Jerald F. Lawless

The Analysis of Panel Data under a Markov Assumption

Abstract Methods for the analysis of panel data under a continuous-time Markov model are proposed. We present procedures for obtaining maximum likelihood estimates and associated asymptotic covariance matrices for transition intensity parameters in time homogeneous models, and for other process characteristics such as mean sojourn times and equilibrium distributions. Generalizations to handle covariance analysis and to the fitting of certain nonhomogeneous models are presented, and an example based on a longitudinal study of the smoking habits of school children is discussed. Questions of embeddability and estimation are examined.

Some Simple Robust Methods for the Analysis of Recurrent Events

Nelson discussed a method of estimating the cumulative mean function for identically distributed processes of recurrent events. We show that a similar approach can be used with more general models, including regression, The key idea is to use point estimates based on Poisson models and to develop robust variance estimates that are valid more generally. The methods are illustrated on reliability and warranty data.

Regression Methods for Poisson Process Data

Abstract This article is directed toward situations where individuals can experience repeated events, and data on an individual consist of the number and occurrence times of events, along with concomitant variables. Methods of regression analysis are presented, based on Poisson process and proportional intensity assumptions. These include parametric and semi-parametric approaches to model fitting, model assessment, and the treatment of random effects. In addition, insight is gained as to the central role of Poisson and mixed Poisson regression analysis of counts in these methods, and as to the effects of unobserved heterogeneity on semi-parametric analyses. The methods in the article are based on the proportional intensity Poisson process model, for which an individual with given fixed covariate vector x has repeated events occur according to a nonhomogeneous Poisson process with intensity function λx(t) = λ0(t)exp(x′β). Estimation of β and the baseline intensity λ0(t) are considered when λ0(t) is specifi...

MARGINAL ANALYSIS OF RECURRENT EVENTS AND A TERMINATING EVENT

Chronic medical conditions are often manifested by the incidence of recurrent adverse clinical events. In clinical trials designed to investigate therapeutic interventions for such conditions it is natural to make treatment comparisons on the basis of event occurrence. However, when there is a more serious, possibly related, event that terminates the occurrence of the recurrent events, the problem of dependent censoring arises. Here, we consider robust modelling strategies for expressing covariate effects on the recurrent event process that address the possible dependence between the recurrent and terminal events. The various methods differ in the way the dependence is addressed, and hence in the interpretation of covariate effects. The methods are applied to a data set from a kidney transplant study and simulated data chosen for illustrative purposes.

Semiparametric methods for response‐selective and missing data problems in regression

Suppose that data are generated according to the model f(y|x; θ) g(x), where y is a response and x are covariates. We derive and compare semiparametric likelihood and pseudolikelihood methods for estimating θ for situations in which units generated are not fully observed and in which it is impossible or undesirable to model the covariate distribution. The probability that a unit is fully observed may depend on y, and there may be a subset of covariates which is observed only for a subsample of individuals. Our key assumptions are that the probability that a unit has missing data depends only on which of a finite number of strata that (y, x) belongs to and that the stratum membership is observed for every unit. Applications include case–control studies in epidemiology, field reliability studies and broad classes of missing data and measurement error problems. Our results make fully efficient estimation of θ feasible, and they generalize and provide insight into a variety of methods that have been proposed for specific problems.

Analysis of repeated events

Events that may occur repeatedly for individual subjects are of interest in many medical studies. We review methods of analysis for repeated events, emphasizing that the approach taken in a given study should allow clinical questions to be addressed as directly as possible. Methods based on full models for event processes as well as on simpler ‘marginal’ assumptions are considered. The treatment of dependent terminating events related to the recurrent events is also discussed. We apply various methods of analysis to studies involving pulmonary exacerbations in persons with cystic fibrosis, and the occurrence of bone metastases and skeletal events in cancer patients, respectively. Most of the methodology considered can be implemented with existing software.

Statistical Methods in Reliability

Some of the advances made during the past 25 years in the statistical treatment of reliability problems are reviewed. The impact of statistical methods on reliability is discussed, and some areas where work is needed are suggested.

ROBUST TESTS FOR TREATMENT COMPARISONS BASED ON RECURRENT EVENT RESPONSES

Robust nonparametric tests are considered for use in longitudinal studies in which the response of interest is a recurrent event. The tests are robust in the sense that they do not rely on distributional assumptions regarding the processes generating the events. The methods we describe are presented in the context of a clinical trial with attention initially directed at the two-sample problem in which a single experimental treatment is compared to a control. We investigate a family of generalized pseudo-score statistics (Lawless and Nadeau, 1995, Technometrics 37, 158-168) in which weight functions may be chosen to generate tests sensitive to various types of departure from the null hypothesis that the mean functions for the treatment and control groups are identical. All tests we consider are evaluated by simulation with respect to the type I error rate and power under a variety of practical scenarios. An application involving data from a kidney transplant study illustrates these procedures. For trials with multiple treatment arms, we generalize these approaches and indicate test statistics appropriate for unstructured alternatives and tests based on linear contrasts of the treatment-specific mean functions. Extensions of this methodology for stratified designs are also indicated.

On a scheme for predictive maintenance

An operating system contains a replaceable unit whose wear (i.e. accumulated amount of damage) can be observed over time. When the wear reaches a certain level the unit is no longer able to function satisfactorily and needs to be replaced. Although units are produced to the same nominal specification there is still some random variation among them in their wear rates. This will be expressed by incorporating a random effect, or frailty term, in the model for individual degradation. There are costs for observing the wear on a unit, for replacing a unit, and for allowing a unit to fail before being replaced. When the last cost is comparatively large replacement before failure is preferable. For some standard examples of wear processes the lifetime distributions are obtained and the cost consequences of particular maintenance schemes are investigated.

Nonparametric estimation of a lifetime distribution when censoring times are missing

We consider datasets for which lifetimes associated with the units in a population are observed if they occur within certain time intervals but for which lengths of the time intervals, or censoring times of unfailed units, are missing. We consider nonparametric estimation of the lifetime distribution for the population from such data; a maximum likelihood estimator and a simple moment estimator are obtained. An example involving automobile warranty data is discussed at some length.