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.     

Variable selection in subdistribution hazard frailty models with competing risks data

The proportional subdistribution hazards model (i.e. Fine‐Gray model) has been widely used for analyzing univariate competing risks data. Recently, this model has been extended to clustered competing risks data via frailty. To the best of our knowledge, however, there has been no literature on variable selection method for such competing risks frailty models. In this paper, we propose a simple but unified procedure via a penalized h‐likelihood (HL) for variable selection of fixed effects in a general class of subdistribution hazard frailty models, in which random effects may be shared or correlated. We consider three penalty functions, least absolute shrinkage and selection operator (LASSO), smoothly clipped absolute deviation (SCAD) and HL, in our variable selection procedure. We show that the proposed method can be easily implemented using a slight modification to existing h‐likelihood estimation approaches. Numerical studies demonstrate that the proposed procedure using the HL penalty performs well, providing a higher probability of choosing the true model than LASSO and SCAD methods without losing prediction accuracy. The usefulness of the new method is illustrated using two actual datasets from multi‐center clinical trials. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

Estimating and testing for center effects in competing risks

The problems of fitting Gaussian frailties proportional hazards models for the subdistribution of a competing risk and of testing for center effects are considered. In the analysis of competing risks data, Fine and Gray proposed a proportional hazards model for the subdistribution to directly assess the effects of covariates on the marginal failure probabilities of a given failure cause. Katsahianbiet al. extended their model to clustered time to event data, by including random center effects or frailties in the subdistribution hazard. We first introduce an alternate estimation procedure to the one proposed by Katsahian et al. This alternate estimation method is based on the penalized partial likelihood approach often used in fitting Gaussian frailty proportional hazards models in the standard survival analysis context, and has the advantage of using standard survival analysis software. Second, four hypothesis tests for the presence of center effects are given and compared via Monte-Carlo simulations. Statistical and numerical considerations lead us to formulate pragmatic guidelines as to which of the four tests is preferable. We also illustrate the proposed methodology with registry data from bone marrow transplantation for acute myeloid leukemia (AML).     

Penalized likelihood estimation for semiparametric mixed models,with application to alcohol treatment research

In this article, we implement a practical computational method for various semiparametric mixed effects models, estimating nonlinear functions by penalized splines. We approximate the integration of the penalized likelihood with respect to random effects with the use of adaptive Gaussian quadrature, which we can conveniently implement in SAS procedure NLMIXED. We carry out the selection of smoothing parameters through approximated generalized cross‐validation scores. Our method has two advantages: (1) the estimation is more accurate than the current available quasi‐likelihood method for sparse data, for example, binary data; and (2) it can be used in fitting more sophisticated models. We show the performance of our approach in simulation studies with longitudinal outcomes from three settings: binary, normal data after Box–Cox transformation, and count data with log‐Gamma random effects. We also develop an estimation method for a longitudinal two‐part nonparametric random effects model and apply it to analyze repeated measures of semicontinuous daily drinking records in a randomized controlled trial of topiramate. Copyright © 2012 John Wiley & Sons, Ltd.     

A joint model of recurrent events and a terminal event with a nonparametric covariate function

We extend the shared frailty model of recurrent events and a dependent terminal event to allow for a nonparametric covariate function. We include a Gaussian random effect (frailty) in the intensity functions of both the recurrent and terminal events to capture correlation between the two processes. We employ the penalized cubic spline method to describe the nonparametric covariate function in the recurrent events model. We use Laplace approximation to evaluate the marginal penalized partial likelihood without a closed form. We also propose the variance estimates for regression coefficients. Numerical analysis results show that the proposed estimates perform well for both the nonparametric and parametric components. We apply this method to analyze the hospitalization rate of patients with heart failure in the presence of death.     

