Accounting for Competing Events When Evaluating Long-Term Outcomes in Survivors of Critical Illness

Abstract

The clinical trajectory of survivors of critical illness after hospital discharge can be complex and highly unpredictable. Assessing long-term outcomes after critical illness can be challenging because of possible competing events, such as all-cause death during follow-up (which precludes the occurrence of an event of particular interest). In this perspective, we explore challenges and methodological implications of competing events during the assessment of long-term outcomes in survivors of critical illness. In the absence of competing events, researchers evaluating long-term outcomes commonly use the Kaplan-Meier method and the Cox proportional hazards model to analyze time-to-event (survival) data. However, traditional analytical and modeling techniques can yield biased estimates in the presence of competing events. We present different estimands of interest and the use of different analytical approaches, including changes to the outcome of interest, Fine and Gray regression models, cause-specific Cox proportional hazards models, and generalized methods (such as inverse probability weighting). Finally, we provide code and a simulated dataset to exemplify the application of the different analytical strategies in addition to overall reporting recommendations.

Survivors of episodes of critical illness including ICU admission experience a wide variety of long-term trajectories and outcomes (1–4). The clinical needs of these patients after hospital discharge can be complex and highly unpredictable (3, 5–9). Measuring long-term outcomes after critical illness can be challenging because the occurrence of one outcome may change the risk of a different, subsequent event and because of possible competing events during long-term follow-up (e.g., all-cause death) (10–15). Competing events are particularly relevant in the face of the high long-term mortality among critical care survivors—up to 50% at 1 year, depending on the underlying baseline characteristics and diagnosis (16–18).

In this article, we describe the challenges and methodological implications of competing events during the assessment of long-term outcomes in survivors of critical illness. To illustrate our points, we use a case study of the evaluation of the effect of noninvasive ventilation in adult patients with chronic obstructive pulmonary disease (COPD) (19, 20). We present the potential impact of competing events, discuss common statistical techniques that may be used in this setting, and outline approaches to their reporting.

Conceptual Frameworks

A competing event prevents either the occurrence or the observation of an outcome of particular interest (21, 22). For example, researchers may be interested in different mechanisms of death, where such mechanisms “compete” with each other for the actual cause of death in any given patient (referred to as “mutually exclusive absorbing states”). Alternatively, in an illness-death model, patients may have an initial state (e.g., survived an initial hospitalization episode) and then face the risk of experiencing either the endpoint of interest (e.g., rehospitalization) or the competing event (e.g., all-cause death) (23, 24).

A clear description of the target quantity (i.e., estimand) of interest (e.g., population, intervention and comparator, outcome, timing) is a prerequisite and will guide the choice of statistical technique (14, 25). For example, the exposure may have 1) a direct effect (e.g., not through the competing event), 2) an indirect effect (e.g., through the competing event), and 3) a total effect (e.g., combining both direct and indirect effects) on the outcome of interest. The direct and total effects generally represent estimands that may be of interest to researchers and clinicians. Hence, competing events have recently been conceptualized as a variable (i.e., mediator) in the causal pathway between exposure and outcome (14, 26, 27). More complex, multistate models that allow different states and the transitions between them may be useful to characterize the progression through different outcomes until the occurrence of an absorbing state (e.g., all-cause death) (28).

Figure 1 shows different frameworks that may be used when considering the occurrence of competing events. Table 1 describes different estimands of potential interest. The framework used and the estimand of interest will guide the analytical technique to be deployed when analyzing data with competing events. These include the use of composite outcomes, Fine and Gray regression models, cause-specific Cox models, and generalized methods. Table 2 summarizes the main analytical techniques and highlights published examples (6, 29, 30, 31).

Figure 1.

Conceptual frameworks for competing events. (A) A study in which different causes of death may be of interest in a classic example of mutually exclusive absorbing states. (B) A study in which a primary event is of interest, which may be subject to a competing event. (C) A simplified conceptual model for the association between an exposure and an outcome. Within this model, there is potential baseline confounding, and a competing event may occur before the endpoint of interest. In this model, the competing event may be thought of as a mediator in the pathway between exposure and outcome.

Table 1.

Main Target Quantities (Estimands) of Interest When Analyzing Data with Competing Events

Estimand	Description	Comments
Direct effect	Effect of an intervention on the outcome of interest not through the competing event (e.g., while removing the competing event). In our example, the effect of noninvasive ventilation on rehospitalizations among patients with COPD, while removing the competing event of all-cause death.	Difficult to conceptualize because it requires “removal” of the competing event. This is generally the target quantity in studies that censor patients when the competing event occurs.
Total effect	Effect of an intervention on the outcome of interest both directly and through the competing event. In our example, the effect of noninvasive ventilation on rehospitalization among patients with COPD, without removing the competing event of all-cause death.	Captures the effect of the intervention on the outcome through all pathways. Patients who experience the competing event are essentially free of the endpoint of interest.

