ML and REML estimation in survival analysis with time dependent correlated frailty

In the study of multiple failure times for the same subjects, for example, recurrent infections for patients with a given disease, there are often subject effects, that is, subjects have different risks that cannot be explained by known covariates. Standard methods, which ignore subject effects, lead to overestimation of precision. The frailty model for subject effects is better, but can be insufficient, because it assumes that subject effects are constant over time. Experience has shown that the dependence between different time periods often decreases with distance in time. Such a model is presented here, assuming that the frailty is no longer constant, but time varying, with one value for each spell. The main example is a first-order autoregressive process. This is applied to a data set of 128 patients with chronic granulomatous disease (CGD), participating in a placebo controlled randomized trial of gamma interferon (γ-IFN), suffering between 0 and 7 infections. It is shown that the time varying frailty model gives a significantly better fit than the constant frailty model. © 1998 John Wiley & Sons, Ltd.     

Nonproportional hazards and unobserved heterogeneity in clustered survival data: When can we tell the difference?

Multivariate survival data are frequently encountered in biomedical applications in the form of clustered failures (or recurrent events data). A popular way of analyzing such data is by using shared frailty models, which assume that the proportional hazards assumption holds conditional on an unobserved cluster-specific random effect. Such models are often incorporated in more complicated joint models in survival analysis. If the random effect distribution has finite expectation, then the conditional proportional hazards assumption does not carry over to the marginal models. It has been shown that, for univariate data, this makes it impossible to distinguish between the presence of unobserved heterogeneity (eg, due to missing covariates) and marginal nonproportional hazards. We show that time-dependent covariate effects may falsely appear as evidence in favor of a frailty model also in the case of clustered failures or recurrent events data, when the cluster size or number of recurrent events is small. When true unobserved heterogeneity is present, the presence of nonproportional hazards leads to overestimating the frailty effect. We show that this phenomenon is somewhat mitigated as the cluster size grows. We carry out a simulation study to assess the behavior of test statistics and estimators for frailty models in such contexts. The gamma, inverse Gaussian, and positive stable shared frailty models are contrasted using a novel software implementation for estimating semiparametric shared frailty models. Two main questions are addressed in the contexts of clustered failures and recurrent events: whether covariates with a time-dependent effect may appear as indication of unobserved heterogeneity and whether the additional presence of unobserved heterogeneity can be detected in this case. Finally, the practical implications are illustrated in a real-world data analysis example.     

Nested frailty models using maximum penalized likelihood estimation

The frailty model is a random effect survival model, which allows for unobserved heterogeneity or for statistical dependence between observed survival data. The nested frailty model accounts for the hierarchical clustering of the data by including two nested random effects. Nested frailty models are particularly appropriate when data are clustered at several hierarchical levels naturally or by design. In such cases it is important to estimate the parameters of interest as accurately as possible by taking into account the hierarchical structure of the data. We present a maximum penalized likelihood estimation (MPnLE) to estimate non-parametrically a continuous hazard function in a nested gamma-frailty model with right-censored and left-truncated data. The estimators for the regression coefficients and the variance components of the random effects are obtained simultaneously. The simulation study demonstrates that this semi-parametric approach yields satisfactory results in this complex setting. In order to illustrate the MPnLE method and the nested frailty model, we present two applications. One is for modelling the effect of particulate air pollution on mortality in different areas with two levels of geographical regrouping. The other application is based on recurrent infection times of patients from different hospitals. We illustrate that using a shared frailty model instead of a nested frailty model with two levels of regrouping leads to inaccurate estimates, with an overestimation of the variance of the random effects. We show that even when the frailty effects are fairly small in magnitude, they are important since they alter the results in a systematic pattern.     

A relaxation of the gamma frailty (Burr) model

Frailty models are used in univariate data to account for individual heterogeneity. In the popular gamma frailty model the marginal hazard has the form of a Burr model. Although the Burr model is very useful and can offer insight on the data, it is far from perfect. The estimation of the covariate effects is linked to the baseline hazard and this makes the model coefficients hard to interpret. At the same time, the frailties are assumed constant over time, while biological reasoning in some cases may indicate that frailties may be time dependent. In this paper we present a relaxation of the Burr model which is based on loosening the link between the estimation of the covariate effects and the baseline hazard. This can be achieved by replacing the cumulative baseline hazard in the Burr model by a set of time functions, and the frailty variance by a vector of coefficients directly estimated from the data using a partial likelihood. We illustrate the similarities of the model with the Burr model and a further extension of the latter, a model with an autoregressive stochastic process for the frailty. We compare the models on simulated data sets with constant and time-dependent frailties and show how the relaxed Burr models performs on two different real data sets. We show that the relaxed Burr model serves as a good approximation to the Burr model when the frailty is constant, and furthermore it gives better results when the frailty is time dependent.     

