Competing risks data arise naturally in medical research, when subjects under study are at risk of more than one mutually exclusive event such as death from different causes. The competing risks framework also includes settings where different possible events are not mutually exclusive but the interest lies on the first occurring event. For example, in HIV studies where seropositive subjects are receiving highly active antiretroviral therapy (HAART), treatment interruption and switching to a new HAART regimen act as competing risks for the first major change in HAART. This article introduces competing risks data and critically reviews the widely used statistical methods for estimation and modelling of the basic (estimable) quantities of interest. We discuss the increasingly popular Fine and Gray model for subdistribution hazard of interest, which can be readily fitted using standard software under the assumption of administrative censoring. We present a simulation study, which explores the robustness of inference for the subdistribution hazard to the assumption of administrative censoring. This shows a range of scenarios within which the strictly incorrect assumption of administrative censoring has a relatively small effect on parameter estimates and confidence interval coverage. The methods are illustrated using data from HIV-1 seropositive patients from the collaborative multicentre study CASCADE (Concerted Action on SeroConversion to AIDS and Death in Europe).
Collaborative Group on AIDS Incubation and HIV Survival including the CASCADE EU Concerted Action. Concerted Action on SeroConversion to AIDS and Death in Europe. Time from HIV-1 seroconversion to AIDS and death before widespread use of highly-active antiretroviral therapy: a collaborative re-analysis. Lancet2000; 355: 1131–1137.
2.
PorterKBabikerABhaskaranK. Determinants of survival following HIV-1 seroconversion after the introduction of HAART. Lancet2003; 362: 1267–1274.
3.
CoxDROakesD. Analysis of survival data. London: Chapman & Hall, 1984.
4.
KalbfleischJDPrenticeRL. The statistical analysis of failure time data, 2nd ed. New York: Wiley, 2002.
5.
ColletD. Modeling survival data in medical research. New York: Chapman & Hall/CRC, 1994.
6.
TsiatisAA. Competing risks. Encyclopedia of biostatistics. New York: Wiley, 1999.
7.
PepeMSMoriM. Kaplan-Meier, marginal or conditional probability curves in summarizing competing risks failure time data?Stat Med1993; 12: 737–751.
8.
PutterHFioccoMGeskusRB. Tutorial in biostatistics: competing risks and multi-state models. Stat Med2007; 26: 2389–2430.
9.
BakoyannisGTouloumiG, on behalf of CASCADE collaboration. A practical guide on modeling competing risk data, www.cascade-collaboration.org (accessed June 2010).
10.
TouloumiGPantazisNAntoniouAStirnadelHAWalkerSAPorterKCASCADE Collaboration. Highly active antiretroviral therapy interruption: predictors and virological and immunologic consequences. J Acquir Immune Defic Syndr2006; 42: 554–561.
11.
PrenticeRKalbfleischJPetersonAFlournoyNFarewellVBreslowN. The analysis of failure times in the presence of competing risks. Biometrics1978; 34: 541–554.
12.
CASCADE Collaboration. Changes in the uptake of antiretroviral therapy and survival in people with known duration of HIV infection in Europe: results from CASCADE. HIV Med2000; 1: 224–231.
13.
TsiatisAA. A nonidentifiability aspect of the problem of competing risks. Proc Nat Acad Sci USA1975; 72: 20–22.
14.
AndersenPKAbildstromSZRosthojS. Competing risks as a multi-state model. Stat Methods Med Res2002; 11: 203–215.
15.
GailM. A review and critique of some models used in competing risk analysis. Biometrics1975; 31: 209–222.
16.
AndersenPKBorganOGillRDKeidingN. Statistical models based on counting processes. New York: Springer, 1996.
17.
GaynorJJFeuerEJTanCC. On the use of cause-specific failure and conditional failure probabilities: examples from clinical oncology data. J Am Stat Assoc1993; 88: 400–409.
18.
AalenOJohansenS. An empirical transition matrix for non-homogeneous Markov chains based on censored observations. Scand J Stat1978; 5: 141–150.
19.
GrayRJ. A class of k-sample tests for comparing the cumulative incidence of a competing risk. Ann Stat1988; 16: 1141–1154.
20.
AalenO. Nonparametric estimation of partial transition probabilities in multiple decrement models. Ann Stat1978; 6: 534–545.
21.
LinDY. Non-parametric inference for cumulative incidence functions in competing risks studies. Stat Med1997; 16: 901–910.
22.
