Event Time Data Research Articles

A Copula Approach to Joint Modeling of Longitudinal Measurements and Survival Times Using Monte Carlo Expectation-Maximization with Application to AIDS Studies

Joint modeling of longitudinal measurements and time to event data is often performed by fitting a shared parameter model. Another method for joint modeling that may be used is a marginal model. As a marginal model, we use a Gaussian model for joint modeling of longitudinal measurements and time to event data. We consider a regression model for longitudinal data modeling and a Weibull proportional hazard model for event time data modeling. A Gaussian copula is used to consider the association between these two models. A Monte Carlo expectation-maximization approach is used for parameter estimation. Some simulation studies are conducted in order to illustrate the proposed method. Also, the proposed method is used for analyzing a clinical trial dataset.

Case-base methods for studying vaccination safety.

Pooling of controls under nested-case control settings can produce substantial efficiency gains compared to standard time-matched analysis using the Mantel-Haenszel method or conditional logistic regression. In the context of possible adverse effects of early childhood vaccinations, we propose pooling of the information from the controls to estimate the population exposure prevalence as a parametric or nonparametric function of time, and possibly other factors. This function in turn may be used as a plug-in estimate to control for confounding in the subsequent estimation of rate ratios. We derive standard errors for the resulting two-step estimators, demonstrate through simulations the efficiency gains compared to standard matched analysis, and propose a novel graphical presentation of the vaccination and adverse event time data. We formulate the methods in the general framework of case-base sampling, which subsumes the different case-control and case-only methods.

Jointly Modeling Event Time and Skewed-Longitudinal Data with Missing Response and Mismeasured Covariate for AIDS Studies

In longitudinal studies it is often of interest to investigate how a repeatedly measured marker in time is associated with a time to an event of interest. This type of research question has given rise to a rapidly developing field of biostatistics research that deals with the joint modeling of longitudinal and time-to-event data. Normality of model errors in longitudinal model is a routine assumption, but it may be unrealistically obscuring important features of subject variations. Covariates are usually introduced in the models to partially explain between- and within-subject variations, but some covariates such as CD4 cell count may be often measured with substantial errors. Moreover, the responses may encounter nonignorable missing. Statistical analysis may be complicated dramatically based on longitudinal-survival joint models where longitudinal data with skewness, missing values, and measurement errors are observed. In this article, we relax the distributional assumptions for the longitudinal models using skewed (parametric) distribution and unspecified (nonparametric) distribution placed by a Dirichlet process prior, and address the simultaneous influence of skewness, missingness, covariate measurement error, and time-to-event process by jointly modeling three components (response process with missing values, covariate process with measurement errors, and time-to-event process) linked through the random-effects that characterize the underlying individual-specific longitudinal processes in Bayesian analysis. The method is illustrated with an AIDS study by jointly modeling HIV/CD4 dynamics and time to viral rebound in comparison with potential models with various scenarios and different distributional specifications.

Joint modeling of multivariate censored longitudinal and event time data with application to the Genetic Markers of Inflammation Study

The Genetic Markers of Inflammation Study (GenIMS) was conceived to investigate the role of severe sepsis, which is typically defined as system-wide multi-organ failure, on survival. One major hypothesis for this systemic collapse, and reduction in survival, is a cascade of pro-inflammatory and anti-inflammatory cytokines. In this paper, we devised a novel joint modeling strategy to evaluate the joint effect of longitudinal anti-inflammatory marker IL-6 and pro-inflammatory marker IL-10 on 90-day survival. We found that, on average, patients with high initial values of both IL-6 and IL-10, that tend to increase over time, are associated with a reduction in survival expectancy and that accounting for their assumed correlation was justified.

