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REVIEW 3 major objections 6 minor 49 references

Simplifying Causal Mediation Analysis for Time-to-Event Outcomes using Pseudo-Values

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Pseudo-values transform censored survival outcomes into continuous-like responses, so the familiar product-method mediation formulas apply to survival probability, restricted mean survival time, and cumulative incidence.

desk verdict Useful applied extension of pseudo-value mediation to survival probability and competing risks, but the 'unbiased' claim rests on simulations where the linear outcome model is misspecified. read the letter →

arxiv 2411.17533 v2 pith:RJSPJDNQ submitted 2024-11-26 stat.ME

Alex Ocampo , Enrico Giudice , Dieter A. Häring , Baldur Magnusson , Theis Lange , Zachary R. McCaw This is my paper · ORCID
classification stat.ME MSC 62N0162N0262P10
keywords pseudo-valuescausalmediationsurvivalanalysisrestrictedmeantimecumulativeincidencecompetingriskstime-to-eventoutcomesproductmethod
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that pseudo-values—jackknife-style transformations that turn censored survival data into values that behave like continuous outcomes—let researchers run causal mediation analysis on time-to-event endpoints with the same simple product formulas used for linear outcomes. The central claim is that natural direct, indirect, and total effects on survival probability, restricted mean survival time, and cumulative incidence (with competing risks) can be estimated by fitting an ordinary linear model to pseudo-values and multiplying coefficients from a mediator regression. If right, this makes mediation for survival data accessible to standard software and gives an additive, interpretable decomposition instead of decomposing hazard ratios. The simulations across 324 estimands are offered as evidence: the pseudo-value estimates are reported unbiased and type-I error is controlled at the nominal level under the null.

What carries the argument

The central object is the pseudo-value $Y_i = n\hat{\theta} - (n-1)\hat{\theta}_{(-i)}$, where $\hat{\theta}$ is the full-sample estimate of the survival functional and $\hat{\theta}_{(-i)}$ is the leave-one-out estimate. When the influence function of the estimator is known, the pseudo-value is approximated by $\hat{\theta} + \varphi(X_i)$, which the paper validates as nearly identical to the jackknife version and much faster to compute. This machinery converts censored-data estimation into a continuous-outcome regression problem, so the causal estimands are identified by the coefficients of two linear models and combined by the product-method formulas.

What would settle it

Generate data with a treatment-by-mediator interaction, for example a hazard $\lambda = \exp(\beta_0 + A\beta_A + A\beta_{AM}M + M\beta_M)$, compute the true natural indirect effect by numerical integration over the mediator, and compare it with the product $\alpha_A\beta_M$ from the paper's pipeline. A systematic discrepancy would show that the linear no-interaction models are load-bearing rather than a convenience.

Watch

Extended reading notes

Core claim

The paper's central discovery is a simplification: once survival data are converted into pseudo-values for an expectation-type estimand, censoring is accounted for inside those values, and the outcome model can be fit as a linear regression. Combining this with a linear mediator model yields the classical product-method decomposition: the natural direct effect is the treatment coefficient in the pseudo-value outcome model, the natural indirect effect is the product of the treatment coefficient in the mediator model and the mediator coefficient in the outcome model, and the total effect is their sum. The paper claims this pipeline is unbiased for all three estimands across sample sizes, time points, and the presence or absence of direct or indirect effects, and that p-values are uniformly distributed under the null of no mediation.

Load-bearing premise

The approach assumes the mediator model and the pseudo-value outcome model are both correctly specified as linear with no treatment–mediator interaction, and that every mediator–outcome confounder is measured and included; the simulations only exercise this exact structure.

