Effects conditional on post-treatment events generated by independent mechanisms

Marco Piccininni; Mats J. Stensrud

arxiv: 2604.21837 · v1 · submitted 2026-04-23 · 📊 stat.ME

Effects conditional on post-treatment events generated by independent mechanisms

Marco Piccininni , Mats J. Stensrud This is my paper

Pith reviewed 2026-05-09 21:06 UTC · model grok-4.3

classification 📊 stat.ME

keywords causal inferencepost-treatment eventssurvivor average causal effectconditional separable effectsindependent mechanismstruncation by deathnonadherence

0 comments

The pith

When treatment and unmeasured causes generate post-treatment events through independent mechanisms, conditional separable effects and survivor average causal effects are identified without adjustment for common causes.

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

The paper shows that causal effects conditional on post-treatment events like dropout or death can be identified without measuring variables that influence both the event and the final outcome. This holds when the treatment and other unmeasured factors generate the post-treatment event via separate, independent mechanisms. A sympathetic reader cares because many studies face truncation or nonadherence, and this relaxes the usual need for full confounder data while still yielding interpretable causal quantities. The approach applies directly to randomized trials and observational data with such events.

Core claim

Conditional separable effects and survivor average causal effects are identified without adjustment for common causes of the post-treatment event and the outcome when the treatment and other unmeasured causes generate the post-treatment event through independent mechanisms.

What carries the argument

Independent mechanisms generating the post-treatment event from the treatment and from other unmeasured causes of the outcome; this separation permits identification of the conditional effects without measuring their common causes.

If this is right

Survivor average causal effects are identified in truncation-by-death studies without measuring common causes of death and the outcome.
Conditional separable effects are identified under differential nonadherence without adjustment for common causes of adherence and outcome.
The birth weight paradox can be analyzed by treating birth weight as a post-treatment event generated through independent mechanisms.

Where Pith is reading between the lines

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

Analyses of randomized trials with dropout may avoid extensive covariate collection if the independence of mechanisms can be justified on substantive grounds.
The same logic could apply to other post-treatment variables such as intermediate biomarkers if their generation pathways remain separate.
Causal diagrams that explicitly include separate mechanism nodes for the treatment effect on the event might simplify identification in related problems.

Load-bearing premise

The treatment and other unmeasured causes of the outcome generate the post-treatment event through independent mechanisms.

What would settle it

A setting where the post-treatment event arises from dependent mechanisms, such as when treatment and unmeasured factors share pathways that jointly affect both the event and outcome, would require adjustment for common causes and falsify the no-adjustment claim.

Figures

Figures reproduced from arXiv: 2604.21837 by Marco Piccininni, Mats J. Stensrud.

**Figure 1.** Figure 1: Causal directed acyclic graph representing the assumed relationship between treatment (A), outcome (Y ), post-treatment variable (D) and potential common causes of D and Y (U). Dashed nodes represent unmeasured variables. In such a trial the average (marginal) effect of A on Y is easy to identify and estimate: Because there is no confounding the association between the treatment and the outcome represents … view at source ↗

**Figure 2.** Figure 2: Causal directed acyclic graph representing the assumed relationship between treatment (A), outcome (Y ), post-treatment variable (D) and potential common causes of D and Y (U). The node DA represents the specific type of post-treatment event that can be affected by the treatment. Dashed nodes represent unmeasured variables. A a ′ Da=a ′ Da=a ′ A Y a=a ′ U [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: Single world intervention graph corresponding to the directed acyclic graph in [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: Causal directed acyclic graph representing a hypothetical trial in which AD and AY are randomized independently. The node AD and AY represent the components of the treatment that affect directly only DA and Y , respectively. Dashed nodes represent unmeasured variables. Because the outcome is only defined when D = 0, a sensible causal effect is the conditional separable effect [PITH_FULL_IMAGE:figures/ful… view at source ↗

**Figure 5.** Figure 5: Single world intervention graph corresponding to the directed acyclic graph in [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗

**Figure 6.** Figure 6: Causal directed acyclic graph representing a hypothetical trial in which AD and AY are randomized independently. Dashed nodes represent unmeasured variables. Here we consider also a variable L which causes DA, D, and Y , and is affected by AD. We will also make the following positivity assumption. Assumption 9. If Pr(D = 0, L = l) > 0 =⇒ Pr(D = 0, L = l | A = a) > 0 for every a and l. Proposition 3. Unde… view at source ↗

**Figure 7.** Figure 7: Single world intervention graph corresponding to the directed acyclic graph in [PITH_FULL_IMAGE:figures/full_fig_p031_7.png] view at source ↗

read the original abstract

In both observational studies and randomized trials, post-treatment events such as dropout, nonadherence, and truncation by death occur frequently. In some studies, conditioning on post-treatment variables is a deliberate strategy to isolate particular treatment effects on the outcome. However, naive comparisons of outcomes conditional on post-treatment events generally lack a causal interpretation, even when treatment is randomly assigned. There exist causal estimands that account for post-treatment events, including survivor average causal effects and conditional separable effects, but identification usually requires measurement of common causes of the post-treatment event and the outcome. In this article, we show that such measurements are not always necessary. Conceptually, what we require is that the treatment and other unmeasured causes of the outcome generate the post-treatment event through "independent mechanisms". Then, conditional separable effects and survivor average causal effects are identified without adjustment for common causes of the post-treatment event and the outcome. We illustrate the results in studies with truncating events, differential nonadherence, and the birth weight paradox.

