Analyzing left‐truncated and right‐censored infectious disease cohort data with interval‐censored infection onset

In an infectious disease cohort study, individuals who have been infected with a pathogen are often recruited for follow up. The period between infection and the onset of symptomatic disease, referred to as the incubation period, is of interest because of its importance on disease surveillance and control. However, the incubation period is often difficult to ascertain due to the uncertainty associated with asymptomatic infection onset time. An additional complication is that the observed infected subjects are likely to have longer incubation periods due to the prevalent sampling. In this article, we demonstrate how to estimate the distribution of the incubation period with the uncertain infection onset, subject to left‐truncation and right‐censoring. We employ a family of sufficiently general parametric models, the generalized odds‐rate class of regression models, for the underlying incubation period and its correlation with covariates. In simulation studies, we assess the finite sample performance of the model fitting and hazard function estimation. The proposed method is illustrated on data from the HIV/AIDS study on injection drug users admitted to a detoxification program in Badalona, Spain.     

Frailty proportional mean residual life regression for clustered survival data: A hierarchical quasi-likelihood method

Frailty models are widely used to model clustered survival data arising in multicenter clinical studies. In the literature, most existing frailty models are proportional hazards, additive hazards, or accelerated failure time model based. In this paper, we propose a frailty model framework based on mean residual life regression to accommodate intracluster correlation and in the meantime provide easily understand and straightforward interpretation for the effects of prognostic factors on the expectation of the remaining lifetime. To overcome estimation challenges, a novel hierarchical quasi-likelihood approach is developed by making use of the idea of hierarchical likelihood in the construction of the quasi-likelihood function, leading to hierarchical estimating equations. Simulation results show favorable performance of the method regardless of frailty distributions. The utility of the proposed methodology is illustrated by its application to the data from a multi-institutional study of breast cancer.     

Regression with incomplete covariates and left‐truncated time‐to‐event data

Studies of chronic diseases routinely sample individuals subject to conditions on an event time of interest. In epidemiology, for example, prevalent cohort studies aiming to evaluate risk factors for survival following onset of dementia require subjects to have survived to the point of screening. In clinical trials designed to assess the effect of experimental cancer treatments on survival, patients are required to survive from the time of cancer diagnosis to recruitment. Such conditions yield samples featuring left‐truncated event time distributions. Incomplete covariate data often arise in such settings, but standard methods do not deal with the fact that individuals’ covariate distributions are also affected by left truncation. We describe an expectation–maximization algorithm for dealing with incomplete covariate data in such settings, which uses the covariate distribution conditional on the selection criterion. We describe an extension to deal with subgroup analyses in clinical trials for the case in which the stratification variable is incompletely observed. Copyright © 2012 John Wiley & Sons, Ltd.     

Summarising censored survival data using the mean residual life function

The mean residual life function provides a clear and simple summary of the effect of a treatment or a risk factor in units of time, avoiding hazard ratios or probability scales, which require careful interpretation. Estimation of the mean residual life is complicated by the upper tail of the survival distribution not being observed as, for example, patients may still be alive at the end of the follow‐up period. Various approaches have been developed to estimate the mean residual life in the presence of such right censoring. In this work, a novel semi‐parametric method that combines existing non‐parametric methods and an extreme value tail model is presented, where the limited sample information in the tail (prior to study termination) is used to estimate the upper tail behaviour. This approach will be demonstrated with simulated and real‐life examples. Copyright © 2015 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.     

Multiple imputation based on restricted mean model for censored data

Most multiple imputation (MI) methods for censored survival data either ignore patient characteristics when imputing a likely event time, or place quite restrictive modeling assumptions on the survival distributions used for imputation. In this research, we propose a robust MI approach that directly imputes restricted lifetimes over the study period based on a model of the mean restricted life as a linear function of covariates. This method has the advantages of retaining patient characteristics when making imputation choices through the restricted mean parameters and does not make assumptions on the shapes of hazards or survival functions. Simulation results show that our method outperforms its closest competitor for modeling restricted mean lifetimes in terms of bias and efficiency in both independent censoring and dependent censoring scenarios. Survival estimates of restricted lifetime model parameters and marginal survival estimates regain much of the precision lost due to censoring. The proposed method is also much less subject to dependent censoring bias captured by covariates in the restricted mean model. This particular feature is observed in a full statistical analysis conducted in the context of the International Breast Cancer Study Group Ludwig Trial V using the proposed methodology.     

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.     

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.     

Regression methods for cost-effectiveness analysis with censored data

A system of seemingly unrelated regression equations is proposed for prognostic factor adjustment and subgroup analysis when comparing two groups in a cost-effectiveness analysis with censored data. Because of the induced dependent censoring on costs and quality-adjusted survival, inverse probability weighting is employed for parameter estimation. The method is illustrated with data from two recent examples using both survival time and quality-adjusted survival time as the measures of effectiveness.     