Editorial extensions

If this is right

  • Analysts can apply existing continuous-outcome mediation software to survival endpoints without writing specialized survival mediation routines.
  • Effects can be reported as differences in probabilities or restricted mean survival time, with an additive total effect that makes the proportion mediated easy to state.
  • The same procedure covers cumulative incidence in the presence of competing risks, an estimand rarely handled in earlier survival mediation work.
  • Influence-function approximations make bootstrap inference for pseudo-value mediation computationally practical in large samples.
  • The approach sidesteps the non-collapsibility of hazard ratios and the incompatibility of proportional hazards models with and without the mediator.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the linearity assumptions hold even approximately, the same pseudo-value pipeline could extend to other expectation-type estimands such as years of life lost or event-free survival at multiple horizons, but the paper does not test those extensions.
  • The requirement that the mediator is measured after treatment and before the outcome makes the method most defensible for non-terminal recurrent events or early biomarkers; applying it to terminal events where mediator values are observed only among survivors would require additional missing-data assumptions.
  • One could benchmark the pseudo-value product method against nonparametric natural-effect estimates on real trial data to see whether the proportion mediated is stable across modeling choices.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper proposes using pseudo-values to transform survival outcomes into continuous outcomes that can be analyzed with standard linear mediation models, thereby estimating natural direct, indirect, and total effects on survival probability, restricted mean survival time, and cumulative incidence. The method is illustrated with simulations across 324 scenarios and an application to the PARADIGMS multiple sclerosis trial. The central claim is that after constructing pseudo-values, the product-method formulas (NDE = β_A, NIE = α_A β_M, TE = β_A + α_A β_M) yield unbiased estimates and valid inference.

Significance. If the central claim were fully established, the paper would offer a practical simplification: applied analysts could use familiar software (e.g., CMAverse) for mediation analysis of time-to-event outcomes, with interpretable additive decompositions. The paper also contributes a computationally efficient influence-function approximation for pseudo-values and validates it thoroughly (R² = 1.00; reduced runtime). However, the load-bearing assumptions of linearity and independence of the pseudo-value outcome model are not adequately examined, and the simulation evidence does not support the claim of exact unbiasedness as stated. The method is potentially useful but needs substantial additional validation and more careful qualifiers.

major comments (3)
  1. [§4.1–4.3.1, Eq. (20)] The data-generating process T|A,M ~ Exp(exp(β0 + AβA + MβM)) yields a conditional survival probability S(τ|A,M) = exp(−τ exp(β0 + AβA + MβM)) that is not linear in M, and the same holds for the conditional mean of the pseudo-values. Equation (14) is therefore misspecified in the very scenarios used to validate the method. The product-method formulas (15)–(17) are exact only when the pseudo-value outcome model is correctly specified, so the claim in §4.3.1 that the approach is 'unbiased for the true NIE, NDE, and TE' cannot hold exactly under this generative model. The reported results likely reflect small bias on a narrow parameter grid (k=3, τ∈{2,3,4}) rather than exact unbiasedness. The paper should either provide a theoretical argument for why the linear pseudo-value model yields consistent natural effects under this nonlinear data-generating process, or substantially expand the simulations to include stronger nonlinearity, A×M interactions, and non-normal mediators, and qualify the 'unbiased' wording accordingly.
  2. [§4.2, §4.3.2, Fig. 3] The inference in the simulation studies is based on the multivariate delta method applied to standard linear-model output from lm(), which treats pseudo-values as independent observations. Pseudo-values are correlated by construction because each leave-one-out estimate shares the full-sample estimate (Eq. 6). The standard errors from lm() are therefore not valid for pseudo-value regression, and the reported type I error and coverage results do not provide reliable evidence of valid inference. The paper mentions that bootstrap should recalculate pseudo-values, but the simulations do not use this approach as the primary inference. The authors should either use a variance estimator that accounts for the pseudo-value dependence (e.g., bootstrap with recalculation, or a sandwich estimator) or explicitly state that the reported type I error/coverage results are for an ad hoc procedure whose repeated-sampling properties are not guaranteed.
  3. [§3.1, Step 4 and §6] The paper's robustness suggestion—comparing TE from the model-based decomposition with an independent estimate—is not reported in the simulations. Given that the pseudo-value outcome model is misspecified in the simulated scenarios, the degree of agreement between the product-method TE and the direct difference of estimands would directly quantify the bias induced by the linearity assumption. Adding this diagnostic to the simulation results would help readers assess whether the product method is acceptable in practice under realistic misspecification.
minor comments (6)
  1. [§2.1, Fig. 1 caption] The text says 'casual graph' in the caption; should be 'causal graph'.
  2. [§3.1, Eq. (14)] The outcome model includes confounders C but Eq. (1) in the background does not; this is fine but should be noted explicitly to avoid confusion about whether the earlier equations are unadjusted.
  3. [§4.1, Eq. (19)–(20)] The notation 'TTE' is used once and then 'T'; please define consistently.
  4. [§4.3.1, Fig. 2] The figure caption says 'Bias was evaluated across 10,000 simulation replicates', but the text states N ∈ {50,100,200} per arm; it would help to state the total number of replicates for each N and τ.
  5. [§4.3.2] The sentence 'the empirical type I error was 0.05 ... and 0.04 ... averaged across all cases considered' is vague; specify the estimands, time points, and sample sizes included in the average.
  6. [§5, Table 1] The table reports 'Proportion Mediated' with confidence intervals, but the method for obtaining the CI for the proportion is not described; clarify whether it is from the bootstrap or delta method.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the pseudo-value mediation derivation is self-contained and validated against independent benchmarks.

