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A causal framework for classical statistical estimands in failure time settings with competing events

Jessica G. Young, Mats J. Stensrud, Eric J. Tchetgen Tchetgen, Miguel A. Hernán

arXiv:1806.06136v3stat.ME

TL;DR

Classical competing-risks estimands lacked a formal causal framework, complicating interpretation of estimates and analytic recommendations. Using counterfactual definitions, the paper shows that risk contrasts can represent total or direct effects depending on whether competing events are treated as censoring, whereas hazard contrasts generally lack causal interpretation.

  • Problem

    Classical competing-risks estimands had not been characterized within a formal causal framework, limiting interpretation of effect estimates and analytic recommendations.

    Source: §1 · p. 2

  • Method

    The paper uses counterfactual definitions and causal diagrams to formalize classical estimands and their identifying conditions, representing competing events as time-varying covariates.

    Source: §6.2 · p. 14, §10 · p. 27

  • Results

    Risk contrasts can represent total effects or direct effects depending on whether competing events are treated as censoring, while hazard contrasts generally lack causal interpretation.

    Source: §10 · pp. 27, 29

  • Takeaways & Limitations

    For total effects with loss to follow-up, competing events are time-varying covariates needed for exchangeability, even in randomized trials.

    Source: §10 · p. 27

    Identifying direct effects can require censoring independent of changing prognostic factors, an assumption often considered implausible when competing events are treated as censoring.

    Source: §6.1 · p. 14

Abstract

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In failure-time settings, a competing risk event is any event that makes it impossible for the event of interest to occur. For example, cardiovascular disease death is a competing event for prostate cancer death because an individual cannot die of prostate cancer once he has died of cardiovascular disease. Various statistical estimands have been defined as possible targets of inference in the classical competing risks literature. Many reviews have described these statistical estimands and their estimating procedures with recommendations about their use. However, this previous work has not used a formal framework for characterizing causal effects and their identifying conditions, which makes it difficult to interpret effect estimates and assess recommendations regarding analytic choices. Here we use a counterfactual framework to explicitly define each of these classical estimands. We clarify that, depending on whether competing events are defined as censoring events, contrasts of risks can define a total effect of the treatment on the event of interest, or a direct effect of the treatment on the event of interest not mediated through the competing event. In contrast, regardless of whether competing events are defined as censoring events, counterfactual hazard contrasts cannot generally be interpreted as causal effects. We illustrate how identifying assumptions for all of these counterfactual estimands can be represented in causal diagrams in which competing events are depicted as time-varying covariates. We present an application of these ideas to data from a randomized trial designed to estimate the effect of estrogen therapy on prostate cancer mortality.

1 Introduction

The introduction motivates a counterfactual framework for classical competing-risk estimands. It distinguishes risk contrasts, which can represent total or direct effects depending on whether competing events are treated as censoring, from hazard contrasts, which generally cannot be interpreted causally.

1 Introduction

  • A competing event makes the event of interest impossible; for example, cardiovascular disease death precludes later prostate cancer death.Because competing events cannot be prevented by design, they can occur in randomized and nonrandomized studies.Source: §1 · pp. 1–2
  • Prior literature describes classical competing-risk estimands and their estimating procedures, but lacks a formal framework for causal effects and identifying conditions.This gap complicates interpretation of effect estimates and assessment of recommendations about analytic choices.Source: §1 · p. 2
  • Risk contrasts can define total effects or direct effects not mediated through competing events, depending on whether competing events are defined as censoring.The framework gives two counterfactual definitions of risk and three of hazard.Source: §1 · p. 2
  • The framework represents identifying assumptions for all counterfactual estimands in causal diagrams depicting competing events as time-varying covariates.This addresses previous claims to the contrary.Source: §1 · p. 2

2 Observed data structure

The section defines the observed data for a randomized prostate cancer trial using repeated follow-up intervals, event indicators, and time-varying covariates. It assumes measured variables are error-free and follow-up is complete through Section 5.

