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Am J Epidemiol. 2009 July 15; 170(2): 244–256.

Published online 2009 June 3. doi: 10.1093/aje/kwp107

PMCID: PMC2732996

Correspondence to Dr. Bryan Lau, Department of Medicine, Johns Hopkins School of Medicine, 1830 East Monument Street, Room 8070, Baltimore, MD 21287 (e-mail: ude.imhj@1ualb).

Received 2008 August 2; Accepted 2009 April 6.

Copyright American Journal of Epidemiology © The Author 2009. Published by the Johns Hopkins Bloomberg School of Public Health. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org.

This article has been cited by other articles in PMC.

Competing events can preclude the event of interest from occurring in epidemiologic data and can be analyzed by using extensions of survival analysis methods. In this paper, the authors outline 3 regression approaches for estimating 2 key quantities in competing risks analysis: the cause-specific relative hazard (_{cs}RH) and the subdistribution relative hazard (_{sd}RH). They compare and contrast the structure of the risk sets and the interpretation of parameters obtained with these methods. They also demonstrate the use of these methods with data from the Women's Interagency HIV Study established in 1993, treating time to initiation of highly active antiretroviral therapy or to clinical disease progression as competing events. In our example, women with an injection drug use history were less likely than those without a history of injection drug use to initiate therapy prior to progression to acquired immunodeficiency syndrome or death by both measures of association (_{cs}RH=0.67, 95% confidence interval: 0.57, 0.80 and _{sd}RH=0.60, 95% confidence interval: 0.50, 0.71). Moreover, the relative hazards for disease progression prior to treatment were elevated (_{cs}RH=1.71, 95% confidence interval: 1.37, 2.13 and _{sd}RH=2.01, 95% confidence interval: 1.62, 2.51). Methods for competing risks should be used by epidemiologists, with the choice of method guided by the scientific question.

In time-to-event analyses, the occurrence of the event of interest is often precluded by another event. The canonical example is the study predictors of cause-specific mortality, whereby a death due to the primary cause of interest (e.g., cancer-related deaths) is precluded by death due to other causes. In this competing risks setting (1–3), individuals are observed from study entry to the occurrence of the event of interest, a competing event, or censoring. Competing risks are common to epidemiologic research (4–7), and recognition dates to the 1700s when Bernoulli estimated mortality rates (1, 8, 9).

The complement of the Kaplan-Meier survival curve may not appropriately estimate the cumulative incidence when competing events are censored (10–13). Both nonparametric (2, 10, 14, 15) and regression (8, 16, 17) methods exist for analyzing data with competing events. Although the nonparametric approaches have been well described, 2 widely used measures from regression approaches, the cause-specific relative hazard (_{cs}RH) and the subdistribution relative hazard (_{sd}RH), have not been well described in the epidemiology literature (18).

The purpose of this paper is 3-fold. First, we provide intuition for the _{cs}RH and _{sd}RH by considering the construction of risk sets and interpretation of the underlying hazard function. Second, we describe 3 different regression models for the analysis of epidemiologic data with competing risks. Third, we illustrate the use of these methods in an analysis that explores the association of injection drug use with the time to 2 competing outcomes in a cohort of human immunodeficiency virus (HIV)-infected women: initiation of combination antiretroviral therapy and the occurrence of acquired immunodeficiency syndrome (AIDS) or death prior to initiating combination antiretroviral therapy.

Five interrelated building blocks underpin standard (noncompeting) survival analysis: the time scale *t* (e.g., age, calendar time, disease duration, or study duration); the risk set; the hazard function *h*(*t*); the cumulative incidence function (CIF) *F*(*t*); and its complement, the survival function *S*(*t*) = 1 – *F*(*t*). In a competing risks framework, each of these components remains of central importance but modified, depending on how the competing event is handled. We consider an event of interest (event 1) and only 1 competing event (event 2), although one may extend to more events. We assume no measurement error, noninformative censoring, and no unmeasured confounding. Henceforth, *T* will be defined as the minimum time to either event 1 or event 2 (*T*=min(*T*_{1}, *T*_{2}, *C*), where *T*_{1} and *T*_{2} correspond to time to event 1 and event 2, respectively, and *C* corresponds to the censoring time).

The regression approaches described below focus on 2 definitions of hazard, the cause-specific and the subdistribution hazards. The corresponding _{cs}RH may be better suited for studying the etiology of diseases, whereas the _{sd}RH has use in predicting an individual's risk or allocating resources.