full rationale

The central method combines two established, externally supported building blocks: pseudo-values for survival estimands (Andersen et al., 2003; Andersen and Perme, 2010) and the standard linear-model product method for mediation (Baron and Kenny, 1986). The paper does not define pseudo-values in terms of the mediation effect, nor does it fit a parameter and then rename it as a prediction. The simulation truth is computed independently by Gauss-Hermite quadrature from the generative exponential model, and the pseudo-value estimates are compared with that truth, so the reported unbiasedness is not true by construction. The only self-citation of note is the SurvUtils package used for influence-function approximations, but the paper validates that approximation against the jackknife pseudo-values (R^2 = 1.00 across settings), making the citation non-load-bearing. The stated linear pseudo-value outcome model is a modeling assumption rather than a definitional identity; the possibility that it is misspecified under the exponential simulation is a correctness or robustness concern, not circularity. Likewise, the discussion's acknowledged limitation about conditioning on survival when measuring the mediator is a substantive identifiability issue, not a circular step. The derivation chain therefore does not reduce to its own inputs, and no circular pattern from the enumerated categories is present.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The method introduces no new postulates or fitted free parameters. It relies on established pseudo-value regression theory, standard mediation identification assumptions, and linear model specifications. The user-chosen time point tau is part of the estimand definition, not a fitted parameter.

assumptions (5)
  • standard math Pseudo-value regression theory: coefficients from GEE or linear regression on pseudo-values asymptotically target the conditional expectation of the estimand.
    Invoked throughout Section 3; from Andersen et al. 2003, treated as established.
  • standard math The pseudo-value estimator is regular and asymptotically linear with known influence function, so the jackknife and influence-function pseudo-values coincide asymptotically.
    Section 2.4, Eqs. (8)-(10).
  • domain assumption The mediator model E[M|A] = alpha_0 + alpha_A A (Eq. 13) and the pseudo-value outcome model E[Y|A,M,C] = beta_0 + beta_A A + beta_M M + beta_C C (Eq. 14) are correctly specified, with no treatment-mediator interaction.
    Step 3; the product method equals NDE/NIE only under these linear, no-interaction specifications.
  • domain assumption Sequential ignorability: no unmeasured confounding of the mediator-outcome relationship conditional on C, no interference, consistency, positivity.
    Section 2.1 and Step 3; standard mediation identification.
  • domain assumption Censoring is independent of event time and the mediator is measured after treatment but before the outcome without conditioning on survival in a way that biases the estimand.
    Section 2.2 and Section 5; KM/AJ estimators require independent censoring, and the re-baselined application conditions on survival to year 1, which is acknowledged as a limitation for terminal events.

