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A robust hazard ratio for general modeling of survival-times

  • Pablo Martínez-Camblor ORCID logo EMAIL logo , Todd A. MacKenzie und A. James O’Malley
Veröffentlicht/Copyright: 23. August 2021
The International Journal of Biostatistics
Aus der Zeitschrift The International Journal of Biostatistics Band 18 Heft 2

Abstract

Hazard ratios (HR) associated with the well-known proportional hazard Cox regression models are routinely used for measuring the impact of one factor of interest on a time-to-event outcome. However, if the underlying real model does not fit with the theoretical requirements, the interpretation of those HRs is not clear. We propose a new index, gHR, which generalizes the HR beyond the underlying survival model. We consider the case in which the study factor is a binary variable and we are interested in both the unadjusted and adjusted effect of this factor on a time-to-event variable, potentially, observed in a right-censored scenario. We propose non-parametric estimations for unadjusted gHR and semi-parametric regression-induced techniques for the adjusted case. The behavior of those estimators is studied in both large and finite sample situations. Monte Carlo simulations reveal that both estimators provide good approximations of their respective inferential targets. Data from the Health and Lifestyle Study are used for studying the relationship of the tobacco use and the age of death and illustrate the practical application of the proposed technique. gHR is a promising index which can help facilitate better understanding of the association of one study factor on a time-dependent outcome.

Keywords: Cox regression model; hazard ratio; mis-specified model; omitted covariate; robust procedure
Corresponding author: Pablo Martínez-Camblor, Department of Anesthesiology, Dartmouth-Hitchcock Medical Center, 7 Lebanon Street, Suite 309, Hinman Box 7261, Hanover, NH, 03755, USA; and Department of Biomedical Data Science, Geisel School of Medicine at Dartmouth, Hanover, USA, E-mail:

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix: A Technical issues

A.1 Theorem’s proof

Since S x (τ N ) → N 0 (x = 0, 1) and N 1 / 2 ⋅ { S ̂ x ( u ) − S x ( u ) } → L N N ( 0 , σ A ( u ) ) , with σ A (u) < ∞ (for u ∈ R ), then N 1 / 2 ⋅ S ̂ 0 ( τ N ) ⋅ S ̂ x ( τ N ) → P N 0 . On the other hand,

(12) N 1 / 2 ⋅ ∫ S ̂ 1 ( u ) d F ̂ 0 ( u ) − ∫ S 1 ( u ) d F 0 ( u ) = N 1 / 2 ⋅ ∫ S ̂ 1 ( u ) d F ̂ 0 ( u ) − ∫ S 1 ( u ) d F ̂ 0 ( u ) + N 1 / 2 ⋅ ∫ S 1 ( u ) d F ̂ 0 ( u ) − ∫ S 1 ( u ) d F 0 ( u ) = λ 0 , N ⋅ n 0 1 / 2 ⋅ ∫ S 1 ( u ) d { F ̂ 0 ( u ) − F 0 ( u ) } − λ 1 , N ⋅ n 1 1 / 2 ⋅ ∫ S ̂ 0 ( u ) d { F ̂ 1 ( u ) − F 1 ( u ) } .

Stute [43] proved that, for any continuous function ψ(⋅), in the right-censored context,

(13) n x 1 / 2 ⋅ ∫ ψ ( u ) d { F ̂ x ( u ) − F x ( u ) } → L n x N ( 0 , σ x [ ψ ] ) ,

where σ x 2 [ ψ ] = ‖ ψ , S x ‖ G x with G x (⋅) is the CDF of the random censor variable. Eq. (12), the strong law of large number applied on S ̂ 1 ( ⋅ ) , and the Slutski’s Lemma we get the weak convergence. □

A.2 Corollary’s proof

We have that

N 1 / 2 ⋅ log ( gHR ̂ X ) − log ( gHR X ) = N 1 / 2 ⋅ ς ( A ̂ ( X ) ) − ς ( A ( X ) ) ,

where ς (u) = log(1 − u) − log(u). The result is directly derived from the Delta-method and the previous Theorem. □

References

1. Demidenko, E. The p-value you can’t buy. Am Statistician 2016;70:33–8. https://doi.org/10.1080/00031305.2015.1069760.Suche in Google Scholar PubMed PubMed Central

2. Hanley, J, McNeil, B. The meaning and use of the area under a receiver operating characteristic (ROC) curve. Radiology 1982;143:29–36. https://doi.org/10.1148/radiology.143.1.7063747.Suche in Google Scholar PubMed