Estimation of intervention effects using recurrent event time data in the presence of event dependence and a cured fraction

Recurrent event data with a fraction of subjects having zero event are often seen in randomized clinical trials. Those with zero event may belong to a cured (or non-susceptible) fraction. Event dependence refers to the situation that a person's past event history affects his future event occurrences. In the presence of event dependence, an intervention may have an impact on the event rate in the non-cured through two pathways-a primary effect directly on the outcome event and a secondary effect mediated through event dependence. The primary effect combined with the secondary effect is the total effect. We propose a frailty mixture model and a two-step estimation procedure for the estimation of the effect of an intervention on the probability of cure and the total effect on event rate in the non-cured. A summary measure of intervention effects is derived. The performance of the proposed model is evaluated by simulation. Data on respiratory exacerbations from a randomized, placebo-controlled trial are re-analyzed for illustration.

An introduction to survival models: in honor of Ross Prentice

I review some key ideas and models in survival analysis with emphasis on modeling the effects of covariates on survival times. I focus on the proportional hazards model of Cox (J R Stat Soc B 34:187-220, 1972), its extensions and alternatives, including the accelerated life model. I briefly describe some models for competing risks data, multiple and repeated event-time data and multivariate survival data.

Canine Lymphomas

Canine lymphoma is the neoplasm most often treated by chemotherapy, yet there are few data to correlate response to therapy with its different subtypes. This study is based on biopsy specimens from 992 dogs for which lymphoma was the clinical diagnosis. All cases were phenotyped by immunohistochemistry for CD3 and CD79alpha. Cases with histiocytic proliferation were evaluated immunohistochemically for CD18. Clonality was verified in 12 cases by polymerase chain reaction (PCR). Survival (event time) data and complete survival information (cause of death or time to last follow-up) were available on 456 dogs. Additional covariate information when available included size, age, sex, phenotype, stage and grade of lymphoma, mitotic index, and treatment protocol. Because of the many subtypes of B- and T-cell lymphoma, the cases were grouped into 7 diagnostic categories: (1) benign hyperplasia; (2) low-grade B-cell; (3) high-grade B- and T-cell; (4) low-grade T-cell; (5) centroblastic large B-cell of all mitotic grades (subdivided by clinical stage); (6) immunoblastic large B-cell of all mitotic grades, and (7) high-grade peripheral T-cell. Grouping was determined by histological grade (based on mitotic rate/400× field, with low-grade 0-5, intermediate 6-10, and high-grade >10) and stage for survival function estimation. No association with survival was found for size (based on breed of dog) or sex. All diagnostic categories of indolent or low-grade type had low mitotic rates, whereas those with clinically high grades had high mitotic rates. The diagnostic category with the most cases was centroblastic large B-cell lymphoma. Compared with dogs in this largest represented group of lymphomas, dogs with high-grade lymphomas had significantly higher mortality rates, and dogs with low-grade T-cell lymphomas had significantly lower mortality rates. Treatments for high-, intermediate-, and low-grade lymphomas were divided into 4 groups: absence of treatment, chemotherapy with or without hydroxydaunorubicin, and only prednisone. Dogs with low-grade T-cell (T-zone) lymphomas had the longest median survival (622 days), whereas the shortest median survival was in dogs with T-cell high-grade (peripheral T-cell) subtype (162 days). The dogs with centroblastic large B-cell lymphomas had a median survival of 127 days with low stage, 221 days with intermediate stage, and 215 days with advanced stage. Dogs with T-zone lymphoma were probably diagnosed in later stages of disease because of the lack of signs associated with progression. As with human lymphomas, a histological diagnosis with immunophenotyping is a minimal requirement for diagnosis of a specific subtype.