Estimating a time‐dependent concordance index for survival prediction models with covariate dependent censoring

Given a predictive marker and a time‐to‐event response variable, the proportion of concordant pairs in a data set is called concordance index. A specifically useful marker is the risk predicted by a survival regression model. This article extends the existing methodology for applications where the length of the follow‐up period depends on the predictor variables. A class of inverse probability of censoring weighted estimators is discussed in which the estimates rely on a working model for the conditional censoring distribution. The estimators are consistent for a truncated concordance index if the working model is correctly specified and if the probability of being uncensored at the truncation time is positive. In this framework, all kinds of prediction models can be assessed, and time trends in the discrimination ability of a model can be captured by varying the truncation time point. For illustration, we re‐analyze a study on risk prediction for prostate cancer patients. The effects of misspecification of the censoring model are studied in simulated data. Copyright © 2012 John Wiley & Sons, Ltd.     

Logistic‐AFT location‐scale mixture regression models with nonsusceptibility for left‐truncated and general interval‐censored data

In conventional survival analysis there is an underlying assumption that all study subjects are susceptible to the event. In general, this assumption does not adequately hold when investigating the time to an event other than death. Owing to genetic and/or environmental etiology, study subjects may not be susceptible to the disease. Analyzing nonsusceptibility has become an important topic in biomedical, epidemiological, and sociological research, with recent statistical studies proposing several mixture models for right‐censored data in regression analysis. In longitudinal studies, we often encounter left, interval, and right‐censored data because of incomplete observations of the time endpoint, as well as possibly left‐truncated data arising from the dissimilar entry ages of recruited healthy subjects. To analyze these kinds of incomplete data while accounting for nonsusceptibility and possible crossing hazards in the framework of mixture regression models, we utilize a logistic regression model to specify the probability of susceptibility, and a generalized gamma distribution, or a log‐logistic distribution, in the accelerated failure time location‐scale regression model to formulate the time to the event. Relative times of the conditional event time distribution for susceptible subjects are extended in the accelerated failure time location‐scale submodel. We also construct graphical goodness‐of‐fit procedures on the basis of the Turnbull–Frydman estimator and newly proposed residuals. Simulation studies were conducted to demonstrate the validity of the proposed estimation procedure. The mixture regression models are illustrated with alcohol abuse data from the Taiwan Aboriginal Study Project and hypertriglyceridemia data from the Cardiovascular Disease Risk Factor Two‐township Study in Taiwan. Copyright © 2013 John Wiley & Sons, Ltd.     

Nonparametric regression of state occupation,entry, exit,and waiting times with multistate right‐censored data

We construct nonparametric regression estimators of a number of temporal functions in a multistate system based on a continuous univariate baseline covariate. These estimators include state occupation probabilities, state entry, exit, and waiting (sojourn) time distribution functions of a general progressive (e.g., acyclic) multistate model. We subject the data to right censoring, and the censoring mechanism is explainable by observable covariates that could be time dependent. The resulting estimators are valid even if the multistate process is non‐Markov. We study the performance of the estimators in two simulation settings. We establish large sample consistency of these estimators. We illustrate our estimators using a data set on bone marrow transplant recipients. Copyright © 2012 John Wiley & Sons, Ltd.     

A model‐informed rank test for right‐censored data with intermediate states

The generalized Wilcoxon and log‐rank tests are commonly used for testing differences between two survival distributions. We modify the Wilcoxon test to account for auxiliary information on intermediate disease states that subjects may pass through before failure. For a disease with multiple states where patients are monitored periodically but exact transition times are unknown (e.g. staging in cancer), we first fit a multi‐state Markov model to the full data set; when censoring precludes the comparison of survival times between two subjects, we use the model to estimate the probability that one subject will have survived longer than the other given their censoring times and last observed status, and use these probabilities to compute an expected rank for each subject. These expected ranks form the basis of our test statistic. Simulations demonstrate that the proposed test can improve power over the log‐rank and generalized Wilcoxon tests in some settings while maintaining the nominal type 1 error rate. The method is illustrated on an amyotrophic lateral sclerosis data set. Copyright © 2015 John Wiley & Sons, Ltd.     

Regression analysis of mean quality‐adjusted survival time based on pseudo‐observations

Regression models for the mean quality‐adjusted survival time are specified from hazard functions of transitions between two states and the mean quality‐adjusted survival time may be a complex function of covariates. We discuss a regression model for the mean quality‐adjusted survival (QAS) time based on pseudo‐observations, which has the advantage of directly modeling the effect of covariates in the QAS time. Both Monte Carlo simulations and a real data set are studied. Copyright © 2009 John Wiley & Sons, Ltd.     

Restricted mean models for transplant benefit and urgency

The US lung allocation policy estimates each individual's urgency and transplant benefit in defining a lung allocation score (LAS). Transplant benefit, as defined by the Organ Procurement and Transplantation Network Thoracic Committee, is the days of life gained over the following year if transplanted versus not transplanted. Urgency is measured by days of life during the next year without transplant. In both definitions, accurate estimation of wait list days lived, or a wait list restricted mean lifetime, is required. Risk factors are available to estimate patient urgency when listed, with more urgent patients removed from the wait list upon death or transplant. As a patient progresses, priority for transplant (censoring) changes accordingly. Therefore, it is crucial to adjust for dependent censoring in modeling days of life. We develop a model for the restricted mean as a function of covariates, by using pseudo-observations that account for dependent censoring linked to a series of longitudinal measures (LAS). Simulation results show that our method performs well in situations comparable with the LAS setting. Applying wait list and post-transplant model results that account for dependent censoring to wait list patients, we obtain estimates of transplant benefit that are larger for many of the more urgent patients in need of transplant. The difference in LAS for an individual, when properly accounting for dependent censoring, has high impact on the priority and timing of an organ offer for these patients.     