Consider an example. A group of diseased individuals are randomized to treatment A or treatment B, everyone is compliant to treatment protocols, and all are followed until either the disease is cured or individuals have an adverse event requiring discontinuation of treatment. The cumulative incidence for being cured may be estimated as 1 – the Kaplan-Meier product limit estimator stratified by treatment. Both curves will be essentially 1.0 by the end of follow-up, as everyone is followed to 1 of the 2 events. The shift in curves represents the etiologic association between treatment type and being cured, in that it reflects the relative change in the underlying hazard. However, one would not predict that the probability of being cured was 1.0 for either treatment by the end of follow-up, as we know that some individuals have the adverse event and must discontinue therapy. For prediction, one may require a curve that reflects the proportion cured by the end of follow-up. Additionally, if the _{cs}RH_{adverse}>_{cs}RH_{cure}≥1.0 comparing treatment A versus treatment B, then the adverse event is occurring at a greater hazard rate in treatment group A. Thus, the proportion of individuals being cured would be different by treatment status by the end of follow-up even if the cure rates are the same. Therefore, the shift in these observed cumulative incidence curves should represent not only the etiologic association of treatment with being cured but also the influence of having a reduced number of individuals remaining at risk for being cured in group A due to a greater number of adverse events. The lower observed number of individuals being cured because of a greater proportion of adverse events may actually overpower the etiologic association, such that the observed cumulative incidence of being cured is no longer different between treatment groups. The nonparametric estimator for competing risks accounts for these issues. Similarly, the cause-specific and subdistribution hazard approaches reflect these 2 different kinds of comparison.

The manner in which risk sets are defined in standard survival analyses may be modified to allow for competing events. In standard survival analysis, the *risk set* is defined as the group of individuals that have not experienced the outcome and therefore are at risk for the event of interest at time *t*. Individuals who have a competing event can be removed from all later risk sets for the event of interest. Figure 1 illustrates this approach in discrete time. At time 0, there are 30 individuals at risk. At time 1, 1 individual has event 1, and another individual has event 2, such that the risk set for time 2 is now 28=30 –1_{event 1} – 1_{event 2}. Thus, individuals with an event 1 or event 2 prior to time *t* are excluded from the risk set at time *t*.

Cause-specific hazard schematic. The risk set starts with 30 individuals (solid circles). Over time, individuals have either event 1 (square) or event 2 (triangle). As individuals have either event, they are removed from the remaining risk sets. The calculation **...**

An estimate of the hazard for event 1 can be described in the discrete time setting as the number of individuals who experience the event divided by the number at risk at time *t*. For example, at time 3, this would be 3/26 = 0.12, which estimates the *cause-specific hazard*, which is formally defined as ${h}_{j}\left(t\right)=P(T=t,J=j|T\ge t)$, where *J* = *j* indicates whether event 1 (*j* = 1) or event 2 (*j*=2) is being estimated.

The cause-specific hazard can be extended to continuous time (8, 19):

(1)

where *f _{j}^{*}*(

As described in the Web supplement posted on the *Journal*’s website (http://aje.oxfordjournals.org/), a likelihood function can be constructed from the cause-specific hazards, whereby individuals who experienced a competing event are treated as censored (8). Consequently, a proportional hazards model can be constructed for the cause-specific hazard:

(2)

where *h*_{0j} is the arbitrary baseline cause-specific hazard, and β* _{j}*,

A Breslow estimator (21, 22) of the cumulative incidence proportion can be calculated by using the cause-specific hazard under the (untestable) assumption that the competing events are independent of each other (3, 18, 23–25). Models linking covariates to cause-specific hazards as measured by _{cs}RH_{j}_{=1} provide a summary of how a covariate directly impacts the incidence without considering the effect of the competing event. Much has been written about how inferences from this approach need to be evaluated cautiously (8, 26), because the assumption of independent competing events is strongly needed to underpin the inference that the cause-specific hazard and corresponding cumulative incidence functions quantify the risk of the event in hypothetical populations where competing events are eliminated (8). Therefore, caution must be used in interpreting _{cs}RH as an increase (decrease) in apparent risk; it is, however, valid to interpret it as a relative change in the cause-specific hazard rate.