2 Observed data structure

  • Participants are randomized at baseline to estrogen therapy or placebo, and follow-up tracks event-of-interest and competing-event indicators across equally spaced intervals.Both event indicators are zero at baseline; the target follow-up ends no later than the study’s administrative end.Source: §2 · p. 3
  • Time-varying characteristics are updated by interval, with baseline covariates measured before treatment assignment and within-interval ordering set as competing event, event of interest, then covariates.Variables are assumed measured without error; when covariates are not measured each interval, last values and time since measurement may be used [Hern´an et al., 2009].Source: §2 · p. 3
  • After a competing event occurs without a prior event of interest, all future event-of-interest indicators are known to be zero, and loss to follow-up is excluded until Section 5.The notation distinguishes variable histories through an interval from their future values over follow-up.Source: §2 · p. 3

3 Counterfactual estimands when competing events do not exist

Without competing events, potential outcomes define treatment-specific risks and hazards for the event of interest. Risk contrasts can represent causal effects, but hazard contrasts generally cannot because hazards condition on treatment-affected survival.

3 Counterfactual estimands when competing events do not exist

  • The simplified case without competing events applies when death from any cause is the event of interest; prostate cancer death with other-cause death as a competitor is considered next.Source: §3 · p. 3
  • Without competing events, each person’s potential outcome indicates whether the event occurs by a given interval under treatment assignment, defining population counterfactual risk.The population-level causal effect can be expressed as a contrast of risks, including on a non-additive scale.Source: §3 · p. 4
  • The discrete-time hazard is the interval-specific event probability conditional on survival to the interval’s start; it also represents a continuous-time hazard in the limit as interval length approaches zero.The hazard is defined whether the underlying counterfactual failure time is discrete or continuous.Source: §3 · p. 4
  • Unlike risk contrasts, counterfactual hazard contrasts generally are not causal effects because treatment can alter who survives to each interval [Hernán et al., 2004, Hernán, 2010].Hazards may differ because the groups surviving under each treatment contain different individuals; such contrasts can nevertheless be precisely defined and identifiable.Source: §3 · p. 4

4 Counterfactual estimands when competing events exist

The section maps counterfactual risks and hazards under competing-event interventions to classical estimands: risk contrasts can represent direct or total effects, whereas hazard contrasts generally are not causal.

4.1 Direct effects

  • Under hypothetical elimination of competing events, risk (net risk or marginal cumulative incidence [Geskus, 2016]) defines a controlled direct effect not mediated through competing events.The causal diagram allows mediation through the competing event and includes an arrow from that event to the event of interest because experiencing the competing event precludes the event of interest.Source: §4.1 · p. 6

4.2 Total effects

  • Without eliminating competing events, risk is the subdistribution function, cause-specific cumulative incidence, or crude risk [Kalbfleisch and Prentice, 1980, Fine and Gray, 1999, Geskus, 2016], and treatment contrasts capture total effects.When the events are mutually competing, the effect on a composite outcome of either event is the sum of the total effects on the event of interest and the competing event.Source: §4.2 · pp. 6–7

4.3 Counterfactual hazards

  • Counterfactual hazard contrasts—marginal, subdistribution, or cause-specific—cannot generally be interpreted as causal effects [Hern´an et al., 2004, Hern´an, 2010], because treatment may change who survives to each risk set.Source: §4.3 · pp. 8–9

5 Definition of a censoring event

Censoring is defined relative to the estimand: an event is censoring when it makes future counterfactual outcomes of interest unknowable. Thus, whether competing events count as censoring depends on which outcomes the estimand uses.

