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Inference for Causal Path-Specific Effects in Longitudinal Studies with Repeated Treatments and Multiple Mediators

Xiaxian Ou, Xinwei He, Razieh Nabi

arXiv:2610.34349v1stat.MEmath.STstat.MLstat.TH

TL;DR

Longitudinal path-specific effects lack general semiparametric inference when repeated treatments are followed by intermediate-variable sequences that vary across occasions. The paper develops edge g-formula identification and two efficient-influence-function-based estimation approaches, alongside sensitivity analysis for selected identifying-assumption violations. It establishes multiple-robustness properties for the estimators and examines their performance in simulations and a Framingham application.

  • Problem

    General semiparametric inference is not available for edge g-formula path-specific effects when repeated-treatment settings have intermediate-variable sequences that vary across occasions.

    Source: §1 · pp. 2, 4

  • Method

    The paper derives two efficient-influence-function representations and one-step estimators for longitudinal path-specific effects, including a regression-based approach that avoids direct intermediate-variable density estimation.

    Source: §1 · p. 4

  • Results

    The estimators have multiple-robustness properties, and the paper examines finite-sample performance and robustness in simulations and applies the methods to sustained-smoking pathways through repeated BMI measurements in Framingham.

    Source: §1 · pp. 4–5

  • Takeaways & Limitations

    Sensitivity analysis lets investigators examine how estimates vary across scientifically plausible departures from selected cross-world independence restrictions.

    Source: §1 · p. 5

    Identification relies on the NPSEM-IE cross-world restrictions, along with consistency, positivity, and edge consistency.

    Source: §2 · p. 6

Abstract

from arXiv · show

In longitudinal studies, repeated treatments may be followed by different sequences of mediators and time-varying covariates. We develop semiparametric inference for path-specific effects in this setting, allowing the number, type, and ordering of intermediate variables to vary across treatment occasions. Under consistency, positivity, edge consistency, and the cross-world restrictions of a nonparametric structural equation model with independent errors, path-specific means are identified by the edge g-formula. We derive two parameterizations of the efficient influence function of the identifying functional. The first follows the observed-data factorization and uses conditional intermediate-variable densities. The second uses sequential regressions to replace the corresponding integrations and Bayes' rule to express density ratios through treatment-history regressions, thereby avoiding direct estimation of intermediate-variable densities. Each yields a one-step estimator that permits flexible nuisance estimation. We establish product-rate conditions for asymptotic linearity and semiparametric efficiency. The density-based estimator has blockwise multiple robustness, generalizing triply robust estimation, while the regression-based estimator has sequential multiple robustness. We further develop a sensitivity analysis for selected violations of cross-world independence assumptions used to identify these effects. Simulations examine finite-sample performance and robustness to model misspecifications. An application to the Framingham Heart Study examines sustained-smoking effects on systolic blood pressure along pathways through repeated body mass index measurements. The proposed methods are implemented in the R package flexPaths on Gihub at xxou/flexPaths.

1. Introduction

Longitudinal path-specific effects require general inference when repeated treatments are followed by intermediate-variable sequences that vary in number, type, and ordering. The paper develops semiparametric estimators for this setting and a sensitivity analysis for selected violations of identifying assumptions.

