Causal treatment effect decompositions with time-to-event outcomes under competing events

Jenni Lehtisalo; Kari Auranen; Miia Kivipelto; Mikko Valtanen; Tiia Ngandu; Tommi H\"ark\"anen

arxiv: 2605.19693 · v1 · pith:WILREYQJnew · submitted 2026-05-19 · 📊 stat.ME

Causal treatment effect decompositions with time-to-event outcomes under competing events

Mikko Valtanen , Tommi H\"ark\"anen , Jenni Lehtisalo , Tiia Ngandu , Miia Kivipelto , Kari Auranen This is my paper

Pith reviewed 2026-05-20 02:37 UTC · model grok-4.3

classification 📊 stat.ME

keywords causal decompositioncompeting eventstime-to-event outcomestreatment effectssurvival analysiscounterfactual estimandsrandomized trials

0 comments

The pith

A four-way decomposition shows how treatment effects on a target time-to-event outcome arise through four mechanisms involving competing events.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proposes a causal decomposition that breaks down the overall treatment effect on a target event into four distinct mechanisms involving both the target and competing events. It does so by defining cross-world estimands that compare counterfactual scenarios in which the treatment is set to different levels for the target event versus the competing event. This approach clarifies how the competing event can contribute to or distort the apparent treatment effect on the outcome of interest. Sympathetic readers would care because many medical studies involve treatments that influence both the primary event and other events that can prevent it, making standard effect estimates hard to interpret without such a breakdown. The decomposition is estimable from observed data under exchangeability and consistency assumptions, as demonstrated in two randomized trial datasets.

Core claim

The authors introduce a decomposition of the causal treatment effect on the target event that partitions it into four components arising from distinct mechanisms involving both the target and competing events, based on a causal model using cross-world estimands that reflect counterfactual scenarios with the treatment set to conflicting levels for the two event types.

What carries the argument

Cross-world estimands reflecting counterfactual scenarios in which treatment affects the target and competing events at conflicting levels.

If this is right

The decomposition reveals the specific role of the competing event in generating the observed treatment effect on the target.
It supplies a basis for defining causally interpretable estimands when competing events are present.
The four components can be estimated from data collected in randomized trials.
Standard methods for analyzing time-to-event data can be extended to separate direct effects on the target from effects operating through the competing event.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The framework could be used to design interventions that modify only one of the four mechanisms rather than the overall effect.
Similar decompositions might apply to non-randomized observational data if additional identifying assumptions are introduced.
The approach links to existing causal methods for multiple outcomes and could support sensitivity analyses when exchangeability is in doubt.

Load-bearing premise

Exchangeability and consistency assumptions hold so that the decomposition can be estimated from observed data.

What would settle it

A dataset from a randomized trial in which the four estimated components fail to sum to the total treatment effect on the target event.

Figures

Figures reproduced from arXiv: 2605.19693 by Jenni Lehtisalo, Kari Auranen, Miia Kivipelto, Mikko Valtanen, Tiia Ngandu, Tommi H\"ark\"anen.

**Figure 2.** Figure 2: Treatment effects on CVD events with other-cause deaths as a [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

**Figure 3.** Figure 3: Treatment effects on prostate cancer death with other-cause deaths [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

read the original abstract

Inference about treatment effects for time-to-event outcomes is often obscured by the presence of competing events. A particularly complex situation arises when the treatment influences the occurrence of the competing event. A comprehensive assessment should then account for different mechanisms by which the treatment and the competing event together produce the apparent treatment effect. Here, we propose a decomposition of the treatment's effect on the event of interest (target), characterising how it arises due to four distinct mechanisms involving both the target and competing events. Based on a causal model, the decomposition relies on cross-world estimands reflecting counterfactual scenarios in which the treatment affects the two events as if set to conflicting levels. We specify exchangeability and consistency assumptions under which the decomposition can be estimated from observed data. We discuss how the new decomposition reveals the role of the competing event and serves as a basis for defining causally interpretable estimands in the presence of competing events. Finally, we demonstrate the use of the four-way decomposition with datasets from two randomised trials.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper introduces a four-way cross-world decomposition for treatment effects on a target event when treatment also affects competing events, and applies it to two trial datasets.