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Cite this review

Pith. "Pith review of Simplifying Causal Mediation Analysis for Time-to-Event Outcomes using Pseudo-Values." pith.science (2026). https://pith.science/paper/RJSPJDNQ

@misc{pith2026241117533,
  author       = {Pith},
  title        = {Pith review of: Simplifying Causal Mediation Analysis for Time-to-Event Outcomes using Pseudo-Values},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RJSPJDNQ}},
  note         = {Machine review of arXiv:2411.17533}
}
read the original abstract

Mediation analysis for survival outcomes is challenging. Most existing methods quantify the treatment effect using the hazard ratio (HR) and attempt to decompose the HR into the direct effect of treatment plus an indirect, or mediated, effect. However, the HR is not expressible as an expectation, which complicates this decomposition, both in terms of estimation and interpretation. Here, we present an alternative approach which leverages pseudo-values to simplify estimation and inference. Pseudo-values take censoring into account during their construction, and once derived, can be modeled in the same way as any continuous outcome. Thus, pseudo-values enable mediation analysis for a survival outcome to fit seamlessly into standard mediation software (e.g. CMAverse in R). Pseudo-values are easy to calculate via a leave-one-observation-out procedure (i.e. jackknifing) and the calculation can be accelerated when the influence function of the estimator is known. Mediation analysis for causal effects defined by survival probabilities, restricted mean survival time, and cumulative incidence functions - in the presence of competing risks - can all be performed within this framework. Extensive simulation studies demonstrate that the method is unbiased across 324 scenarios/estimands and controls the type-I error at the nominal level under the null of no mediation. We illustrate the approach using data from the PARADIGMS clinical trial for the treatment of pediatric multiple sclerosis using fingolimod. In particular, we evaluate whether an imaging biomarker lies on the causal path between treatment and time-to-relapse, which aids in justifying this biomarker as a surrogate outcome. Our approach greatly simplifies mediation analysis for survival data and provides a decomposition of the total effect that is both intuitive and interpretable.

Figures

Figures reproduced from arXiv: 2411.17533 by the authors.

Figure 1
Figure 1. Causal Graph for Mediation. βA characterizes the direct effect of the treatment (A) on the outcome (Y). αA constitutes the effect of the treatment on the mediator, and βM the effect of the mediator on the outcome. When outcomes are continuous and modeled linearly, the product αAβM constitutes the mediated effect, and the sum of the direct and mediated effects βA + αAβM comprises the total effect the treatment on the… view at source ↗
Figure 2
Figure 2. presents the bias for the three mediation estimands on the survival probability, RMST, and cumulative incidence curve scales, in the case where the indirect and direct effect are both present (βA ̸= 0 and βM ̸= 0 in equation 20). The bias was evaluated across 10,000 simulation replicates. As expected, the variance of the effect size distribution decreases as the sample size increases. Overall, these results confirm … view at source ↗
Figure 3
Figure 3. P-values for pseudo-value based mediation effects are uniformly distributed under the null. Shown are uniform quantile-quantile plots for the observed vs. expected p-value distributions under the null scenario (βA = 0 and βM = 0). Performance is evaluated for the survival probability, RMST, and cumulative incidence curve mediation estimands at τ = 2 and sample sizes per-arm of N ∈ {50, 100, 200}. Adherence to the di… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Influence function approximation to the jackknife accelerates pseudo-value calculation without compromising accuracy. (a) Comparison of influence function versus jackknife calculation of the pseudo-value for the cumulative incidence function across sample sizes and tim…
Figure 5
Figure 5. Figure 5: Time-to-relapse cumulative incidence curves under FTY and INF- [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]

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