1. Introduction

  • Path-specific effects target specified causal pathways, but their identification can require stronger assumptions than total effects and can be obstructed by recanting witnesses or recanting districts.Alternative estimands avoid some cross-world restrictions but answer different questions; the paper focuses on path-specific effects when their assumptions are defensible.Source: §1 · pp. 2–3
  • Existing longitudinal formulations accommodate repeated treatments but do not provide general semiparametric inference for edge g-formula path-specific effects when intermediate-variable sequences vary across treatment occasions.The paper distinguishes this gap from work on more structured longitudinal effects and from approaches addressing unmeasured confounding.Source: §1 · pp. 3–4
  • The framework allows repeated treatments to be followed by varying sequences of discrete, continuous, or multivariate intermediate variables, and includes repeated-mediator longitudinal mediation as a special case.A node-wise assignment map specifies the treatment history associated with each intermediate variable; under the stated identification conditions, the path-specific mean is identified by the edge g-formula.Source: §1 · p. 4
  • Two equivalent efficient-influence-function representations yield one-step estimators: one retains intermediate-variable densities, while the other uses sequential regressions and treatment-history regressions to avoid direct density estimation.The paper gives product-rate conditions for asymptotic linearity and semiparametric efficiency, and establishes multiple robustness to nuisance-model misspecification.Source: §1 · p. 4
  • The paper parameterizes selected violations of cross-world independence restrictions and derives an observed-data representation and augmented estimator for sensitivity analysis.Investigators can examine how estimates vary across scientifically plausible patterns and magnitudes of departure from the identifying assumptions.Source: §1 · p. 5
  • Simulations examine finite-sample performance, efficiency, and robustness to nuisance-model misspecification; the Framingham application examines sustained-smoking effects on blood pressure through repeated BMI measurements.The methods are implemented in the R package flexPaths.Source: §1 · p. 5

2. Target parameter and identification

The section defines longitudinal data with repeated treatments and intermediate variables, then constructs nested counterfactual outcomes by assigning treatment values along selected causal pathways. It distinguishes variable versions by their downstream continuation to the outcome.

2. Target parameter and identification

  • The observed-data sequence comprises baseline covariates, repeated treatments, intermediate variables after each treatment, and the outcome.The number of intermediate variables after treatment A_k is denoted R_k; the notation assumes R_k > 0.Source: §2 · p. 5
  • Under the NPSEM-IE, each node is generated from its parents and an unrestricted structural function, with mutually independent exogenous errors; the observed-data law factorizes by node.The model also encodes the cross-world restrictions used for path-specific effect identification.Source: §2 · pp. 5–6
  • A nested counterfactual assigns each treatment its active or reference value according to whether the complete proper path from that treatment to the outcome belongs to the selected path collection.The assignment is made separately for each treatment occurrence in the recursively expanded outcome.Source: §2 · p. 6
  • When the same intermediate variable occurs on distinct downstream paths, its counterfactual versions are indexed by their continuation to the outcome and need not coincide.The continuation records the variable's position in the nested expression, not an intervention on downstream variables.Source: §2 · pp. 6–7

Appendix S3.1 illustrates this recursion.

The section states the assumptions for edge g-formula identification and explains how edge consistency permits a node-wise treatment assignment for intermediate variables. It also notes a graphical obstruction that can prevent identification.

Appendix S3.1 illustrates this recursion.

  • Path-specific effects are contrasts of path-specific means; additive contrasts relative to a reference treatment history follow by linearity.Source: Appendix S3.1 illustrates this recursion · p. 7
  • A recanting witness can obstruct nonparametric identification when the same intermediate variable must receive conflicting treatment values across paths.Source: Appendix S3.1 illustrates this recursion · p. 8
  • Edge consistency makes continuation-specific versions of each intermediate variable coincide, allowing their common treatment inputs to be encoded by a node-wise assignment map.The map assigns active or reference treatment values according to whether each treatment-to-variable edge lies on a path in the selected collection.Source: Appendix S3.1 illustrates this recursion · p. 8
  • Under consistency, NPSEM-IE cross-world restrictions, edge consistency, and positivity, the path-specific mean is identified by the edge g-formula.Source: Appendix S3.1 illustrates this recursion · pp. 7–8

3. Statistical representations and estimators

The section develops complementary representations and estimators for the identified path-specific-effect functional. Its one-step estimators use equivalent efficient-influence-function representations, with nuisance functions parameterized through either intermediate-variable densities or regressions.

