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Non-proportionality of Hazards in the Competing Risks Framework

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Risk Assessment and Evaluation of Predictions

Part of the book series: Lecture Notes in Statistics ((LNSP,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 (a 1 and a 2) are different from 1. In this chapter, we show that a 1 and a 2 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 a 1 and a 2 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 a 1 and a 2 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.

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References

  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

    Article  MathSciNet  Google Scholar 

  2. Cox, D.R., Oakes, D.: The Analysis of Survival Data, pp. 142–155. Chapman & Hall, New York (1984)

    Google Scholar 

  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

    Article  MATH  Google Scholar 

  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

    Article  MATH  Google Scholar 

  5. Kalbfleisch, J.D., Prentice, R.L.: The Statistical Analysis of Failure Time Data. Wiley, New York (1980)

    MATH  Google Scholar 

  6. Larson, M.G.: Covariate analysis of competing-risks data with log-linear models. Biometrics 40, 459–469 (1984)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  11. Beyersmann, J., Allignol, A., Schumacher, M.: Competing risks and multistate models with R, pp. 89–153. Springer, New York (2012)

    Book  MATH  Google Scholar 

  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

    Article  Google Scholar 

  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

    Article  Google Scholar 

  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

    Article  Google Scholar 

  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

    Article  Google Scholar 

  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

    Article  Google Scholar 

  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

    Article  Google Scholar 

  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

    Article  Google Scholar 

  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

    MathSciNet  Google Scholar 

  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

    Article  MathSciNet  Google Scholar 

  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

    Article  MathSciNet  Google Scholar 

  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

    Article  MathSciNet  Google Scholar 

  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

    Article  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  25. Jeong, J.-H., Fine, J.P.: Parametric regression on cumulative incidence function. Biostatistics 8, 184–196 (2007). doi:10.1093/biostatistics/kxj040

    Article  MATH  Google Scholar 

  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. Klein, J.P.: Modelling competing risks in cancer studies. Stat. Med. 25, 1015–1034 (2006). doi:10.1002/sim.2246

    Article  MathSciNet  Google Scholar 

  28. Zhang, M.J., Fine, J.: Summarizing differences in cumulative incidence functions. Stat. Med. 27, 4939–4949 (2008). doi:10.1002/sim.3339

    Article  MathSciNet  Google Scholar 

  29. Chang, W.-H., Wang, W.: Regression analysis for cumulative incidence probability under competing risks. Statistica Sinica 19, 391–408 (2009)

    MathSciNet  MATH  Google Scholar 

  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

    Article  MATH  Google Scholar 

  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

    Article  MathSciNet  Google Scholar 

  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]

    Article  MathSciNet  Google Scholar 

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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.

Author information

Authors and Affiliations

  1. Johns Hopkins Bloomberg School of Public Health, 615 North Wolfe Street – E7648, Baltimore, MD, 21205, USA

    Alvaro Muñoz

  2. Johns Hopkins Bloomberg School of Public Health, 615 North Wolfe Street – E7640, Baltimore, MD, 21205, USA

    Alison G. Abraham

  3. Johns Hopkins Bloomberg School of Public Health, 615 North Wolfe Street – E7009, Baltimore, MD, 21205, USA

    Matthew Matheson

  4. Johns Hopkins Bloomberg School of Public Health, 615 North Wolfe Street – E7650, Baltimore, MD, 21205, USA

    Nikolas Wada

Authors
  1. Alvaro Muñoz
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  2. Alison G. Abraham
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  3. Matthew Matheson
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  4. Nikolas Wada
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Corresponding author

Correspondence to Alvaro Muñoz .

Editor information

Editors and Affiliations

  1. University of Maryland, College Park, Maryland, USA

    Mei-Ling Ting Lee

  2. National Cancer Institute Div. Cancer Epidemiology & Genetics, Bethesda, Maryland, USA

    Mitchell Gail

  3. National Cancer Institute, Bethesda, Maryland, USA

    Ruth Pfeiffer

  4. Centers for Disease Control and Prevention, Atlanta, Georgia, USA

    Glen Satten

  5. Department of Biostatistics, Harvard School of Public Health, Boston, Massachusetts, USA

    Tianxi Cai

  6. Department of Mathematics, Imperial College London, London, United Kingdom

    Axel Gandy

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Muñoz, A., Abraham, A.G., Matheson, M., Wada, N. (2013). Non-proportionality of Hazards in the Competing Risks Framework. In: Lee, ML., Gail, M., Pfeiffer, R., Satten, G., Cai, T., Gandy, A. (eds) Risk Assessment and Evaluation of Predictions. Lecture Notes in Statistics, vol 215. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8981-8_1

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