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

By Richard J. Cook

This e-book provides types and statistical equipment for the research of recurrent occasion info. The authors supply large, precise insurance of the key techniques to research, whereas emphasizing the modeling assumptions that they're according to. extra basic intensity-based versions also are thought of, in addition to less complicated types that concentrate on expense or suggest features. Parametric, nonparametric and semiparametric methodologies are all lined, with strategies for estimation, checking out and version checking.

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Mary Lou Dufton and Joan Hatton supplied secretarial counsel within the education of this booklet, for which we're thankful. we'd particularly prefer to thank Ker-Ai Lee, whose professional statistical programming helped within the education of the examples, and who presents vital aid to our study. a lot of the paintings right here was once constructed whereas the first writer held an Investigator Award from the Canadian Institutes of wellbeing and fitness study and a Canada examine Chair in Statistical tools for future health study, and whereas the second one writer held an business examine Chair co-sponsored by means of normal Preface ix automobiles Canada and the traditional Sciences and Engineering examine Council of Canada. This aid is gratefully stated. ultimately we want to thank our better halves Alison (R. J. C. ) and Liz (J. F. L. ) for his or her endurance and help in the course of the instruction of this e-book. college of Waterloo December 2006 Richard cook dinner Jerry Lawless Glossary the next is a precis of the notation used all through this e-book. • I(A) is the indicator functionality, equaling 1 if A is right and zero another way • Pr(A) is the chance of occasion A • E(·) denotes expectation, var(·) denotes variance, cov(·) denotes covariance, corr(·) denotes correlation, asvar(·) and ascov(·) denote asymptotic variance and covariance, respectively • mX (t) = E(exp(Xt)) is the instant producing functionality of X • Γ (a) = • B(a, b) = Γ (a)Γ (b)/Γ (a + b) is the beta functionality, the place a > zero and b > zero • g(x) ∼ o(x) skill g(x)/x → zero as x → zero • The transpose of a matrix A is A • Vectors are written in column shape so, for instance, θ = (θ1 , . . . , θr ) • If g(θ) = (g1 (θ), . . . , gk (θ)) is a vector of features, then ∂g(θ)/∂θ is the ok × r matrix with (i, j) point ∂gi (θ)/∂θj • [a,b] {1 ∞ zero ua−1 exp(−u)du is the gamma functionality, the place a > zero + g(u)du} is a product indispensable; see part 2. 1 b a • The imperative • L(θ), (θ), U (θ), I(θ), and I(θ) signify the chance, log-likelihood, ranking, saw details, and anticipated details services, respectively; see Appendix A • θ denotes an estimate of the parameter θ • If θ = (θ1 , θ2 ) , then θ˜1 (θ2 ) is the profile chance estimate of θ1 for fixed θ2 dG(u) is a Riemann–Stieltjes necessary; see part 2. 1. xii word list • Tk is the time of the kth occasion for somebody • Wk = Tk − Tk−1 is the length of time among the (k − 1)st and kth occasions for someone • B(t) = t − TN (t− ) is the backwards recurrence time, or the time because the final occasion earlier than t • Yk (t) = I(Tk < t ≤ Tk+1 ) exhibits no matter if someone is susceptible to a (k + 1)st occasion at time t; if there are J sorts of occasions then Yjk (t) shows even if a person is vulnerable to a (k + 1)st style j occasion at time t, j = 1, . . . , J. • {N (t), zero ≤ t} is a right-continuous counting approach giving the variety of occasions in [0, t] • N (s, t) = N (t) − N (s) files the variety of occasions over (s, t] • ∆N (t) = N (t + ∆t− ) − N (t− ) • dN (t) = lim [N (t + ∆t− ) − N (t− )] • [τ0 , τ ] is an period of statement measured at the scale of technique age t and whilst τ0 = zero, τ denotes the right-censoring or end-of-followup time • Y (t) shows no matter if a person is below statement at time t; frequently Y (t) = I(τ0 ≤ t ≤ τ ) • ¯ (t), zero ≤ t}, the place N ¯ (t) = {N technique • Y¯k (t) = Y (t)Yk (t) exhibits even if someone is saw and liable to a (k + 1)st occasion at time t ; Y¯jk (t) = Y (t)Yjk (t) • x(t) = (x1 (t), x2 (t), .

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