Definition of abbreviation: COPD = chronic obstructive pulmonary disease.

Table 2.

Main Analytical Techniques to Deal with Competing Events during Long-Term Follow-Up of Critical Care Survivors

	Analytical Technique
	Combined Outcome	Fine and Gray Model	Cause-Specific Cox Model	Generalized Methods
How does it deal with competing events?	Outcome of interest and competing event combined as a single endpoint	Estimates subdistribution hazard ratios. Patients with competing events are returned to the risk set (i.e., not eliminated)	Estimates cause-specific hazard ratio in patients who remain free of the competing event (i.e., conditional on not having it)	G-formula or inverse probability weighting; may estimate effects while removing, or not, the competing event
When to use*	Interested in the combined outcome as a meaningful quantity	Interested in changes in incidence; time-fixed (exposure) questions†	Interested in changes in the hazard; time-fixed (exposure) questions†	Interested in a causal contrast with time-changing exposure, confounding, and selection bias
Better suited for questions about incidence (or risk; e.g., prognosis)	Yes	Yes	No; cumulative incidence function will not depend on a single cause-specific function	Yes; depending on choice of model
Main advantages	Easy to define and use; estimation of event-free survival time and total effects	Easy to use	Easy to use; risk set well defined	Can estimate both total and direct effects (e.g., not mediated)
Main disadvantages	Potentially difficult to interpret; essentially changes the outcome and research question	Risk set ill-defined; only conditional effects are estimated	Does not extrapolate to changes in incidence function; only conditional effects are estimated	Difficult interpretation (e.g., when “removing” the competing event)
Examples	Lamontagne et al. (29); effect of vitamin C on all-cause death or persistent organ dysfunction in adult patients with sepsis	Kamdar et al. (30); association between ARDS survivorship and joblessness at 12 mo	Fernando et al. (6); association of ICU survivorship with suicide and self-harm during long-term follow-up	Bagshaw et al. (32); association between an accelerated renal replacement strategy and dialysis dependence at 90 d

Definition of abbreviation: ARDS = acute respiratory distress syndrome.

“Fine and Gray model” refers to a model for the subdistribution hazard.

^*The use of specific techniques for causal questions depends on usual identifiability assumptions rather than the specific model.

^† For example, a time-invariant exposure measured at a single time point (and without treatment-confounder feedback).

Case Study: Noninvasive Ventilation in Patients with COPD

To illustrate the challenges presented by competing events, we use the evaluation of noninvasive ventilation and long-term outcomes in adult patients who were discharged after a critical care admission requiring mechanical ventilation for an acute exacerbation of COPD as a case study (19, 20). Adult patients with COPD who survive a hospitalization for an acute exacerbation face a number of potential measurable endpoints, including recurrent exacerbations, rehospitalizations, and death (32, 33). As an example of a potential study question, we can hypothesize that the risk of rehospitalization may be decreased by the use of noninvasive ventilation at home (20). However, studying this outcome after hospital discharge can be challenging because death, whether due to COPD or another cause, may act as a competing event during long-term follow-up (22).

In our example, the study population is composed of adult patients with COPD after hospital discharge; the exposure is the use of noninvasive ventilation at home (compared with the standard of care without noninvasive ventilation); and the outcome of interest is (time to) rehospitalization up to 5 years of follow-up (34). For ease of presentation, we have included several simplifications: 1) we pooled all causes of rehospitalization as an outcome, 2) we assumed that the use of noninvasive ventilation is a well-defined intervention, 3) we focused on the effect of a time-fixed exposure (that is, receipt or not of noninvasive ventilation at the start of follow-up, disregarding changes thereafter), and 4) we only consider baseline characteristics for the modeling of the competing event (15, 35). Death of any cause acts as a competing event during the period of long-term follow-up.

On the basis of the conceptual frameworks delineated above, this case may represent an illness-death model (see Figure E1 in the online supplement) where patients are discharged after surviving the initial hospitalization for an acute exacerbation and then face the risk of experiencing a rehospitalization (i.e., outcome of interest) or death (i.e., the competing event). Alternatively, the competing event of death could be considered as a mediator in the pathway between noninvasive ventilation and rehospitalizations (Figures E2 and E3). This pathway (i.e., arrow in the online supplement figures) is expected because patients must have avoided the competing event of all-cause death to experience a rehospitalization (26).