5 Definition of a censoring event

  • An event is censoring when it makes future counterfactual outcomes of interest under a and an intervention eliminating selected events unknowable, even for someone receiving a.The outcomes of interest are the counterfactual outcomes on which the investigator-chosen estimand depends.Source: §5 · p. 9
  • Competing events censor for direct-effect and hazard contrasts indexed to eliminating them, but not for total-effect, subdistribution-hazard, or cause-specific-hazard contrasts.The latter estimands use outcomes indexed only by treatment; people who experience a competing event cannot subsequently experience the event of interest.Source: §5 · p. 10
  • Loss to follow-up meets the definition; with loss to follow-up, the next section’s identification results generally apply only to estimands indexed by an intervention eliminating it.If loss to follow-up is assumed not to affect future events, the results also apply to estimands without that intervention index.Source: §5 · p. 10
  • Surviving all causes of failure through administrative study end is not censoring under this definition; left-censored failures require particularly strong assumptions and may be avoided by restricting enrollment to those alive and at risk.Source: §5 · p. 10
  • When only loss to follow-up—not competing events—is censoring, no individual potential censoring time is required; after a competing event, no later censoring event can occur.For example, after cardiovascular death, prostate cancer death cannot occur and no subsequent event can prevent knowledge of that fact.Source: §5 · pp. 10–11

6 Identification of estimands when competing events exist

Identification depends on whether competing events are treated as censoring: eliminating them targets a direct effect, while retaining them targets a total effect. Counterfactual hazard contrasts can be identified under specified assumptions but generally do not have a causal interpretation.

6.1 Direct effects

  • Identifying direct effects requires untestable exchangeability, positivity, and consistency assumptions for the risk under elimination of competing events.When only baseline covariates are measured, stronger exchangeability is needed; it may be implausible because censoring must be independent of changing prognostic factors during follow-up.Source: §6.1 · pp. 11, 12, 14

6.2 Total effects

  • Total effects require weaker identifying assumptions than direct effects because only loss to follow-up, not competing events, is treated as censoring.In a randomized trial with no loss to follow-up, the total effect is identified by the difference in observed risks between treatment groups.Source: §6.2 · pp. 14, 16

6.3 Counterfactual hazards and the “hazards of hazard ratios” revisited

  • Direct-effect assumptions identify hazards under elimination of competing events, total-effect assumptions identify subdistribution hazards, and cause-specific hazards require assumptions for total effects on both event types.The direct-effect assumptions are consistent with Figure 1 but fail under Figure 2; the total-effect assumptions are consistent with both figures.Source: §6.3 · pp. 16–17
  • Weighted observed cause-specific hazards can target contrasts in hazards under elimination of competing events, although those hazard contrasts are not generally causal effects.The observed function identifying the counterfactual hazard under elimination is a weighted version of observed cause-specific hazards among those free of loss to follow-up.Source: §6.3 · p. 17
  • Even when identified, counterfactual hazard contrasts generally lack a causal interpretation because conditioning on prior event-free status can open a non-causal path between treatment and outcome.For subdistribution hazards, conditioning on prior survival from the event of interest opens the path A → Yk ← U → Yk+1; similar arguments apply to other hazard contrasts.Source: §6.3 · p. 17

7 Choosing a definition of causal effect when competing events exist

Choosing between total and direct effects involves balancing the total effect’s weaker identification requirements against the risk that it reflects treatment effects on competing events. A direct effect excludes mediation through the competing event but requires well-defined interventions to eliminate it.

7 Choosing a definition of causal effect when competing events exist

  • The total effect requires weaker untestable assumptions than the direct effect and is expected to be identified by design in a randomized trial without loss to follow-up.Its interpretation may nevertheless be uncertain when treatment affects the competing event.Source: §7 · p. 18
  • A treatment can appear protective against the event of interest through the total effect solely because it increases the competing event, so both risks may be informative.In the extreme example, the total effect on prostate cancer death is protective because estrogen therapy instantly causes cardiovascular death; the competing-event risk effect is positive.Source: §7 · pp. 18–19
  • The direct effect captures treatment’s effect on the event of interest not mediated through the competing event, but requires well-defined interventions to eliminate competing events.When eliminating a competing event is unrealistic, the plausibility of assumptions for identifying the direct effect may be especially questionable; near-zero probabilities of avoiding discharge among recovered patients can also violate positivity.Source: §7 · p. 19
  • Alternative definitions that exclude treatment effects on competing events rely on cross-world counterfactuals, requiring knowledge of the same individual’s outcomes under different interventions.Source: §7 · p. 19
  • If treatment has no effect on the competing event, the total effect quantifies its effect on the event of interest without capturing an effect on the competing event.The importance of mediation may also vary by setting, such as whether discharge-mediated infection risk is as consequential as cardiovascular-death-mediated prostate cancer risk.Source: §7 · p. 19
  • A composite outcome avoids interventions on competing events and is identified by design in randomized trials without loss to follow-up, but may not answer the event-specific question.For mutually competing events, the treatment effects on the event of interest and the competing event sum to the composite effect, which is closer to zero when they have opposite signs.Source: §7 · pp. 19–20

