Estimating the Expected Total Number of Events in a Process

Abstract

We consider estimation of the cumulative mean function of a process recurring in time, such as the numbers of arrests or migrations accrued by an individual, as a function of their age. We call this the age pro le of a series of events. In some situations we can expect a nite value for the total number of events experienced by an individual, for example, when the distribution of the interevent times is improper, so that the process may cease at a nite time with positive probability. We propose and analyze a new estimator, constructed from Kaplan-Meier (KM) estimators of the interevent time distributions, for such an age pro le, and compare it with the Nelson-Aalen (NA) estimator of the cumulative mean function of a process. The KM estimator is proved to be uniformly consistent for the age pro le if and only if follow-up in the sample is suf cient in a sense that is manifested in practice by the leveling off of the pro les at their right-side ends, and is asymptotically normally distributed around its true value under mild conditions. Simulation results suggest that it is generally a better estimator than the NA estimator for the total number of events. It also appears to be more stable when applied to real data examples. For accurate estimation, it seems to be important to select a cohort of individuals whose ages at the rst event are as similar as possible. The estimators are illustrated on some time-to-arrest data.

Recurrent Events Analysis in the Presence of Terminal Event and Zero-recurrence Subjects

“…Its focus was on the estimation of the cumulative distribution functions of event times using the KaplanMeier estimators. The zero-recurrence subjects in our proposed model are those who will never experience any recurrence, rather than the next recurrence as in Maller et al (2002). Essentially, our model (2.2) is semiparametric and can easily deal with the effect of covariates and information censoring.…”

“…Before closing this section, we mention an alternative recurrent event data model proposed by Maller et al (2002), referred to as the age-profile model. Its focus was on the estimation of the cumulative distribution functions of event times using the KaplanMeier estimators.…”

“…Ignoring their presence may yield incorrect or biased parameter estimates from a conventional survival analysis. Little has been done, however, to model recurrent event data with long-term survivors, apart from McDonald and Rosina (2001) and Maller et al (2002) (without terminal events).…”

“…This assumption, however, is untenable in many applications where a proportion of subjects may never experience the recurrence. Examples include migrants who never migrate again, released prisoners who never return to prison (Maller et al, 2002), and leukemia patients who never suffer a relapse after successful treatment. Such individuals are termed as cured, long-term survivors, or immunes in the Downloaded by [Imperial College London Library] at 10:21 06 April 2015 literature (cf.…”

The Statistical Analysis of Recurrent Events

“…Different methods of estimating transition probabilities in multistate models, as used in Section 5.3, were discussed by Couper and Pepe (1997), Aalen et al (2001), Datta and Satten (2001), Glidden (2002), , and others. Maller et al (2002) consider the Kaplan-Meier method (Pepe, 1991) for estimation of µ(t) via (5.18). Andersen et al (1993) is an authoritative reference on modulated Markov models as well as on estimation for Markov multistate models.…”