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A Solution to the Problem of Monotone Likelihood in Cox Regression

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  • Georg Heinze
  • Michael Schemper

Abstract

Summary. The phenomenon of monotone likelihood is observed in the fitting process of a Cox model if the likelihood converges to a finite value while at least one parameter estimate diverges to ±∞. Monotone likelihood primarily occurs in small samples with substantial censoring of survival times and several highly predictive covariates. Previous options to deal with monotone likelihood have been unsatisfactory. The solution we suggest is an adaptation of a procedure by Firth (1993, Biometrika80, 27–38) originally developed to reduce the bias of maximum likelihood estimates. This procedure produces finite parameter estimates by means of penalized maximum likelihood estimation. Corresponding Wald‐type tests and confidence intervals are available, but it is shown that penalized likelihood ratio tests and profile penalized likelihood confidence intervals are often preferable. An empirical study of the suggested procedures confirms satisfactory performance of both estimation and inference. The advantage of the procedure over previous options of analysis is finally exemplified in the analysis of a breast cancer study.

Suggested Citation

  • Georg Heinze & Michael Schemper, 2001. "A Solution to the Problem of Monotone Likelihood in Cox Regression," Biometrics, The International Biometric Society, vol. 57(1), pages 114-119, March.
    • Handle: RePEc:bla:biomet:v:57:y:2001:i:1:p:114-119
    DOI: 10.1111/j.0006-341X.2001.00114.x
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    1. Wei Wang & Shou‐En Lu & Jerry Q. Cheng & Minge Xie & John B. Kostis, 2022. "Multivariate survival analysis in big data: A divide‐and‐combine approach," Biometrics, The International Biometric Society, vol. 78(3), pages 852-866, September.
    2. Chau, Nancy H. & Qin, Yu & Zhang, Weiwen, 2015. "Networked Leaders in the Shadow of the Market – A Chinese Experiment in Allocating Land Conversion Rights," Working Papers 250022, Cornell University, Department of Applied Economics and Management.
    3. Everett, Bethany G. & Wall, Melanie & Shea, Eileen & Hughes, Tonda L., 2021. "Mortality risk among a sample of sexual minority women: A focus on the role of sexual identity disclosure," Social Science & Medicine, Elsevier, vol. 272(C).
    4. Negreiros, Ana Cláudia Souza Vidal de & Lins, Isis Didier & Moura, Márcio José das Chagas & Droguett, Enrique López, 2020. "Reliability data analysis of systems in the wear-out phase using a (corrected) q-Exponential likelihood," Reliability Engineering and System Safety, Elsevier, vol. 197(C).
    5. Frederico Machado Almeida & Enrico Antônio Colosimo & Vinícius Diniz Mayrink, 2021. "Firth adjusted score function for monotone likelihood in the mixture cure fraction model," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 27(1), pages 131-155, January.
    6. Maalouf, Maher & Trafalis, Theodore B., 2011. "Robust weighted kernel logistic regression in imbalanced and rare events data," Computational Statistics & Data Analysis, Elsevier, vol. 55(1), pages 168-183, January.
    7. Krüger, Jens, 2015. "Survival analysis in product life cycle investigations: An assessment of robustness for the German automobile industry," Darmstadt Discussion Papers in Economics 223, Darmstadt University of Technology, Department of Law and Economics.
    8. Pianto, Donald M. & Cribari-Neto, Francisco, 2011. "Dealing with monotone likelihood in a model for speckled data," Computational Statistics & Data Analysis, Elsevier, vol. 55(3), pages 1394-1409, March.
    9. Frederico M. Almeida & Vinícius D. Mayrink & Enrico A. Colosimo, 2023. "Bayesian solution to the monotone likelihood in the standard mixture cure model," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 77(3), pages 365-390, August.
    10. Lior Rennert & Zichen Ma & Christopher S. McMahan & Delphine Dean, 2022. "Effectiveness and protection duration of Covid-19 vaccines and previous infection against any SARS-CoV-2 infection in young adults," Nature Communications, Nature, vol. 13(1), pages 1-8, December.
    11. Eun Ju Cho & Jeong-Hoon Lee & Yuri Cho & Yun Bin Lee & Jeong-Ju Yoo & Minjong Lee & Dong Hyeon Lee & Su Jong Yu & Yoon Jun Kim & Jung-Hwan Yoon & Hyo-Suk Lee, 2015. "Comparison of the Efficacy of Entecavir and Tenofovir in Nucleos(T)ide Analogue-Experienced Chronic Hepatitis B Patients," PLOS ONE, Public Library of Science, vol. 10(6), pages 1-10, June.
    12. Chau, Nancy H. & Qin, Yu & Zhang, Weiwen, 2016. "Leader Networks and Transaction Costs: A Chinese Experiment in Interjurisdictional Contracting," IZA Discussion Papers 9641, Institute of Labor Economics (IZA).
    13. Il Do Ha & Liming Xiang & Mengjiao Peng & Jong-Hyeon Jeong & Youngjo Lee, 2020. "Frailty modelling approaches for semi-competing risks data," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 26(1), pages 109-133, January.
    14. Fontana, Roberto & Vezzulli, Andrea, 2016. "Technological leadership and persistence in product innovation in the Local Area Network industry 1990–1999," Research Policy, Elsevier, vol. 45(8), pages 1604-1619.
    15. John E. Kolassa & Juan Zhang, 2023. "Inference in the presence of likelihood monotonicity for proportional hazards regression," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 77(3), pages 322-339, August.

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