3. Martínez-Camblor, P, Peréz-Fernández, S, Díaz-Coto, S. Area under the ROC curve as effect size measure. Pak J Statistics 2020;36:13–238.Suche in Google Scholar

4. Kaplan, E, Meier, P. Nonparametric estimation from incomplete observations. J Am Stat Assoc 1958;53:457–81. https://doi.org/10.1080/01621459.1958.10501452.Suche in Google Scholar

5. Cox, DR. Regression models and life-tables. J Roy Stat Soc B 1972;34:187–220. https://doi.org/10.1111/j.2517-6161.1972.tb00899.x.Suche in Google Scholar

6. Pearl, J. Causality: models, reasoning, and inference. New York, NY, USA: Cambridge University Press; 2000.Suche in Google Scholar

7. Martinussen, T, Vansteelandt, S, Andersen, P. Subtleties in the interpretation of hazard contrasts. Lifetime Data Anal 2020;26:833–55. https://doi.org/10.1007/s10985-020-09501-5.Suche in Google Scholar PubMed

8. Royston, P, Parmar, M. Restricted mean survival time: an alternative to the hazard ratio for the design and analysis of randomized trials with a time-to-event outcome. BMC Med Res Methodol 2013;13:1–15. https://doi.org/10.1186/1471-2288-13-152.Suche in Google Scholar PubMed PubMed Central

9. Xu, R, O’Quigley, J. Estimating average regression effect under non-proportional hazards. Biostatistics 2000;1:423–39. https://doi.org/10.1093/biostatistics/1.4.423.Suche in Google Scholar PubMed

10. Hernán, M. The hazards of hazard ratios. Epidemiology 2010;21:13–5. https://doi.org/10.1097/ede.0b013e3181c1ea43.Suche in Google Scholar PubMed PubMed Central

11. Monnickendam, G, Zhu, M, McKendrick, J, Su, Y. Measuring survival benefit in health technology assessment in the presence of nonproportional hazards. Value Health 2019;22:431–8. https://doi.org/10.1016/j.jval.2019.01.005.Suche in Google Scholar PubMed

12. Nieto, F, Coresh, J. Adjusting survival curves for confounders: a review and a new method. Am J Epidemiol 1996;143:1059–68. https://doi.org/10.1093/oxfordjournals.aje.a008670.Suche in Google Scholar PubMed

13. De Neve, J, Gerds, T. On the interpretation of the hazard ratio in Cox regression. Biom J 2020;62:742–50. https://doi.org/10.1002/bimj.201800255.Suche in Google Scholar PubMed

14. Martinussen, T, Vansteelandt, S. On collapsibility and confounding bias in Cox and Aalen regression models. Lifetime Data Anal 2013;19:279–96. https://doi.org/10.1007/s10985-013-9242-z.Suche in Google Scholar PubMed

15. Cox, B, Blaxter, M, Buckle, A, Fenner, N, Golding, J, Gore, M, et al.. The health and lifestyle survey. Preliminary report of a nationwide survey of the physical and mental health, attitudes and lifestyle of a random sample of 9,003 British adults. Health Promotion Research Trust; 1987.Suche in Google Scholar

16. Gonzalez-Manteiga, W, Cadarso-Suarez, C. Asymptotic properties of a generalized Kaplan-Meier estimator with some applications. J Nonparametric Statistics 1994;4:65–78. https://doi.org/10.1080/10485259408832601.Suche in Google Scholar

17. Cox, DR. Partial likelihood. Biometrika 1975;62:269–76. https://doi.org/10.1093/biomet/62.2.269.Suche in Google Scholar

18. Keil, A, Edwards, J, Richardson, D, Naimi, A, Cole, S. The parametric g-formula for time-to-event data: intuition and a worked example. Epidemiology 2014;25:889–97. https://doi.org/10.1097/ede.0000000000000160.Suche in Google Scholar PubMed PubMed Central

19. Martínez-Camblor, P, Mackenzie, T, Staiger, D, Goodney, P, O’Malley, A. Summarizing causal differences in survival curves in the presence of unmeasured confounding. Int J Biostat 2021;17:223–40.10.1515/ijb-2019-0146Suche in Google Scholar PubMed

20. Aalen, OO. Nonparametric inference for a family of counting processes. Ann Stat 1978;6:701–26. https://doi.org/10.1214/aos/1176344247.Suche in Google Scholar

21. Martínez-Camblor, P. Fully non-parametric receiver operating characteristic curve estimation for random-effects meta-analysis. Stat Methods Med Res 2017;26:5–20. https://doi.org/10.1177/0962280214537047.Suche in Google Scholar PubMed

