Impact of nonignorable coarsening on Bayesian inference

The coarse data model of Heitjan and Rubin (1991) generalizes the missing data model of Rubin (1976) to cover other forms of incompleteness such as censoring and grouping. The model has 2 components: an ideal data model describing the distribution of the quantity of interest and a coarsening mechanism that describes a distribution over degrees of coarsening given the ideal data. The coarsening mechanism is said to be nonignorable when the degree of coarsening depends on an incompletely observed ideal outcome, in which case failure to properly account for it can spoil inferences. A theme in recent research is to measure sensitivity to nonignorability by evaluating the effect of a small departure from ignorability on the maximum likelihood estimate (MLE) of a parameter of the ideal data model. One such construct is the "index of local sensitivity to nonignorability" (ISNI) (Troxel and others, 2004), which is the derivative of the MLE with respect to a nonignorability parameter evaluated at the ignorable model. In this paper, we adapt ISNI to Bayesian modeling by instead defining it as the derivative of the posterior expectation. We propose the application of ISNI as a first step in judging the robustness of a Bayesian analysis to nonignorable coarsening. We derive formulas for a range of models and apply the method to evaluate sensitivity to nonignorable coarsening in 2 real data examples, one involving missing CD4 counts in an HIV trial and the other involving potentially informatively censored relapse times in a leukemia trial.     

Inference for set-based effects in genetic association studies with interval-censored outcomes

The rapid acceleration of genetic data collection in biomedical settings has recently resulted in the rise of genetic compendiums filled with rich longitudinal disease data. One common feature of these data sets is their plethora of interval-censored outcomes. However, very few tools are available for the analysis of genetic data sets with interval-censored outcomes, and in particular, there is a lack of methodology available for set-based inference. Set-based inference is used to associate a gene, biological pathway, or other genetic construct with outcomes and is one of the most popular strategies in genetics research. This work develops three such tests for interval-censored settings beginning with a variance components test for interval-censored outcomes, the interval-censored sequence kernel association test (ICSKAT). We also provide the interval-censored version of the Burden test, and then we integrate ICSKAT and Burden to construct the interval censored sequence kernel association test—optimal (ICSKATO) combination. These tests unlock set-based analysis of interval-censored data sets with analogs of three highly popular set-based tools commonly applied to continuous and binary outcomes. Simulation studies illustrate the advantages of the developed methods over ad hoc alternatives, including protection of the type I error rate at very low levels and increased power. The proposed approaches are applied to the investigation that motivated this study, an examination of the genes associated with bone mineral density deficiency and fracture risk.     

Nonparametric estimation of the joint distribution of a survival time subject to interval censoring and a continuous mark variable

This article considers three nonparametric estimators of the joint distribution function for a survival time and a continuous mark variable when the survival time is interval censored and the mark variable may be missing for interval-censored observations. Finite and large sample properties are described for the nonparametric maximum likelihood estimator (NPMLE) as well as estimators based on midpoint imputation (MIDMLE) and coarsening the mark variable (CMLE). The estimators are compared using data from a simulation study and a recent phase III HIV vaccine efficacy trial where the survival time is the time from enrollment to infection and the mark variable is the genetic distance from the infecting HIV sequence to the HIV sequence in the vaccine. Theoretical and empirical evidence are presented indicating the NPMLE and MIDMLE are inconsistent. Conversely, the CMLE is shown to be consistent in general and thus is preferred.     

Generalized log-rank tests for partly interval-censored failure time data

In this paper, we consider incomplete survival data: partly interval-censored failure time data where observed data include both exact and interval-censored observations on the survival time of interest. We present a class of generalized log-rank tests for this type of survival data and establish their asymptotic properties. The method is evaluated using simulation studies and illustrated by a set of real data from a diabetes study.     

Regression analysis of multivariate interval-censored failure time data with application to tumorigenicity experiments

This paper discusses multivariate interval-censored failure time data that occur when there exist several correlated survival times of interest and only interval-censored data are available for each survival time. Such data occur in many fields. One is tumorigenicity experiments, which usually concern different types of tumors, tumors occurring in different locations of animals, or together. For regression analysis of such data, we develop a marginal inference approach using the additive hazards model and apply it to a set of bivariate interval-censored data arising from a tumorigenicity experiment. Simulation studies are conducted for the evaluation of the presented approach and suggest that the approach performs well for practical situations.     