In light of the strong assumption of independence between events to allow interpretation of the cause-specific cumulative incidence function (_{cs}CIF), the competing risk literature has focused on an alternative measure of risk: the subdistribution cumulative incidence function (_{sd}CIF). This function is defined as the joint probability of an event prior to time *t* and that the event is of type *j*: *F _{j}^{*}*(

Interpretation of this measure can be understood by returning to the construction of risk sets and hazard functions. In contrast to the construction of risk sets that eliminate individuals who have the competing cause, risk sets were constructed so that they include both individuals without any event and those who have had the competing event. It may be counterintuitive to maintain individuals who had a competing event in the risk set. However, one can think of these individuals as a “placeholder” for the proportion of the population that cannot have the event of interest and place a constraint on this hazard function definition (16). Figure 2 illustrates this construction with the same population as in Figure 1. For example, one individual had the competing event at time 1 and is therefore maintained in the subsequent risk sets. Therefore, at *t* = 2, the risk set comprised 29 individuals; at *t* = 3, a total of 3 individuals by this time have previously experienced event 2 and are maintained in the risk set. With increasing *t*, the risk set comprised an increasing proportion of individuals who have had event 2.

Subdistribution hazard schematic. The risk set starts with 30 individuals (solid circles). Over time, individuals have either event 1 (square) or event 2 (triangle). As individuals have the competing event (event 2, triangle), they are maintained in the **...**

With this structure, a different hazard function is defined as the probability of the event given that an individual has survived up to time *t* without any event or has had the competing event prior to time *t*. This is the *subdistribution hazard* (16). For example at *t* = 3, the subdistribution hazard is 3/29 = 0.103, which is smaller than the cause-specific hazard of 0.12 because of the larger risk set.

For the discrete time setting, the subdistribution hazard is ${\text{\lambda}}_{j}\left(t\right)=P(T=t,J=j|T\ge t\text{\hspace{0.5em}}\mathrm{or}\text{\hspace{0.5em}}(T<t\text{\hspace{0.5em}}\mathrm{and}\text{\hspace{0.5em}}J\ne j))$. In continuous time, the subdistribution hazard is the following (16):

(3)

where *F _{j}^{*}*(

An alternative proportional hazards model may be constructed from the subdistribution hazard, which is useful because the cause-specific hazard approach does not necessarily reflect what occurs with the _{sd}CIFs (16). This occurs because the _{sd}CIF is a function of the cause-specific hazards for both events 1 and 2 (29, 30) (Web supplement). The proportional subdistribution hazards model is then:

(4)

where λ_{0j} is the unspecified baseline subdistribution hazard. The proportionality assumption may be assessed by plotting the log(–log(1 – *F _{j}^{*}*(

The interpretation of _{sd}RH* _{j}* = exp(

The relation between the _{cs}RH_{j}_{=1} and _{sd}RH_{j}_{=1} is a function of the _{cs}RH for the competing event (_{cs}RH_{j}_{=2}), the unspecified baseline cause-specific hazard for both events (*h*_{01}(*t*) and *h*_{02}(*t*)), and time (refer to Appendix):

(5)

Therefore, a situation in which the _{cs}RH_{j}_{=1}=_{sd}RH_{j}_{=1} is when *h*_{02}=0. This also suggests that the _{cs}RH_{j}_{=1} will be similar to _{sd}RH_{j}_{=1} when *h*_{02} is small but that generally _{cs}RH* _{j}* ≠

Given that the _{cs}RH* _{j}* and

In contrast, the _{sd}RH is useful for comparing the cumulative incidence for those with and without exposure because of the direct modeling of the _{sd}CIF. For instance, a situation could arise where the _{cs}RH_{j}_{=1} = 1, suggesting no difference in the cause-specific hazard rate comparing exposed versus unexposed individuals. However, because the exposed individuals are more likely to have the competing event (_{cs}RH_{j}_{=2} > 1), the _{sd}RH_{j}_{=1} will be <1 (Table 2) because of the differential modification of the risk sets as caused by the association between exposure and the competing event. This drives the subdistribution hazard lower for those with exposure relative to unexposed individuals, causing _{sd}RH_{j}_{=1} < 1. While _{cs}RH_{j}_{=1} = 1 suggests no association, exposed individuals will be less likely to have the event because of the association of the exposure with the competing event. Therefore, the _{cs}RH_{j}_{=1} directly measures the association of an exposure on event 1 as the competing event contributes only passively by removing individuals from the risk set, whereas the _{sd}RH_{j}_{=1} is a measure of association that reflects both the association of exposure with event 1 and the contribution of event 2 by actively maintaining individuals in the risk sets for exposed and unexposed individuals. Should the association of the exposure with event 1 be in direct opposition with the contribution of event 2, the _{sd}RH may be quite different from the _{cs}RH (Table 2).