22. Csorgő, S. Universal Gaussian approximations under random censorship. Ann Stat 1996;24:2744–78.10.1214/aos/1032181178Suche in Google Scholar

23. Peña, EA, Rohatgi, VK. Small sample and efficiency results for the nelson-aalen estimator. J Stat Plann Inference 1993;37:193–202. https://doi.org/10.1016/0378-3758(93)90088-n.Suche in Google Scholar

24. van der Vaart, A. Asymptotic statistics. Asymptotic Statistics. Cambridge University Press; 2000.Suche in Google Scholar

25. Dabrowska, DM. Uniform consistency of the kernel conditional Kaplan–Meier estimate. Ann Stat 1989;17:1157–67. https://doi.org/10.1214/aos/1176347261.Suche in Google Scholar

26. MacKenzie, T, Brown, J, Likosky, D, Wu, Y, Grunkemeier, G. Review of case-mix corrected survival curves. Ann Thorac Surg 2012;93:1416–25. https://doi.org/10.1016/j.athoracsur.2011.12.094.Suche in Google Scholar PubMed

27. Rodríguez-Álvarez, MX, Tahoces, PG, Cadarso-Suárez, C, Lado, MJ. Comparative study of ROC regression technique-applications for the computer-aided diagnostic system in breast cancer detection. Comput Stat Data Anal 2011;55:888–902. https://doi.org/10.1016/j.csda.2010.07.018.Suche in Google Scholar

28. Pardo-Fernandez, JC, Rodríguez-Álvarez, MX, Keilegom, IV. A review on ROC curves in the presence of covariates. Revstat-Statistical J 2014;12:21–41.Suche in Google Scholar

29. Wei, L. The accelerated failure time model: a useful alternative to the Cox regression model in survival analysis. Stat Med 1992;11:1871–9. https://doi.org/10.1002/sim.4780111409.Suche in Google Scholar PubMed

30. Therneau, T. A package for survival analysis in S and R. R package version 2.38; 2015.Suche in Google Scholar

31. Truthers, C, Kalbfleisch, J. Misspecified proportional hazard models. Biometrika 1986;73:363–9. https://doi.org/10.1093/biomet/73.2.363.Suche in Google Scholar

32. Combescure, C, Perneger, T, Weber, D, Daurés, J, Foucher, Y. Prognostic ROC curves: a method for representing the overall discriminative capacity of binary markers with right-censored time-to-event endpoints. Epidemiology 2014;25:103–9. https://doi.org/10.1097/ede.0000000000000004.Suche in Google Scholar PubMed

33. Gönen, M, Heller, G. Lehmann family of ROC curves. Med Decis Making 2010;30:509–17. https://doi.org/10.1177/0272989x09360067.Suche in Google Scholar PubMed PubMed Central

34. Martínez-Camblor, P, Corral, N, Rey, C, Pascual, J, Cernuda-Morollón, E. Receiver operating characteristic curve generalization for non-monotone relationships. Stat Methods Med Res 2017;26:113–23. https://doi.org/10.1177/0962280214541095.Suche in Google Scholar PubMed

35. Martínez-Camblor, P. Testing the equality among distribution functions from independent and right censored samples via Cramér-von mises criterion. J Appl Stat 2011;38:1117–31. https://doi.org/10.1080/02664763.2010.484486.Suche in Google Scholar

36. Chen, Q, Zeng, D, Ibrahim, J, Chen, M-H, Pan, Z, Xue, X. Quantifying the average of the time-varying hazard ratio via a class of transformations. Lifetime Data Anal 2015;21:259–79. https://doi.org/10.1007/s10985-014-9301-0.Suche in Google Scholar PubMed PubMed Central

37. Diao, G, Ibrahim, J. Quantifying time-varying cause-specific hazard and subdistribution hazard ratios with competing risks data. Clin Trials 2019;16:363–74. https://doi.org/10.1177/1740774519852708.Suche in Google Scholar PubMed PubMed Central

38. Yang, S, Prentice, R. Semiparametric analysis of short-term and long-term hazard ratios with two-sample survival data. Biometrika 2005;92:1–17. https://doi.org/10.1093/biomet/92.1.1.Suche in Google Scholar

39. Diao, G, Zeng, D, Yang, S. Efficient semiparametric estimation of short-term and long-term hazard ratios with right-censored data. Biometrics 2013;69:840–9. https://doi.org/10.1111/biom.12097.Suche in Google Scholar PubMed PubMed Central

