Defective regression models for cure rate modeling with interval-censored data

Regression models in survival analysis are most commonly applied for right-censored survival data. In some situations, the time to the event is not exactly observed, although it is known that the event occurred between two observed times. In practice, the moment of observation is frequently taken as the event occurrence time, and the interval-censored mechanism is ignored. We present a cure rate defective model for interval-censored event-time data. The defective distribution is characterized by a density function whose integration assumes a value less than one when the parameter domain differs from the usual domain. We use the Gompertz and inverse Gaussian defective distributions to model data containing cured elements and estimate parameters using the maximum likelihood estimation procedure. We evaluate the performance of the proposed models using Monte Carlo simulation studies. Practical relevance of the models is illustrated by applying datasets on ovarian cancer recurrence and oral lesions in children after liver transplantation, both of which were derived from studies performed at A.C. Camargo Cancer Center in São Paulo, Brazil.

Nonparametric comparison of survival functions based on interval-censored data with unequal censoring

Nonparametric comparison of survival functions is one of the most commonly required task in failure time studies such as clinical trials, and for this, many procedures have been developed under various situations. This paper considers a situation that often occurs in practice but has not been discussed much: the comparison based on interval-censored data in the presence of unequal censoring. That is, one observes only interval-censored data, and the distributions of or the mechanisms behind censoring variables may depend on treatments and thus be different for the subjects in different treatment groups. For the problem, a test procedure is developed that takes into account the difference between the distributions of the censoring variables, and the asymptotic normality of the test statistics is given. For the assessment of the performance of the procedure, a simulation study is conducted and suggests that it works well for practical situations. An illustrative example is provided. Copyright © 2017 John Wiley & Sons, Ltd.

Weighted logrank tests for interval censored data when assessment times depend on treatment

We consider weighted logrank tests for interval censored data when assessment times may depend on treatment, and for each individual, we only use the two assessment times that bracket the event of interest. It is known that treating finite right endpoints as observed events can substantially inflate the type I error rate under assessment-treatment dependence (ATD), but the validity of several other implementations of weighted logrank tests (score tests, permutation tests, multiple imputation tests) has not been studied in this situation. With a bounded number of unique assessment times, the score test under the grouped continuous model retains the type I error rate asymptotically under ATD; however, although the approximate permutation test based on the permutation central limit theorem is not asymptotically valid under every ATD scenario, we show through simulation that in many ATD scenarios, it retains the type I error rate better than the score test. We show a case where the approximate permutation test retains the type I error rate when the exact permutation test does not. We study and modify the multiple imputation logrank tests of Huang, Lee, and Yu (2008, Statistics in Medicine, 27: 3217-3226), showing that the distribution of the rank-like scores asymptotically does not depend on the assessment times. We show through simulations that our modifications of the multiple imputation logrank tests retain the type I error rate in all cases studied, even with ATD and a small number of individuals in each treatment group. Simulations were performed using the interval R package.

Tutorial on methods for interval-censored data and their implementation in R

Interval censoring is encountered in many practical situations when the event of interest cannot be observed and it is only known to have occurred within a time window. The theory for the analysis of interval-censored data has been developed over the past three decades and several reviews have been written. However, it is still a common practice in medical and reliability studies to simplify the interval censoring structure of the data into a more standard right censoring situation by, for instance, imputing the midpoint of the censoring interval. The availability of software for right censoring might well be the main reason for this simplifying practice. In contrast, several methods have been developed to deal with interval-censored data and the corresponding algorithms to make the procedures feasible are scattered across the statistical software or remain behind the personal computers of many researchers. The purpose of this tutorial is to present, in a pedagogical and unified manner, the methodology and the available software for analyzing interval-censored data. The paper covers frequentist non-parametric, parametric and semiparametric estimating approaches, non-parametric tests for comparing survival curves and a section on simulation of interval-censored data. The methods and the software are described using the data from a dental study.

Linear transformation models for interval-censored data

In statistical analysis, when the value of a random variable is only known to be between two bounds, we say that this random variable is interval censored. This complicated censoring pattern is a common problem in research fields such as clinical trials or actuarial studies and raises challenges for statistical analysis. In this paper, we focus on regression analysis of case 2 interval-censored data. We first briefly review existing regression methods and an estimation approach under the class of linear transformation models developed by Zhang et al. We then propose a method for survival probability prediction via generalized estimating equations. We also consider a graphical model checking technique and a model selection tool. Some theoretical properties are established and the performance of our procedures is evaluated and illustrated by numerical studies including a real-life data analysis.

