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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 (csRH) and the subdistribution relative hazard (sdRH). 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 (csRH=0.67, 95% confidence interval: 0.57, 0.80 and sdRH=0.60, 95% confidence interval: 0.50, 0.71). Moreover, the relative hazards for disease progression prior to treatment were elevated (csRH=1.71, 95% confidence interval: 1.37, 2.13 and sdRH=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 (csRH) and the subdistribution relative hazard (sdRH), have not been well described in the epidemiology literature (18).
The purpose of this paper is 3-fold. First, we provide intuition for the csRH and sdRH 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(T1, T2, C), where T1 and T2 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 csRH may be better suited for studying the etiology of diseases, whereas the sdRH 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 csRHadverse>csRHcure≥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 –1event 1 – 1event 2. Thus, individuals with an event 1 or event 2 prior to time t are excluded from the risk set at time t.
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 , where J = j indicates whether event 1 (j = 1) or event 2 (j=2) is being estimated.
where fj*(t)=P(T=t, J=j) is a “sub”-density function (“*” indicates an improper, i.e., “sub”-density function that integrates to <1), and S(t) reflects the net survival function of both events 1 and 2, that is, , where h(t) is the net hazard for having either event 1 or event 2 (8).
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:
where h0j is the arbitrary baseline cause-specific hazard, and βj, j=1, 2 are the corresponding regression coefficients, where exp(βj)=csRHj is interpretable as the relative change in the cause-specific hazard for the jth event corresponding to a 1-unit increase in the corresponding covariate. No assumptions of the relation between the competing outcomes are needed for estimation (2, 8). Estimation may be accomplished by using standard software. A proportional hazards model is constructed separately for each event type in which individuals who experience the competing event are treated as censored observations. Because the likelihood may be written such that the competing event is treated as a censored event, this proportional hazards model is exactly the same as what some investigators model when “ignoring” competing events. Alternatively, rather than separate models, a joint model could be used (20) (refer to Web supplement).
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 csRHj=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 csRH 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 (csCIF), the competing risk literature has focused on an alternative measure of risk: the subdistribution cumulative incidence function (sdCIF). This function is defined as the joint probability of an event prior to time t and that the event is of type j: Fj*(t)=P(T<t, J=j). Although the sdCIF may be estimated from the csRHj=1, extra steps are required as the sdCIF is a function of the net survivor function and therefore directly impacted by the competing event (27, 28). The sdCIF may be modeled directly.
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.
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 . In continuous time, the subdistribution hazard is the following (16):
where Fj*(t)=P(T<t, J=j), Sj*(t)=P(T>t, J=j), and are the subdistribution cumulative incidence, subsurvivor, and subdensity functions (note that P(J=j)=Sj*(t)+Fj*(t)).
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 sdCIFs (16). This occurs because the sdCIF 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:
where λ0j is the unspecified baseline subdistribution hazard. The proportionality assumption may be assessed by plotting the log(–log(1 – Fj*(t)) against log(time) stratified by the covariate, where Fj*(t) can be estimated from a nonparametric estimator for competing risks (2, 10, 14, 15). In the presence of noninformative censoring, it has been recommended to use a weighted score function to obtain an unbiased estimating equation from the partial likelihood (Web supplement) (16). This has been implemented in the CMPRSK library in the R statistical program.
The interpretation of sdRHj = exp(j) is the relative change in the subdistribution hazard for a 1-unit increase in the corresponding covariate. The sdRHj is directly interpretable as a measure of association for the jth sdCIF, and it is straightforward to estimate the subdistribution cumulative incidence by using a Breslow-type estimator to obtain the cumulative subdistribution hazard and evaluate 1 – exp(cumulative subdistribution hazard) (16).
The relation between the csRHj=1 and sdRHj=1 is a function of the csRH for the competing event (csRHj=2), the unspecified baseline cause-specific hazard for both events (h01(t) and h02(t)), and time (refer to Appendix):
Therefore, a situation in which the csRHj=1=sdRHj=1 is when h02=0. This also suggests that the csRHj=1 will be similar to sdRHj=1 when h02 is small but that generally csRHj ≠ sdRHj. When csRHj=2 ≠ 1, the risk sets for the event of interest among exposed and unexposed individuals are modified differentially. When csRHj=2>1, a larger proportion of the risk set (for event J=1) for exposed compared with unexposed individuals have had the competing event (and vice versa for csRHj=2<1). Consequently, the ratio between the subdistribution hazards for exposed and unexposed individuals for the event of interest will not be equivalent to the csRH.
Given that the csRHj and sdRHj are generally different, how do we use these measures (Table 1)? In noncompeting risk settings, the impact of a high (low) relative hazard will directly translate to an increase (decrease) in cumulative incidence of the event for the exposed individuals as compared with unexposed individuals. In a competing risk framework, this is not necessarily true for the csRHj. The csRH is a measure of association that does not necessarily directly translate into a measure of risk without the assumption of independence between the competing events. Without the assumption of independence or conducting extra steps to obtain the sdCIF (Web supplement), the csRH does not allow comparison of the cumulative incidence of the event in exposed versus unexposed individuals. Rather, the csRH is a valid measure of the apparent effect of a covariate on the relative instantaneous hazard rate given that individuals have survived both events until time t. However, in that same instant, individuals may have a stronger (or weaker) relative hazard rate for the competing event.