8 Estimation of direct and total effects

The section reviews parametric g-formula and inverse probability weighting approaches for estimating direct and total effects, highlighting how their inputs and modeling requirements differ.

8.1 Direct effects

  • For direct effects, the parametric g-formula models its components and uses Monte Carlo simulation over time-varying covariate histories; under exchangeability given L0, it reduces to observed event hazards.In that restricted case, no Monte Carlo simulation is required.Source: §8.1 · p. 20
  • For direct effects, IPW estimates risk as the complement of a weighted Kaplan–Meier estimator; consistency requires correct weight-denominator models, and stabilized weighting may reduce variability.Unlike the parametric g-formula, IPW complexity is not substantially reduced by adjustment for only L0.Source: §8.1 · pp. 20–21

8.2 Total effects

  • For total effects, the parametric g-formula also models competing-event cause-specific hazards; under exchangeability given L0, it uses both event-of-interest and competing-event hazards.The direct-effect g-formula does not require the competing-event hazards.Source: §8.2 · pp. 21–22
  • For total effects, IPW can use weighted subdistribution hazards or weighted cause-specific hazards; the latter yields a weighted Aalen–Johansen estimator, and both require fewer model assumptions than the parametric g-formula.Their algorithmic complexity does not substantially change under adjustment for only L0.Source: §8.2 · p. 22

9 Example: A randomized trial on prostate cancer therapy

In this randomized-trial application, total risk estimates suggested estrogen reduced prostate-cancer death, but increased competing-cause mortality could account for that pattern. The hypothetical direct-effect estimate was smaller and depended on strong identifying assumptions.

9 Example: A randomized trial on prostate cancer therapy

  • The example used 125 high-dose estrogen and 127 placebo patients followed for 60 months; only baseline covariates were available, and continuous covariates were categorized.The authors note that categorization limited possible model forms and made exchangeability less reasonable, particularly for the direct effect.Source: §9 · p. 23
  • The analysis estimated total and direct effects using parametric g-formula and IPW estimators with pooled logistic models for cause-specific event and loss-to-follow-up hazards.R code for all analyses was provided in the supplementary materials.Source: §9 · p. 23, §9.1 · p. 23, §9.2 · p. 25

9.1 Total effect of estrogen therapy on prostate cancer death

  • By 60 months, point estimates indicated a total protective effect of estrogen on prostate-cancer-death risk, with wide confidence intervals; Figure 4 shows risks diverging shortly after follow-up began.The risk estimates compared high-dose estrogen therapy with placebo.Source: §9.1 · p. 24, Figure 4 · p. 25
  • Estimates suggested a harmful total effect on other-cause death, while the composite risk of death from any cause stayed close to null through most follow-up.The authors suggest the protective prostate-cancer-death effect may be attributable to treatment's effect on other causes of death.Source: §9.1 · p. 25, Figure 6 · p. 27

9.2 Direct effect of estrogen therapy on prostate cancer death

  • By 60 months, the direct effect on prostate-cancer-death risk under hypothetical elimination of competing deaths was smaller than the total effect, with some variation over time.Interpretation is cautious because confidence intervals were wide and the effect relies on strong identifying assumptions for an impossible-to-verify scenario.Source: §9.2 · p. 26

10 Discussion

The paper uses a counterfactual framework to define competing-risks estimands and clarify their causal interpretation and identification. Risk contrasts can represent total or direct effects depending on whether competing events are treated as censoring, whereas counterfactual hazard contrasts generally lack a causal interpretation.