read the letter

The core contribution is a decomposition that splits the total effect on the target time-to-event outcome into four mechanisms, each tied to how treatment influences the target event, the competing event, and their interactions under conflicting treatment levels in counterfactual worlds. This targets a common headache in medical survival data where the treatment changes the chance of the competing event, making standard effect measures harder to interpret causally. They lay out exchangeability and consistency assumptions and claim these let you estimate the pieces from observed data, then illustrate with randomized trial examples. That demonstration on real data is useful and shows the idea can be put to work without needing new software from scratch. The approach extends existing causal survival tools rather than reinventing them, and the four-mechanism split gives a clearer breakdown than the usual cause-specific or subdistribution hazards alone. The soft spot is the identification step for the cross-world quantities. Standard no-confounding assumptions identify single-world counterfactuals cleanly, but when the target and competing events are allowed to respond to opposite treatment values at the same time, dependence between the latent event times or unmeasured factors that hit the two events differently can leave some pieces non-identified without extra restrictions. The abstract states the assumptions suffice, but the letter would benefit from seeing the explicit identification formulas and any sensitivity checks in the full text. If those hold up without hidden parametric assumptions, the decomposition is recoverable; if not, the practical value shrinks. This is for methodologists and applied statisticians in epidemiology or clinical trials who already work with competing risks and want finer causal language. A reader who needs to report treatment effects on a primary event while acknowledging how the treatment shifts competing mortality will get direct value. It is worth sending for peer review because the problem is recurring, the proposal is specific, and the empirical examples provide a starting point for checking the claims.

Referee Report

1 major / 2 minor

Summary. The manuscript proposes a four-way causal decomposition of the total treatment effect on the incidence of a target time-to-event outcome into distinct mechanisms involving both the target event and a competing event. The decomposition is defined via cross-world counterfactuals in which treatment is set to conflicting levels along the target versus competing-event pathways. The authors state that standard exchangeability and consistency assumptions suffice for identification and estimation from observed data, and they illustrate the approach using data from two randomized trials.

Significance. If the identification of the cross-world quantities holds under the stated assumptions, the decomposition would offer a principled way to attribute apparent treatment effects on the target event to direct versus competing-event-mediated pathways, addressing a common interpretive challenge in survival analysis. The empirical examples from actual trials provide a useful demonstration of practical application.

major comments (1)

[§3] §3 (Identification): The claim that exchangeability and consistency alone identify the four cross-world estimands is load-bearing for the central contribution. Standard no-unmeasured-confounding assumptions identify single-world counterfactuals but do not automatically extend to quantities in which treatment is set to opposite levels for the target and competing latent times. The manuscript should supply the explicit identification formulas (e.g., the g-formula or inverse-probability expressions for each of the four components) together with any auxiliary assumptions (such as conditional independence of the latent failure times or absence of treatment-by-event interactions on the latent scale) that are required to close the identification gap.

minor comments (2)

[Abstract] The abstract states that the decomposition 'can be estimated from observed data' under the listed assumptions; this phrasing should be qualified to reflect that identification requires the additional structure discussed in §3.
[Figure 1] Figure 1 (causal diagram): the arrows from treatment to the two latent times should be labeled to indicate that the diagram encodes the cross-world intervention, not a single-world intervention.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive and detailed review of our manuscript. The comments help clarify the presentation of our identification results, and we address the major comment below.

read point-by-point responses

Referee: [§3] §3 (Identification): The claim that exchangeability and consistency alone identify the four cross-world estimands is load-bearing for the central contribution. Standard no-unmeasured-confounding assumptions identify single-world counterfactuals but do not automatically extend to quantities in which treatment is set to opposite levels for the target and competing latent times. The manuscript should supply the explicit identification formulas (e.g., the g-formula or inverse-probability expressions for each of the four components) together with any auxiliary assumptions (such as conditional independence of the latent failure times or absence of treatment-by-event interactions on the latent scale) that are required to close the identification gap.