Urn sampling, interval censoring and proportional hazard models

This paper proposes a new distribution-free statistical method for testing hypotheses about covariates for survival data having simultaneously right-, left- and interval-censored survival times. The new test is motivated by the analogue between urn sampling and the Cox’s proportional hazard models. Investigations of the significance levels and power as a function of the proportion of interval-censored observations and interval length show that the test performs well for most censoring situations encountered in practice. Simulation results also suggest that there is negligible loss of power in the practical situation in which the mean interval length for interval-censored observations is less than the mean survival time. This holds even with heavy interval censoring. Comparison with the widely used Mantel’s method for comparing two groups shows that the power of the new method appears to be superior. Furthermore, the test is relatively simple to carry out and generalizes to comparing k populations as well as the testing of general linear hypothesis for arbitrary covariates.

An overview of methods for interval-censored data with an emphasis on applications in dentistry

Interval-censored time-to-event data occur in many medical areas, with dentistry or AIDS research being typical representatives. This article reviews methods for the analysis of such data, with an emphasis on the use of the accelerated failure time (AFT) model. A flexible AFT model (avoiding parametric assumptions on the distribution of the error term) is described in greater detail and is used to solve a typical dental question in a longitudinal oral health study.

Simulation study and implementation of the tests based on weighted Turnbull's estimators for interval-censored data

Several tests for comparing k groups of interval-censored data based on Turnbull's estimator of a survival function are compared by a Monte Carlo simulation study. The tests under consideration include the IWD test proposed by Petroni and Wolfe, the score test derived by Finkelstein, and a logrank type test suggested by Sun. With a large sample size and in a two-sample comparison, simulation results suggest that the IWD test is applicable for early and crossing hazard difference alternatives, whereas Finkelstein's score test, Sun's test and the logrank test are slightly more powerful than the IWD tests under late hazard difference alternatives. The IWD test is designed for the two-sample testing problem. With a large sample size and in a k-sample comparison, Finkelstein's score test is slightly better than the other tests under a proportional hazards model, while Sun's test and the logrank test are slightly better than the score test under early and crossing hazard difference alternatives. However, none of the test statistics is uniformly better than the others for the alternatives considered in this paper with both non-overlapping and overlapping interval-censored data.

Using conditional logistic regression to fit proportional odds models to interval censored data

An easily implemented approach to fitting the proportional odds regression model to interval-censored data is presented. The approach is based on using conditional logistic regression routines in standard statistical packages. Using conditional logistic regression allows the practitioner to sidestep complications that attend estimation of the baseline odds ratio function. The approach is applicable both for interval-censored data in settings in which examinations continue regardless of whether the event of interest has occurred and for current status data. The methodology is illustrated through an application to data from an AIDS study of the effect of treatment with ZDV+ddC versus ZDV alone on 50% drop in CD4 cell count from baseline level. Simulations are presented to assess the accuracy of the procedure.

Comparing several score tests for interval censored data

I create a general model to perform score tests on interval censored data. Special cases of this model are the score tests of Finkelstein, Sun and Fay. Although Sun's was derived as a test for discrete data and Finkelstein's and Fay's tests were derived under a grouped continuous model, by writing all tests under one general model we see that as long as the regularity conditions hold, any of these three classes of tests may be applied to either grouped continuous or discrete data. I show the equivalence between the weighted logrank form of the general test and the form with a term for each individual, the form often used with permutation tests. From the weighted logrank form of the tests, we see that Sun's and Finkelstein's test are similar, giving constant (or approximately constant) weights to differences in survival distributions over time. In contrast, the proportional odds model (Fay's model with logistic error) gives more weight to early differences.

Linear rank tests for doubly censored data

This paper establishes the asymptotic properties of the non-parametric maximum likelihood estimator of the survival function based on data censored both from the left and the right. We show that the sample mean based on the survival estimator converges to a normal distribution, and we derive non-parametric tests for the differences in survival functions. The methods are applied to a study of dementia associated with Parkinson's disease.