Neural network on interval-censored data with application to the prediction of Alzheimer's disease

Alzheimer's disease (AD) is a progressive and polygenic disorder that affects millions of individuals each year. Given that there have been few effective treatments yet for AD, it is highly desirable to develop an accurate model to predict the full disease progression profile based on an individual's genetic characteristics for early prevention and clinical management. This work uses data composed of all four phases of the Alzheimer's Disease Neuroimaging Initiative (ADNI) study, including 1740 individuals with 8 million genetic variants. We tackle several challenges in this data, characterized by large-scale genetic data, interval-censored outcome due to intermittent assessments, and left truncation in one study phase (ADNIGO). Specifically, we first develop a semiparametric transformation model on interval-censored and left-truncated data and estimate parameters through a sieve approach. Then we propose a computationally efficient generalized score test to identify variants associated with AD progression. Next, we implement a novel neural network on interval-censored data (NN-IC) to construct a prediction model using top variants identified from the genome-wide test. Comprehensive simulation studies show that the NN-IC outperforms several existing methods in terms of prediction accuracy. Finally, we apply the NN-IC to the full ADNI data and successfully identify subgroups with differential progression risk profiles. Data used in the preparation of this article were obtained from the ADNI database.     

Adjusting for Nonignorable Missingness When Estimating Generalized Additive Models

Generalized additive models (GAMs) have been widely used for flexible modeling of various types of outcomes. When the outcome in a GAM is subject to missing, practical analyses often assume that missingness is missing at random (MAR). This assumption can be of suspicion when the missingness is not by design. Evaluating the potential effects of alternative nonignorable missing data mechanism on the MAR inference from a GAM can be important but often challenging due to the complicatedness of alternative nonignorable models. We apply the index approach to local sensitivity (Troxel, Ma, and Heitjan 2004 (2004). Statistica Sinica 14 , 1221–1237) to evaluate the potential changes of the GAM estimates in the neighborhood of the MAR model. The approach avoids fitting any complicated nonignorable GAM. Only MAR estimates are required to calculate the resulting sensitivity index and adjust the GAM estimates to account for nonignorable missingness. Thus the proposed approach is considerably simpler to conduct, as compared with the alternative methods. The simulation study shows that the index provides valid assessment of the local sensitivity of the GAM estimates to nonignorable missingness. We then illustrate the method using a rheumatoid arthritis clinical trial data set.     

Bayesian nonparametric analysis of restricted mean survival time

The restricted mean survival time (RMST) evaluates the expectation of survival time truncated by a prespecified time point, because the mean survival time in the presence of censoring is typically not estimable. The frequentist inference procedure for RMST has been widely advocated for comparison of two survival curves, while research from the Bayesian perspective is rather limited. For the RMST of both right- and interval-censored data, we propose Bayesian nonparametric estimation and inference procedures. By assigning a mixture of Dirichlet processes (MDP) prior to the distribution function, we can estimate the posterior distribution of RMST. We also explore another Bayesian nonparametric approach using the Dirichlet process mixture model and make comparisons with the frequentist nonparametric method. Simulation studies demonstrate that the Bayesian nonparametric RMST under diffuse MDP priors leads to robust estimation and under informative priors it can incorporate prior knowledge into the nonparametric estimator. Analysis of real trial examples demonstrates the flexibility and interpretability of the Bayesian nonparametric RMST for both right- and interval-censored data.     

Fast approximations of pseudo-observations in the context of right censoring and interval censoring

In the context of right-censored and interval-censored data, we develop asymptotic formulas to compute pseudo-observations for the survival function and the restricted mean survival time (RMST). These formulas are based on the original estimators and do not involve computation of the jackknife estimators. For right-censored data, Von Mises expansions of the Kaplan–Meier estimator are used to derive the pseudo-observations. For interval-censored data, a general class of parametric models for the survival function is studied. An asymptotic representation of the pseudo-observations is derived involving the Hessian matrix and the score vector. Theoretical results that justify the use of pseudo-observations in regression are also derived. The formula is illustrated on the piecewise-constant-hazard model for the RMST. The proposed approximations are extremely accurate, even for small sample sizes, as illustrated by Monte Carlo simulations and real data. We also study the gain in terms of computation time, as compared to the original jackknife method, which can be substantial for a large dataset.     

Parametric survival models for interval-censored data with time-dependent covariates

We present a parametric family of regression models for interval-censored event-time (survival) data that accomodates both fixed (e.g. baseline) and time-dependent covariates. The model employs a three-parameter family of survival distributions that includes the Weibull, negative binomial, and log-logistic distributions as special cases, and can be applied to data with left, right, interval, or non-censored event times. Standard methods, such as Newton-Raphson, can be employed to estimate the model and the resulting estimates have an asymptotically normal distribution about the true values with a covariance matrix that is consistently estimated by the information function. The deviance function is described to assess model fit and a robust sandwich estimate of the covariance may also be employed to provide asymptotically robust inferences when the model assumptions do not apply. Spline functions may also be employed to allow for non-linear covariates. The model is applied to data from a long-term study of type 1 diabetes to describe the effects of longitudinal measures of glycemia (HbA1c) over time (the time-dependent covariate) on the risk of progression of diabetic retinopathy (eye disease), an interval-censored event-time outcome.     