3.1 Plug-in estimators

  • Direct plug-in estimation can require difficult density estimation and high-dimensional integration for continuous or multivariate intermediates, while retaining first-order sensitivity to nuisance-estimation error.The sequential-regression plug-in avoids direct intermediate-variable density estimation but remains first-order sensitive to nuisance error.Source: §3.1 · p. 10

3.2 Inverse probability weighted estimators

  • Weighting representations can use treatment-history regressions instead of conditional intermediate-variable densities, including when intermediates are continuous or multivariate.The density-ratio representation requires estimating the conditional density of the intermediate variable itself.Source: §3.2 · pp. 11–12

3.3 One-step estimators

  • The EIF combines the outcome residual and successive sequential-regression residuals, each with its corresponding weight.Cross-fitting fits nuisance functions outside each observation’s validation fold and controls the empirical-process term without Donsker conditions when the fitted EIF converges in L2(P).Source: §3.3 · pp. 13–14
  • The EIF has equivalent representations that support one-step estimation using either density-based or regression-based nuisance parameterizations.The density-based parameterization follows the observed-data factorization; the regression-based parameterization directly estimates sequential outcome and treatment-history regressions.Source: §3 · p. 10, §3.3 · p. 13

4. Inference and asymptotic properties

The section gives conditions for asymptotic linearity and efficiency and describes distinct robustness patterns for the two one-step estimators. Density-based robustness is organized by nuisance blocks; regression-based robustness can alternate across the longitudinal ordering.

4. Inference and asymptotic properties

  • Under the stated regularity conditions, op(n^-1/4) convergence for all nuisances is sufficient for the density-based estimator’s product-rate requirements.Unequal nuisance-estimation rates are also allowed when each pairwise rate condition holds.Source: §4 · p. 15
  • For the density-based one-step estimator, consistency holds when any S + 1 of its S + 2 nuisance blocks are consistently estimated.Thus, any one block may be misspecified while all others are consistently estimated; for S = 1 this is the two-out-of-three, or triply robust, property.Source: §4 · p. 15
  • The regression-based one-step estimator allows sequential multiple robustness, with correctly estimated treatment-history or sequential regressions alternating across intermediate variables.At each intermediate variable and block transition, at least one of the corresponding regression pair must be consistent, with a matching condition at the initial step.Source: §4 · pp. 16–17

5. Unmeasured confounding and sensitivity analysis

The sensitivity analysis examines selected departures from cross-world independence restrictions used to identify path-specific effects. It parameterizes departures through sensitivity functions and constructs an augmented estimator for the resulting sensitivity-indexed target.

5. Unmeasured confounding and sensitivity analysis

  • The illustration restricts attention to one non-outcome assignment pattern and discrete nodes in L, assigning compatible mediators to M.The approach retains the remaining identification conditions while relaxing selected restrictions linking intermediate variables to the outcome.Source: §5 · p. 17
  • Sensitivity functions describe how the conditional mean of the outcome counterfactual varies across levels of relevant intermediate-variable counterfactuals.Under cross-world independence, the two conditional means coincide and the sensitivity function is zero.Source: §5 · pp. 17–18
  • For fixed sensitivity functions, the augmented sensitivity estimator is consistent if all entering intermediate-variable laws are consistent and either the outcome regression or all treatment mechanisms are consistent.This is the stated double-robustness condition for the sensitivity functional’s estimator.Source: §5 · pp. 18–19

6. Simulation studies

The simulations assess finite-sample behavior, flexible nuisance estimation, and robustness to misspecification. Under correct specification, both estimators performed similarly; under misspecification, flexible estimation and protected robustness regimes improved centering.

Simulation design

  • The correct-specification experiment examined the estimators’ asymptotic behavior across sample sizes from 500 to 8000 using 1000 Monte Carlo samples per size.The target assignment pattern set M1, M2, and the outcome to reference treatment, and L1 and L2 to active treatment.Source: §6 · p. 19

Simulation 1: Finite-sample behavior under correct specification

  • Under correct specification, both estimators had negligible bias, decreasing standard deviations and mean squared errors, near-nominal 95% coverage, and nearly indistinguishable performance.The n-scaled variance approached the theoretical efficiency bound.Source: §6 · pp. 19–20, Table 1 · p. 20

Simulation 2: Misspecified parametric models vs. flexible estimation

  • With misspecified GLMs, bias persisted for both estimators, while Super Learner bias generally decreased toward zero; their Monte Carlo standard deviations were similar.Super Learner coverage rose from about 92% for n ≤ 2000 to 94% at n = 8000, while GLM coverage fell to about 88%.Source: §6 · p. 21, Table 2 · p. 21