General Direction of the Time-averaged Subdistribution Relative Hazard for a Given Direction of the Time-averaged Cause-specific Relative Hazards for Both Events 1 and 2^{a}

A caveat when applying the _{sd}CIF to other populations is that the transportability of the estimate may be questionable if the distribution of the competing events differs from the original population. This is because the risk sets for exposed and unexposed individuals will be impacted differently by a change in the distribution of competing events.

The 2 primary models developed for estimating _{cs}RH and _{sd}RH depend upon proportional hazard assumptions. Because equations 1 and 3 are not equivalent, a proportional cause-specific hazards model does not necessarily imply a proportional subdistribution hazards model (29, 30). Although time interactions could be included in the model to account for nonproportionality, this can complicate interpretation.

An alternative approach is to consider more general models that do not constrain any of the hazard functions to be proportional. A mixture of distributions for competing risks was proposed by Cox in 1959 (31) and was later expanded through decomposing the _{sd}CIF as follows (17):

(6)

and constructing likelihood contributions for the *i*th individual:

(7)

where for *j*=1, 2 corresponds to a probability density function to model the *j*th event, *S _{j}*(

Under this formulation, parametric distributions can be utilized to impose structure for *f* and π with parameters that can be linked to covariates. To model *f* (and *S*), a flexible parametric distribution, such as the generalized gamma distribution, can accommodate various shapes of the hazard function (32). A binary model can be constructed for the *P*(*J*=*j*) term to describe the occurrence for 2 events. Regression analysis can proceed by linking covariates to the parameters of these distributions (Web supplement).

This mixture model approach has a distinct advantage over other models: Both the cause-specific and subdistribution relative hazards _{cs}RH* _{j}* and

Prior studies have shown that HIV-infected individuals with past injection drug use are less likely to initiate effective therapy than those without (35–38) and are more likely to die in the era of highly active antiretroviral therapy (5, 38, 39). Yet, the comparison of treatment initiation by history of injection drug use when it has the potential to be the most effective (prior to AIDS or death) has not been undertaken.

The Women's Interagency HIV Study (WIHS) was established in August 1993 to investigate the impact of HIV infection on US women at 6 sites in New York (2 sites); Washington, DC; Los Angeles and San Francisco, California; and Chicago, Illinois. Details are provided elsewhere (40–43). In 1994–1995, 2,054 HIV-positive and 569 HIV-negative women were enrolled. Follow-up visits occur at 6-month intervals in which data are collected by structured interviews, physical examinations, and laboratory testing.

The study sample consisted of 1,164 women enrolled in WIHS, who were alive, infected with HIV, and free of clinical AIDS on December 6, 1995 (baseline), when the first protease inhibitor (saquinavir mesylate) was approved by the Federal Drug Administration. Women were followed until the first of the following: treatment initiation, AIDS diagnosis, death, or administrative censoring (September 28, 2006). Covariates included history of injection drug use at WIHS enrollment, whether an individual was African American, age, and CD4 nadir prior to baseline.

Individuals with and without an injection drug use history had similar nadir CD4 counts prior to baseline (Table 3). Women with an injection drug use history were more likely to be African American and older than those without an injection drug use history. Although the majority of women initiated treatment prior to clinical AIDS or death, this proportion was lower among those with a history of injection drug use. The proportion with AIDS or death prior to treatment was higher among those with injection drug use.

Characteristics for Women Enrolled in the Women's Interagency HIV Study on December 6, 1995, and Followed Through September 2006, United States

Figure 3 shows both the estimated cause-specific and subdistribution cumulative incidences by outcome and injection drug use status. To illustrate the difference between the cause-specific and _{sd}CIFs, we estimated the _{cs}CIF directly from the _{cs}RH under the assumption of independence between events (Figure 3, A and C). However, the _{sd}CIF (Figure 3, B and D) can be estimated from the cause-specific proportional hazards model by taking extra steps (Web supplement) (27, 28). In addition, a nonparametric estimation of the subdistribution CIF was obtained by using an extension of the Kaplan-Meier methods to competing risks (2, 44). Regardless of the method (cause-specific proportional hazards model, subdistribution proportional hazards model, or mixture model) used for obtaining the subdistribution CIF, the estimated subdistributions were essentially equivalent to the extended Kaplan-Meier method for competing risks (*P* < 0.001 for both events comparing those with past injection drug use vs. those without) (2, 44).

