects, which is suitable for exploring the nonlinear interactions between covariates, time and some exposure variable. The authors propose the local ordinary and weighted least squares estimators for the varying‐coefficient functions and establish the corresponding asymptotic normality properties. Simulation studies are conducted to empirically examine the finite‐sample performance of the new methods, and a real data example from a recent breast cancer study is used as an illustration. The Canadian Journal of Statistics 37: 659–674; 2009 © 2009 Statistical Society of Canada     

Smoothed empirical likelihood confidence intervals for the relative distribution with left‐truncated and right‐censored data

The study of differences among groups is an interesting statistical topic in many applied fields. It is very common in this context to have data that are subject to mechanisms of loss of information, such as censoring and truncation. In the setting of a two‐sample problem with data subject to left truncation and right censoring, we develop an empirical likelihood method to do inference for the relative distribution. We obtain a nonparametric generalization of Wilks' theorem and construct nonparametric pointwise confidence intervals for the relative distribution. Finally, we analyse the coverage probability and length of these confidence intervals through a simulation study and illustrate their use with a real data set on gastric cancer. The Canadian Journal of Statistics 38: 453–473; 2010 © 2010 Statistical Society of Canada