A multistate model for bivariate interval-censored failure time data

SUMMARY: Interval-censored life-history data arise when the events of interest are only detectable at periodic assessments. When interest lies in the occurrence of two such events, bivariate-interval censored event time data are obtained. We describe how to fit a four-state Markov model useful for characterizing the association between two interval-censored event times when the assessment times for the two events may be generated by different inspection processes. The approach treats the two events symmetrically and enables one to fit multiplicative intensity models that give estimates of covariate effects as well as relative risks characterizing the association between the two events. An expectation-maximization (EM) algorithm is described for estimation in which the maximization step can be carried out with standard software. The method is illustrated by application to data from a trial of HIV patients where the events are the onset of viral shedding in the blood and urine among individuals infected with cytomegalovirus.     

Bayesian modeling of multiple lesion onset and growth from interval-censored data

In studying rates of occurrence and progression of lesions (or tumors), it is typically not possible to obtain exact onset times for each lesion. Instead, data consist of the number of lesions that reach a detectable size between screening examinations, along with measures of the size/severity of individual lesions at each exam time. This interval-censored data structure makes it difficult to properly adjust for the onset time distribution in assessing covariate effects on rates of lesion progression. This article proposes a joint model for the multiple lesion onset and progression process, motivated by cross-sectional data from a study of uterine leiomyoma tumors. By using a joint model, one can potentially obtain more precise inferences on rates of onset, while also performing onset time-adjusted inferences on lesion severity. Following a Bayesian approach, we propose a data augmentation Markov chain Monte Carlo algorithm for posterior computation.     

Fitting semi-Markov models to interval-censored data with unknown initiation times

In a semi-Markov model, the hazard of making a transition between stages depends on the time spent in the current stage but is independent of time spent in other stages. If the initiation time (time of entry into the network) is not known for some persons and if transition time data are interval censored (i.e., if transition times are not known exactly but are known only to have occurred in some interval), then the length of time these persons spent in any stage is not known. We show how a semi-Markov model can still be fit to interval-censored data with missing initiation times. For the special case of models in which all persons enter the network at the same initial stage and proceed through the same succession of stages to a unique absorbing stage, we present discrete-time nonparametric maximum likelihood estimators of the waiting-time distributions for this type of data.     

A regularized estimation approach for case-cohort periodic follow-up studies with an application to HIV vaccine trials

This paper discusses regression analysis of the failure time data arising from case-cohort periodic follow-up studies, and one feature of such data, which makes their analysis much more difficult, is that they are usually interval-censored rather than right-censored. Although some methods have been developed for general failure time data, there does not seem to exist an established procedure for the situation considered here. To address the problem, we present a semiparametric regularized procedure and develop a simple algorithm for the implementation of the proposed method. In addition, unlike some existing procedures for similar situations, the proposed procedure is shown to have the oracle property, and an extensive simulation is conducted and it suggests that the presented approach seems to work well for practical situations. The method is applied to an HIV vaccine trial that motivated this study.     

Nonparametric estimation of a Markov 'illness-death' process from interval-censored observations, with application to diabetes survival data

The nonparametric estimation of the cumulative transition intensityfunctions in a threestate time-nonhomogeneous Markov processwith irreversible transitions, an ‘illness-death’model, is considered when times of the intermediate transition,e.g. onset of a disease, are interval-censored. The times of‘death’ are assumed to be known exactly or to beright-censored. In addition the observed process may be left-truncated.Data of this type arise when the process is sampled periodically.For example, when the patients are monitored through periodicexaminations the observations on times of change in their diseasestatus will be interval-censored. Under the sampling schemeconsidered here the Nelson–Aalen estimator (Aalen, 1978)for a cumulative transition intensity is not applicable. Inthe proposed method the maximum likelihood estimators of someof the transition intensities are derived from the estimatorsof the corresponding subdistribution functions. The maximumlikelihood estimators are shown to have a self-consistency property.The self-consistency algorithm is developed for the computationof the estimators. This approach generalises the results fromTurnbull (1976) and Frydman (1992). The methods are illustratedwith diabetes survival data.     