The use of Gaussian quadrature for estimation in frailty proportional hazards models

In this paper, we propose a novel Gaussian quadrature estimation method in various frailty proportional hazards models. We approximate the unspecified baseline hazard by a piecewise constant one, resulting in a parametric model that can be fitted conveniently by Gaussian quadrature tools in standard software such as SAS Proc NLMIXED. We first apply our method to simple frailty models for correlated survival data (e.g. recurrent or clustered failure times), then to joint frailty models for correlated failure times with informative dropout or a dependent terminal event such as death. Simulation studies show that our method compares favorably with the well-received penalized partial likelihood method and the Monte Carlo EM (MCEM) method, for both normal and Gamma frailty models. We apply our method to three real data examples: (1) the time to blindness of both eyes in a diabetic retinopathy study, (2) the joint analysis of recurrent opportunistic diseases in the presence of death for HIV-infected patients, and (3) the joint modeling of local, distant tumor recurrences and patients survival in a soft tissue sarcoma study. The proposed method greatly simplifies the implementation of the (joint) frailty models and makes them much more accessible to general statistical practitioners.     

Design and analysis of nested case–control studies for recurrent events subject to a terminal event

The process by which patients experience a series of recurrent events, such as hospitalizations, may be subject to death. In cohort studies, one strategy for analyzing such data is to fit a joint frailty model for the intensities of the recurrent event and death, which estimates covariate effects on the two event types while accounting for their dependence. When certain covariates are difficult to obtain, however, researchers may only have the resources to subsample patients on whom to collect complete data: one way is using the nested case–control (NCC) design, in which risk set sampling is performed based on a single outcome. We develop a general framework for the design of NCC studies in the presence of recurrent and terminal events and propose estimation and inference for a joint frailty model for recurrence and death using data arising from such studies. We propose a maximum weighted penalized likelihood approach using flexible spline models for the baseline intensity functions. Two standard error estimators are proposed: a sandwich estimator and a perturbation resampling procedure. We investigate operating characteristics of our estimators as well as design considerations via a simulation study and illustrate our methods using two studies: one on recurrent cardiac hospitalizations in patients with heart failure and the other on local recurrence and metastasis in patients with breast cancer.     

Proportional hazards models with frailties and random effects

We discuss some of the fundamental concepts underlying the development of frailty and random effects models in survival. One of these fundamental concepts was the idea of a frailty model where each subject has his or her own disposition to failure, their so-called frailty, additional to any effects we wish to quantify via regression. Although the concept of individual frailty can be of value when thinking about how data arise or when interpreting parameter estimates in the context of a fitted model, we argue that the concept is of limited practical value. Individual random effects (frailties), whenever detected, can be made to disappear by elementary model transformation. In consequence, unless we are to take some model form as unassailable, beyond challenge and carved in stone, and if we are to understand the term 'frailty' as referring to individual random effects, then frailty models have no value. Random effects models on the other hand, in which groups of individuals share some common effect, can be used to advantage. Even in this case however, if we are prepared to sacrifice some efficiency, we can avoid complex modelling by using the considerable power already provided by the stratified proportional hazards model. Stratified models and random effects models can both be seen to be particular cases of partially proportional hazards models, a view that gives further insight. The added structure of a random effects model, viewed as a stratified proportional hazards model with some added distributional constraints, will, for group sizes of five or more, provide no more than modest efficiency gains, even when the additional assumptions are exactly true. On the other hand, for moderate to large numbers of very small groups, of sizes two or three, the study of twins being a well known example, the efficiency gains of the random effects model can be far from negligible. For such applications, the case for using random effects models rather than the stratified model is strong. This is especially so in view of the good robustness properties of random effects models. Nonetheless, the simpler analysis, based upon the stratified model, remains valid, albeit making a less efficient use of resources.     