40. MacKenzie, T, Tosteson, T, Morden, N, Stukel, T, O’Malley, J. Using instrumental variables to estimate a cox’s proportional hazards regression subject to additive confounding. Health Serv Outcome Res Methodol 2014;14:54–68. https://doi.org/10.1007/s10742-014-0117-x.Suche in Google Scholar PubMed PubMed Central

41. MacKenzie, T, Martínez-Camblor, P, O’Malley, A. Time dependent hazard ratio estimation using instrumental variables without conditioning on an omitted covariate. BMC Med Res Methodol 2021;56:1–11. https://doi.org/10.1186/s12874-021-01245-6.Suche in Google Scholar PubMed PubMed Central

42. Martínez-Camblor, P, Mackenzie, T, Staiger, D, Goodney, P, O’Malley, J. Adjusting for bias introduced by instrumental variable estimation in the Cox proportional hazards model. Biostatistics 2017;20:80–96. https://doi.org/10.1093/biostatistics/kxx062.Suche in Google Scholar PubMed

43. Stute, W. The statistical analysis of Kaplan–Meier integrals. Analysis of censored data. IMS Lecture Notes-Monograph Series 1995;27:231–54. https://doi.org/10.1214/lnms/1215452223.Suche in Google Scholar

Supplementary Material

As supplementary material we provide the R package generalizedHR which contains the functions used in the Monte Carlo simulation study and in the real-world data example. This package also provides functions for emulating the considered simulation models. Original data can be obtained, under request, at https://beta.ukdataservice.ac.uk/datacatalogue/studies/study?id=2218.

The online version of this article offers supplementary material (https://doi.org/10.1515/ijb-2021-0003).

Received: 2021-01-06
Revised: 2021-05-11
Accepted: 2021-08-03
Published Online: 2021-08-23

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Research Articles
  3. Doubly robust adaptive LASSO for effect modifier discovery
  4. Increasing the efficiency of randomized trial estimates via linear adjustment for a prognostic score
  5. Review
  6. Review and comparison of treatment effect estimators using propensity and prognostic scores
  7. Research Articles
  8. Error rate control for classification rules in multiclass mixture models
  9. Regression trees and ensembles for cumulative incidence functions
  10. Causal inference under over-simplified longitudinal causal models
  11. Causal inference under interference with prognostic scores for dynamic group therapy studies
  12. Bayesian multi-response nonlinear mixed-effect model: application of two recent HIV infection biomarkers
  13. A Bayesian semiparametric accelerate failure time mixture cure model
  14. Quantifying the extent of visit irregularity in longitudinal data
  15. An improved method for analysis of interrupted time series (ITS) data: accounting for patient heterogeneity using weighted analysis
  16. A robust hazard ratio for general modeling of survival-times
  17. Penalized likelihood estimation of the proportional hazards model for survival data with interval censoring
  18. A parametric approach to relaxing the independence assumption in relative survival analysis
  19. The number of response categories in ordered response models
  20. A comparison of joint dichotomization and single dichotomization of interacting variables to discriminate a disease outcome
  21. Spike detection for calcium activity
https://doi.org/10.1515/ijb-2021-0003
Schlagwörter für diesen Artikel
Cox regression model; hazard ratio; mis-specified model; omitted covariate; robust procedure

Artikel in diesem Heft

  1. Frontmatter
  2. Research Articles
  3. Doubly robust adaptive LASSO for effect modifier discovery
  4. Increasing the efficiency of randomized trial estimates via linear adjustment for a prognostic score
  5. Review
  6. Review and comparison of treatment effect estimators using propensity and prognostic scores
  7. Research Articles
  8. Error rate control for classification rules in multiclass mixture models
  9. Regression trees and ensembles for cumulative incidence functions
  10. Causal inference under over-simplified longitudinal causal models
  11. Causal inference under interference with prognostic scores for dynamic group therapy studies
  12. Bayesian multi-response nonlinear mixed-effect model: application of two recent HIV infection biomarkers
  13. A Bayesian semiparametric accelerate failure time mixture cure model
  14. Quantifying the extent of visit irregularity in longitudinal data
  15. An improved method for analysis of interrupted time series (ITS) data: accounting for patient heterogeneity using weighted analysis
  16. A robust hazard ratio for general modeling of survival-times
  17. Penalized likelihood estimation of the proportional hazards model for survival data with interval censoring
  18. A parametric approach to relaxing the independence assumption in relative survival analysis
  19. The number of response categories in ordered response models
  20. A comparison of joint dichotomization and single dichotomization of interacting variables to discriminate a disease outcome
  21. Spike detection for calcium activity
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