Semiparametric varying‐coefficient model for interval censored data with a cured proportion

Varying‐coefficient models have claimed an increasing portion of statistical research and are now applied to censored data analysis in medical studies. We incorporate such flexible semiparametric regression tools for interval censored data with a cured proportion. We adopted a two‐part model to describe the overall survival experience for such complicated data. To fit the unknown functional components in the model, we take the local polynomial approach with bandwidth chosen by cross‐validation. We establish consistency and asymptotic distribution of the estimation and propose to use bootstrap for inference. We constructed a BIC‐type model selection method to recommend an appropriate specification of parametric and nonparametric components in the model. We conducted extensive simulations to assess the performance of our methods. An application on a decompression sickness data illustrates our methods. Copyright © 2013 John Wiley & Sons, Ltd.     

A test for informative censoring in clustered survival data

Frailty models are frequently used to analyse clustered survival data. The assumption of non-informative censoring is commonly used by these models, even though it may not be true in many situations. This article proposes a test for this assumption. It uses the estimated correlation between two types of martingale residuals, one from a model for failure and the other from a model for censoring. It distinguishes two types of censoring, namely withdrawal and the end of the study. Simulation studies show that the proposed test works well under various scenarios. For illustration, the test is applied to a data set for kidney disease patients from multiple dialysis centres.     

Modelling clustered survival data from multicentre clinical trials

In randomized clinical trials, subjects are recruited at multiple study centres. Factors that vary across centres may exert a powerful independent influence on study outcomes. A common problem is how to incorporate these centre effects into the analysis of censored time-to-event data. We survey various methods and find substantial advantages in the gamma frailty model. This approach compares favourably with competing methods and appears minimally affected by violation of the assumption of a gamma-distributed frailty. Recent computational advances make use of the gamma frailty model a practical and appealing tool for addressing centre effects in the analysis of multicentre trials.     

On the equivalence of some medical cost estimators with censored data

In clinical trials comparing different treatments and in health economics and outcomes research, medical costs are frequently analysed to evaluate the economical impacts of new treatment options and economic values of health-care utilization. Since Lin et al.'s first finding in the problem of applying the survival analysis techniques to the cost data, many new methods have been proposed. In this report, we establish analytic relationships among several widely adopted medical cost estimators that are seemingly different. Specifically, we report the equivalence among various estimators that were introduced by Lin et al., Bang and Tsiatis, and Zhao and Tian. Lin's estimators are formerly known to be asymptotically unbiased in some discrete censoring situations and biased otherwise, whereas all other estimators discussed here are consistent for the expected medical cost. Thus, we identify conditions under which these estimators become identical and, consequently, the biased estimators achieve consistency. We illustrate these relationships using an example from a clinical trial examining the effectiveness of implantable cardiac defibrillators in preventing death among people who had prior myocardial infarctions.     

Estimation of survival functions in interval and right censored data using STD behavioural diaries

In investigations of sexually transmitted diseases (STDs), true infection times are often either interval censored or right censored. For such data, reliable estimation of the survival function is difficult to obtain. In this research, we propose a resampling based method for the estimation of the survival function using auxiliary behavioural information provided by daily diaries. By imputing the unknown infection time from a list of sexual encounter times recorded in the diaries, the proposed procedure can be easily implemented using existing estimation procedures for right censored data. Our simulations show that the proposed procedure consistently outperforms the existing ad hoc estimation methods and produces smaller mean integrated squared errors (MISE). We illustrate this method by analysing infection data obtained from an ongoing STD investigation of adolescent females.     

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. Published 2012. This article is a US Government work and is in the public domain in the USA.     