Cumulative incidence of treatment initiation prior to acquired immunodeficiency syndrome (AIDS) or death (A and B) and the cumulative incidence of AIDS or death prior to treatment (C and D) by injection drug use status and type of cumulative incidence **...**

The estimates of the _{cs}RH and _{sd}RH from the semiparametric and parametric approaches are shown in Table 4 stratified by competing events. The parametric mixture model provided inferences essentially identical to the proportional hazards models. The _{sd}RH had a stronger association than the _{cs}RH for both events. The _{cs}RH_{treatment} was equal to 0.67; however, the _{cs}RH_{AIDS/death} was 1.7. Therefore, as the subdistribution hazard maintains individuals who develop the competing event in the risk set (equation 3), this implies that individuals with an injection drug use history are maintained in the risk set in a greater proportion than those without a history of injection drug use. Thus, a greater relative change between the cause-specific and subdistribution hazards would be expected for those with an injection drug use history than for those without (i.e., a larger denominator among the injection drug use group because of a higher AIDS/death hazard rate). Therefore, the _{sd}RH_{treatment} should be less than the _{cs}RH_{treatment}, which was observed (Table 4).

In this paper, we have discussed the 2 common methods for handling competing risks and their applications to regression settings. The _{cs}RH and the _{cs}CIF are familiar quantities because they reflect measures that are estimated when individuals with the competing event are censored. However, we have illustrated the utility of the subdistribution hazard and CIF as complementary measures of risk.

Should the _{cs}RH_{AIDS/death} have been greater (e.g., 3.0), the arbitrary baseline hazard for AIDS/death > 0.2 (e.g., a constant 1.1) per year, and the observed _{cs}RH_{treatment} = 0.67, then the _{sd}RH_{treatment} would have been lower than the 0.67. This would imply that, despite a direct association between injection drug use status and treatment initiation (_{cs}RH_{treatment} = 0.67), individuals with an injection drug use (IDU) history were less likely to initiate treatment before disease progression (_{sd}CIF_{IDU} < _{sd}CIF_{not-IDU} as indicated by _{sd}RH_{treatment} < 1.0) and more likely to have HIV disease progression before therapy. However, the _{cs}RH_{treatment} = 0.67 and the _{cs}RH_{AIDS/death} = 1.71, which suggests that AIDS/death should contribute to an even lower _{sd}RH_{treatment} because those with past injection drug use had a higher cause-specific hazard rate for AIDS/death. The similarity of the _{cs}RH_{treatment} and _{sd}RH_{treatment} implies that disease progression to AIDS/death did not greatly contribute to a further reduction in the association between injection drug use history and treatment initiation. This was due to the relatively low baseline hazard rate for AIDS/death; *h*_{02}(*t*) ranged from 0.157 to 0.224 per year. Thus, the _{sd}RH_{treatment} is only slightly stronger than that from the cause-specifc proportional hazards model (0.60 vs. 0.67, respectively). Beyersmann et al. (29) recently provide an alternative example where the difference between _{cs}RH and _{sd}RH is large and they are in opposite directions.

The properties of the _{cs}RH* _{j}* (no interpretation to

For the second question, the _{sd}RH is a better measure of association. This can be illustrated by assuming that abacavir is not directly associated with myocardial infarction (_{cs}RH = 1 for association of abacavir with myocardial infarction). It remains possible that investigators may still expect a higher probability of myocardial infarction among those taking abacavir if individuals not on abacavir were more likely to die prior to a myocardial infarction. Consequently, the _{sd}RH for myocardial infarction would be >1 for those on abacavir, but it is by reducing mortality and keeping individuals alive to be able to experience a myocardial infarction. The latter knowledge may be useful in policy decisions.

We recognize that important issues such as left truncation and causality (46) as they pertain to competing risks have not been addressed here. Our goals were to describe and illustrate 2 common measures of association that may be used in the competing risk setting but that epidemiologists have avoided. The cause-specific hazard ratio and subdistribution hazard ratio are distinct, and the choice of approach should be driven by the scientific question. Future research should continue to explore the differences in approaches and expand the tools to understand and implement competing risk methods for epidemiologic data.