Joint modeling of longitudinal and event time data: application to HIV study

Abstract Many clinical studies generate a dataset having longitudinal repeated biomarker measurement data and time to an event data, which often depend on each other. In such studies, characteristics of the pattern of a biomarker

A FRAILTY MODEL APPROACH FOR REGRESSION ANALYSIS OF BIVARIATE INTERVAL-CENSORED SURVIVAL DATA

Owing to the fact that general semiparametric inference procedures are still underdeveloped for multivariate interval-censored event time data, we pro- pose semiparametric maximum likelihood estimation for the gamma-frailty Cox model under mixed-case interval censoring. We establish the consistency of the semiparametric maximum likelihood estimator (SPMLE) for the model parame- ters, including the regression coefficients and the cumulative hazard functions in the Cox model, and the variance of the gamma frailty. The SPMLEs of the cumu- lative hazard functions are shown to have a n 1=3 -rate of convergence, while those of the regression coefficients and the frailty variance have a n 1=2 -rate of convergence; here n denotes the number of study units. The asymptotic normality of the regres- sion coefficients and the frailty variance is also established, with the asymptotic variance given by the inverse of the efficient Fisher information matrix. A profile- likelihood approach is proposed for estimating the asymptotic variance. Based on the self-consistency equations and the contraction principle, we propose a stable and efficient computation algorithm. Simulation results reveal that the large sam- ple theories work quite well in finite samples. We analyze a dataset from an AIDS clinical trial by the proposed methods to assess the effects of the baseline CD4 cell counts on the times to CMV shedding in blood and urine.

Goodness-of-fit tests for the frailty distribution in proportional hazards models with shared frailty

Frailty models account for the clustering present in event time data. A proportional hazards model with shared frailties expresses the hazard for each subject. Often a one-parameter gamma distribution is assumed for the frailties. In this paper, we construct formal goodness-of-fit tests to test for gamma frailties. We construct a new class of frailty models that extend the gamma frailty model by using certain polynomial expansions that are orthogonal with respect to the gamma density. For this extended family, we obtain an explicit expression for the marginal likelihood of the data. The order selection test is based on finding the best fitting model in such a series of expanded models. A bootstrap is used to obtain p-values for the tests. Simulations and data examples illustrate the test's performance.

A unified inference procedure for a class of measures to assess improvement in risk prediction systems with survival data

Risk prediction procedures can be quite useful for the patient's treatment selection, prevention strategy, or disease management in evidence-based medicine. Often, potentially important new predictors are available in addition to the conventional markers. The question is how to quantify the improvement from the new markers for prediction of the patient's risk in order to aid cost-benefit decisions. The standard method, using the area under the receiver operating characteristic curve, to measure the added value may not be sensitive enough to capture incremental improvements from the new markers. Recently, some novel alternatives to area under the receiver operating characteristic curve, such as integrated discrimination improvement and net reclassification improvement, were proposed. In this paper, we consider a class of measures for evaluating the incremental values of new markers, which includes the preceding two as special cases. We present a unified procedure for making inferences about measures in the class with censored event time data. The large sample properties of our procedures are theoretically justified. We illustrate the new proposal with data from a cancer study to evaluate a new gene score for prediction of the patient's survival.

Joint modeling of censored longitudinal and event time data

Censoring of a longitudinal outcome often occurs when data are collected in a biomedical study and where the interest is in the survival and or longitudinal experiences of a study population. In the setting considered herein, we encountered upper and lower censored data as the result of restrictions imposed on measurements from a kinetic model producing “biologically implausible” kidney clearances. The goal of this paper is to outline the use of a joint model to determine the association between a censored longitudinal outcome and a time to event endpoint. This paper extends Guo and Carlin's [6] paper to accommodate censored longitudinal data, in a commercially available software platform, by linking a mixed effects Tobit model to a suitable parametric survival distribution. Our simulation results showed that our joint Tobit model outperforms a joint model made up of the more naïve or “fill-in” method for the longitudinal component. In this case, the upper and/or lower limits of censoring are replaced by the limit of detection. We illustrated the use of this approach with example data from the hemodialysis (HEMO) study [3] and examined the association between doubly censored kidney clearance values and survival.

