TY - JOUR

T1 - Fifty years with the Cox proportional hazards model

T2 - history, influence, and future

AU - Andersen, Per Kragh

PY - 2023

Y1 - 2023

N2 - At the symposium, entitled ‘A celebration of 50 years of the Cox model in memory of Sir David Cox’, held in London in November 2022, the article by Andersen (2022) was presented. That article (and the lecture) summarised the fundamental (Cox, 1972) paper, emphasising its impact on both statistical and medical literature and reviewing the way in which the mathematical framework of counting processes and martingales has clarified the statistical properties of the Cox model (Andersen & Gill, 1982).The Cox model has been influential within survival analysis where its multiplicative structure has become the standard for regression analysis of, e.g. transition rates in multi-state models (Andersen et al., 1993), competing risks cumulative incidences (Fine & Gray, 1999), and mean number of recurrent events (Ghosh & Lin, 2002). However, more generally, the Cox paper has led to entirely new fields of statistical research, including partial likelihood and semi-parametric inference. In medical research, from which the majority of the vast number of references to the Cox paper has come, the model has become the standard choice when analysing survival data.In spite of this success, the Cox model has been criticised for being too simple and rarely fitting data properly, for estimating a non-collapsible parameter (the hazard ratio), for being inferior to machine learning for prediction purposes, and for lacking a causal interpretation (for references, see Andersen, 2022). In view of this criticism, the question ‘What is the future of the Cox model?’ was asked. The conclusion was that the model will likely still play an important role because it is, indeed, a simple method that provides a one-number summary of survival curves, it is very well established in the medical world, it is frequently used in machine learning as a benchmark against which other methods are compared, and it is a much used tool in causal inference—not as the sole analysis for the purpose of estimating hazard ratios—but rather as one possible means for estimating absolute risks.

AB - At the symposium, entitled ‘A celebration of 50 years of the Cox model in memory of Sir David Cox’, held in London in November 2022, the article by Andersen (2022) was presented. That article (and the lecture) summarised the fundamental (Cox, 1972) paper, emphasising its impact on both statistical and medical literature and reviewing the way in which the mathematical framework of counting processes and martingales has clarified the statistical properties of the Cox model (Andersen & Gill, 1982).The Cox model has been influential within survival analysis where its multiplicative structure has become the standard for regression analysis of, e.g. transition rates in multi-state models (Andersen et al., 1993), competing risks cumulative incidences (Fine & Gray, 1999), and mean number of recurrent events (Ghosh & Lin, 2002). However, more generally, the Cox paper has led to entirely new fields of statistical research, including partial likelihood and semi-parametric inference. In medical research, from which the majority of the vast number of references to the Cox paper has come, the model has become the standard choice when analysing survival data.In spite of this success, the Cox model has been criticised for being too simple and rarely fitting data properly, for estimating a non-collapsible parameter (the hazard ratio), for being inferior to machine learning for prediction purposes, and for lacking a causal interpretation (for references, see Andersen, 2022). In view of this criticism, the question ‘What is the future of the Cox model?’ was asked. The conclusion was that the model will likely still play an important role because it is, indeed, a simple method that provides a one-number summary of survival curves, it is very well established in the medical world, it is frequently used in machine learning as a benchmark against which other methods are compared, and it is a much used tool in causal inference—not as the sole analysis for the purpose of estimating hazard ratios—but rather as one possible means for estimating absolute risks.

U2 - 10.1093/jrsssa/qnad114

DO - 10.1093/jrsssa/qnad114

M3 - Journal article

JO - Journal of the Royal Statistical Society. Series A: Statistics in Society

JF - Journal of the Royal Statistical Society. Series A: Statistics in Society

SN - 0964-1998

ER -