Author affiliations: Department of Medicine, Johns Hopkins School of Medicine, Baltimore, Maryland (Bryan Lau); Department of Epidemiology, Johns Hopkins Bloomberg School of Public Health, Baltimore, Maryland (Bryan Lau, Stephen R. Cole, Stephen J. Gange); and Department of Epidemiology, University of North Carolina, Chapel Hill, North Carolina (Stephen R. Cole).

This research was supported by the National Institutes of Health (K01-AI071754 for Dr. Lau; U01-AI069918 for the North American AIDS Cohort Collaboration on Research and Design that is a part of the International Epidemiologic Databases to Evaluate AIDS (IEDEA); and U01-AI-42590 for the Women's Interagency HIV Study).

The authors thank Dr. Michael Silverberg for his insightful comments and suggestions.

The funding sources have had no involvement with this manuscript.

Conflict of interest: none declared.

- AIDS
- acquired immunodeficiency syndrome
- CIF
- cumulative incidence function
_{cs}CIF- cause-specific cumulative incidence function
_{cs}RH- cause-specific relative hazard
- HIV
- human immunodeficiency virus
_{sd}CIF- subdistribution cumulative incidence function
_{sd}RH- subdistribution relative hazard
- WIHS
- Women's Interagency HIV Study

This appendix further details the relation between the cause-specific hazard and the subdistribution hazard. Further details regarding the methods outlined within the main text may be found in the Web supplement data, which provide more rigorous details regarding the methods that may be useful for some readers but felt to be too technical such that the main points would be obscured to others. Additionally, the data used in the application are provided a long with code to implement competing risk analyses in R or SAS.

Beyersmann et al. (29) noted that the _{cs}RH is in good agreement with the _{sd}RH when there is no association of exposure and the competing event. To illustrate, let *X* be a binary exposure variable for 2 competing events. Let the _{cs}RH(*t*) for event 1 and event 2 be equal to some constant, _{cs}RH_{1} and _{cs}RH_{2}, respectively, and thus both events have proportional hazards across exposure status. Let the arbitrary baseline cause-specific hazard (i.e., when *X* = 0) for event 1 and event 2 be *h*_{01}(*t*) and *h*_{02}(*t*), respectively. Then, the hazards for those with *X* = 1 are *h*_{11}(*t*) = *h*_{01}(*t*) exp(β*X*) = *h*_{01}(*t*) × _{cs}RH_{1} and *h*_{12}(*t*) = *h*_{02}(*t*) exp(β*X*) = *h*_{02}(*t*) × _{cs}RH_{2} for events 1 and 2, respectively. Let λ_{1}(*t*) be the subdistribution hazard for event 1 and λ_{01}(*t*) and λ_{11}(*t*) be the subdistribution hazard for unexposed and exposed individuals, respectively. Beyersmann et al. (29) showed that the cause-specific hazard has the following general relation (not considering covariate *X*) with the subdistribution hazard for event 1:

(A1)

where ${F}_{2}^{\mathrm{*}}\left(t\right)$ is the subdistribution function for event 2, and *S*(*t*) is the net survival function. Thus, the cause-specific hazard for exposed individuals may be written as follows:

(A2)

Thus, the _{cs}RH_{1} is as follows:

(A3)

Note that the subdistribution equals the net survival multiplied by the cause-specific hazard (i.e., ${F}_{02}\left(t\right)={\displaystyle {\int}_{0}^{t}S\left(u|X=0\right){h}_{02}\left(u\right)\text{\hspace{0.17em}}du\text{\hspace{0.17em}and\hspace{0.17em}}{F}_{12}\left(t\right)}={\displaystyle {\int}_{0}^{t}S\left(u|X=1\right){h}_{12}\left(u\right)du}$). Thus, when *h*_{02}(*t*) is close to 0, the fractions within the parentheses in the numerator and in the denominator both tend toward 0 and _{cs}RH_{1}=_{sd}RH_{1}. Therefore, a low cause-specific hazard for the competing event can mitigate the effect of a large _{cs}RH_{2} that would contribute to the numerator in both the subdistribution, ${F}_{12}^{*}\left(t\right)$ and net survival among exposed individuals, *S*(*t*|*X*=1). Additionally, Latouche et al. (47) showed through simulation the _{cs}RH_{1}≈_{sd}RH_{1} when _{cs}RH_{2}=1.