Recurrent injury event-time analysis

Public health decision making based on data sources that are characterized by a lack of independence and other complicating factors requires the development of innovative statistical techniques. Studies of injuries in occupational cohorts require methods to account for recurrent injuries to workers over time and the temporary removal of workers from the 'risk set' while recuperating. In this study, the times until injury events are modelled in an occupational cohort of employees in a large power utility company where employees are susceptible to recurrent events. The injury history over a ten-year period is used to compare the hazards of specific jobs, adjusted for age when first hired, and race/ethnicity differences. Subject-specific random effects and multiple event-times are accommodated through the application of frailty models which characterize the dependence of recurrent events over time. The counting process formulation of the proportional hazards regression model is used to estimate the effects of covariates for subjects with discontinuous intervals of risk. In this application, subjects are not at risk of injury during recovery periods or other illness, changes in jobs, or other reasons. Previous applications of proportional hazards regression in frailty models have not needed to account for the changing composition of the risk set which is required to adequately model occupational injury data. Published in 1999 by John Wiley & Sons, Ltd. This article is a US Government work and is in the public domain in the United States.     

A comparison of frailty models for multivariate survival data

This paper reviews some of the main approaches to the analysis of multivariate censored survival data. Such data typically have correlated failure times. The correlation can be a consequence of the observational design, for example with clustered sampling and matching, or it can be a focus of interest as in genetic studies, longitudinal studies of recurrent events and other studies involving multiple measurements. We assume that the correlation between the failure or survival times can be accounted for by fixed or random frailty effects. We then compare the performance of conditional and mixture likelihood approaches to estimating models with these frailty effects in censored bivariate survival data. We find that the mixture methods are surprisingly robust to misspecification of the frailty distribution. The paper also contains an illustrative example on the times to onset of chest pain brought on by three endurance exercise tests during a drug treatment trial of heart patients.     

Coping with time and space in modelling malaria incidence: a comparison of survival and count regression models

To study the effect of a mega hydropower dam in southwest Ethiopia on malaria incidence, we have set up a longitudinal study. To gain insight in temporal and spatial aspects, that is, in time (period = year–season combination) and location (village), we need models that account for these effects. The frailty model with periodwise constant baseline hazard (a constant value for each period) and a frailty term that models the clustering in villages provides an appropriate tool for the analysis of such incidence data. Count data can be obtained by aggregating for each period events at the village level. The mixed Poisson regression model can be used to model the count data. We show the similarities between the two models. The risk factor in both models is the distance to the dam, and we study the effect of the risk factor on malaria incidence. In the frailty model, each subject has its own risk factor, whereas in the Poisson regression model, we also need to average the risk factors of all subjects contributing to a particular count. The power loss caused by using village averaged distance instead of individual distance is studied and quantified. The loss in the malaria data example is rather small. In such a setting, it might be advantageous to use less labor‐intensive sampling schemes than the weekly individual follow‐up scheme used in this study; the proposed alternative sampling schemes might also avoid community fatigue, a typical problem in such research projects. Copyright © 2013 John Wiley & Sons, Ltd.     

On robustness of marginal regression coefficient estimates and hazard functions in multivariate survival analysis of family data when the frailty distribution is mis-specified

The shared frailty model is an extension of the Cox model to correlated failure times and, essentially, a random effects model for failure time outcomes. In this model, the latent frailty shared by individual members in a cluster acts multiplicatively as a factor on the hazard function and is typically modelled parametrically. One commonly used distribution is gamma, where both shape and scale parameters are set to be the same to allow for unique identification of baseline hazard function. It is popular because it is a conjugate prior, and the posterior distribution possesses the same form as gamma. In addition, the parameter can be interpreted as a time-independent cross-ratio function, a natural extension of odds ratio to failure time outcomes. In this paper, we study the effect of frailty distribution mis-specification on the marginal regression estimates and hazard functions under assumed gamma distribution with an application to family studies. The simulation results show that the biases are generally 10% and lower, even when the true frailty distribution deviates substantially from the assumed gamma distribution. This suggests that the gamma frailty model can be a practical choice in real data analyses if the regression parameters and marginal hazard function are of primary interest and individual cluster members are exchangeable with respect to their dependencies.     

A semiparametric recurrent events model with time‐varying coefficients

We consider a recurrent events model with time‐varying coefficients motivated by two clinical applications. We use a random effects (Gaussian frailty) model to describe the intensity of recurrent events. The model can accommodate both time‐varying and time‐constant coefficients. We use the penalized spline method to estimate the time‐varying coefficients. We use Laplace approximation to evaluate the penalized likelihood without a closed form. We estimate the smoothing parameters in a similar way to variance components. We conduct simulations to evaluate the performance of the estimates for both time‐varying and time‐independent coefficients. We apply this method to analyze two data sets: a stroke study and a child wheeze study. Copyright © 2012 John Wiley & Sons, Ltd.     

