Analysis of Recurrent Event Data

doi:10.1016/S0169-7161(03)23034-0

Handbook of Statistics

Volume 23, 2003, Pages 603-623

https://doi.org/10.1016/S0169-7161(03)23034-0 Get rights and content

Publisher Summary

This chapter describes non- and semiparametric methods of analyzing recurrent event data. It focuses on the functions that are modeled in the analysis of recurrent event data. Recurrent event data are often encountered in biomedicine (e.g., opportunistic infections among AIDS patients), demography (e.g., birth patterns among woman of child-bearing age), and quality control (e.g., automobile repairs). The data structure for recurrent events represents a special case of multivariate survival data, where the failure times for a subject are ordered. As such, recurrent event data have often been analyzed using methods of multivariate survival analysis. Semiparametric regression methods are described in the chapter, with special attention given to elucidating the sometimes subtle differences between the methods, with respect to interpretation of parameter estimates. The methods are also illustrated using an example analysis of a preschool asthma data set.

Introduction

In many research settings, the event of interest can be experienced more than once per subject. Such outcomes have been termed recurrent events. Recurrent event data are often encountered in biomedicine (e.g., opportunistic infections among AIDS patients), demography (e.g., birth patterns among woman of child-bearing age) and quality control (e.g., automobile repairs). Typically, the study will have a fixed termination date, such that the occurrence times are potentially censored. The data structure for recurrent events represents a special case of multivariate survival data, where the failure times for a subject are ordered. As such, recurrent event data have often been analyzed using methods of multivariate survival analysis (e.g., Wei et al., 1989, Prentice et al., 1981, Andersen and Gill, 1982). However, fundamental characteristics of recurrent event data mean that care must be exercised in the application of methods designed for a larger class of general data structures amenable to multivariate survival analysis. Correspondingly, the analysis of recurrent event data has recently been the subject of much methodological research.

In this chapter, we describe non- and semi-parametric methods of analyzing recurrent event data. The chapter is organized as follows. In the second section, we set up essential notation, then focus on the functions which are modelled in the analysis of recurrent event data. In the third section, semiparametric regression methods are described, with special attention given to elucidating the sometimes subtle differences between the methods, with respect to interpretation of parameter estimates. The methods will be illustrated using an example analysis of a preschool asthma data set. In Section 4, we describe various nonparametric estimation methods which can be used for recurrent event data. The final section contains some concluding remarks, and lists some very recent work dealing with data structures which lie beyond the scope of this chapter.

Section snippets

Notation and basic functions of interest

Throughout this chapter, the notation will be as follows. N_i(t)=∫₀^tdN_i(s) will represent the number of events in [0,t] for subject i, for i=1,…,n, where dN_i(s) denotes the number of events in the small time interval [s,s+ds). We assume that subject i is observed over the [0,C_i] interval, where C_i denotes censoring time, and is observed to experience events at times T_i,1,…,T_{i,m_i}. We assume that C_i is determined independently of ${d N_{i} (t); t⩾0}$ . In the presence of covariates, the assumption is

Semiparametric models for recurrent event data

The effects of certain covariates on the recurrent event process are often of interest to investigators. In this section, we describe various semiparametric models which often form the basis for regression analysis of recurrent event data. Several of the models are illustrated through the analysis of a retrospective cohort study on preschool asthma (Schaubel et al., 1996). For this study, children born during a one-year period, April 1, 1984 to March 31, 1985, were followed retrospectively from

Nonparametric estimation of the recurrent event survival and distribution functions

Each of the methods described in this chapter thus far has involved semiparametric modelling. We now consider nonparametric estimation of the gap time distribution, survival and related functions.