A multiple imputation approach for clustered interval‐censored survival data

Multivariate interval‐censored failure time data arise commonly in many studies of epidemiology and biomedicine. Analysis of these type of data is more challenging than the right‐censored data. We propose a simple multiple imputation strategy to recover the order of occurrences based on the interval‐censored event times using a conditional predictive distribution function derived from a parametric gamma random effects model. By imputing the interval‐censored failure times, the estimation of the regression and dependence parameters in the context of a gamma frailty proportional hazards model using the well‐developed EM algorithm is made possible. A robust estimator for the covariance matrix is suggested to adjust for the possible misspecification of the parametric baseline hazard function. The finite sample properties of the proposed method are investigated via simulation. The performance of the proposed method is highly satisfactory, whereas the computation burden is minimal. The proposed method is also applied to the diabetic retinopathy study (DRS) data for illustration purpose and the estimates are compared with those based on other existing methods for bivariate grouped survival data. Copyright © 2010 John Wiley & Sons, Ltd.     

A multivariate cure model for left‐censored and right‐censored data with application to colorectal cancer screening patterns

We develop a multivariate cure survival model to estimate lifetime patterns of colorectal cancer screening. Screening data cover long periods of time, with sparse observations for each person. Some events may occur before the study begins or after the study ends, so the data are both left‐censored and right‐censored, and some individuals are never screened (the ‘cured’ population). We propose a multivariate parametric cure model that can be used with left‐censored and right‐censored data. Our model allows for the estimation of the time to screening as well as the average number of times individuals will be screened. We calculate likelihood functions based on the observations for each subject using a distribution that accounts for within‐subject correlation and estimate parameters using Markov chain Monte Carlo methods. We apply our methods to the estimation of lifetime colorectal cancer screening behavior in the SEER‐Medicare data set. Copyright © 2016 John Wiley & Sons, Ltd.     

A shared random effects model for censored medical costs and mortality

In this paper, we propose a model for medical costs recorded at regular time intervals, e.g. every month, as repeated measures in the presence of a terminating event, such as death. Prior models have related monthly medical costs to time since entry, with extra costs at the final observations at the time of death. Our joint model for monthly medical costs and survival time incorporates two important new features. First, medical cost and survival may be correlated because more 'frail' patients tend to accumulate medical costs faster and die earlier. A joint random effects model is proposed to account for the correlation between medical costs and survival by a shared random effect. Second, monthly medical costs usually increase during the time period prior to death because of the intensive care for dying patients. We present a method for estimating the pattern of cost prior to death, which is applicable if the pattern can be characterized as an additive effect that is limited to a fixed time interval, say b units of time before death. This 'turn back time' method for censored observations censors cost data b units of time before the actual censoring time, while keeping the actual censoring time for the survival data. Time-dependent covariates can be included. Maximum likelihood estimation and inference are carried out through a Monte Carlo EM algorithm with a Metropolis-Hastings sampler in the E-step. An analysis of monthly outpatient EPO medical cost data for dialysis patients is presented to illustrate the proposed methods.     

Regression analysis of failure time data with informative interval censoring

Interval censoring arises when a subject misses prescheduled visits at which the failure is to be assessed. Most existing approaches for analysing interval-censored failure time data assume that the censoring mechanism is independent of the true failure time. However, there are situations where this assumption may not hold. In this paper, we consider such a situation in which the dependence structure between the censoring variables and the failure time can be modelled through some latent variables and a method for regression analysis of failure time data is proposed. The method makes use of the proportional hazards frailty model and an EM algorithm is presented for estimation. Finite sample properties of the proposed estimators of regression parameters are examined through simulation studies and we illustrate the method with data from an AIDS study.     

Joint analysis of multi-level repeated measures data and survival: an application to the end stage renal disease (ESRD) data

Shared random effects models have been increasingly common in the joint analyses of repeated measures (e.g. CD4 counts, hemoglobin levels) and a correlated failure time such as death. In this paper we study several shared random effects models in the multi-level repeated measures data setting with dependent failure times. Distinct random effects are used to characterize heterogeneity in repeated measures at different levels. The hazard of death may be dependent on random effects from various levels. To simplify the estimation procedure, we adopt the Gaussian quadrature technique with a piecewise log-linear baseline hazard for the death process, which can be conveniently implemented in the freely available software aML. As an example, we analyze repeated measures of hematocrit level and survival for end stage renal disease patients clustered within a randomly selected 126 dialysis centers in the U.S. renal data system data set. Our model is very comprehensive yet easy to implement, making it appealing to general statistical practitioners.     

