Advertisement

Non-proportionality of Hazards in the Competing Risks Framework

  • Alvaro MuñozEmail author
  • Alison G. Abraham
  • Matthew Matheson
  • Nikolas Wada
Conference paper
First Online:
Part of the Lecture Notes in Statistics book series (LNS, volume 215)

Abstract

The simplest means of determining the effect of an exposure on the frequency and timing of two competing events is to contrast the cumulative incidences between the exposed and unexposed groups for each event type. Methods and software are widely available to semi-parametrically model the sub-hazards of the cumulative incidences as proportional and to test whether the constant relative sub-hazards (a1 and a2) are different from 1. In this chapter, we show that a1 and a2 are tethered by a strong relationship which is independent of the timing of the competing events; the relationship is fully determined by the overall frequencies of events, and a1 and a2 must be on opposite sides of 1. When violations of proportionality occur, separate analyses for the two competing events often yield an inadmissible result in which the estimates of a1 and a2 are on the same side of 1, and may even exhibit statistical significance. We further characterize the compatibility of concurrent proportionality of cause-specific hazards and sub-hazards, and show that strong tethering also occurs among these quantities; and that, of the sub-hazards and cause-specific hazards, at most two of the four can be proportional, but without restriction on which two. Because proportionality rarely holds in practice, the default analytical approach should allow the relative hazards to depend on time, which can be easily carried out with widely available software. However, the statistical power of this approach is limited in the case of large numbers of event-free observations. An application using data from a North American cohort study of children with kidney disease is presented.

Keywords

Chronic Kidney Disease Cumulative Incidence Event Type Exposed Group Unexposed Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

This work and the Chronic Kidney Disease in Children Study are supported by grants from the National Institute of Diabetes and Digestive and Kidney Diseases (NIDDK) at the National Institutes of Health (NIH), funded in collaboration the National Institute of Child Health and Human Development (NICHD) and the National Heart, Lung and Blood Institute (NHLBI) of NIH: Grant numbers U01-DK-66116, U01-DK-66143, U01-DK-66174, and U01-DK-82194.