A complication in the estimation of gap time functions is induced dependent censoring. That is, even if total times are censored independently (e.g., loss to follow-up, administrative censoring), the gap times for the second and subsequent events will be subject to induced dependent censoring, except

Conclusion

In this chapter, we have reviewed several methods useful in the analysis of recurrent event data. The example analysis of the preschool asthma data set illustrates the differences between the methods, and resulting parameter estimators. Ultimately, the choice of which model to fit depends on the objectives of the investigator, and possibly the specifics of the data set of interest. Methodological interest in recurrent event data persists. Methods have recently been developed for data structures

Acknowledgements

This work was partially supported by National Institutes of Heath grant R01 HL-57444.

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Sufficient dimension reduction on marginal regression for gaps of recurrent events
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A semiparametric linear transformation of gap time is proposed to model recurrent event data with high-dimensional covariates and informative censoring. It is derived from a proportional hazards model for the conditional intensity function of a renewal process. To overcome the difficulty arising from high-dimensional covariates, we develop a modified sliced regression for censored data and use a sufficient dimension reduction procedure to transform them to a lower dimensional space. Simulation studies are performed to confirm and evaluate the theoretical findings, and to compare the proposed method with existing methods in the literature. An example of application on a set of medical data is demonstrated as well. The proposed model together with the dimension reduction method offers an effective alternative for the analysis of recurrent event with high-dimensional covariates and informative censoring.
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By panel count data, we mean the data that concern occurrence rates of certain recurrent events and give only the numbers of the events that occur between the observation times, but not their occurrence times. Such data naturally occur in longitudinal follow-up studies on recurrent events in which study subjects can be observed only at discrete time points rather than continuously (Cai and Schaubel, 2004; Cook and Lawless, 2007; Sun, 2006). Many authors have discussed the analysis of panel count data when the recurrent event process of interest and the observation process are independent completely or conditional on covariates.
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Analysis of Recurrent Event Data

Publisher Summary

Introduction

Section snippets

Notation and basic functions of interest

Semiparametric models for recurrent event data

Nonparametric estimation of the recurrent event survival and distribution functions

Conclusion

Acknowledgements

Statistical Models Based on Counting Processes

Cox's regression model for counting processes: A large sample study

Ann. Statist.

Multiple statistics for multiple events, with application to repeated infections in the growth factor studies

Statist. Medicine

Contribution to the discussion of the paper by D.R. Cox

J. Roy. Statist. Soc. Ser. B

Estimating equations for hazard ratio parameters based on correlated failure time data

Biometrika

Regression estimation using multivariate failure time data and a common baseline hazard function model

Lifetime Data Anal.

Marginal means and rates models for multiple type recurrent event data

Lifetime Data Anal

Large sample properties of nonparametric statistical inference

Information and asymptotic efficiency in some generalized proportional hazards models for counting processes

Ann. Statist.

Conditional regression analysis for recurrence time data

J. Amer. Statist. Assoc.

Introduction to Stochastic Processes in Biostatistics

Some approaches to the analysis of recurrent event data

Statist. Methods Medical Res.

Discussion of paper by Wei and Glidden

Statist. Medicine

Marginal analysis of recurrent events and a terminating event

Statist. Medicine

Regression models and life-tables (with discussion)

J. Roy. Statist. Soc. Ser. B

Partial likelihood

Biometrika

Point Processes

Kaplan–Meier estimate on the plane

Ann. Statist.

Censored data and the bootstrap

J. Amer. Statist. Assoc.

Counting Processes and Survival Analysis

An empirical comparison of two semi-parametric approaches for the estimation of covariate effects from multivariate failure time data

Statist. Medicine

Nonparametric analysis of recurrent events and death

Biometrics

The behaviour of maximum likelihood estimates under nonstandard conditions

The Statistical Analysis of Failure Time Data

Nonparametric estimation from incomplete samples

J. Amer. Statist. Assoc.

Survival analysis for recurrent event data: An application to childhood infectious diseases

Statist. Medicine

Covariance analysis of censored survival data using log-linear analysis techniques

J. Amer. Statist. Assoc.

The analysis of recurrent events for multiple subjects

Appl. Statist.

Some simple robust methods for the analysis of recurrent events

Technometrics