Multilevel model with random effects for clustered survival data with multiple failure outcomes

We present a multilevel frailty model for handling serial dependence and simultaneous heterogeneity in survival data with a multilevel structure attributed to clustering of subjects and the presence of multiple failure outcomes. One commonly observes such data, for example, in multi-institutional, randomized placebo-controlled trials in which patients suffer repeated episodes (eg, recurrent migraines) of the disease outcome being measured. The model extends the proportional hazards model by incorporating a random covariate and unobservable random institution effect to respectively account for treatment-by-institution interaction and institutional variation in the baseline risk. Moreover, a random effect term with correlation structure driven by a first-order autoregressive process is attached to the model to facilitate estimation of between patient heterogeneity and serial dependence. By means of the generalized linear mixed model methodology, the random effects distribution is assumed normal and the residual maximum likelihood and the maximum likelihood methods are extended for estimation of model parameters. Simulation studies are carried out to evaluate the performance of the residual maximum likelihood and the maximum likelihood estimators and to assess the impact of misspecifying random effects distribution on the proposed inference. We demonstrate the practical feasibility of the modeling methodology by analyzing real data from a double-blind randomized multi-institutional clinical trial, designed to examine the effect of rhDNase on the occurrence of respiratory exacerbations among patients with cystic fibrosis.     

Dispersion frailty models and HGLMs

In medical research recurrent event times can be analysed using a frailty model in which the frailties for different individuals are independent and identically distributed. However, such a homogeneous assumption about frailties could sometimes be suspect. For modelling heterogeneity in frailties we describe dispersion frailty models arising from a new class of models, namely hierarchical generalized linear models. Using the kidney infection data we illustrate how to detect and model heterogeneity among frailties. Stratification of frailty models is also investigated.     

Multilevel mixed effects parametric survival models using adaptive Gauss–Hermite quadrature with application to recurrent events and individual participant data meta‐analysis

Multilevel mixed effects survival models are used in the analysis of clustered survival data, such as repeated events, multicenter clinical trials, and individual participant data (IPD) meta‐analyses, to investigate heterogeneity in baseline risk and covariate effects. In this paper, we extend parametric frailty models including the exponential, Weibull and Gompertz proportional hazards (PH) models and the log logistic, log normal, and generalized gamma accelerated failure time models to allow any number of normally distributed random effects. Furthermore, we extend the flexible parametric survival model of Royston and Parmar, modeled on the log‐cumulative hazard scale using restricted cubic splines, to include random effects while also allowing for non‐PH (time‐dependent effects). Maximum likelihood is used to estimate the models utilizing adaptive or nonadaptive Gauss–Hermite quadrature. The methods are evaluated through simulation studies representing clinically plausible scenarios of a multicenter trial and IPD meta‐analysis, showing good performance of the estimation method. The flexible parametric mixed effects model is illustrated using a dataset of patients with kidney disease and repeated times to infection and an IPD meta‐analysis of prognostic factor studies in patients with breast cancer. User‐friendly Stata software is provided to implement the methods. Copyright © 2014 John Wiley & Sons, Ltd.     

The partly Aalen's model for recurrent event data with a dependent terminal event

Recurrent event data are commonly observed in biomedical longitudinal studies. In many instances, there exists a terminal event, which precludes the occurrence of additional repeated events, and usually there is also a nonignorable correlation between the terminal event and recurrent events. In this article, we propose a partly Aalen's additive model with a multiplicative frailty for the rate function of recurrent event process and assume a Cox frailty model for terminal event time. A shared gamma frailty is used to describe the correlation between the two types of events. Consequently, this joint model can provide the information of temporal influence of absolute covariate effects on the rate of recurrent event process, which is usually helpful in the decision‐making process for physicians. An estimating equation approach is developed to estimate marginal and association parameters in the joint model. The consistency of the proposed estimator is established. Simulation studies demonstrate that the proposed approach is appropriate for practical use. We apply the proposed method to a peritonitis cohort data set for illustration. Copyright © 2015 John Wiley & Sons, Ltd.     