Generalized survival models for correlated time‐to‐event data

Our aim is to develop a rich and coherent framework for modeling correlated time‐to‐event data, including (1) survival regression models with different links and (2) flexible modeling for time‐dependent and nonlinear effects with rich postestimation. We extend the class of generalized survival models, which expresses a transformed survival in terms of a linear predictor, by incorporating a shared frailty or random effects for correlated survival data. The proposed approach can include parametric or penalized smooth functions for time, time‐dependent effects, nonlinear effects, and their interactions. The maximum (penalized) marginal likelihood method is used to estimate the regression coefficients and the variance for the frailty or random effects. The optimal smoothing parameters for the penalized marginal likelihood estimation can be automatically selected by a likelihood‐based cross‐validation criterion. For models with normal random effects, Gauss‐Hermite quadrature can be used to obtain the cluster‐level marginal likelihoods. The Akaike Information Criterion can be used to compare models and select the link function. We have implemented these methods in the R package rstpm2. Simulating for both small and larger clusters, we find that this approach performs well. Through 2 applications, we demonstrate (1) a comparison of proportional hazards and proportional odds models with random effects for clustered survival data and (2) the estimation of time‐varying effects on the log‐time scale, age‐varying effects for a specific treatment, and two‐dimensional splines for time and age.     

Model selection for multi-component frailty models

Various frailty models have been developed and are now widely used for analysing multivariate survival data. It is therefore important to develop an information criterion for model selection. However, in frailty models there are several alternative ways of forming a criterion and the particular criterion chosen may not be uniformly best. In this paper, we study an Akaike information criterion (AIC) on selecting a frailty structure from a set of (possibly) non-nested frailty models. We propose two new AIC criteria, based on a conditional likelihood and an extended restricted likelihood (ERL) given by Lee and Nelder (J. R. Statist. Soc. B 1996; 58:619-678). We compare their performance using well-known practical examples and demonstrate that the two criteria may yield rather different results. A simulation study shows that the AIC based on the ERL is recommended, when attention is focussed on selecting the frailty structure rather than the fixed effects.     

A general frailty model to accommodate individual heterogeneity in the acquisition of multiple infections: An application to bivariate current status data

The analysis of multivariate time-to-event (TTE) data can become complicated due to the presence of clustering, leading to dependence between multiple event times. For a long time, (conditional) frailty models and (marginal) copula models have been used to analyze clustered TTE data. In this article, we propose a general frailty model employing a copula function between the frailty terms to construct flexible (bivariate) frailty distributions with the application to current status data. The model has the advantage to impose a less restrictive correlation structure among latent frailty variables as compared to traditional frailty models. Specifically, our model uses a copula function to join the marginal distributions of the frailty vector. In this article, we considered different copula functions, and we relied on marginal gamma distributions due to their mathematical convenience. Based on a simulation study, our novel model outperformed the commonly used additive correlated gamma frailty model, especially in the case of a negative association between the frailties. At the end of the article, the new methodology is illustrated on real-life data applications entailing bivariate serological survey data.     

Generalized gamma frailty model

In this article, we present a frailty model using the generalized gamma distribution as the frailty distribution. It is a power generalization of the popular gamma frailty model. It also includes other frailty models such as the lognormal and Weibull frailty models as special cases. The flexibility of this frailty distribution makes it possible to detect a complex frailty distribution structure which may otherwise be missed. Due to the intractable integrals in the likelihood function and its derivatives, we propose to approximate the integrals either by Monte Carlo simulation or by a quadrature method and then determine the maximum likelihood estimates of the parameters in the model. We explore the properties of the proposed frailty model and the computation method through a simulation study. The study shows that the proposed model can potentially reduce errors in the estimation, and that it provides a viable alternative for correlated data. The merits of proposed model are demonstrated in analysing the effects of sublingual nitroglycerin and oral isosorbide dinitrate on angina pectoris of coronary heart disease patients based on the data set in Danahy et al. (sustained hemodynamic and antianginal effect of high dose oral isosorbide dinitrate. Circulation 1977; 55:381-387).     

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.     

A multiparameter regression model for interval-censored survival data

We develop flexible multiparameter regression (MPR) survival models for interval-censored survival data arising in longitudinal prospective studies and longitudinal randomised controlled clinical trials. A multiparameter Weibull regression survival model, which is wholly parametric, and has nonproportional hazards, is the main focus of the article. We describe the basic model, develop the interval-censored likelihood, and extend the model to include gamma frailty and a dispersion model. We evaluate the models by means of a simulation study and a detailed reanalysis of data from the Signal Tandmobiel study. The results demonstrate that the MPR model with frailty is computationally efficient and provides an excellent fit to the data.     