1. Crowder MJ. Classical Competing Risks. Boca Raton, FL: Chapman & Hall/CRC; 2001.

2. Kalbfleisch JD, Prentice RL. The Statistical Analysis of Failure Time Data. New York, NY: John Wiley & Sons, Inc; 1980.

3. Pintilie M. Competing Risks: A Practical Perspective. Chichester, England: John Wiley & Sons, Ltd; 2006.

4. Babiker A, Darbyshire J, Pezzotti P, 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. 2002;31(5):951–958. [PubMed]

5. Lau B, Gange SJ, Moore RD. Risk of non-AIDS-related mortality may exceed risk of AIDS-related mortality among individuals enrolling into care with CD4+ counts greater than 200 cells/mm^{3}. J Acquir Immune Defic Syndr. 2007;44(2):179–187. [PubMed]

6. Tsai HK, D'Amico AV, Sadetsky N, et al. Androgen deprivation therapy for localized prostate cancer and the risk of cardiovascular mortality. J Natl Cancer Inst. 2007;99(20):1516–1524. [PubMed]

7. Xue QL, Fried LP, Glass TA, et al. Life-space constriction, development of frailty, and the competing risk of mortality: the Women's Health And Aging Study I. Am J Epidemiol. 2008;167(2):240–248. [PubMed]

8. Prentice RL, Kalbfleisch JD, Peterson AV, Jr, et al. Analysis of failure times in presence of competing risks. Biometrics. 1978;34(4):541–554. [PubMed]

9. Seal HL. Studies in the history of probability and statistics. XXXV: Multiple decrements or competing risks. Biometrika. 1977;64(3):429–439.

10. Gaynor JJ, Feuer EJ, Tan CC, et al. On the use of cause-specific failure and conditional failure probabilities—examples from clinical oncology data. J Am Stat Assoc. 1993;88(422):400–409.

11. Korn EL, Dorey FJ. Applications of crude incidence curves. Stat Med. 1992;11(6):813–829. [PubMed]

12. Lin DY. Non-parametric inference for cumulative incidence functions in competing risks studies. Stat Med. 1997;16(8):901–910. [PubMed]

13. Pepe MS, Mori M. Kaplan-Meier, marginal or conditional-probability curves in summarizing competing risks failure time data. Stat Med. 1993;12(8):737–751. [PubMed]

14. Ghani AC, Donnelly CA, Cox DR, et al. Methods for estimating the case fatality ratio for a novel, emerging infectious disease. Am J Epidemiol. 2005;162(5):479–486. [PubMed]

15. Satagopan JM, Ben-Porat L, Berwick M, et al. A note on competing risks in survival data analysis. Br J Cancer. 2004;91(7):1229–1235. [PMC free article] [PubMed]

16. Fine JP, Gray RJ. A proportional hazards model for the subdistribution of a competing risk. J Am Stat Assoc. 1999;94(446):496–509.

17. Larson MG, Dinse GE. A mixture model for the regression-analysis of competing risks data. J R Stat Soc Ser C Appl Stat. 1985;34(3):201–211.

18. Latouche A, Beyersmann J, Fine JP. Comments on ‘Analysing and interpreting competing risk data.’ Stat Med. 2007;26(19):3676–3679. [PubMed]

19. Chiang CL. Competing risks and conditional probabilities. Biometrics. 1970;26(4):767–776.

20. Lunn M, McNeil N. Applying Cox regression to competing risks. Biometrics. 1995;51(2):524–532. [PubMed]

21. Breslow NE. Contribution to the discussion of the paper by D. R. Cox. J R Stat Soc Ser B Stat Methodol. 1972;34(2):187–220.

22. Hanley JA. The Breslow estimator of the nonparametric baseline survivor function in Cox's regression model—some heuristics. Epidemiology. 2008;19(1):101–102. [PubMed]

23. Pintilie M. Analysing and interpreting competing risk data. Stat Med. 2007;26(6):1360–1367. author reply 3523. [PubMed]

24. Wolbers M, Koller M. Comments on ‘Analysing and interpreting competing risk data’ (original article and author's reply) Stat Med. 2007;26(18):3521–3523. [PubMed]

25. Andersen PK, Abildstrom SZ, Rosthøj S. Competing risks as a multi-state model. Stat Methods Med Res. 2002;11(2):203–215. [PubMed]

26. Cornfield J. The estimation of the probability of developing a disease in the presence of competing risks. Am J Public Health Nations Health. 1957;47(5):601–607. [PubMed]

