The Cox proportional hazards model and its role in survival analysis and reliability

The proportional hazards model was proposed by Sir David Cox 40 years ago. Today the Cox model is the most important model in survival analysis and reliability. We discuss the influence of this model on the development of the statistical inference in Biostatistics, Demography, Econometrics, Finance, Pharmacology, Reliability, etc., and on the process of modeling in these sciences. The role of the Cox model in these sciences one can compare with the influence of the Laws of Newton on the development of mechanics, physics, mathematics, philosophy, $\dots$ The popularity and the success of the Cox model is based on the fact that there exist simple semi-parametric estimation procedures which can be used when the form of survival distribution function is not specified. It is well known that under the Cox model the ratios of hazard rates under different fixed covariates are constant in time. But in practice the hazard rates may approach, go away or even intersect. In some cases it can be the cause of the cross-effect of the survival functions.
The proportional hazards model is generalized by assuming that at any moment the ratio of hazard rates is depending on values of time varying covariates. Relations with generalized proportional hazards, frailty, linear transformation, Sedyakin and Degradation models and models with cross-effects of survivals functions are considered.
Using some new flexible regression models models with dynamic environment we may analyse the survival data (Stablein & Koutrouvelis, 1985) of the Gastrointestinal Tumor Study Group concerning effects of chemotherapy and radiotherapy on the survival times of gastric cancer patients, and the data of Piantadosi (1997) on lung cancer patients. These models can be used in many other similar clinical trials, see for example Wu & Hsieh (2002). Another goal of this talk is to show some new interesting applications of these models in reliability, Ceci & Mazliak (2004).
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D. Cox (1972). Regression models and life tables, J. Roy. Statist. Soc., B 34, 187–220.
V. Bagdonavicius & M. Nikulin (2001). Accelerated Life Models. Chapman & Hall/CRC.
V.Bagdonavicius, J.Kruopis, M.Nikulin (2011). Nonparametric Tests for Censored data, Wiley-ISTE.
W. Kahle, A. Lehmann (2010). The Wiener Process as a Degradation Model: Modeling and Parameter Estimation. In: Advances in Degradation Modeling: Applications to Reliability, Survival Analysis and Finance (Eds. M. Nikulin, N. Limnios, N. Balakrishnan, C. Huber, W. Kahle), Birkhauser: Boston, 127–146.
C. Huber-Carol, M. S. Nikulin (2008). Extended Cox and Accelerated Models in Reliability with General Censoring and Truncation. In: Statistical Models and Methods for Biomedical and Technical Systems, (Eds. F. Vonta, M. Nikulin, N. Limnios, C. Huber), Birkhauser : Boston, 3–22.
T. Martinussen, T. Scheike (2006). Dynamic Regression Models for Survival Data. Springer: New York.