10 Discussion

  • Risk contrasts quantify a direct effect when competing events are defined as censoring, and otherwise quantify a total effect that includes pathways through the competing event.The direct effect does not capture treatment’s effect on the competing event.Source: §10 · p. 27
  • For total effects with loss to follow-up, competing events must be treated as time-varying covariates for exchangeability, even in randomized trials.Identification functions for total and direct effects coincide with different versions of Robins’s g-formula [1986], and assumptions can be assessed using established graphical rules once the estimand is explicit [Pearl, 1995, Richardson and Robins, 2013].Source: §10 · pp. 27–28
  • The trial application had only baseline covariates, making exchangeability less plausible, though the framework also applies to observational studies with nonrandom treatment assignment.The analysis considered intention-to-treat effects; under per-protocol effects, nonadherence would be a censoring event [Hern´an and Robins, 2017, Hern´an and Hern´andez-D´ıaz, 2012].Source: §10 · p. 28
  • Counterfactual contrasts of cause-specific, subdistribution, or other hazards generally lack a causal interpretation, so the authors advise against routinely using hazards as causal-effect measures.This remains true even when assumptions identify the contrasts and even when assumptions identify a direct effect on risk; observed hazard choices may affect misspecification bias and precision while targeting a total effect.Source: §10 · p. 29
  • Because competing events generally mediate treatment effects, total effects can be hard to interpret; controlled direct effects avoid this pathway but require strong assumptions and potentially unrealistic interventions.The survivor average causal effect avoids interventions on competing events but concerns an unknown subset whose members would never experience the event under either treatment.Source: §10 · p. 29
  • The framework makes treatment effects, identifying assumptions, and method choices explicit, but available effects excluding treatment’s impact on competing events require ill-defined interventions or an unknowable subpopulation.Developing effect definitions that avoid these problems is an essential direction for future work.Source: §10 · p. 29

A Single-world intervention graphs

Single World Intervention Graphs (SWIGs) transform an assumed causal DAG into a counterfactual graph for evaluating estimand identification; the resulting exchangeability conditions depend on how competing events are treated.

A Single-world intervention graphs

  • Given the assumed data-generating DAG and appropriate consistency, d-separation on a SWIG can imply exchangeability for a time-varying treatment effect.The Figure 1 SWIG template implies exchangeability (20) holds.Source: A Single-world intervention graphs · p. 36, Figure 8 · p. 37
  • SWIGs evaluate estimand identification by splitting intervention nodes into natural and fixed values, indexing post-intervention variables counterfactually, and redirecting arrows.Interventions include treatment and censoring, where censoring may include competing events or only loss to follow-up.Source: A Single-world intervention graphs · p. 36
  • When the event of interest and competing event share an unmeasured cause, an unblocked backdoor path remains and exchangeability (20) fails.Figure 9 depicts the SWIG template for this DAG and failure.Source: A Single-world intervention graphs · p. 37, Figure 9 · p. 38
  • When an intervention eliminates only loss to follow-up and leaves competing events in place, the Figure 10 SWIG implies exchangeability (26) holds.This is the alternative intervention applied to the Figure 2 causal DAG.Source: A Single-world intervention graphs · p. 37, Figure 10 · p. 38

A Identification Proofs

The section proves that counterfactual risks and hazards in competing-risk settings equal specified g-formula or alternative expressions under identifying assumptions, with distinct results for eliminating versus retaining competing events.