Simulation 3: Robustness against misspecification

  • Both estimators remained well centered in their protected misspecification regimes, whereas appreciable bias remained when all nuisance models were misspecified.For the regression-based estimator, this pattern held at both endpoints and all four cut points.Source: §6 · pp. 22–23, Figure 2 · p. 23

7. Real data application

The application uses Framingham Heart Study data to examine sustained-smoking effects on Exam 7 systolic blood pressure through repeated BMI measurements. The estimated BMI-path effect was negative under both nuisance-estimation strategies, while total and complementary pathway intervals included zero.

Study setting

  • The analysis examined sustained smoking and systolic blood pressure at Exam 7 along pathways through repeated BMI measurements, using baseline covariates from Exam 1 and repeated measurements from later exams.The observed sequence included four smoking indicators and four BMI measurements.Source: §7 · pp. 23–24

Analytic cohort

  • The analysis included 2,850 participants who survived and remained under observation through Exam 7 and met the cohort’s baseline and analytic-variable criteria.The results should not be interpreted as effects in the baseline cohort without additional assumptions about survival and retention.Source: §7 · p. 25

Estimands

  • The BMI-path contrast compares its counterfactual mean with sustained nonsmoking, while the complementary-path contrast is the sustained-smoking mean minus the BMI-path mean.The two contrasts sum to the total-effect contrast.Source: §7 · p. 25

Estimated effects

  • The estimated BMI-path effect was negative under both strategies, with reductions of 1.82 mm Hg using GLMs and 0.65 mm Hg using Super Learner; total and complementary pathway confidence intervals included zero.GLM confidence intervals were substantially wider than Super Learner intervals.Source: §7 · p. 25, Table 4 · p. 25

8. Discussion

The discussion presents a node-wise formulation for longitudinal path-specific effects and contrasts two estimator parameterizations with different nuisance requirements and robustness properties. It identifies limits in sensitivity analysis and estimation for longer sequences.

Contribution

  • The node-wise formulation indexes treatment assignments to individual intermediate variables, accommodating changing numbers, types, and orderings across treatment occasions.It does not rely on labeling variables as mediators or time-varying confounders.Source: §8 · p. 26

Estimator properties

  • The density-based estimator has blockwise multiple robustness, whereas the regression-based estimator relies on combinations of treatment-history and sequential outcome regressions.The regression-based approach avoids conditional-density estimation, especially useful for continuous or multivariate intermediates; the density-based approach remains useful when its conditional laws are readily estimable or its robustness conditions are easier to satisfy.Source: §8 · p. 26

Limitations and future work

  • The sensitivity analysis is limited to one non-outcome assignment pattern and discrete intermediates; longer sequences may require unstable treatment-history regressions for sparsely represented patterns.Pooling related regressions could improve stability but would need to preserve the established robustness properties.Source: §8 · p. 26

Supplementary Materials for

The supplementary materials provide notation, proofs, additional examples, and further details for the simulations and real-data application. They also include dedicated materials on sensitivity analysis and robustness.

Appendices S1–S2

  • Appendix S1 summarizes notation, and Appendix S2 contains proofs covering identification, estimation, inference, and sensitivity analysis.The sensitivity-analysis proofs include identification and a doubly robust estimator.Source: Supplementary Materials for · p. 30

Appendix S3

  • Appendix S3 provides additional illustrative examples, including recursive counterfactual definitions, minimal treatment-assignment partitions, multiple robustness, and sensitivity analysis..Source: Supplementary Materials for · p. 30

Appendices S4–S5

S1. Glossary of terms and notations

Table S1 collects the manuscript’s notation for longitudinal treatments, intermediate variables, path-specific counterfactuals, estimators, sensitivity analysis, and the Framingham application. Supplement-only notation is listed separately in Table S2.

