Biometrika-1993-LIN-557-72.pdf

Biometrika-1993-LIN-557-72.pdf - Biometrika(1993 80 3 pp...

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Biometrika (1993), 80, 3, pp. 557-72 Printed in Great Britain Checking the Cox model with cumulative sums of martingale-based residuals BY D. Y. LIN Department of Biostatistics, SC-32, University of Washington, Seattle, Washington 98195, U.S.A. L. J. WEI Department of Biostatistics, Harvard University, 677 Huntington Ave., Boston, Massachusetts 02115, U.S.A. AND Z. YING Department of Statistics, 101 Illini Hall, University of Illinois, Champaign, Illinois 61820, U.S.A. SUMMARY This paper presents a new class of graphical and numerical methods for checking the adequacy of the Cox regression model. The procedures are derived from cumulative sums of martingale-based residuals over follow-up time and/or covariate values. The dis- tributions of these stochastic processes under the assumed model can be approximated by zero-mean Gaussian processes. Each observed process can then be compared, both visually and analytically, with a number of simulated realizations from the approximate null distribution. These comparisons enable the data analyst to assess objectively how unusual the observed residual patterns are. Special attention is given to checking the functional form of a covariate, the form of the link function, and the validity of the pro- portional hazards assumption. An omnibus test, consistent against any model misspecifi- cation, is also studied. The proposed techniques are illustrated with two real data sets. Some key words: Censoring; Goodness of fit; Link function; Omnibus test; Proportional hazards; Regression diagnostic; Residual plot; Survival data. 1. INTRODUCTION The proportional hazards model (Cox, 1972) with the partial likelihood principle (Cox, 1975) has become exceedingly popular for the analysis of failure time obser- vations. This model specifies that the hazard function for the failure time T associated with a p x 1 vector of covariates Z takes the form of \(t;Z) = \(t)exp(P' 0 Z), (1-1) where A<,(.) is an unspecified baseline hazard function, and /3 0 is a p x 1 vector of unknown regression parameters. Let C denote the censoring time. Assume that Z is bounded and that T and C are independent conditional on Z. Suppose that the data consist of n independent repli- cates of (X,A,Z), where X = min(T, C), A = I(T^C), and /(.) is the indicator at North Carolina State University on October 15, 2012 http://biomet.oxfordjournals.org/ Downloaded from
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558 D. Y. LIN, L. J. WEI AND Z. YING function. Then the partial likelihood score function for 0 O is U(0) = £A,{Z 1 -Z(P,X I )}, (1-2) where 2(0,1) = t Y l (t)tx V (0'Z^Z l /f J r,(0exp(/?'Z,), Y,(t) = I(X,> t). / For future reference, we denote the denominator of 2(0, t) by S m (0, t). The maximum partial likelihood estimator 0 is the solution to the estimating equation U(0) = 0. Under some mild regularity conditions (Andersen & Gill, 1982), the random vector J*(P)(P 0 O ) is asymptotically zero-mean normal with an identity covariance matrix, where J(0) is minus the derivative matrix of U(0).
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