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A counting process approach to the regression analysis of grouped survival data. (English) Zbl 0659.62122

The additive risk model of O. O. Aalen [Mathematical statistics and probability theory, Proc. 6th int. Conf., Wisla/Pol. 1978, Lect. Notes Stat. 2, 1-25 (1980; Zbl 0445.62095)] specifies the hazard rate \(h_ i(t)\) for the survival time \(T_ i\) of an individual i with covariate vector \(Y_ i=(Y_{i1},...,Y_{ip})'\) by \(h_ i(t)=Y_ i'\alpha (t)\) where \(\alpha =(\alpha_ 1,...,\alpha_ p)'\) is a p-vector of unknown time-dependent hazard functions, \(i=1,...,n\). In this paper a weighted least squares estimator \({\tilde \alpha}\) for \(\alpha\) is derived when only grouped data from successive calendar periods \(\phi_ 1,...,\phi_ d\) are available (e.g. total number of deaths and person-years at risk for the various levels of the covariates in epidemiological cohort studies).
Using a multivariate counting process formulation, the author develops an asymptotic distribution theory for the integrated weighted least squares estimator \[ \tilde A(t)=\int^{t}_{t_ 0}{\tilde \alpha}(s)ds,\quad 0\leq t_ 0<1, \] defined by componentwise integration of \({\tilde \alpha}\) and proves the weak convergence of \(\sqrt{n}(\tilde A-\bar A)\) to a p-variate Gaussian process where \(\bar A(t)\) denotes the piecewise linear approximation to the vector of integrated hazard functions \(A(t)=\int^{t}_{t_ 0}\alpha (s)ds\) based on the grouping \(\phi_ 1^{(n)},...,\phi_ d^{(n)}\), and depending on the total number of individuals n. The proofs are based on the theory of counting processes and use CLT for martingales.
A Kolmogorov-Smirnov type test statistic is introduced for testing \(H_ 0: \alpha_ j=\alpha_ 0\), where \(\alpha_ 0\) is a known function and its asymptotic null distribution is obtained as Brownian bridge. Simultaneous asymptotic confidence bands for \(\bar A_ j\) are given and it is shown how the results transfer to the analysis of grouped survival data.
Reviewer: L.Edler

MSC:

62P10 Applications of statistics to biology and medical sciences; meta analysis
62E20 Asymptotic distribution theory in statistics
62J99 Linear inference, regression
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
60F05 Central limit and other weak theorems
60G44 Martingales with continuous parameter

Citations:

Zbl 0445.62095
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References:

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