Predictive accuracy of markers or risk scores for interval censored survival data

Methods for the evaluation of the predictive accuracy of biomarkers with respect to survival outcomes subject to right censoring have been discussed extensively in the literature. In cancer and other diseases, survival outcomes are commonly subject to interval censoring by design or due to the follow up schema. In this article, we present an estimator for the area under the time-dependent receiver operating characteristic (ROC) curve for interval censored data based on a nonparametric sieve maximum likelihood approach. We establish the asymptotic properties of the proposed estimator and illustrate its finite-sample properties using a simulation study. The application of our method is illustrated using data from a cancer clinical study. An open-source R package to implement the proposed method is available on Comprehensive R Archive Network.     

An Expectation Maximization algorithm for fitting the generalized odds‐rate model to interval censored data

The generalized odds‐rate model is a class of semiparametric regression models, which includes the proportional hazards and proportional odds models as special cases. There are few works on estimation of the generalized odds‐rate model with interval censored data because of the challenges in maximizing the complex likelihood function. In this paper, we propose a gamma‐Poisson data augmentation approach to develop an Expectation Maximization algorithm, which can be used to fit the generalized odds‐rate model to interval censored data. The proposed Expectation Maximization algorithm is easy to implement and is computationally efficient. The performance of the proposed method is evaluated by comprehensive simulation studies and illustrated through applications to datasets from breast cancer and hemophilia studies. In order to make the proposed method easy to use in practice, an R package ‘ICGOR’ was developed. Copyright © 2016 John Wiley & Sons, Ltd.     

Evaluating surrogate marker information using censored data

Given the long follow‐up periods that are often required for treatment or intervention studies, the potential to use surrogate markers to decrease the required follow‐up time is a very attractive goal. However, previous studies have shown that using inadequate markers or making inappropriate assumptions about the relationship between the primary outcome and surrogate marker can lead to inaccurate conclusions regarding the treatment effect. Currently available methods for identifying and validating surrogate markers tend to rely on restrictive model assumptions and/or focus on uncensored outcomes. The ability to use such methods in practice when the primary outcome of interest is a time‐to‐event outcome is difficult because of censoring and missing surrogate information among those who experience the primary outcome before surrogate marker measurement. In this paper, we propose a novel definition of the proportion of treatment effect explained by surrogate information collected up to a specified time in the setting of a time‐to‐event primary outcome. Our proposed approach accommodates a setting where individuals may experience the primary outcome before the surrogate marker is measured. We propose a robust non‐parametric procedure to estimate the defined quantity using censored data and use a perturbation‐resampling procedure for variance estimation. Simulation studies demonstrate that the proposed procedures perform well in finite samples. We illustrate the proposed procedures by investigating two potential surrogate markers for diabetes using data from the Diabetes Prevention Program. Copyright © 2017 John Wiley & Sons, Ltd.     

The power of testing a semi-parametric shared gamma frailty parameter in failure time data

Frailty is usually modeled as an unobserved random variable acting multiplicatively on the baseline hazard function, where a shared unobserved quantity in the intensity induces a positive correlation among the observed failure times. Using the asymptotic properties of the nonparametric maximum likelihood estimator for the gamma frailty model, we derive a power function for testing the shared frailty parameter and evaluate the effect of number of groups and number of individuals per groups on the power of the test by simulation studies. The results show that choosing between 8 and 25 groups with sample sizes between 200 and 500 individuals will be enough to get a high power.     