S1. Glossary of terms and notations

  • Table S1 is the comprehensive notation reference, while Table S2 lists notation introduced only in the supplementary materials.The glossary spans causal notation, estimation, sensitivity analysis, and application terms.Source: S1. Glossary of terms and notations · p. 31, Table S2 · p. 33, Table S1
  • Treatment occasions k and intermediate-node indices r organize the repeated-treatment history; C denotes baseline covariates and Y the final outcome.A_k is the binary treatment at occasion k, and X_k,r is intermediate node r in treatment block k.Source: S1. Glossary of terms and notations · p. 31
  • The path-specific intervention assigns each node a treatment input through τ(Z), defining nested counterfactuals and the target mean ψρ.The assignments depend on whether the complete proper path through a node belongs to the selected pathway set.Source: S1. Glossary of terms and notations · p. 31
  • The nuisance collections distinguish a density-based parameterization Q1=(π,f,μK,RK) from a regression-based parameterization Q2=(π,g,μseq).Here π denotes treatment mechanisms, f node densities, g treatment-history regressions, and μseq sequential regressions.Source: S1. Glossary of terms and notations · p. 31
  • The one-step estimator adds the estimated efficient influence function to the plug-in estimate, with cross-fitting training nuisance functions outside each validation fold.The glossary also names the second-order remainder and nuisance-rate indices used for inference.Source: S1. Glossary of terms and notations · p. 32
  • Sensitivity notation defines node-specific sensitivity functions and an adjusted target, while application notation maps treatments and BMI nodes to the Framingham study.In that application, A_k is smoking status at selected exams, L_k is BMI, and Y is Exam 7 systolic blood pressure.Source: S1. Glossary of terms and notations · p. 32

S2.1 Identification results

Under the stated assumptions, the path-specific counterfactual mean is identified by the edge g-formula. The identification argument obtains counterfactual node laws from observed conditional laws and combines them recursively.

S2.1 Identification results

  • Consistency and positivity are required for the relevant node-specific treatment assignments and histories.Positivity for the additional Bayes-ratio regressions is required only for the regression-based reparameterization, not for edge g-formula identification.Source: S2.1 Identification results · p. 34
  • The NPSEM-IE supplies the needed conditional independence through mutually independent exogenous errors, while edge consistency rules out incompatible treatment inputs along continuations.The stated no-recanting-witness condition is equivalent to excluding incompatible counterfactual values along included and excluded pathways.Source: S2.1 Identification results · pp. 34–35
  • Consistency, positivity, and independent errors let each counterfactual node law match an observed conditional law, which is then combined across nodes by the chain rule.The recursive construction applies the same argument to intermediate variables and the outcome before forming their joint distribution.Source: S2.1 Identification results · pp. 36–37
  • The edge g-formula gives the observed-data functional identifying the joint edge-intervention distribution of the intermediate variables and outcome.The identification result links the edge-intervention distribution to the functional in equation (4).Source: S2.1 Identification results · p. 37

S2.2 Estimation results

The section develops inference for the proposed estimators, covering equivalent density- and regression-based efficient influence function representations, efficiency and robustness properties, treatment-assignment partitions, and sensitivity analysis for cross-world dependence. It also reports simulation results under nuisance-model misspecification and additional Framingham cohort characteristics.

S2.2 Estimation results

  • Bayes-rule reparameterization makes the density-based and regression-based EIF representations algebraically equivalent, expressing density ratios through treatment-history probabilities.The equivalence holds for treatment histories for which the ratios are defined and denominators are positive.Source: S2.2 Estimation results · pp. 37, 41, 42
  • The EIF derivative includes contributions from the outcome, intermediate-variable, and baseline laws, but not directly from the treatment mechanism.The identifying functional does not directly involve the treatment mechanism.Source: S2.2 Estimation results · pp. 38, 41