Authors: We thank the referee for this important observation on the identification strategy. The manuscript states that the four cross-world estimands are identified from the observed data under exchangeability (no unmeasured confounding for treatment) and consistency. To address the request for greater explicitness, we will revise §3 to include the full g-formula expressions for each of the four components. These expressions are obtained by integrating the observed conditional distributions of the event times and covariates, with treatment fixed at the appropriate level for the target-event pathway and the competing-event pathway in each counterfactual world. We maintain that no auxiliary assumptions beyond the stated exchangeability and consistency are required; the separate definition of the latent failure times for the target and competing events in the causal model permits direct application of the g-formula without invoking conditional independence of the latent times or absence of treatment-by-event interactions on the latent scale. The revised section will present these formulas and the corresponding identification argument in detail. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivation introduces new estimands

full rationale

The paper defines a novel four-way decomposition of treatment effects on the target event via cross-world counterfactuals under competing events. It explicitly states exchangeability and consistency assumptions to identify the decomposition from observed data. No load-bearing step reduces by construction to a fitted parameter, self-citation chain, or renamed input; the central contribution is the proposal and interpretation of new causally interpretable estimands rather than a statistical prediction derived from prior fits. The derivation chain is self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on a causal model whose identification requires stated exchangeability and consistency assumptions; no free parameters or invented entities are mentioned in the abstract.

axioms (1)

domain assumption Exchangeability and consistency assumptions allow estimation of the cross-world estimands from observed data.
Explicitly invoked in the abstract as the conditions under which the decomposition is identifiable.

pith-pipeline@v0.9.0 · 5726 in / 1143 out tokens · 55127 ms · 2026-05-20T02:37:44.120842+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the decomposition relies on cross-world estimands reflecting counterfactual scenarios in which the treatment affects the two events as if set to conflicting levels

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

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= kX s=1 P(Y a′,a′′ s = 1, Da′,a′′ s =Y a′,a′′ s−1 =· · ·=D a′,a′′ 1 =Y a′,a′′ 1 = 0)

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[26]

= kX s=1 P(Y a′,∅ s = 1, D∅,a′′ s =Y a′,∅ s−1 =· · ·=D ∅,a′′ 1 =Y a′,∅ 1 = 0)

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= X w kX s=1 P(Y a′,∅ s = 1, D∅,a′′ s =Y a′,∅ s−1 =· · ·=D ∅,a′′ 1 =Y a′,∅ 1 = 0, w)

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Next, we employ assumptions A2 and A3 to change treatments levels under which the survival from the controlled processes are conditioned on (step 5)

= X w kX s=1 P(Y a′,∅ s = 1|D ∅,a′′ s =Y a′,∅ s−1 = 0, w) × sY j=1 P(D ∅,a′′ j = 0|D ∅,a′′ j−1 =Y a′,∅ j−1 = 0, w) ×P(Y a′,∅ j−1 = 0|Y a′,∅ j−2 =D ∅,a′′ j−1 = 0, w)P(w). Next, we employ assumptions A2 and A3 to change treatments levels under which the survival from the controlled processes are conditioned on (step 5). Assumption A1 along with positivity P...

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[29]

= X w kX s=1 P(Y a′,∅ s = 1|D ∅,a′ s =Y a′,∅ s−1 = 0, w) × sY j=1 P(D ∅,a′′ j = 0|D ∅,a′′ j−1 =Y a′′,∅ j−1 = 0, w) ×P(Y a′,∅ j−1 = 0|Y a′,∅ j−2 =D ∅,a′ j−1 = 0, w)P(w)

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(11) To incorporate censoring, we include the trivial conditionD 0 =Y 0 =C 0 = 0

= X w kX s=1 P(Y a′,∅ s = 1|D ∅,a′ s =Y a′,∅ s−1 = 0, A=a ′, w) × sY j=1 P(D ∅,a′′ j = 0|D ∅,a′′ j−1 =Y a′′,∅ j−1 = 0, A=a ′′, w) ×P(Y a′,∅ j−1 = 0|Y a′,∅ j−2 =D ∅,a′ j−1 = 0, A=a ′, w)P(w). (11) To incorporate censoring, we include the trivial conditionD 0 =Y 0 =C 0 = 0. We then invoke the assumptions P1, S1 and C1–C3 repeatedly to express each term usin...