In contrast, the sdRH is useful for comparing the cumulative incidence for those with and without exposure because of the direct modeling of the sdCIF. For instance, a situation could arise where the csRHj=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 (csRHj=2 > 1), the sdRHj=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 sdRHj=1 < 1. While csRHj=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 csRHj=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 sdRHj=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 sdRH may be quite different from the csRH (Table 2).
A caveat when applying the sdCIF 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 csRH and sdRH 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 sdCIF as follows (17):
and constructing likelihood contributions for the ith individual:
where for j=1, 2 corresponds to a probability density function to model the jth event, Sj(t) is the corresponding survivor function P(T>t|J=j), πi is the mixture probability P(J=1), and γi and θi are indicator functions for J=1 and J=2, respectively.
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 csRHj and sdRHj may be derived and are not constrained to be constant over time. If a summary (over time) measure is desired, a time-weighted estimate can be constructed with confidence intervals obtained by bootstrap (33). Another advantage of the mixture model is that it is relatively easy to compare the subdistribution CIF, cause-specific hazards, or subdistribution hazards stratified by exposure and over time (34). Estimation of the parameters for the mixture model can be performed in SAS software by using the NLMIXED procedure and the log-likelihood function from equation 7.
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.
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 sdCIFs, we estimated the csCIF directly from the csRH under the assumption of independence between events (Figure 3, A and C). However, the sdCIF (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).
The estimates of the csRH and sdRH 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 sdRH had a stronger association than the csRH for both events. The csRHtreatment was equal to 0.67; however, the csRHAIDS/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 sdRHtreatment should be less than the csRHtreatment, 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 csRH and the csCIF 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 csRHAIDS/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 csRHtreatment = 0.67, then the sdRHtreatment would have been lower than the 0.67. This would imply that, despite a direct association between injection drug use status and treatment initiation (csRHtreatment = 0.67), individuals with an injection drug use (IDU) history were less likely to initiate treatment before disease progression (sdCIFIDU < sdCIFnot-IDU as indicated by sdRHtreatment < 1.0) and more likely to have HIV disease progression before therapy. However, the csRHtreatment = 0.67 and the csRHAIDS/death = 1.71, which suggests that AIDS/death should contribute to an even lower sdRHtreatment because those with past injection drug use had a higher cause-specific hazard rate for AIDS/death. The similarity of the csRHtreatment and sdRHtreatment 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; h02(t) ranged from 0.157 to 0.224 per year. Thus, the sdRHtreatment 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 csRH and sdRH is large and they are in opposite directions.
The properties of the csRHj (no interpretation to sdCIF without assumption) and sdRHj (translatable to sdCIF) illustrate the circumstances in which the 2 measures of association may be most useful and therefore suggest a general guideline for use. The csRH might be more applicable for studying the etiology of diseases, whereas the sdRH might be more appropriate for predicting an individual's risk for an outcome or resource allocation. For example, the use of the antiretroviral drug abacavir has recently been associated with increased risk of myocardial infarction (45). Two competing questions can be framed: 1) Is the use of abacavir directly associated with myocardial infarction, and 2) regardless of the direct association, are individuals taking abacavir more likely to experience a myocardial infarction? For the first question, the csRH may be more appropriate, as this measure will assess at any given time whether the individuals on abacavir have an increased instantaneous hazard rate for myocardial infarction among all individuals that have survived all events to this time point.
For the second question, the sdRH is a better measure of association. This can be illustrated by assuming that abacavir is not directly associated with myocardial infarction (csRH = 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 sdRH 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.
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 csRH is in good agreement with the sdRH 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 csRH(t) for event 1 and event 2 be equal to some constant, csRH1 and csRH2, 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 h01(t) and h02(t), respectively. Then, the hazards for those with X = 1 are h11(t) = h01(t) exp(βX) = h01(t) × csRH1 and h12(t) = h02(t) exp(βX) = h02(t) × csRH2 for events 1 and 2, respectively. Let λ1(t) be the subdistribution hazard for event 1 and λ01(t) and λ11(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:
where 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:
Thus, the csRH1 is as follows:
Note that the subdistribution equals the net survival multiplied by the cause-specific hazard (i.e., ). Thus, when h02(t) is close to 0, the fractions within the parentheses in the numerator and in the denominator both tend toward 0 and csRH1=sdRH1. Therefore, a low cause-specific hazard for the competing event can mitigate the effect of a large csRH2 that would contribute to the numerator in both the subdistribution, and net survival among exposed individuals, S(t|X=1). Additionally, Latouche et al. (47) showed through simulation the csRH1≈sdRH1 when csRH2=1.