Estimation of summary protective efficacy using a frailty mixture model for recurrent event time data

Recurrent event time data are common in experimental and observational studies. The analytic strategy needs to consider three issues: within-subject event dependence, between-subject heterogeneity in event rates, and the possibility of a nonsusceptible fraction. Motivated by the need to estimate the summary protective efficacy from recurrent event time data as seen in many infectious disease clinical trials, we propose a two-part frailty mixture model that simultaneously accommodates all the three issues. In terms of vaccine action models, the proposed model is a combination of the 'all-or-none' and the 'leaky' models, and the summary protective efficacy is a unified measure of the vaccine's twofold effects in completely or partially protecting the vaccinated individuals against the study event. The model parameters of interest are estimated using the expectation-maximization algorithm with their respective variances estimated using Louis's formula for the expectation-maximization algorithm. The summary protective efficacy is estimated by a composite estimand with its variance estimated using the delta method. The performance of the proposed estimation approach is investigated by a simulation study. Data from a trial of malaria prophylaxis conducted in Ghana are reanalyzed.

Goodness-of-Fit Tests for the Frailty Distribution in Proportional Hazards Models with Shared Frailty

Frailty models account for the clustering present in grouped event time data. A proportional hazards model with shared frailties expresses the hazard for each subject. Often a one-parameter gamma distribution is assumed for the frailties. In this paper we construct formal goodness-of-fit tests to test for gamma frailties. We construct a new class of frailty models that extend the gamma frailty model by using certain polynomial expansions that are orthogonal with respect to the gamma density. For this extended family we obtain an explicit expression for the marginal likelihood of the data. The order selection test is based on finding the best fitting model in such a series of expanded models. A bootstrap is used to obtain p-values for the tests. Simulations and data examples illustrate the test's performance.

Joint Modeling of Longitudinal Change and Survival

Joint longitudinal-survival models are useful when repeated measures and event time data are available and possibly associated. The application of this joint model in aging research is relatively rare, albeit particularly useful, when there is the potential for nonrandom dropout. In this article we illustrate the method and discuss some issues that may arise when fitting joint models of this type. Using prose recall scores from the Swedish OCTO-Twin Longitudinal Study of Aging, we fitted a joint longitudinal-survival model to investigate the association between risk of mortality and individual differences in rates of change in memory. A model describing change in memory scores as following an accelerating decline trajectory and a Weibull survival model was identified as the best fitting. This model adjusted for random effects representing individual variation in initial memory performance and change in rate of decline as linking terms between the longitudinal and survival models. Memory performance and change in rate of memory decline were significant predictors of proximity to death. Joint longitudinal-survival models permit researchers to gain a better understanding of the association between change functions and risk of particular events, such as disease diagnosis or death. Careful consideration of computational issues may be required because of the complexities of joint modeling methodologies.

Estimating Progression-Free Survival in Paediatric Brain Tumour Patients When Some Progression Statuses are Unknown

In oncology, progression-free survival time, which is defined as the minimum of the times to disease progression or death, often is used to characterize treatment and covariate effects. We are motivated by the desire to estimate the progression time distribution on the basis of data from 780 paediatric patients with choroid plexus tumours, which are a rare brain cancer where disease progression always precedes death. In retrospective data on 674 patients, the times to death or censoring were recorded but progression times were missing. In a prospective study of 106 patients, both times were recorded but there were only 20 non-censored progression times and 10 non-censored survival times. Consequently, estimating the progression time distribution is complicated by the problems that, for most of the patients, either the survival time is known but the progression time is not known, or the survival time is right censored and it is not known whether the patient's disease progressed before censoring. For data with these missingness structures, we formulate a family of Bayesian parametric likelihoods and present methods for estimating the progression time distribution. The underlying idea is that estimating the association between the time to progression and subsequent survival time from patients having complete data provides a basis for utilizing covariates and partial event time data of other patients to infer their missing progression times. We illustrate the methodology by analysing the brain tumour data, and we also present a simulation study.

