A semiparametric cure model for interval‐censored data

There is a growing interest in the analysis of survival data with a cured proportion particularly in tumor recurrences studies. Biologically, it is reasonable to assume that the recurrence time is mainly affected by the overall health condition of the patient that depends on some covariates such as age, sex, or treatment type received. We propose a semiparametric frailty‐Cox cure model to quantify the overall health condition of the patient by a covariate‐dependent frailty that has a discrete mass at zero to characterize the cured patients, and a positive continuous part to characterize the heterogeneous health conditions among the uncured patients. A multiple imputation estimation method is proposed for the right‐censored case, which is further extended to accommodate interval‐censored data. Simulation studies show that the performance of the proposed method is highly satisfactory. For illustration, the model is fitted to a set of right‐censored melanoma incidence data and a set of interval‐censored breast cosmesis data. Our analysis suggests that patients receiving treatment of radiotherapy with adjuvant chemotherapy have a significantly higher probability of breast retraction, but also a lower hazard rate of breast retraction among those patients who will eventually experience the event with similar health conditions. The interpretation is very different to those based on models without a cure component that the treatment of radiotherapy with adjuvant chemotherapy significantly increases the risk of breast retraction.     

Semiparametric transformation models for interval‐censored data in the presence of a cure fraction

Mixed case interval‐censored data arise when the event of interest is known only to occur within an interval induced by a sequence of random examination times. Such data are commonly encountered in disease research with longitudinal follow‐up. Furthermore, the medical treatment has progressed over the last decade with an increasing proportion of patients being cured for many types of diseases. Thus, interest has grown in cure models for survival data which hypothesize a certain proportion of subjects in the population are not expected to experience the events of interest. In this article, we consider a two‐component mixture cure model for regression analysis of mixed case interval‐censored data. The first component is a logistic regression model that describes the cure rate, and the second component is a semiparametric transformation model that describes the distribution of event time for the uncured subjects. We propose semiparametric maximum likelihood estimation for the considered model. We develop an EM type algorithm for obtaining the semiparametric maximum likelihood estimators (SPMLE) of regression parameters and establish their consistency, efficiency, and asymptotic normality. Extensive simulation studies indicate that the SPMLE performs satisfactorily in a wide variety of settings. The proposed method is illustrated by the analysis of the hypobaric decompression sickness data from National Aeronautics and Space Administration.     

Normal frailty probit model for clustered interval‐censored failure time data

Clustered interval‐censored data commonly arise in many studies of biomedical research where the failure time of interest is subject to interval‐censoring and subjects are correlated for being in the same cluster. A new semiparametric frailty probit regression model is proposed to study covariate effects on the failure time by accounting for the intracluster dependence. Under the proposed normal frailty probit model, the marginal distribution of the failure time is a semiparametric probit model, the regression parameters can be interpreted as both the conditional covariate effects given frailty and the marginal covariate effects up to a multiplicative constant, and the intracluster association can be summarized by two nonparametric measures in simple and explicit form. A fully Bayesian estimation approach is developed based on the use of monotone splines for the unknown nondecreasing function and a data augmentation using normal latent variables. The proposed Gibbs sampler is straightforward to implement since all unknowns have standard form in their full conditional distributions. The proposed method performs very well in estimating the regression parameters as well as the intracluster association, and the method is robust to frailty distribution misspecifications as shown in our simulation studies. Two real‐life data sets are analyzed for illustration.     

Hazard ratio estimation for two‐sample case under interval‐censored failure time data

This paper discusses two‐sample comparison in the case of interval‐censored failure time data. For the problem, one common approach is to employ some nonparametric test procedures, which usually give some p‐values but not a direct or exact quantitative measure of the survival or treatment difference of interest. In particular, these procedures cannot provide a hazard ratio estimate, which is commonly used to measure the difference between the two treatments or samples. For interval‐censored data, a few nonparametric test procedures have been developed, but it does not seem to exist as a procedure for hazard ratio estimation. Corresponding to this, we present two procedures for nonparametric estimation of the hazard ratio of the two samples for interval‐censored data situations. They are generalizations of the corresponding procedures for right‐censored failure time data. An extensive simulation study is conducted to evaluate the performance of the two procedures and indicates that they work reasonably well in practice. For illustration, they are applied to a set of interval‐censored data arising from a breast cancer study.     

Cox model with interval‐censored covariate in cohort studies

In cohort studies the outcome is often time to a particular event, and subjects are followed at regular intervals. Periodic visits may also monitor a secondary irreversible event influencing the event of primary interest, and a significant proportion of subjects develop the secondary event over the period of follow‐up. The status of the secondary event serves as a time‐varying covariate, but is recorded only at the times of the scheduled visits, generating incomplete time‐varying covariates. While information on a typical time‐varying covariate is missing for entire follow‐up period except the visiting times, the status of the secondary event are unavailable only between visits where the status has changed, thus interval‐censored. One may view interval‐censored covariate of the secondary event status as missing time‐varying covariates, yet missingness is partial since partial information is provided throughout the follow‐up period. Current practice of using the latest observed status produces biased estimators, and the existing missing covariate techniques cannot accommodate the special feature of missingness due to interval censoring. To handle interval‐censored covariates in the Cox proportional hazards model, we propose an available‐data estimator, a doubly robust‐type estimator as well as the maximum likelihood estimator via EM algorithm and present their asymptotic properties. We also present practical approaches that are valid. We demonstrate the proposed methods using our motivating example from the Northern Manhattan Study.     

A quantile‐slicing approach for sufficient dimension reduction with censored responses

Sufficient dimension reduction (SDR) that effectively reduces the predictor dimension in regression has been popular in high‐dimensional data analysis. Under the presence of censoring, however, most existing SDR methods suffer. In this article, we propose a new algorithm to perform SDR with censored responses based on the quantile‐slicing scheme recently proposed by Kim et al. First, we estimate the conditional quantile function of the true survival time via the censored kernel quantile regression (Shin et al.) and then slice the data based on the estimated censored regression quantiles instead of the responses. Both simulated and real data analysis demonstrate promising performance of the proposed method.     

Analysing censored data in agricultural research: A review with examples and software tips

Metric data are usually assessed on a continuous scale with good precision, but sometimes agricultural researchers cannot obtain precise measurements of a variable. Values of such a variable cannot then be expressed as real numbers (e.g., 1.51 or 2.56), but often can be represented by intervals into which the values fall (e.g., from 1 to 2 or from 2 to 3). In this situation, statisticians talk about censoring and censored data, as opposed to missing data, where no information is available at all. Traditionally, in agriculture and biology, three methods have been used to analyse such data: (a) when intervals are narrow, some form of imputation (e.g., mid‐point imputation) is used to replace the interval and traditional methods for continuous data are employed (such as analyses of variance [ANOVA] and regression); (b) for time‐to‐event data, the cumulative proportions of individuals that experienced the event of interest are analysed, instead of the individual observed times‐to‐event; (c) when intervals are wide and many individuals are collected, non‐parametric methods of data analysis are favoured, where counts are considered instead of the individual observed value for each sample element. In this paper, we show that these methods may be suboptimal: The first one does not respect the process of data collection, the second leads to unreliable standard errors (SEs), while the third does not make full use of all the available information. As an alternative, methods of survival analysis for censored data can be useful, leading to reliable inferences and sound hypotheses testing. These methods are illustrated using three examples from plant and crop sciences.