Teoría del cuidado humano

In this section, we review previous research on joint modeling of recurrent and termi- nal events. Statistical methodology and theory for analyzing recurrent event data are typically developed based on non-informative censoring times. In many applications, however, when a failure event serves as a part of the censoring mechanism, mean- ing that the failure event terminates observing further recurrent events (so-called informative censoring), the independent censoring assumption can be violated. For example, if the rate of recurrent tumors is high in a patient, this patient is also sub- ject to increased risk of death. The most popular solution to model or control the dependence of recurrent events with a terminal event or informative censoring is a joint modeling approach.

Joint (or shared) frailty (or random effects) models have been studied by several authors. In these models, the dependence between recurrent and terminal events were specified via a common frailty variable allowed to have a multiplicative effect on their respective rates. The most popular distributional assumption on the frailty was a gamma distribution with unit mean to avoid the non-identifiability issue (Lancaster and Intrator, 1998; Liu et al., 2004; Ye et al., 2007; Huang and Liu, 2007). Lancaster and Intrator (1998) considered joint parametric modeling of recurrent event and sur- vival data, using Poisson processes for the rate functions of the recurrent and terminal events. Liu et al. (2004) considered proportional hazards frailty models where the re- current and terminal event processes were jointly modeled by a shared gamma frailty.

The frailty effect was allowed to be different for the two processes and time-dependent covariates could be incorporated separately in both processes. A Monte Carlo EM algorithm with a Metropolis-Hastings sampler in the E-step was adapted to obtain the maximum likelihood estimators. However, the Monte Carlo EM algorithm is of- ten computationally inefficient and less accurate than the standard EM algorithm. Instead, Rondeau et al. (2007) studied a penalized likelihood approach, to estimate parameters in the model proposed by Liu et al. (2004), adopting the sum of squared norms of the second derivatives of the intensity and hazard functions as the roughness penalty.

Without distributional assumptions on the latent variables and censoring time, Wang et al. (2001) modeled the occurrence rate function for recurrent events with informative censoring in semiparametric and nonparametric ways. They assumed a subject-specific nonstationary Poisson process via a latent variable. However, the proposed model is not applicable to situations where inferences for both the recurrent and terminal events are of interest. To overcome this limitation, Huang and Wang (2004) presented a joint model for recurrent event process and a failure time, while informative censoring is allowed for observing both the recurrent events and failure times. They assumed that the recurrent, failure, and censoring events are mutually independent conditioning on the covariates and latent variables. They proposed a “borrow-strength estimation” procedure, in which first the value of the latent variable was estimated from recurrent event data, then the estimated value was used in the failure time model. Since the proposed approach did not utilize the information of the failure times in estimating the latent effect, it might be less efficient than it could be. The semiparametric models proposed by Wang et al. (2001) and Huang and Wang (2004) are flexible in that no parametric assumption was imposed on the frailty by treating as a nuisance parameter, however, these models are not applicable

to time-dependent covariates.

Most of the existing work required the proportional intensity or hazards assump- tion and assumed time-independent covariates. Recently, Zeng and Lin (2009) de- veloped transformation models for the recurrent and terminal events that can deal with non-proportional hazards as well as time-varying covariates. Their proposed models are flexible enough as one can choose different forms of transformation for the respective events and as the class of transformations includes a variety of models of interest such as the proportional hazards model and the proportional odds model. Also, there is a wide range of choices open for the distribution of the shared random effects as long as satisfying the imposed conditions.

Chapter 3

Joint Models of Longitudinal Data

and Recurrent Events with

Informative Terminal Event

3.1

Introduction

In many biomedical studies, data are collected from patients who experience the same type of event multiple times, such as repeated hospital admissions or medical emer- gency episodes, recurrent strokes, multiple infection episodes, or tumor recurrences. At the same time, some longitudinal biomarkers are observed either at the time of occurrence of the event or at regular clinic visits. In addition, some subjects may experience a terminal event such as death. As longitudinal markers, recurrent events, and death are dependent on and informative of one another, analyzing one or two of these processes but ignoring the dependence from the other processes may lead to bias or result in inefficient inference. Therefore, it is important to jointly model longitudinal markers, recurrent events, and death altogether. In this way, we will be able to make the most efficient use of all data and identify the effects of variables

after correctly controlling the interplay among these processes.

There is scant literature considering the dependence of a terminal event in mod- eling both repeated measures and recurrent event processes. Most recently, Liu and Huang (2009) have developed a joint model for repeatedly measured CD4 cell counts and related opportunistic infection recurrences while associating their relationship with the mortality of HIV patients in the CPCRA (AIDS) study. In this study, since the CD4 cell counts were observed at scheduled visits, the observation times were non-informative. However, when the CD4 cell counts are measured at emergency admissions or unexpected hospitalizations, the information of the number and times of observations is critical, and hence it should be taken into account in modeling. By treating the hospital visits as recurrent events, Liu et al. (2008) presented a joint model of the medical cost process for chronic heart failure patients in the presence of informative observation times and a dependent death event. The joint modeling ap- proaches by Liu et al. (2008) and Liu and Huang (2009) required the proportionality assumption for both recurrent and terminal events. In cases where the proportion- ality assumption does not hold, their joint models may yield biased estimators. In their inference procedures, the piecewise constant functions were adopted for estimat- ing the underlying baseline intensity and hazards functions; however, there was no general rule for selecting the number of knots that led to the best reflection of the underlying baseline intensity functions. Moreover, the theoretical properties of the suggested estimators have not been established.

In this paper, we will use general transformation models for both the recurrent events and the terminal event, while accounting for the dependence among these two event processes and longitudinal data. Our transformation models include the proportional hazards models and the proportional odds models as special cases. We will propose efficient estimates and establish their asymptotic properties. The rest

of the chapter is organized as follows. In Section 3.2, we introduce joint models for longitudinal measurements and recurrent events in the presence of a terminal event. In Section 3.3, we estimate all the parameters using the nonparametric maximum likelihood estimation (NPMLE) and provide computationally simple algorithms to implement the proposed inference procedure. The theoretical work that shows the weak convergence and efficiency of the proposed NPMLEs is given in Section 3.4. Section 3.5 evaluates the performance of the proposed method through extensive simulation studies, and the application to the Atherosclerosis Risk in Communities (ARIC) data is reported in Section 3.6. We conclude with some remarks in Section 3.7. In Section 3.8, the EM algoritm is described in more detail, and the proof of the established asymptotic properties are provided in Section 3.9.