Improved Doubly Robust Estimation When Data Are Monotonely Coarsened,with Application to Longitudinal Studies with Dropout

Summary A routine challenge is that of making inference on parameters in a statistical model of interest from longitudinal data subject to dropout, which are a special case of the more general setting of monotonely coarsened data. Considerable recent attention has focused on doubly robust (DR) estimators, which in this context involve positing models for both the missingness (more generally, coarsening) mechanism and aspects of the distribution of the full data, that have the appealing property of yielding consistent inferences if only one of these models is correctly specified. DR estimators have been criticized for potentially disastrous performance when both of these models are even only mildly misspecified. We propose a DR estimator applicable in general monotone coarsening problems that achieves comparable or improved performance relative to existing DR methods, which we demonstrate via simulation studies and by application to data from an AIDS clinical trial.     

A semiparametric model for regression analysis of interval-censored failure time data

Left-, right-, and interval-censored response time data arise in a variety of settings, including the analyses of data from laboratory animal carcinogenicity experiments, clinical trials, and longitudinal studies. For such incomplete data, the usual regression techniques such as the Cox (1972, Journal of the Royal Statistical Society, Series B 34, 187-220) proportional hazards model are inapplicable. In this paper, we present a method for regression analysis which accommodates interval-censored data. We present applications of this methodology to data sets from a study of breast cancer patients who were followed for cosmetic response to therapy, a small animal tumorigenicity study, and a clinical trial.     

An information ratio-based goodness-of-fit test for copula models on censored data

Copula is a popular method for modeling the dependence among marginal distributions in multivariate censored data. As many copula models are available, it is essential to check if the chosen copula model fits the data well for analysis. Existing approaches to testing the fitness of copula models are mainly for complete or right-censored data. No formal goodness-of-fit (GOF) test exists for interval-censored or recurrent events data. We develop a general GOF test for copula-based survival models using the information ratio (IR) to address this research gap. It can be applied to any copula family with a parametric form, such as the frequently used Archimedean, Gaussian, and D-vine families. The test statistic is easy to calculate, and the test procedure is straightforward to implement. We establish the asymptotic properties of the test statistic. The simulation results show that the proposed test controls the type-I error well and achieves adequate power when the dependence strength is moderate to high. Finally, we apply our method to test various copula models in analyzing multiple real datasets. Our method consistently separates different copula models for all these datasets in terms of model fitness.     

Regression modeling of semicompeting risks data

Semicompeting risks data are often encountered in clinical trials with intermediate endpoints subject to dependent censoring from informative dropout. Unlike with competing risks data, dropout may not be dependently censored by the intermediate event. There has recently been increased attention to these data, in particular inferences about the marginal distribution of the intermediate event without covariates. In this article, we incorporate covariates and formulate their effects on the survival function of the intermediate event via a functional regression model. To accommodate informative censoring, a time-dependent copula model is proposed in the observable region of the data which is more flexible than standard parametric copula models for the dependence between the events. The model permits estimation of the marginal distribution under weaker assumptions than in previous work on competing risks data. New nonparametric estimators for the marginal and dependence models are derived from nonlinear estimating equations and are shown to be uniformly consistent and to converge weakly to Gaussian processes. Graphical model checking techniques are presented for the assumed models. Nonparametric tests are developed accordingly, as are inferences for parametric submodels for the time-varying covariate effects and copula parameters. A novel time-varying sensitivity analysis is developed using the estimation procedures. Simulations and an AIDS data analysis demonstrate the practical utility of the methodology.     

Nonparametric estimation in a Markov "illness-death" process from interval censored observations with missing intermediate transition status

Summary .  In many clinical trials patients are intermittently assessed for the transition to an intermediate state, such as occurrence of a disease-related nonfatal event, and death. Estimation of the distribution of nonfatal event free survival time, that is, the time to the first occurrence of the nonfatal event or death, is the primary focus of the data analysis. The difficulty with this estimation is that the intermittent assessment of patients results in two forms of incompleteness: the times of occurrence of nonfatal events are interval censored and, when a nonfatal event does not occur by the time of the last assessment, a patient's nonfatal event status is not known from the time of the last assessment until the end of follow-up for death. We consider both forms of incompleteness within the framework of an "illness–death" model. We develop nonparametric maximum likelihood (ML) estimation in an "illness–death" model from interval-censored observations with missing status of intermediate transition. We show that the ML estimators are self-consistent and propose an algorithm for obtaining them. This work thus provides new methodology for the analysis of incomplete data that arise from clinical trials. We apply this methodology to the data from a recently reported cancer clinical trial ( Bonner et al., 2006 , New England Journal of Medicine 354, 567–578) and compare our estimation results with those obtained using a Food and Drug Administration recommended convention.