Doubly robust estimation,optimally truncated inverse‐intensity weighting and increment‐based methods for the analysis of irregularly observed longitudinal data

Longitudinal data arising from routine follow‐up of patients will often have irregular measurement times. Existing methods for analysis include joint modelling of the outcome and measurement processes, and inverse‐intensity weighting (IIW). This work extends previously proposed analysis of increments to the case of irregular follow‐up, yielding a model for the increments that can be used as a stand‐alone method. Furthermore, we propose two ways of combining the increments and IIW estimators. First, we use the increment model to select the truncation point for the inverse‐intensity weights that minimises the mean squared error of the IIW estimator. Second, we use the increment model to augment the usual IIW estimating equations to form a doubly robust estimator. We evaluate the methods through simulation and apply these to a recent study of juvenile dermatomyositis. Copyright © 2012 John Wiley & Sons, Ltd.     

Frailty models for pneumonia to death with a left‐censored covariate

Frailty models are multiplicative hazard models for studying association between survival time and important clinical covariates. When some values of a clinical covariate are unobserved but known to be below a threshold called the limit of detection (LOD), naive approaches ignoring this problem, such as replacing the undetected value by the LOD or half of the LOD, often produce biased parameter estimate with larger mean squared error of the estimate. To address the LOD problem in a frailty model, we propose a flexible smooth nonparametric density estimator along with Simpson's numerical integration technique. This is an extension of an existing method in the likelihood framework for the estimation and inference of the model parameters. The proposed new method shows the estimators are asymptotically unbiased and gives smaller mean squared error of the estimates. Compared with the existing method, the proposed new method does not require distributional assumptions for the underlying covariates. Simulation studies were conducted to evaluate the performance of the new method in realistic scenarios. We illustrate the use of the proposed method with a data set from Genetic and Inflammatory Markers of Sepsis study in which interlekuin‐10 was subject to LOD. Copyright © 2015 John Wiley & Sons, Ltd.     

Inverse probability weighting to control confounding in an illness‐death model for interval‐censored data

Multistate models with interval‐censored data, such as the illness‐death model, are still not used to any considerable extent in medical research regardless of the significant literature demonstrating their advantages compared to usual survival models. Possible explanations are their uncommon availability in classical statistical software or, when they are available, by the limitations related to multivariable modelling to take confounding into consideration. In this paper, we propose a strategy based on propensity scores that allows population causal effects to be estimated: the inverse probability weighting in the illness semi‐Markov model with interval‐censored data. Using simulated data, we validated the performances of the proposed approach. We also illustrated the usefulness of the method by an application aiming to evaluate the relationship between the inadequate size of an aortic bioprosthesis and its degeneration or/and patient death. We have updated the R package multistate to facilitate the future use of this method.     

Incorporation of the time aspect into the liability‐threshold model for case‐control‐family data

Familial aggregation and the role of genetic and environmental factors can be investigated through family studies analysed using the liability‐threshold model. The liability‐threshold model ignores the timing of events including the age of disease onset and right censoring, which can lead to estimates that are difficult to interpret and are potentially biased. We incorporate the time aspect into the liability‐threshold model for case‐control‐family data following the same approach that has been applied in the twin setting. Thus, the data are considered as arising from a competing risks setting and inverse probability of censoring weights are used to adjust for right censoring. In the case‐control‐family setting, recognising the existence of competing events is highly relevant to the sampling of control probands. Because of the presence of multiple family members who may be censored at different ages, the estimation of inverse probability of censoring weights is not as straightforward as in the twin setting but requires consideration. We propose to employ a composite likelihood conditioning on proband status that markedly simplifies adjustment for right censoring. We assess the proposed approach using simulation studies and apply it in the analysis of two Danish register‐based case‐control‐family studies: one on cancer diagnosed in childhood and adolescence, and one on early‐onset breast cancer. Copyright © 2017 John Wiley & Sons, Ltd.     

Estimating the cumulative mean function for history process with time‐dependent covariates and censoring mechanism

In this paper, an approach to estimating the cumulative mean function for history process with time dependent covariates and right censored time‐to‐event variable is developed using the combined technique of joint modeling and inverse probability weighting method. The consistency of proposed estimator is derived. Theoretical analysis and simulation studies indicate that the estimator given in this paper is quite recommendable to practical applications because of its simplicity and accuracy. A real data set from a multicenter automatic defibrillator implantation trial is used to illustrate the proposed methodology. Copyright © 2016 John Wiley & Sons, Ltd.