MocroftAYouleMMooreA. Reasons for modification and discontinuation of antiretrovirals: results from a single treatment centre. AIDS2001; 25: 185–194.
23.
O’BrienMEClarkRABeschCL. Patterns and correlates of discontinuation of the initial HAART regimen in an urban outpatient cohort. J Acquir Immune Defic Syndr2003; 34: 407–414.
24.
HoltJD. Competing risk analyses with special reference to matched pair experiments. Biometrika1978; 65: 159–165.
25.
AndersenPKBorganO. Counting process models for life history data: a review. Scand J Stat1985; 12: 97–158.
26.
LunnMMcNeilD. Applying Cox regression to competing risks. Biometrics1995; 51: 524–532.
27.
AndersenPKHansenLSKeidingN. Nonparametric and semiparametric estimation of transition-probabilities from censored observation of a nonhomogeneous Markov process. Scand J Stat1991; 18: 153–167.
28.
FineJPGrayRJ. A proportional hazards model for the subdistribution of a competing risk. J Am Stat Assoc1999; 94: 496–509.
29.
RobinsJMRotnitzkyA. Recovery of information and adjustment for dependent censoring using surrogate markers. In: JewellNDietzKFarewellV (eds) AIDS epidemiology-methodological issues. Boston: Birkhauser, 1992, pp. 24–33.
30.
RuanPKGrayRJ. Analyses of cumulative incidence functions via non-parametric multiple imputation. Stat Med2008; 27: 5709–5724.
31.
TaylorJMGMurraySHsuC-H. Survival estimation and testing via multiple imputations. Stat Prob Lett2002; 58: 221–232.
32.
GeskusRB. Cause-specific cumulative incidence estimation and the Fine and Gray model under both left truncation and right censoring. Biometrics2010; DOI: 10.1111/j.1541-0420.2010.01420.x.
33.
HeckmanJJHonoréBE. The identifiability of the competing risks model. Biometrika1989; 76: 325–330.
34.
PintilieM. Analysing and interpreting competing risk data. Stat Med2007; 26: 1360–1367.
35.
LatoucheABeyersmannJFineJP. Letter to the editor: Comments on ‘Analysing and interpreting competing risk data’. Stat Med2007; 26: 3676–3680.
36.
WolbersMKollerM. Letter to the editor: Comments on ‘Analysing and interpreting competing risk data’. Stat Med2007; 26: 3521–3523.
37.
PepeMS. Inference for events with dependent risks in multiple endpoint studies. J Am Stat Assoc1991; 86: 770–778.
38.
GeskusR. Censoring strategies when using competing risks with time-dependent covariates. Letter to the editor. J Acquir Immune Defic Syndr2007; 46: 512–512.
39.
LatoucheABoissonVPorcherRChevretS. Misspecified regression model for the subdistribution hazard of a competing risk. Stat Med2007; 26: 965–974.
40.
ZhangMJFineJ. Summarizing differences in cumulative incidence functions. Stat Med2008; 27: 4939–4949.
41.
CorteseGAnderenPK. Competing risks and time-dependent covariates. Biometrical J2009; 51: 138–158.
42.
BeyersmannJSchumacherM. Time-dependent covariates in the proportional subdistribution hazards model for competing risks. Biostatistics2008; 9: 765–776.
43.
BeyersmannJDettenkoferMBertzHSchumacherM. A competing risks analysis of bloodstream infection after stem-cell transplantation using subdistribution hazards and cause-specific hazards. Stat Med2007; 26: 5360–5369.
44.
SolomonPJ. Effect of misspecification of regression models in the analysis of survival data. Biometrika1984; 71: 291–298.
GrambauerNSchumacherMBeyersmannJ. Proportional subdistribution hazards modelling offers a summary analysis, even if misspecified. Stat Med2010; 29: 875–884.
47.
BayersmannJLatoucheABuchholzASchumacherM. Simulating competing risks data in survival analysis. Stat Med2009; 28: 956–971.
48.
KleinJP. Modelling competing risks in cancer studies. Stat Med2006; 25: 1015–1034.
49.
MillerRG. What price Kaplan-Meier?Biometrics1983; 39: 1077–1081.
50.
JeongJHFineJP. Parametric regression on cumulative incidence function. Biostatistics2007; 8: 184–196.
51.
LuKTsiatisAA. Multiple imputation methods for estimating regression coefficients in the competing risks model with missing cause of failure. Biometrics2001; 57: 1191–1197.
52.
MatsuuraMEguchiS. Modeling late entry bias in survival analysis. Biometrics2005; 61: 559–566.