Semiparametric regression analysis of failure time data with dependent interval censoring

Interval‐censored failure‐time data arise when subjects are examined or observed periodically such that the failure time of interest is not examined exactly but only known to be bracketed between two adjacent observation times. The commonly used approaches assume that the examination times and the failure time are independent or conditionally independent given covariates. In many practical applications, patients who are already in poor health or have a weak immune system before treatment usually tend to visit physicians more often after treatment than those with better health or immune system. In this situation, the visiting rate is positively correlated with the risk of failure due to the health status, which results in dependent interval‐censored data. While some measurable factors affecting health status such as age, gender, and physical symptom can be included in the covariates, some health‐related latent variables cannot be observed or measured. To deal with dependent interval censoring involving unobserved latent variable, we characterize the visiting/examination process as recurrent event process and propose a joint frailty model to account for the association of the failure time and visiting process. A shared gamma frailty is incorporated into the Cox model and proportional intensity model for the failure time and visiting process, respectively, in a multiplicative way. We propose a semiparametric maximum likelihood approach for estimating model parameters and show the asymptotic properties, including consistency and weak convergence. Extensive simulation studies are conducted and a data set of bladder cancer is analyzed for illustrative purposes. Copyright © 2017 John Wiley & Sons, Ltd.     

Joint modeling longitudinal semi‐continuous data and survival,with application to longitudinal medical cost data

It has been increasingly common to analyze simultaneously repeated measures and time to failure data. In this paper we propose a joint model when the repeated measures are semi‐continuous, characterized by the presence of a large portion of zero values, as well as right skewness of non zero (positive) values. Examples include monthly medical costs, car insurance annual claims, or annual number of hospitalization days. A random effects two‐part model is used to describe respectively the odds of being positive and the level of positive values. The random effects from the two‐part model are then incorporated in the hazard of the failure time to form the joint model. The estimation can be carried out by Gaussian quadrature techniques conveniently implemented in SAS Proc NLMIXED. Our model is applied to longitudinal (monthly) medical costs of 1455 chronic heart‐failure patients from the clinical data repository at the University of Virginia. Copyright © 2008 John Wiley & Sons, Ltd.     

Semi-parametric regression models for cost-effectiveness analysis: improving the efficiency of estimation from censored data

In a cost-effectiveness analysis using clinical trial data, estimates of the between-treatment difference in mean cost and mean effectiveness are needed. Several methods for handling censored data have been suggested. One of them is inverse-probability weighting, and has the advantage that it can also be applied to estimate the parameters from a linear regression of the mean. Such regression models can potentially estimate the treatment contrast more precisely, since some of the residual variance can be explained by baseline covariates. The drawback, however, is that inverse-probability weighting may not be efficient. Using existing results on semi-parametric efficiency, this paper derives the semi-parametric efficient parameter estimates for regression of mean cost, mean quality-adjusted survival time and mean survival time. The performance of these estimates is evaluated through a simulation study. Applying both the new estimators and the inverse-probability weighted estimators to the results of the EVALUATE trial showed that the new estimators achieved a halving of the variance of the estimated treatment contrast for cost. Some practical suggestions for choosing an estimator are offered.     

Longitudinal analysis of censored medical cost data

This paper applies the inverse probability weighted (IPW) least-squares method to estimate the effects of treatment on total medical cost, subject to censoring, in a panel-data setting. IPW pooled ordinary-least squares (POLS) and IPW random effects (RE) models are used. Because total medical cost might not be independent of survival time under administrative censoring, unweighted POLS and RE cannot be used with censored data, to assess the effects of certain explanatory variables. Even under the violation of this independency, IPW estimation gives consistent asymptotic normal coefficients with easily computable standard errors. A traditional and robust form of the Hausman test can be used to compare weighted and unweighted least squares estimators. The methods are applied to a sample of 201 Medicare beneficiaries diagnosed with lung cancer between 1994 and 1997.     

Precision of rate estimation under uniform interval censoring

Medical events are often recorded not at their time of occurrence but rather at a subsequent examination. In survival analysis, this is called interval censoring. This paper presents a relationship between the sampling error of the estimated event rate and the number of examinations, which are assumed to be evenly spaced throughout the study. This relationship should help balance the cost of additional examination rounds against the precision gained.