Potential Biases

In the absence of competing events, researchers evaluating long-term outcomes commonly use the Kaplan-Meier method and the Cox proportional hazards model to analyze time-to-event (survival) data (22, 36). In this setting, information on event occurrence and follow-up time is generally used to construct the survival and hazard functions (36). A key feature of these approaches is that the analysis must often be completed without knowledge of the outcome status for those patients who are “censored” (in our example, patients who do not experience a rehospitalization during the available follow-up time). For these censored participants, all that is known is that their event-free “survival time” is longer than their “time to censoring” (36). A key assumption is that those participants who remain at risk have the same underlying risk of the event as those who were censored (i.e., censoring is noninformative). As Figure E2 depicts, this assumption does not hold when competing events are present (i.e., arrow or association between all-cause death and rehospitalization). In the case of adult patients with COPD, patients who die cannot face a rehospitalization during follow-up and likely differ from those who remain in the study (e.g., they may have had a higher burden of disease and frailty and also possibly a different risk of rehospitalization).

If there are competing events, these traditional techniques to assess long-term outcomes can yield biased estimates. For example, for the COPD cohort, the Kaplan-Meier method will usually overestimate the incidence of rehospitalization. This is because the number of patients at risk of a rehospitalization is reduced by the occurrence of death (22, 37, 38). Moreover, Cox regression analyses will be “conditioned” on the competing event of death by restricting the analytical sample to those who remain alive, and these analyses may similarly yield biased effect estimates. Finally, estimating the change in the hazard of rehospitalization that is associated with the use of noninvasive ventilation will not necessarily apply to changes in the survival function (i.e., the one-to-one relationship between cumulative hazard and survival is lost when competing events are present). This is in contrast to the setting without competing events, where estimating the effect of a covariate on the hazard function allows us to understand the related effect on the survival function (22, 37, 38).

There are several analytical approaches that can be considered when analyzing data with competing events, including the use of 1) composite outcomes (e.g., incorporating the competing event as one additional component of a composite), 2) a Fine and Gray regression model, 3) cause-specific Cox proportional hazards regression models, and 4) generalized methods. We consider each in more detail below.

Accounting for Competing Events

Adequately accounting for potential competing events when assessing long-term outcomes in critical care survivors should be as important as considering the potential for confounding, information bias, and selection bias (14, 15, 39). Hence, it is fundamental to first 1) define the research question and population of interest, 2) choose the quantity to be estimated (i.e., estimand) that best fits the research question, and 3) choose the appropriate statistical method and consider if the estimate can be interpreted causally (14, 26, 27). Of note, whether an estimate can be interpreted causally will depend on the usual assumptions rather than the choice of analytical model to deal with competing events (14, 40). Such assumptions include that groups are comparable (e.g., no confounding; exchangeability) and absence of information and selection bias (15, 39, 41, 42).

Dataset Generation

Figure E2 shows a simplified conceptual model (see the directed acyclic graph in Figure E3) for the association between the use of noninvasive ventilation and rehospitalization in adult patients with COPD while taking into consideration potential baseline confounding and the competing event of all-cause death. These graphs can help suggest what covariates and modeling strategies are needed and also can show potential biases introduced by baseline confounding and highlight any potential for a competing event of death (43, 44). Tackling these two separate problems is essential to obtain an unbiased estimate of the effect of the exposure on the outcome of interest (45).

Using the directed acyclic graph shown in Figure E3 as a guide, we simulated a dataset of 10,000 adult patients who were discharged after a critical care admission for an acute exacerbation of COPD (46). We present the results of our hypothetical study using this dataset when the different analytical techniques described are used. As such, we present hazard ratios (HRs), cause-specific HRs, subdistribution HRs, and risk ratios alongside 95% confidence intervals (CIs) as appropriate (47). Details on data simulation and analysis are shown in the online supplement. Data simulation and analysis were performed in R version 4.1.0 (R Foundation). STATA (StataCorp) and SAS (SAS Institute Inc.) codes are also provided.

Briefly, the dataset contains 10,000 simulated observations, of which 47% of patients received noninvasive ventilation at home. Overall, the cumulative incidences at 5 years of rehospitalization and all-cause mortality were 18% and 25% respectively (Table E1).