References

  1. 1.
    Putter, H., Fiocco, M., Geskus, R.B.: Tutorial in biostatistics: competing risks and multi-state models. Stat. Med. 26, 2389–2430 (2007). doi: 10.1002/sim.2712 CrossRefMathSciNetGoogle Scholar
  2. 2.
    Cox, D.R., Oakes, D.: The Analysis of Survival Data, pp. 142–155. Chapman & Hall, New York (1984)Google Scholar
  3. 3.
    Gaynor, J.J., Feurer, E.J., Tan, C., Wu, D.H., Little, C.R., Straus, D.J., Clarkson, B.D.,Brennan, M.F.: On the use of cause-specific failure and conditional failure probabilities: examples from clinical oncology data. J. Am. Stat. Assoc. 88, 400–409 (1993). doi: 10.1080/01621459.1993.10476289 CrossRefzbMATHGoogle Scholar
  4. 4.
    Holt, J.D.: Competing risk analyses with special reference to matched pair experiments. Biometrika 65, 159–165 (1978). doi: 10.1093/biomet/65.1.159 CrossRefzbMATHGoogle Scholar
  5. 5.
    Kalbfleisch, J.D., Prentice, R.L.: The Statistical Analysis of Failure Time Data. Wiley, New York (1980)zbMATHGoogle Scholar
  6. 6.
    Larson, M.G.: Covariate analysis of competing-risks data with log-linear models. Biometrics 40, 459–469 (1984)CrossRefGoogle Scholar
  7. 7.
    Prentice, R.L., Kalbfleisch, J.D., Peterson Jr., A.V., Flournoy, N., Farewell, V.T., Breslow, N.E.: The analysis of failure times in the presence of competing risks. Biometrics 34, 541–554 (1978)CrossRefzbMATHGoogle Scholar
  8. 8.
    Fine, J.P., Gray, R.J.: A proportional hazards model for the subdistribution of a competing risk. J. Am. Stat. Assoc. 94, 496–509 (1999). doi: 10.1080/01621459.1999.10474144 CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Gray, R.J.: A class of K-sample tests for comparing the cumulative incidence of a competing risk. Ann. Stat. 16, 1141–1154 (1988). doi: 10.1214/aos/1176350951 CrossRefzbMATHGoogle Scholar
  10. 10.
    Pepe, M.S.: Inference for events with dependent risks in multiple endpoint studies. J. Am. Stat. Assoc. 86, 770–778 (1991). doi: 10.1080/01621459.1991.10475108 CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Beyersmann, J., Allignol, A., Schumacher, M.: Competing risks and multistate models with R, pp. 89–153. Springer, New York (2012)CrossRefzbMATHGoogle Scholar
  12. 12.
    Babiker, A., Darbyshire, J., Pezzotti, P., Porter, K., Rezza, G., Walker, S.A., et al.: Changes over calendar time in the risk of specific first AIDS-defining events following HIV seroconversion, adjusting for competing risks. Int. J. Epidemiol. 31, 951–958 (2002). doi: 10.1093/ije/31.5.951 CrossRefGoogle Scholar
  13. 13.
    del Amo, J., Perez-Hoyos, S., Moreno, A., Quintana, M., Ruiz, I., Cisneros, J.M., Ferreros, I., Gonzalez, C., de Olalla, P.G., Perez, R., Hernandez, I.: Trends in AIDS and mortality in HIV-infected subjects with hemophilia from 1985 to 2003: the competing risks for death between AIDS and liver disease. J. Acquir. Immune Defic. Syndr. 41, 624–631 (2006). doi: 10.1097/01.qai.0000194232.85336.dc CrossRefGoogle Scholar
  14. 14.
    Lim, H.J., Zhang, X., Dyck, R., Osgood, N.: Methods of competing risks analysis of end-stage renal disease and mortality among people with diabetes. BMC Med. Res. Methodol. 10, 97 (2010). doi: 10.1186/1471-2288-10-97 CrossRefGoogle Scholar
  15. 15.
    Pacheco, A.G., Tuboi, S.H., May, S.B., Moreira, L.F.S., Ramadas, L., Nunes, E.P., Merçon, M., Faulhaber, J.C., Harrison, L.H., Schechter, M.: Temporal changes in causes of death among HIV-infected patients in the HAART era in Rio de Janeiro, Brazil. J. Acquir. Immune Defic. Syndr. 51, 624–630 (2009). doi: 10.1097/QAI.0b013e3181a4ecf5 CrossRefGoogle Scholar
  16. 16.
    Wada, N., Jacobson, L.P., Cohen, M., French, A.L., Phair, J., Muñoz, A.: Cause-specific life expectancies after 35 years of age for human immunodeficiency syndrome-infected and human immunodeficiency syndrome-negative individuals followed simultaneously in long-term cohort studies: 1984–2008. Am. J. Epidemiol. 15, 116–125 (2013). doi: 10.1093/aje/kws321 CrossRefGoogle Scholar
  17. 17.
    Checkley, W., Brower, R.G., Muñoz, A.: Inference for mutually exclusive competing events through a mixture of generalized gamma distributions. Epidemiology 21, 557–565 (2010). doi: 10.1097/EDE.0b013e3181e090ed CrossRefGoogle Scholar
