# The Statistical Analysis of Recurrent Events (Statistics for Biology and Health) # The Statistical Analysis of Recurrent Events (Statistics for Biology and Health)

## Richard J. Cook

Language: English

Pages: 404

ISBN: 0387698094

Format: PDF / Kindle (mobi) / ePub

This book presents models and statistical methods for the analysis of recurrent event data. The authors provide broad, detailed coverage of the major approaches to analysis, while emphasizing the modeling assumptions that they are based on. More general intensity-based models are also considered, as well as simpler models that focus on rate or mean functions. Parametric, nonparametric and semiparametric methodologies are all covered, with procedures for estimation, testing and model checking. , Ym (s)} = dµ(s) regardless of the underlying process. The estimates dµ(s) and (3.17) are Poisson maximum likelihood estimates, but are also therefore valid quite generally. In order to make use of this result, we require a variance estimate that is also valid generally. A robust variance estimate for µ(t) is obtained by noting that √ var{ m(µ(t) − µ(t))} = m · var ¯· (u) dN Y· (u) t 0 t m t = m· i=1 0 0 Yi (u) Yi (v) cov{dNi (u), dNi (v)} . Y· (u) Y· (v) It can be shown that m t t m

periods or at diﬀerent plants. Medical studies may be directed at examining event rates for patients under diﬀerent therapeutic conditions. One natural framework for making comparisons of two groups, 1 and 2, is via models ρi (t) = ρ0 (t) exp(xi β) where xi = 1 if individual i is in group 2 and xi = 0 otherwise. Wald tests of H0 : β = 0 may be carried out under parametric or semiparametric Poisson models of Sections 3.2 and 3.4, respectively, and such tests will be most powerful for detecting

and Lawless (1998) discuss questions of eﬃciency. Scheike (2002) considers additive, and Scheike and Zhang (2003) consider multiplicative-additive, rate function models; see also Martinussen and Scheike (2006). Methods for assessing plausibility of assumptions for counting processes can be based on techniques developed for survival analysis and good reviews are given by Andersen et al. (1993), Grambsch and Therneau (1994), Therneau and Grambsch (2000), and Martinussen and Scheike (2006). Various

considering the multistate Markov model depicted in Figure 5.3. In this model, states represent the cumulative number of events experienced, and only transitions from state k to k + 1 are possible. Thus the counting process {Ni (t), 0 ≤ t} also indicates the state occupied over time. If Hi (t) = {N (s) : 0 ≤ s < t} then 172 5 General Intensity-Based Models Pr(∆Ni (t) = 1|Ni (t− ) = k, Hi (t)) ∆t↓0 ∆t αk (t) = lim (5.14) is the intensity function for transitions from state k to k + 1 under

caution. The eﬀects of adaptive censoring are discussed in Section 7.2. The mean function µ(t) can be estimated with the Nelson–Aalen estimate (3.17), which is valid for Poisson processes but also more generally, provided censoring is completely independent of the event processes. A second approach is to use estimates of p0k (t) in (5.18); if the Aalen–Johansen estimates from (5.20) are used, this has the added advantage of being valid under eventdependent censoring, provided the Markov process