A Identification Proofs

  • Theorem 1 establishes that, under its identifying assumptions, counterfactual risk under treatment a with censoring eliminated is equivalent to the stated expression.The proof treats censoring indicators as including all sources of censoring and uses measured time-varying covariates.Source: A Identification Proofs · p. 39
  • Under assumptions (20)–(22), risk after eliminating competing events equals g-formula (23); under (26)–(28), risk without elimination equals g-formula (29).These are the counterfactual risks defined in the main text.Source: A Identification Proofs · p. 41
  • The g-formulas for risk with and without elimination of competing events equal alternative expressions (24) and (31), respectively.Corollaries 3 and 4 establish these equivalences.Source: A Identification Proofs · p. 44
  • The g-formula (30) is equivalent to expression (33).Theorem 3 proves this identity.Source: A Identification Proofs · p. 47
  • Under assumptions (20)–(22), the counterfactual hazard with competing events eliminated equals expression (25); under (26)–(28), the hazard without elimination equals expression (32).Both equivalences hold for each follow-up interval k.Source: A Identification Proofs · pp. 48–49
  • The proofs also identify competing-event risk under elimination of loss to follow-up and establish expressions for hazards conditioned on competing events and for the competing event itself.Corollary 8 is agnostic to whether the events are mutually competing or the setting is semi-competing risks; the hazard results require assumptions (55)–(57) and (26)–(28).Source: A Identification Proofs · pp. 49–50

A Estimators of the risk of the competing event itself and data application implementation details … A Cross-world alternatives to controlled direct effects

The appendix describes estimators and data-analysis implementation for the risk of the competing event, then considers cross-world alternatives to controlled direct effects and their limitations.

A.1 Parametric g-formula and IPW estimators of the risk of the competing event

  • The competing-event risk can be estimated by a parametric g-formula or IPW using cause-specific hazards, or by an IPW estimator that swaps the event and competing-event roles in subdistribution hazards.The g-formula uses estimated observed cause-specific hazards and adjusts only for baseline L0 under the stated temporal-order assumptions.Source: §A.1 · p. 51
  • In semi-competing-risk settings, the original competing event can be treated as the event of interest, with no competing events and Dk+1 set to 0 for all k.Methods for settings without competing events can then estimate its risk; the original event-of-interest status may be included among time-varying covariates when needed for exchangeability.Source: §A.1 · p. 51

A.2.1 Structure of input data sets

  • The application used separate person-time data sets for cause-specific-hazard estimators, event-of-interest subdistribution hazards, and competing-event subdistribution hazards.The first data set ends records at an event; the second retains people after a competing event, and the third uses that structure with the event roles reversed. R code is provided in the supplementary materials.Source: §A.2.1 · pp. 51–52, §A.2 · p. 51

A.2.2 Model assumptions

  • The application modeled monthly prostate-cancer and other-death hazards with pooled logistic regressions, using a less flexible other-death model after bootstrap convergence failures.Point estimates from more- and less-flexible other-death models were very similar in sensitivity analysis.Source: §A.2.2 · pp. 52–53
  • For IPW, loss-to-follow-up hazards were set to 1 before month 50 and modeled thereafter; risk-set data required excluding prior competing-event failures from model fitting.For those already experiencing a competing event, the loss-to-follow-up hazard was set to 1; hemoglobin was omitted from the final censoring model after convergence failures, with similar sensitivity estimates when included.Source: §A.2.2 · p. 53

A Cross-world alternatives to controlled direct effects

  • A survivor average causal effect (SACE) concerns the event-of-interest risk among people who would avoid the competing event under either treatment, but that subgroup’s size is unknown.Unlike the total effect, it targets a different population; competing events are not censoring events for this estimand. SACE is discussed as a principal-stratum alternative [Robins, 1986, Frangakis and Rubin, 2002; Rubin, 2000].Source: A Cross-world alternatives to… · pp. 53–54
  • The natural direct effect targets the original study population but requires cross-world assumptions and cannot be identified without untestable assumptions.It sets the competing-event history to what it would have been under control; unlike the controlled direct effect, it has an indirect analogue that sums to the total effect [Robins and Greenland, 1992, Pearl, 2001; Vanderweele, 2015].Source: A Cross-world alternatives to… · p. 54
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