Semiparametric regression analysis for alternating recurrent event data

Alternating recurrent event data arise frequently in clinical and epidemiologic studies, where 2 types of events such as hospital admission and discharge occur alternately over time. The 2 alternating states defined by these recurrent events could each carry important and distinct information about a patient's underlying health condition and/or the quality of care. In this paper, we propose a semiparametric method for evaluating covariate effects on the 2 alternating states jointly. The proposed methodology accounts for the dependence among the alternating states as well as the heterogeneity across patients via a frailty with unspecified distribution. Moreover, the estimation procedure, which is based on smooth estimating equations, not only properly addresses challenges such as induced dependent censoring and intercept sampling bias commonly confronted in serial event gap time data but also is more computationally tractable than the existing rank‐based methods. The proposed methods are evaluated by simulation studies and illustrated by analyzing psychiatric contacts from the South Verona Psychiatric Case Register.     

A varying-coefficient model for the evaluation of time-varying concomitant intervention effects in longitudinal studies

Concomitant interventions are often introduced during a longitudinal clinical trial to patients who respond undesirably to the pre-specified treatments. In addition to the main objective of evaluating the pre-specified treatment effects, an important secondary objective in such a trial is to evaluate whether a concomitant intervention could change a patient's response over time. Because the initiation of a concomitant intervention may depend on the patient's general trend of pre-intervention outcomes, regression approaches that treat the presence of the intervention as a time-dependent covariate may lead to biased estimates for the intervention effects. Borrowing the techniques of Follmann and Wu (Biometrics 1995; 51:151-168) for modeling informative missing data, we propose a varying-coefficient mixed-effects model to evaluate the patient's longitudinal outcome trends before and after the patient's starting time of the intervention. By allowing the random coefficients to be correlated with the patient's starting time of the intervention, our model leads to less biased estimates of the intervention effects. Nonparametric estimation and inferences of the coefficient curves and intervention effects are developed using B-splines. Our methods are demonstrated through a longitudinal clinical trial in depression and heart disease and a simulation study.     

Individualizing drug dosage with longitudinal data

We propose a two‐step procedure to personalize drug dosage over time under the framework of a log‐linear mixed‐effect model. We model patients' heterogeneity using subject‐specific random effects, which are treated as the realizations of an unspecified stochastic process. We extend the conditional quadratic inference function to estimate both fixed‐effect coefficients and individual random effects on a longitudinal training data sample in the first step and propose an adaptive procedure to estimate new patients' random effects and provide dosage recommendations for new patients in the second step. An advantage of our approach is that we do not impose any distribution assumption on estimating random effects. Moreover, the new approach can accommodate more general time‐varying covariates corresponding to random effects. We show in theory and numerical studies that the proposed method is more efficient compared with existing approaches, especially when covariates are time varying. In addition, a real data example of a clozapine study confirms that our two‐step procedure leads to more accurate drug dosage recommendations. Copyright © 2016 John Wiley & Sons, Ltd.     

A time-dependent discrimination index for survival data

To derive models suitable for outcome prediction, a crucial aspect is the availability of appropriate measures of predictive accuracy, which have to be usable for a general class of models. The Harrell's C discrimination index is an extension of the area under the ROC curve to the case of censored survival data, which owns a straightforward interpretability. For a model including covariates with time-dependent effects and/or time-dependent covariates, the original definition of C would require the prediction of individual failure times, which is not generally addressed in most clinical applications. Here we propose a time-dependent discrimination index Ctd where the whole predicted survival function is utilized as outcome prediction, and the ability to discriminate among subjects having different outcome is summarized over time. Ctd is based on a novel definition of concordance: a subject who developed the event should have a less predicted probability of surviving beyond his/her survival time than any subject who survived longer. The predicted survival function of a subject who developed the event is compared to: (1) that of subjects who developed the event before his/her survival time, and (2) that of subjects who developed the event, or were censored, after his/her survival time. Subjects who were censored are involved in comparisons with subjects who developed the event before their observed times. The index reduces to the previous C in the presence of separation between survival curves on the whole follow-up. A confidence interval for Ctd is derived using the jackknife method on correlated one-sample U-statistics.The proposed index is used to evaluate the discrimination ability of a model, including covariates having time-dependent effects, concerning time to relapse in breast cancer patients treated with adjuvant tamoxifen. The model was obtained from 596 patients entered prospectively at Istituto Nazionale per lo Studio e la Cura dei Tumori di Milano (INT). The model discrimination ability was validated on an independent testing data set of 175 patients provided by Centro Regionale Indicatori Biochimici di Tumore (CRIBT) in Venice.