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[31]

= P(Y a′,∅ k =y, D ∅,a′ k =Y a′,∅ k−1 =· · ·=D ∅,a′ 1 =Y a′,∅ 1 = 0|C 0 =D 0 =Y 0 = 0, A=a ′, w) P(D ∅,a′ k =Y a′,∅ k−1 =· · ·=D ∅,a′ 1 =Y a′,∅ 1 = 0|C 0 =D 0 =Y 0 = 0, A=a ′, w)

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[32]

= P(Y a′,∅ k =y, D ∅,a′ k =Y a′,∅ k−1 =· · ·=D ∅,a′ 1 =Y a′,∅ 1 = 0| ¯C1 =D 0 =Y 0 = 0, A=a ′, w) P(D ∅,a′ k =Y a′,∅ k−1 =· · ·=D ∅,a′ 1 =Y a′,∅ 1 = 0| ¯C1 =D 0 =Y 0 = 0, A=a ′, w)

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[33]

= P(Y a′,∅ k =y, D ∅,a′ k =Y a′,∅ k−1 =· · ·=D ∅,a′ 2 =Y a′,∅ 1 = 0| ¯C1 =D 1 =Y 0 = 0, A=a ′, w) P(D ∅,a′ k =Y a′,∅ k−1 =· · ·=D ∅,a′ 2 =Y a′,∅ 1 = 0| ¯C1 =D 1 =Y 0 = 0, A=a ′, w)

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[34]

Step 8 used S1 to include conditioning on ¯C1 = 0

= P(Y a′,∅ k =y, D ∅,a′ k =Y a′,∅ k−1 =· · ·=D ∅,a′ 2 =Y a′,∅ 2 = 0| ¯C1 =D 1 =Y 1 = 0, A=a ′, w) P(D ∅,a′ k =Y a′,∅ k−1 =· · ·=D ∅,a′ 2 =Y a′,∅ 2 = 0| ¯C1 =D 1 =Y 1 = 0, A=a ′, w) Step 7 used the positivity assumption to ensure the denominator is not zero. Step 8 used S1 to include conditioning on ¯C1 = 0. Steps 9 and 10 used the consistency assumptions ...

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The Finnish Geriatric Intervention Study to Prevent Cognitive Impairment and Disability (FINGER): Study design and progress,

M. Kivipelto, A. Solomon, S. Ahtiluoto, T. Ngandu, J. Lehtisalo, R. Antikainen, L. B¨ ackman, T. H¨ anninen, A. Jula, T. Laatikainen, J. Lindstr¨ om, F. Mangialasche, A. Nissinen, T. Paajanen, S. Pajala, M. Peltonen, R. Rauramaa, A. Stigsdotter-Neely, T. Strandberg, J. Tuomilehto, and H. Soininen, “The Finnish Geriatric Intervention Study to Prevent Cogni...

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Effect of a multi-domain lifestyle intervention on cardiovascular risk in older people: the FINGER trial,

J. Lehtisalo, M. Rusanen, A. Solomon, R. Antikainen, T. Laatikainen, M. Peltonen, T. Strandberg, J. Tuomilehto, H. Soininen, M. Kivipelto, and T. Ngandu, “Effect of a multi-domain lifestyle intervention on cardiovascular risk in older people: the FINGER trial,”European Heart Journal, vol. 43, pp. 2054–2061, Jan. 2022

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The choice of treatment for cancer patients based on covariate information: Application to prostate cancer,

D. P. Byar and S. B. Green, “The choice of treatment for cancer patients based on covariate information: Application to prostate cancer,”Bulletin Cancer, Paris, vol. 67, p. 477–488, 1980

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Restricted mean survival time: an alternative to the hazard ratio for the design and analysis of randomized trials with a time-to-event outcome,

P. Royston and M. K. Parmar, “Restricted mean survival time: an alternative to the hazard ratio for the design and analysis of randomized trials with a time-to-event outcome,”BMC Medical Research Methodology, vol. 13, Dec. 2013. 20 A Empirical total effect decomposition Here, we provide details on the derivation of the expression (4) in the main text for ...