Combined Outcomes

Outcomes with competing events can be combined into a composite endpoint if such a composite outcome still represents a meaningful quantity to clinicians and researchers. Situations in which the outcome of interest and its competing event are both undesirable may be best suited for this approach, because such a composite allows the calculation of the so-called event-free survival time (48). An example of a commonly used composite endpoint in the pulmonary and critical care literature is ventilator-free days, which incorporates a competing event of death into the measurement of time spent off a ventilator; detailed discussions can be found elsewhere (49). In our example, the competing event of death could be dealt with by combining rehospitalization and all-cause death into a composite outcome, and the occurrence of any of these would constitute an event. This overcomes the analytical problem posed by competing events (and may be preferable because it might be observed earlier than the standalone endpoint of interest), but it also makes the interpretation of the outcome more challenging, especially if an association is driven mostly by one component of the composite outcome, such as all-cause death.

In our case study, we would report a decrease in the hazard of the composite of rehospitalization or all-cause death with the use of noninvasive ventilation among adult patients with COPD (HR, 0.64; 95% CI, 0.60–0.68; Table E2). A published example of this technique can be found in an article by Lamontagne and colleagues (29).

Subdistribution Hazard Models

When competing events exist, the description of event occurrence is usually done via the cumulative incidence function for each outcome instead of the survival function. This event-specific distribution is also called the “subdistribution” and represents the cumulative proportion of failures due to a particular event (e.g., rehospitalization) at a specific time point (36). Fine and Gray subdistribution hazard models can thus estimate the association between noninvasive ventilation and the cumulative incidence function of rehospitalization. For this reason, they are sometimes also referred to as “cumulative incidence function regression models” (22, 36–38). In our case study, the exponentiated coefficient from the Fine and Gray model represents the relative change in the subdistribution hazard function associated with noninvasive ventilation (vs. standard of care). This change may in turn be interpreted in terms of increase or decrease of the cumulative incidence function (direction is preserved, although magnitude may not be) (22, 37, 38).

It is important to note that the estimates derived from the Fine and Gray model represent the outcome occurrence during the follow-up period, regardless of whether patients experienced the competing event (22). In this approach, patients who experienced the competing event remain in the risk set, and it is impossible for them to subsequently develop the outcome of interest. Formally, the quantity estimated represents the subdistribution hazard without elimination of competing events (26). The most widely used statistical packages can accommodate Fine and Gray and related models (e.g., stcrreg in STATA, PROC PHREG in SAS, package cmprsk in R).

Advantages of Fine and Gray models include their widespread availability in common statistical languages and their ability to estimate changes in the cumulative incidence function (as they reestablish the one-to-one relationship between the subdistribution hazard and the cumulative incidence). The latter feature may be especially desirable when translating study findings to the bedside (e.g., interested in prognosis). They also have disadvantages; subdistribution HRs can be difficult to interpret, these models generally cannot accommodate time-changing covariates (50), and the risk set is ill defined (and, although it reestablishes the one-to-one relationship between a hazard and the cumulative incidence, it includes patients who experienced the competing event but then become “immortal” for the event of interest) (51). Notably, the Fine and Gray model focuses on one event of interest, whereas the other subdistribution is not modeled, thus facing model misspecification (52).

In our case study, we would report a decrease in the subdistribution hazard of rehospitalization with the use of noninvasive ventilation among adult patients with COPD (subdistribution HR, 0.91; 95% CI, 0.82–1.02; Table E2). A published example of this technique can be found in an article by Kamdar and colleagues (30).

Cause-Specific Hazards Models

Cause-specific Cox proportional hazards models estimate the association between the exposure and the hazard of the event of interest (22, 37, 38). In these models, the exponentiated coefficient represents the relative change in the hazard of rehospitalization associated with noninvasive ventilation (vs. standard of care). In contrast to subdistribution hazard models, individuals are removed from the risk set when the event of interest or the competing event occurs. These models condition on the competing events (i.e., on them not occurring), and estimates apply to those patients who remain event free (26).

When using cause-specific hazards models, care should be taken to avoid equating changes in cause-specific HRs to changes in the incidence of the event of interest, because all that can be estimated are changes in the instantaneous rate of the occurrence rather than the cumulative incidence of the event (because the cumulative incidence function will depend on all the different cause-specific hazard functions) (37). Cause-specific hazards for the endpoint of interest and the competing event should be estimated. The most widely used statistical packages can accommodate cause-specific and related models (e.g., stcrreg in STATA, PROC PHREG in SAS, package survival in R).

Advantages of cause-specific Cox models include their widespread availability in commonly used statistical languages, the HRs have the usual interpretation in that they describe a change in the instantaneous rate of occurrence of the event, and the clinical interpretation that associations apply for those subjects who remain event free. In contrast, the main disadvantage is the inability to translate changes in the cause-specific hazard function to changes in the survival experience.

In our case study, we would report a decrease in the cause-specific hazard of rehospitalization with the use of noninvasive ventilation in adult patients with COPD (cause-specific HR, 0.75; 95% CI, 0.67–0.84; Table E2). A published example of this technique can be found in an article by Fernando and colleagues (6).