  18. 18.
    Cole, S.R., Li, R., Anastos, K., Detels, R., Young, M., Chmiel, J.S., Muñoz, A.: Accounting for leadtime in cohort studies: evaluating when to initiate HIV therapies. Stat. Med. 23, 3351–3363 (2004). doi: 10.1002/sim.1579 CrossRefGoogle Scholar
  19. 19.
    Larson, M.G., Dinse, G.E.: A mixture model for the regression analysis of competing risks data. J. R. Stat. Soc. Ser. C (Appl. Stat.) 34, 201–211 (1985). doi: 10.2307/2347464 MathSciNetGoogle Scholar
  20. 20.
    Beyersmann, J., Latouche, A., Buchholz, A., Schumacher, M.: Simulating competing risks data in survival analysis. Stat. Med. 28, 956–971 (2009). doi: 10.1002/sim.3516 CrossRefMathSciNetGoogle Scholar
  21. 21.
    Latouche, A., Boisson, V., Chevret, S., Porcher, R.: Misspecified regression model for the subdistribution hazard of a competing risk. Stat. Med. 26, 965–974 (2007). doi: 10.1002/sim.2600 CrossRefMathSciNetGoogle Scholar
  22. 22.
    Zhang, X., Zhang, M.J., Fine, J.: A proportional hazard regression model for the subdistribution with right-censored and left-truncated competing risks data. Stat. Med. 30, 1933–1951 (2011). doi: 10.1002/sim.4264 CrossRefMathSciNetGoogle Scholar
  23. 23.
    Furth, S.L., Cole, S.R., Moxey-Mims, M., Kaskel, F., Mak, R., Schwartz, G., Wong, C., Muñoz, A., Warady, B.A.: Design and methods of the Chronic Kidney Disease in Children (CKiD) prospective cohort study clinical. J. Am. Soc. Nephrol. 1, 1006–1015 (2006). doi: 10.2215/CJN.01941205 CrossRefGoogle Scholar
  24. 24.
    Geskus, R.B.: Cause-specific cumulative incidence estimation and the Fine and Gray model under both left truncation and right censoring. Biometrics 67, 39–49 (2011). doi: 10.1111/j.1541-0420.2010.01420.x CrossRefMathSciNetzbMATHGoogle Scholar
  25. 25.
    Jeong, J.-H., Fine, J.P.: Parametric regression on cumulative incidence function. Biostatistics 8, 184–196 (2007). doi: 10.1093/biostatistics/kxj040 CrossRefzbMATHGoogle Scholar
  26. 26.
    del Amo, J., Jarring, I., May, M., Dabis, F., Crane, H., Podzamczer, D., et al.: Influence of geographical origin and ethnicity on mortality in patients on antiretroviral therapy in Canada, Europe and the United States. Clin. Infect. Dis. 54, 1800–1809 (2013). doi: 10.1093/cid/cit111 Google Scholar
  27. 27.
    Klein, J.P.: Modelling competing risks in cancer studies. Stat. Med. 25, 1015–1034 (2006). doi: 10.1002/sim.2246 CrossRefMathSciNetGoogle Scholar
  28. 28.
    Zhang, M.J., Fine, J.: Summarizing differences in cumulative incidence functions. Stat. Med. 27, 4939–4949 (2008). doi: 10.1002/sim.3339 CrossRefMathSciNetGoogle Scholar
  29. 29.
    Chang, W.-H., Wang, W.: Regression analysis for cumulative incidence probability under competing risks. Statistica Sinica 19, 391–408 (2009)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Fine, J.P.: Analysing competing risks data with transformation models. J. R. Stat. Soc. Ser. B 61, 817–830 (1999). doi: 10.1111/1467-9868.00204 CrossRefzbMATHGoogle Scholar
  31. 31.
    Grambauer, N., Schumacher, M., Beyersmann, J.: Proportional subdistribution hazards modeling offers a summary analysis, even if misspecified. Stat. Med. 29, 875–884 (2010). doi: 10.1002/sim.3786 CrossRefMathSciNetGoogle Scholar
  32. 32.
    Lau, B., Cole, S.R., Gange, S.J.: Parametric mixture models to evaluate and summarize hazard ratios in the presence of competing risks with time-dependent hazards and delayed entry. Stat. Med. 30, 654–665 (2011). doi: 10.1002/sim.4123 [correction in Statistics in Medicine 2012;31:1777–8. doi: 10.1002/sim.4468]CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Alvaro Muñoz
    • 1
    Email author
  • Alison G. Abraham
    • 2
  • Matthew Matheson
    • 3
  • Nikolas Wada
    • 4
  1. 1.Johns Hopkins Bloomberg School of Public HealthBaltimoreUSA
  2. 2.Johns Hopkins Bloomberg School of Public HealthBaltimoreUSA
  3. 3.Johns Hopkins Bloomberg School of Public HealthBaltimoreUSA
  4. 4.Johns Hopkins Bloomberg School of Public HealthBaltimoreUSA

Personalised recommendations

Cite paper

Buy options