27. Andersen PK. Statistical Models Based on Counting Processes. New York, NY: Springer-Verlag; 1993.

28. Cheng SC, Fine JP, Wei LJ. Prediction of cumulative incidence function under the proportional hazards model. Biometrics. 1998;54(1):219–228. [PubMed]

29. Beyersmann J, Dettenkofer M, Bertz H, et al. A competing risks analysis of bloodstream infection after stem-cell transplantation using subdistribution hazards and cause-specific hazards. Stat Med. 2007;26(30):5360–5369. [PubMed]

30. Beyersmann J, Schumacher M. Misspecified regression model for the subdistribution hazard of a competing risk. Stat Med. 2007;26(7):1649–1651. [PubMed]

31. Cox DR. The analysis of exponentially distributed lifetimes with 2 types of failure. J R Stat Soc Series B Stat Methodol. 1959;21(2):411–421.

32. Cox C, Chu H, Schneider MF, et al. Parametric survival analysis and taxonomy of hazard functions for the generalized gamma distribution. Stat Med. 2007;26(23):4352–4374. [PubMed]

33. Efron B. The Jackknife, the Bootstrap, and Other Resampling Plans. Philadelphia, PA: Society for Industrial and Applied Mathematics; 1982.

34. Lau B, Cole SR, Moore RD, et al. Evaluating competing adverse and beneficial outcomes using a mixture model. Stat Med. 2008;27(21):4313–4327. [PMC free article] [PubMed]

35. Celentano DD, Vlahov D, Cohn S, et al. Self-reported antiretroviral therapy in injection drug users. JAMA. 1998;280(6):544–546. [PubMed]

36. Celentano DD, Galai N, Sethi AK, et al. Time to initiating highly active antiretroviral therapy among HIV-infected injection drug users. AIDS. 2001;15(13):1707–1715. [PubMed]

37. Lucas GM, Cheever LW, Chaisson RE, et al. Detrimental effects of continued illicit drug use on the treatment of HIV-1 infection. J Acquir Immune Defic Syndr. 2001;27(3):251–259. [PubMed]

38. Rodriguez-Arenas MA, Jarrin I, del Amo J, et al. Delay in the initiation of HAART, poorer virological response, and higher mortality among HIV-infected injecting drug users in Spain. AIDS Res Hum Retroviruses. 2006;22(8):715–723. [PubMed]

39. Keiser O, Taffé P, Zwahlen M, et al. All cause mortality in the Swiss HIV Cohort Study from 1990 to 2001 in comparison with the Swiss population. AIDS. 2004;18(13):1835–1843. [PubMed]

40. Bacon MC, von Wyl V, Alden C, et al. The Women's Interagency HIV Study: an observational cohort brings clinical sciences to the bench. Clin Diagn Lab Immunol. 2005;12(9):1013–1019. [PMC free article] [PubMed]

41. Barkan SE, Melnick SL, Preston-Martin S, et al. The Women's Interagency HIV Study. WIHS Collaborative Study Group. Epidemiology. 1998;9(2):117–125. [PubMed]

42. Golub ET, Benning L, Sharma A, et al. Patterns, predictors, and consequences of initial regimen type among HIV-infected women receiving highly active antiretroviral therapy. Clin Infect Dis. 2008;46(2):305–312. [PubMed]

43. Hessol NA, Kalinowski A, Benning L, et al. Mortality among participants in the Multicenter AIDS Cohort Study and the Women's Interagency HIV Study. Clin Infect Dis. 2007;44(2):287–294. [PubMed]

44. Gray RJ. A class of K-sample tests for comparing the cumulative incidence of a competing risk. Ann Stat. 1988;16(3):1141–1154.

45. D:A:D Study Group. Sabin CA, Worm SW, et al. Use of nucleoside reverse transcriptase inhibitors and risk of myocardial infarction in HIV-infected patients enrolled in the D:A:D study: a multi-cohort collaboration. Lancet. 2008;371(9622):1417–1426. [PMC free article] [PubMed]

46. Robins JM, Greenland S. Causal inference without counterfactuals—comment. J Am Stat Assoc. 2000;95(450):431–435.

47. Latouche A, Boisson V, Chevret S, et al. Misspecified regression model for the subdistribution hazard of a competing risk. Stat Med. 2007;26(5):965–74. [PubMed]

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