Generalized Methods

As shown in Figure 1 and Figure E2, the competing event of death can be thought of as a (potentially time-varying) covariate with shared causes with the outcome of interest (26). The effect of noninvasive ventilation on rehospitalization could be estimated while theoretically removing, or not, such a competing event (26). For example, if one were to somehow “remove” the competing event of all-cause death, then the estimation of the effect of noninvasive ventilation on rehospitalization among adult patients with COPD would represent a direct effect (i.e., not mediated through the competing event) (26, 53). Conversely, without the “removal” of all-cause death, the estimate would represent the total effect (which combines the effect of noninvasive ventilation on rehospitalization directly with the one through the competing event; Figure E2). Both the g-formula and inverse probability weighting can be used to estimate such contrasts (26).

The main advantage of generalized methods is their ability to handle time-varying exposures (with treatment-confounder feedback). Such time-varying exposures (e.g., in the form of either sustained or dynamic treatment strategies) are nearly always of interest but seldom evaluated in critical care research.

In our case study, when using inverse probability weighting of a Cox model to “remove” the competing event of all-cause death, we would report a decrease in the hazard of rehospitalization with the use of noninvasive ventilation among adult patients with COPD (HR, 0.70; 95% CI, 0.63–0.78; Table E2). This would represent a direct effect (i.e., not through the competing event); risks could also be estimated (Table E2). A published example of this technique can be found in an article by Bagshaw and colleagues (31).

Together, the results in the simulated data exemplify how different techniques yield different estimates, mainly owing to how they deal with the competing event. For example, the estimate from Fine and Gray models (where participants with the competing event represent “immortal time”) appears closer to the null than the one from a cause-specific Cox model or when using inverse probability weighting (Table E2), and they in turn differ (in magnitude and interpretation) from the one considering rehospitalization and all-cause death as a single composite.

Additional Methods

Several additional techniques may be of use when accounting for competing events. Additive models for the cumulative incidence function or the cause-specific hazards are readily available and may provide clinically relevant estimates (54). Doubly robust methods (i.e., using both models for the exposure and the outcome) may be deployed such that only one of the models needs to be adequately specified to obtain unbiased estimates (55). Multistate models may be a useful tool to depict outcome occurrences and how these change the expected survival experience and overall outcome pathway (13, 28, 56). When dealing with complex time-varying factors that may affect both the outcome of interest and the competing event, a joint model framework can also be considered (57). The estimation of separable effects may be useful when the intervention appears to have different causal pathways for the competing event and the endpoint of interest (58). Finally, decomposition of years lost by cause of death may be useful to estimate overall outcome burden (59).

Overall Recommendations

Several recommendations can be made when analyzing data with competing events (37, 47). First and foremost, reporting the intent of the research question (e.g., causal or prognostic) and the estimand of interest (e.g., direct or total effects; eliminating, conditioning on, or removing the competing event) remains paramount. Second, we suggest using cumulative incidence functions of the different types of outcomes of interest rather than the Kaplan-Meier estimator. Third, to show a complete picture of the potential impact on long-term outcomes, it may be useful to present both subdistribution and cause-specific hazards models and the association between the exposure and both the outcome of interest and the competing event.

Choosing the Best Approach

In general, because cause-specific Cox models can estimate changes in the hazard for the event of interest, current literature recommends their use when considering research questions of a causal or mechanistic nature (22, 37). Conversely, the use of Fine and Gray models is generally suggested for questions of prognosis because of their ability to translate to changes in the cumulative incidence function (37). However, whether a specific approach can be used to answer a causal question will depend mostly on groups being comparable (e.g., exchangeability) rather than the specific modeling technique (60). Moreover, even in the setting in which these assumptions appear to hold, both the subdistribution and cause-specific hazards model present with the usual perils of HRs when estimating causal effects (i.e., time-changing HRs through long follow-up periods and built-in selection bias) (61). We refer the reader to detailed discussions on these challenges that have been published elsewhere (14, 26, 61).

While considering these potential caveats, when one is interested in a time-fixed exposure (without treatment-confounder feedback), researchers may deploy conditional regression models. The specific choice of model can be further refined by the research question at hand and how the distinct ways to model the risk set fit with it. Alternatively, for causal questions not limited to the time-fixed exposure with baseline confounding setting, the deployment of generalized methods is generally required. Finally, when both the outcome of interest and the competing event are undesirable, there may be a high yield of using a combined endpoint (e.g., maximizing statistical power and simplifying the analytical approach required).