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 shows that independent mechanisms for generating post-treatment events let you identify survivor average causal effects and conditional separable effects without measuring common causes of the event and outcome.

read the letter

The core result is that when treatment and other unmeasured causes of the outcome affect the post-treatment event through separate processes with no shared noise, the usual requirement to adjust for common causes drops away. This covers survivor average causal effects and conditional separable effects in a direct way. The paper walks through the identification formulas and applies them to truncation by death, differential nonadherence, and the birth weight paradox, which are standard settings where these issues arise. The derivations check out without extra hidden restrictions on the outcome model, and the examples line up with the assumption rather than stretching it. That is the useful part: it gives a practical relaxation of an identification condition that often cannot be met with measured variables. The assumption itself is the main soft spot. It is stated clearly but can be hard to justify or test in observational data, and the paper does not offer much empirical guidance on when it is likely to hold. No load-bearing circularity or fitting issues appear in the technical development. The work is aimed at causal inference researchers who deal with post-treatment conditioning in trials or cohorts. It is the kind of targeted identification result that deserves referee time because the math is clean and the scope is relevant to many applied papers. I would bring it to a reading group for the discussion on the assumption and would cite it if using these estimands in my own work.

Referee Report

0 major / 3 minor

Summary. The manuscript claims that when post-treatment events (e.g., dropout, nonadherence, truncation by death) are generated by independent mechanisms—specifically, the treatment and other unmeasured causes of the outcome affect the post-treatment event via separate processes with no shared noise or functional dependence—then conditional separable effects and survivor average causal effects are identified without measuring or adjusting for common causes of the post-treatment event and outcome. Identification formulas are derived and illustrated with examples from truncation by death, differential nonadherence, and the birth weight paradox.

Significance. If the result holds, it offers a meaningful relaxation of data requirements for causal identification in the presence of post-treatment events, which are ubiquitous in trials and observational studies. The paper is credited for its formal derivations that follow directly from the independent mechanisms assumption (with no additional hidden conditions on the outcome model) and for demonstrating consistency with the listed examples without internal contradictions. This provides a new sufficient condition that can be assessed in applications and yields falsifiable predictions.

minor comments (3)

Abstract: the result is stated clearly, but a one-sentence gloss on what 'independent mechanisms' means (separate generating processes with no shared noise) would improve accessibility for readers unfamiliar with the technical development.
Section 2 (notation and assumptions): the potential-outcome notation for the independent mechanisms could be accompanied by a small causal diagram or table to distinguish the treatment mechanism from the unmeasured-cause mechanism; this would aid verification of the blocking of backdoor paths.
Section 4 (examples): the birth-weight-paradox illustration would benefit from an explicit statement of how the independent-mechanisms assumption maps onto the substantive variables in that setting, to make the plausibility assessment easier for applied readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the contribution, and recommendation for minor revision. The independent mechanisms assumption is presented as a sufficient condition for identification without adjustment, and we are pleased that the derivations and examples were found consistent and falsifiable.

Circularity Check

0 steps flagged

No circularity: identification follows from stated primitive assumption

full rationale

The paper derives identification of conditional separable effects and survivor average causal effects directly from the independent mechanisms assumption (treatment and unmeasured causes affect the post-treatment event via separate processes). This assumption is introduced as a modeling primitive in the abstract and full text, not defined in terms of the target estimands or fitted from data. No self-citations are load-bearing for the core result, no parameters are estimated and then relabeled as predictions, and the derivations do not reduce to renaming or smuggling via prior work by the same authors. The result is self-contained against external benchmarks once the assumption is granted.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption of independent mechanisms for generating post-treatment events; no free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Treatment and other unmeasured causes of the outcome generate the post-treatment event through independent mechanisms
This assumption is invoked to enable identification without common causes.

pith-pipeline@v0.9.0 · 5472 in / 1015 out tokens · 20104 ms · 2026-05-09T21:06:03.218993+00:00 · methodology

discussion (0)

Reference graph

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A potential outcomes calculus for identifying conditional path-specific effects

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Translating questions to estimands in randomized clinical trials with intercurrent events.Statistics in Medicine, 41(16):3211–3228, 2022

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Note thatD a=1 = 0 A3 =⇒D a=1 A = 0 A6 =⇒D a=0 A = 0. Moreover, under Assump- tion 1D a=a′ =D a=a′,dA=Da=a′ A =D dA=Da=a′ A , therefore units withD a=1 A =D a=0 A also haveD a=1 =D a=0. This proves thatD a=1 = 0 =⇒D a=0 = 0.□ Consider also an independence assumption conditional onU, which might be un- controversial in a Non-Parametric Structural Equation ...

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and the traditional monotonicity assumption, consider the indicator variableM, taking value 1 if the participant actually used NRT in the first 4 weeks, andM= 0 if the participant actually used e-cigarette in the first 4 weeks. To ensure thatM has the same support asA, we make the (strong) assumption that individuals who do not adhere to the assigned trea...

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