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= kX s=1 P(Y a′,a′′ s = 1, Da′,a′′ s =Y a′,a′′ s−1 =· · ·=D a′,a′′ 1 =Y a′,a′′ 1 = 0)

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[26] [26]

= kX s=1 P(Y a′,∅ s = 1, D∅,a′′ s =Y a′,∅ s−1 =· · ·=D ∅,a′′ 1 =Y a′,∅ 1 = 0)

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[27] [27]

= X w kX s=1 P(Y a′,∅ s = 1, D∅,a′′ s =Y a′,∅ s−1 =· · ·=D ∅,a′′ 1 =Y a′,∅ 1 = 0, w)

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Next, we employ assumptions A2 and A3 to change treatments levels under which the survival from the controlled processes are conditioned on (step 5)

= X w kX s=1 P(Y a′,∅ s = 1|D ∅,a′′ s =Y a′,∅ s−1 = 0, w) × sY j=1 P(D ∅,a′′ j = 0|D ∅,a′′ j−1 =Y a′,∅ j−1 = 0, w) ×P(Y a′,∅ j−1 = 0|Y a′,∅ j−2 =D ∅,a′′ j−1 = 0, w)P(w). Next, we employ assumptions A2 and A3 to change treatments levels under which the survival from the controlled processes are conditioned on (step 5). Assumption A1 along with positivity P...

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[29] [29]

= X w kX s=1 P(Y a′,∅ s = 1|D ∅,a′ s =Y a′,∅ s−1 = 0, w) × sY j=1 P(D ∅,a′′ j = 0|D ∅,a′′ j−1 =Y a′′,∅ j−1 = 0, w) ×P(Y a′,∅ j−1 = 0|Y a′,∅ j−2 =D ∅,a′ j−1 = 0, w)P(w)

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[30] [30]

(11) To incorporate censoring, we include the trivial conditionD 0 =Y 0 =C 0 = 0

= X w kX s=1 P(Y a′,∅ s = 1|D ∅,a′ s =Y a′,∅ s−1 = 0, A=a ′, w) × sY j=1 P(D ∅,a′′ j = 0|D ∅,a′′ j−1 =Y a′′,∅ j−1 = 0, A=a ′′, w) ×P(Y a′,∅ j−1 = 0|Y a′,∅ j−2 =D ∅,a′ j−1 = 0, A=a ′, w)P(w). (11) To incorporate censoring, we include the trivial conditionD 0 =Y 0 =C 0 = 0. We then invoke the assumptions P1, S1 and C1–C3 repeatedly to express each term usin...

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= P(Y a′,∅ k =y, D ∅,a′ k =Y a′,∅ k−1 =· · ·=D ∅,a′ 1 =Y a′,∅ 1 = 0|C 0 =D 0 =Y 0 = 0, A=a ′, w) P(D ∅,a′ k =Y a′,∅ k−1 =· · ·=D ∅,a′ 1 =Y a′,∅ 1 = 0|C 0 =D 0 =Y 0 = 0, A=a ′, w)

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[32] [32]

= P(Y a′,∅ k =y, D ∅,a′ k =Y a′,∅ k−1 =· · ·=D ∅,a′ 1 =Y a′,∅ 1 = 0| ¯C1 =D 0 =Y 0 = 0, A=a ′, w) P(D ∅,a′ k =Y a′,∅ k−1 =· · ·=D ∅,a′ 1 =Y a′,∅ 1 = 0| ¯C1 =D 0 =Y 0 = 0, A=a ′, w)

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= P(Y a′,∅ k =y, D ∅,a′ k =Y a′,∅ k−1 =· · ·=D ∅,a′ 2 =Y a′,∅ 1 = 0| ¯C1 =D 1 =Y 0 = 0, A=a ′, w) P(D ∅,a′ k =Y a′,∅ k−1 =· · ·=D ∅,a′ 2 =Y a′,∅ 1 = 0| ¯C1 =D 1 =Y 0 = 0, A=a ′, w)

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[34] [34]

Step 8 used S1 to include conditioning on ¯C1 = 0

= P(Y a′,∅ k =y, D ∅,a′ k =Y a′,∅ k−1 =· · ·=D ∅,a′ 2 =Y a′,∅ 2 = 0| ¯C1 =D 1 =Y 1 = 0, A=a ′, w) P(D ∅,a′ k =Y a′,∅ k−1 =· · ·=D ∅,a′ 2 =Y a′,∅ 2 = 0| ¯C1 =D 1 =Y 1 = 0, A=a ′, w) Step 7 used the positivity assumption to ensure the denominator is not zero. Step 8 used S1 to include conditioning on ¯C1 = 0. Steps 9 and 10 used the consistency assumptions ...

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