Fast fitting of joint models for longitudinal and event time data using a pseudo-adaptive Gaussian quadrature rule

Joint models for longitudinal and time-to-event data have recently attracted a lot of attention in statistics and biostatistics. Even though these models enjoy a wide range of applications in many different statistical fields, they have not yet found their rightful place in the toolbox of modern applied statisticians mainly due to the fact that they are rather computationally intensive to fit. The main difficulty arises from the requirement for numerical integration with respect to the random effects. This integration is typically performed using Gaussian quadrature rules whose computational complexity increases exponentially with the dimension of the random-effects vector. A solution to overcome this problem is proposed using a pseudo-adaptive Gauss–Hermite quadrature rule. The idea behind this rule is to use information for the shape of the integrand by separately fitting a mixed model for the longitudinal outcome. Simulation studies show that the pseudo-adaptive rule performs excellently in practice, and is considerably faster than the standard Gauss–Hermite rule.

Graphical Procedures for Evaluating Overall and Subject-Specific Incremental Values from New Predictors with Censored Event Time Data

Quantitative procedures for evaluating added values from new markers over a conventional risk scoring system for predicting event rates at specific time points have been extensively studied. However, a single summary statistic, for example, the area under the receiver operating characteristic curve or its derivatives, may not provide a clear picture about the relationship between the conventional and the new risk scoring systems. When there are no censored event time observations in the data, two simple scatterplots with individual conventional and new scores for "cases" and "controls" provide valuable information regarding the overall and the subject-specific level incremental values from the new markers. Unfortunately, in the presence of censoring, it is not clear how to construct such plots. In this article, we propose a nonparametric estimation procedure for the distributions of the differences between two risk scores conditional on the conventional score. The resulting quantile curves of these differences over the subject-specific conventional score provide extra information about the overall added value from the new marker. They also help us to identify a subgroup of future subjects who need the new predictors, especially when there is no unified utility function available for cost-risk-benefit decision making. The procedure is illustrated with two data sets. The first is from a well-known Mayo Clinic primary biliary cirrhosis liver study. The second is from a recent breast cancer study on evaluating the added value from a gene score, which is relatively expensive to measure compared with the routinely used clinical biomarkers for predicting the patient's survival after surgery.

HIV DYNAMICS AND NATURAL HISTORY STUDIES: JOINT MODELING WITH DOUBLY INTERVAL-CENSORED EVENT TIME AND INFREQUENT LONGITUDINAL DATA.

Hepatitis C virus (HCV) coinfection has become one of the most challenging clinical situations to manage in HIV-infected patients. Recently the effect of HCV coinfection on HIV dynamics following initiation of highly active antiretroviral therapy (HAART) has drawn considerable attention. Post-HAART HIV dynamics are commonly studied in short-term clinical trials with frequent data collection design. For example, the elimination process of plasma virus during treatment is closely monitored with daily assessments in viral dynamics studies of AIDS clinical trials. In this article instead we use infrequent cohort data from long-term natural history studies and develop a model for characterizing post-HAART HIV dynamics and their associations with HCV coinfection. Specifically, we propose a joint model for doubly interval-censored data for the time between HAART initiation and viral suppression, and the longitudinal CD4 count measurements relative to the viral suppression. Inference is accomplished using a fully Bayesian approach. Doubly interval-censored data are modeled semiparametrically by Dirichlet process priors and Bayesian penalized splines are used for modeling population-level and individual-level mean CD4 count profiles. We use the proposed methods and data from the HIV Epidemiology Research Study (HERS) to investigate the effect of HCV coinfection on the response to HAART.

A frailty modeling approach for parental effects in animal breeding

Survival models involving frailties are commonly applied in studies where correlated event time data arise due to natural or artificial clustering. In this paper we present an application of such models in the animal breeding field. Specifically, a mixed survival model with a multivariate correlated frailty term is proposed for the analysis of data from over 3611 Brazilian Nellore cattle. The primary aim is to evaluate parental genetic effects on the trait length in days that their progeny need to gain a commercially specified standard weight gain. This trait is not measured directly but can be estimated from growth data. Results point to the importance of genetic effects and suggest that these models constitute a valuable data analysis tool for beef cattle breeding.