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Using martingale residuals to assess goodness-of-fit for sampled risk set data

University of Oslo
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Dept. of Math. University of Oslo Statistical Research Report No. 8 ISSN 0806–3842 September 2005 Using martingale residuals to assess goodness-of-fit for sampled risk set data Ørnulf Borgan and Bryan Langholz University of Oslo and University of Southern California Department of Mathematics University of Oslo P.O. Box 1053 Blindern N-0316 Oslo, Norway e-mail: [email protected] Department of Preventive Medicine University of Southern California 1540 Alcazar Street CHP-220 Los Angeles, Ca 90033, U.S.A. e-mail: [email protected] Abstract: Standard use of Cox’s regression model and other relative risk regression models for censored survival data requires collection of covariate information on all individuals under study even when only a small fraction of them die or get diseased. For such situations risk set sampling designs offer useful alternatives. For cohort data, methods based on martingale residuals are useful for assessing the fit of a model. Here we introduce grouped martingale residual processes for sampled risk set data, and show that plots of these processes provide a useful tool for checking model-fit. Further we study the large sample properties of the grouped martingale residual processes, and use these to derive a formal goodness-of-fit test to go along with the plots. The methods are illustrated using data on lung cancer deaths in a cohort of uranium miners. AMS 2000 subject classifications: Primary 62N03, 62P10; secondary 62D05, 62F05, 62G20, 62M99. Keywords and phrases: Chi-squared tests, cohort sampling, counter- matching, counting processes, martingales, matching, nested case-control studies, Cox’s regression model, relative risk regression, survival analy- sis. 1 Ø. Borgan and B. Langholz/Martingale residuals for sampled risk set data 2 1. Introduction Cox regression is central to modern survival analysis, and it is the method of choice when one wants to assess the influence of risk factors and other covariates on mortality or morbidity. A number of met

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