Abstract
The 1972 paper introducing the Cox proportional hazards regression model is one of the most widely cited statistical articles. In the present article, we give an account of the model, with a detailed description of its properties, and discuss the marked influence that the model has had on both statistical and medical research. We will also review points of criticism that have been raised against the model.
Similar content being viewed by others
References
Andersen PK, Borgan Ø (1985) Counting process models for life history data: a review (with discussion). Scand J Statist 12:97–158
Andersen PK, Borgan Ø, Gill RD, Keiding N (1993) Statistical models based on counting processes. Springer, New York
Andersen PK, Geskus RB, de Witte T, Putter H (2012) Competing risks in epidemiology: possibilities and pitfalls. Int J Epidemiol 41:861–70
Andersen PK, Gill RD (1982) Cox’s regression model for counting processes: a large sample study. Ann Statist 10:1100–1120
Andersen PK, Skovgaard LT (2006) Regression with linear predictors. Springer-Verlag, New York
Bailey KR (1983) The asymptotic joint distribution of regression and survival parameter estimates in the Cox regression model. Ann Statist 11:39–58
Begun JM, Hall WJ, Huang W-M, Wellner JA (1983) Information and asymptotic efficiency in parametric-nonparametric models. Ann Statist 11:432–452
Bickel PJ, Klaassen CA, Ritov Y, Wellner JA (1998) Efficient and adaptive inference in semiparametric models. Springer, New York
Breslow NE (1974) Covariance analysis of censored survival data. Biometrics 30:89–99
Buckley JD, James IR (1979) Linear regression with censored data. Biometrika 66:429–436
Carroll RJ, Ruppert D, Stefanski LA, Crainiceanu CM (2006) Measurement error in nonlinear models, 2nd edn. Chapman and Hall/CRC, Boca Raton
Clayton DG, Cuzick J (1985) Multivariate generalizations of the proportional hazards model (with discussion). J R Statist Soc A 148:82–117
Cox DR (1972) Regression models and life-tables (with discussion). J Roy Statist Soc B 34:187–220
Cox DR (1975) Partial likelihood. Biometrika 62:269–276
Duchateau L, Janssen P (2008) The frailty model. Springer, New York
Efron B (1977) The efficiency of Cox’s likelihood function for censored data. J Amer Statist Assoc 72:557–565
Fang EX, Ning Y, Liu H (2017) Testing and confidence intervals for high dimensional proportional hazards models. J R Statist Soc B 79:1415–1437
Fine JP, Gray RJ (1999) A proportional hazards model for the subdistribution of a competing risk. J Amer Statist Assoc 94:496–509
Gail MH, Wieand S, Piantadosi S (1984) Biased estimates of treatment effect in randomized experiments with nonlinear regressions and omitted covariates. Biometrika 71:431–444
Ghosh D, Lin DY (2002) Marginal regression models for recurrent and terminal events. Statistica Sinica 12:663–688
Gill RD (1992) Marginal partial likelihood. Scand J Statist 19:133–137
Goeman JJ (2010) L1 penalized estimation in the Cox proportional hazards model. Biom J 52:70–84
Hao L, Kim J, Kwon S, Ha ID (2021) Deep learning-based survival analysis for high-dimensional survival data. Mathematics 9:1244
Hernan MA (2010) The hazards of hazard ratios. Epidemiology 21:13–15
Hernan MA, Robins JM (2020) Causal inference: what if. Chapman and Hall/CRC, Boca Raton
Jacobsen M (1984) Maximum likelihood estimation in the multiplicative intensity model: a survey. Internat Statist Rev 52:193–207
Johansen S (1983) An extension of Cox’s regression model. Internat Statist Rev 51:258–262
Kalbfleisch JD, Prentice RL (1973) Marginal likelihoods based on Cox’s regression and life model. Biometrika 60:267–278
Kalbfleisch JD, Prentice RL (1980) The statistical analysis of failure time data (2nd edn 2002). Wiley, New York
Kaplan EL, Meier P (1958) Non-parametric estimation from incomplete observations. J Amer Statist Assoc 53(457–481):562–563
Kim WJ, Sung JM, Sung D, Chae M, An SK, Namkoong K, Lee E, Chang H (2019) Cox proportional hazard regression versus a deep learning algorithm in the prediction of dementia: an analysis based on periodic health examination. JMIR Med Inform 7(3):e13139
Kvamme H, Borgan Ø, Scheel I (2019) Time-to-event prediction with neural networks and Cox regression. J Mach Learn Res 20:1–30
Leger S, Zwanenburg A, Pilz K, Lohau F, Linge A, Zöphel K, Kotzerke J, Schreiber A, Tinhofer I, Budach V, Sak A, S M, Balermpas P, Rödel C, Ganswindt U, Belka C, Pigorsch S, Combs SE, Mönnich D, Zips D, Krause M, Baumann M, Troost EGC, Löck S, Richter C (2017) A comparative study of machine learning methods for time-to-event survival data for radiomics risk modelling. Nat Sci Rep 7:13206
Lin DY (2000) Proportional means regression for censored medical costs. Biometrics 56:775–778
Lin DY, Wei LJ, Yang I, Ying Z (2000) Semiparametric regression for the mean and rate functions of recurrent events. J R Statist Soc Ser B 62:711–730
Lin DY, Wei LJ, Ying Z (1993) Checking the Cox model with cumulative sums of martingale-based residuals. Biometrika 80:557–572
Lombard M, Portmann B, Neuberger J, Williams R, Tygstrup N, Ranek L, Ring-Larsen H, Rodes J, Navasa M, Trepo C, Pape G, Schou G, Badsberg JH, Andersen PK (1993) Cyclosporin A treatment in primary biliary cirrhosis: results of a long-term placebo controlled trial. Gastroenterol 104:519–526
Martinussen T, Scheike TH, Skovgaard IM (2002) Efficient estimation of fixed and time-varying covariate effects in multiplicative intensity models. Scand J Statist 28:57–74
Martinussen T, Vansteelandt S, Andersen PK (2020) Subtleties in the interpretation of hazard contrasts. Lifetime Data Anal 26:833–855
Oakes D (1977) The asymptotic information in censored survival data. Biometrika 59:472–474
Prentice RL, Kalbfleisch JD, Peterson AV, Flournoy N, Farewell VT, Breslow N (1978) The analysis of failure time data in the presence of competing risks. Biometrics 34:541–554
Schemper M (1992) Cox analysis of survival data with non-proportional hazard functions. J R Statist Soc Ser D (The Statistician) 41:455–465
Slud EV (1986) Inefficiency of inferences with the partial likelihood. Commun Statist Theory Methods 15:3333–3351
Slud EV (1992) Partial likelihood for continuous-time stochastic processes. Scand J Statist 19:97–109
Stensrud MJ, Hernan MA (2020) Why test for proportional hazards? JAMA 323:1401–1402
Tsiatis AA (1981) A large sample study of Cox’s regression model. Ann Statist 9:93–108
Tsiatis AA (2006) Semiparametric theory and missing data. Springer, New York
van Houwelingen HC, Putter H (2012) Dynamic prediction in clinical survival analysis. Chapman and Hall/CRC, Boca Raton
Witten DW, Tibshirani R (2010) Survival analysis with high-dimensional covariates. Statist Methods Med Res 19:29–51
Wong WH (1986) Theory of partial likelihood. Ann Statist 14:88–123
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author declares to have no conflicts of interest,
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Andersen, P.K. Fifty Years with the Cox Proportional Hazards Regression Model. J Indian Inst Sci 102, 1135–1144 (2022). https://doi.org/10.1007/s41745-021-00283-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s41745-021-00283-9