The added value of new covariates to the brier score in cox survival models
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The added value of new covariates to the brier score in cox survival models Glenn Heller1 Received: 29 October 2019 / Accepted: 8 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Calibration is an important measure of the predictive accuracy for a prognostic risk model. A widely used measure of calibration when the outcome is survival time is the expected Brier score. In this paper, methodology is developed to accurately estimate the difference in expected Brier scores derived from nested survival models and to compute an accompanying variance estimate of this difference. The methodology is applicable to time invariant and time-varying coefficient Cox survival models. The nested survival model approach is often applied to the scenario where the full model consists of conventional and new covariates and the subset model contains the conventional covariates alone. A complicating factor in the methodologic development is that the Cox model specification cannot, in general, be simultaneously satisfied for nested models. The problem has been resolved by projecting the properly specified full survival model onto the lower dimensional space of conventional markers alone. Simulations are performed to examine the method’s finite sample properties and a prostate cancer data set is used to illustrate its application. Keywords Brier score · Nested models · Projection theory · Proper score
1 Introduction Risk modeling has emerged as an area of significant interest in clinical research. The models have a direct impact on patient treatment and health. As examples, physicians use them to identify which patients are candidates for further diagnostic testing, such as surgical biopsy; they are also used to assess the likelihood of clinical outcomes,
Electronic supplementary material The online version of this article (https://doi.org/10.1007/s10985020-09509-x) contains supplementary material, which is available to authorized users.
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Glenn Heller [email protected] Department of Epidemiology and Biostatistics, Memorial Sloan Kettering, 485 Lexington Avenue, New York, New York 10017, USA
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such as response to treatment or survival. Due to the complexity of understanding disease etiology, risk models include a combination of risk factors, with new risk factors continually introduced. The utility of these updated models is a function of their predictive accuracy, which include measures of calibration, discrimination, and explained variation. In this work, the prognostic utility of a set of new risk factors will be evaluated by a calibration measure for survival models. Measures of calibration for survival data are often generated through loss functions that are applied to point prediction and survival status prediction (Lawless and Yuan 2010). Loss functions used to produce calibration measures for survival data include: absolute error (Tian et al. 2007; Schemper 1990), entropy (Korn and Simon 1990), missclassification error (Uno et al. 2007), and squared error (Graf et al.
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