Promotion time cure rate model with nonparametric form of covariate effects

Survival data with a cured portion are commonly seen in clinical trials. Motivated from a biological interpretation of cancer metastasis, promotion time cure model is a popular alternative to the mixture cure rate model for analyzing such data. The existing promotion cure models all assume a restrictive parametric form of covariate effects, which can be incorrectly specified especially at the exploratory stage. In this paper, we propose a nonparametric approach to modeling the covariate effects under the framework of promotion time cure model. The covariate effect function is estimated by smoothing splines via the optimization of a penalized profile likelihood. Point‐wise interval estimates are also derived from the Bayesian interpretation of the penalized profile likelihood. Asymptotic convergence rates are established for the proposed estimates. Simulations show excellent performance of the proposed nonparametric method, which is then applied to a melanoma study.     

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.     

Joint modeling of binary response and survival for clustered data in clinical trials

In clinical trials, it is often desirable to evaluate the effect of a prognostic factor such as a marker response on a survival outcome. However, the marker response and survival outcome are usually associated with some potentially unobservable factors. In this case, the conventional statistical methods that model these two outcomes separately may not be appropriate. In this paper, we propose a joint model for marker response and survival outcomes for clustered data, providing efficient statistical inference by considering these two outcomes simultaneously. We focus on a special type of marker response: a binary outcome, which is investigated together with survival data using a cluster-specific multivariate random effect variable. A multivariate penalized likelihood method is developed to make statistical inference for the joint model. However, the standard errors obtained from the penalized likelihood method are usually underestimated. This issue is addressed using a jackknife resampling method to obtain a consistent estimate of standard errors. We conduct extensive simulation studies to assess the finite sample performance of the proposed joint model and inference methods in different scenarios. The simulation studies show that the proposed joint model has excellent finite sample properties compared to the separate models when there exists an underlying association between the marker response and survival data. Finally, we apply the proposed method to a symptom control study conducted by Canadian Cancer Trials Group to explore the prognostic effect of covariates on pain control and overall survival.     

A general semiparametric hazards regression model: efficient estimation and structure selection

We consider a general semiparametric hazards regression model that encompasses the Cox proportional hazards model and the accelerated failure time model for survival analysis. To overcome the nonexistence of the maximum likelihood, we derive a kernel‐smoothed profile likelihood function and prove that the resulting estimates of the regression parameters are consistent and achieve semiparametric efficiency. In addition, we develop penalized structure selection techniques to determine which covariates constitute the accelerated failure time model and which covariates constitute the proportional hazards model. The proposed method is able to estimate the model structure consistently and model parameters efficiently. Furthermore, variance estimation is straightforward. The proposed estimation performs well in simulation studies and is applied to the analysis of a real data set. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

Modelling survival data with a cured fraction using frailty models

Cure models have historically been utilized to analyse time-to-event data with a cured fraction. We consider the use of frailty models as an alternative approach to modelling such data. An attractive feature of the models is the allowance for heterogeneity in risk among those individuals experiencing the event of interest in addition to the incorporation of a cured component. Utilizing maximum likelihood techniques, we fit models to data concerning the recurrence of leukaemia among patients receiving autologous transplantation treatment. The analysis suggests that the gamma frailty mixture model and the compound Poisson improve on the fit of the leukaemia data as compared to the standard cure model.     

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.     

Modeling event count data in the presence of informative dropout with application to bleeding and transfusion events in myelodysplastic syndrome

In many biomedical studies, it is often of interest to model event count data over the study period. For some patients, we may not follow up them for the entire study period owing to informative dropout. The dropout time can potentially provide valuable insight on the rate of the events. We propose a joint semiparametric model for event count data and informative dropout time that allows for correlation through a Gamma frailty. We develop efficient likelihood‐based estimation and inference procedures. The proposed nonparametric maximum likelihood estimators are shown to be consistent and asymptotically normal. Furthermore, the asymptotic covariances of the finite‐dimensional parameter estimates attain the semiparametric efficiency bound. Extensive simulation studies demonstrate that the proposed methods perform well in practice. We illustrate the proposed methods through an application to a clinical trial for bleeding and transfusion events in myelodysplastic syndrome. Copyright © 2017 John Wiley & Sons, Ltd.