S2.3 Inference results

  • Cross-fitting removes the Donsker requirement when the estimated EIF converges in L2, while without sample splitting a Donsker condition suffices for the empirical-process term.The results also assume treatment probabilities in EIF denominators are bounded below and impose likelihood-ratio multiplier and square-integrability conditions.Source: S2.3 Inference results · pp. 42–43
  • Every density-based remainder summand pairs errors from distinct nuisance blocks, and product-rate conditions make the remainder o_p(n^-1/2).The density-based analysis does not require continuous conditional densities to be globally bounded away from zero; it instead controls likelihood-ratio multipliers and remainder terms.Source: S2.3 Inference results · pp. 42, 43, 45
  • The density-based estimator has (S + 1)-out-of-(S + 2) robustness: its remainder vanishes when all but at most one nuisance block are consistently estimated.The blocks comprise treatment mechanisms, the outcome regression and intermediate-variable laws in M, and intermediate-variable laws in each Ls.Source: S2.3 Inference results · pp. 45–46
  • Product-rate conditions yield asymptotic linearity, asymptotic normality, and efficiency, while sequential-regression consistency requirements characterize additional robustness regimes.The sequential conditions require at least one of each specified pair of nuisance estimators to be consistent within blocks, between blocks, and initially; cut-point regimes are included.Source: S2.3 Inference results · p. 48

S2.4 Sensitivity analysis: Identification proof

  • The sensitivity-adjusted identifying functional recovers the original functional when all sensitivity functions are zero.The sensitivity functions represent departures from selected cross-world conditional independence assumptions.Source: S2.4 Sensitivity analysis:… · pp. 49, 50, 52

S2.5 Sensitivity analysis: Doubly robust estimator

  • The sensitivity-adjusted estimator is doubly robust in the outcome regression or treatment mechanisms, conditional on consistently estimated intermediate-variable laws.Both robustness regimes require the intermediate-variable laws to be consistently estimated; either the outcome regression or all treatment mechanisms may be misspecified.Source: S2.5 Sensitivity analysis: Doubly… · pp. 54–55

S3.1 Recursive definition of counterfactuals

  • Counterfactual examples show that mediator versions can differ by pathway-specific treatment assignments while sharing the same structural error.When assignments agree, the versions coincide; when they differ, a single unindexed counterfactual does not distinguish the inputs.Source: S3.1 Recursive definition of… · pp. 56–57

S3.2 Minimal treatment-assignment partitions

  • A K = 3 longitudinal example specifies the observed-data ordering and introduces a node-specific treatment-assignment map for a target satisfying edge consistency.Source: S3.2 Minimal treatment-assignment… · p. 58
  • Processing intermediate variables in reverse order and grouping assignments that are prefixes of existing vectors constructs a minimal collection; the preceding example requires only S = 1 vector.When no existing vector is compatible, the procedure creates one by appending zeros to the current node’s assignment.Source: S3.2 Minimal treatment-assignment… · p. 58

S3.3 Illustrations of multiple robustness

  • Corollary 1 gives two-out-of-three robustness for the density-based estimator: consistency follows when any two of the three blocks are consistently estimated and the remaining block may be misspecified.More generally, Corollary 1 establishes (S +1)-out-of-(S + 2) robustness as a sufficient, not necessary, condition. When BL2 = {f2,2} is misspecified, its last such node is X2,2, before A3; consistency of BM, BL1, bπ1, and bπ2 is sufficient even if bf2,2 and bπ3 are misspecified.Source: S3.3 Illustrations of multiple… · pp. 59, 60, 65
  • Regression-based robustness follows sequential conditions or cut points; in the unique-partition example, Bayes’ rule cancels g1,1 and µ1,1, while the nonunique example retains all five treatment-history regressions.The unique-example cancellations mean those functions need not be consistently estimated when the other requirements of the selected regime hold; the nonunique example has no such cancellation.Source: S3.3 Illustrations of multiple… · pp. 61, 62, 65
  • With nonunique partitions, compatible choices yield invariant estimators and EIFs, and the density-based estimator has four-out-of-five robustness with relaxations that can also permit π3 misspecification.The compatible partition can be chosen to maximize the selected misspecified block; in the example, the largest BL1 ends at X2,1, before A3.Source: S3.3 Illustrations of multiple… · p. 64

S3.4 Sensitivity analysis examples

  • The sensitivity examples cover a single-treatment setting with two ordered intermediates and a longitudinal setting with repeated treatments and two intermediate-variable blocks, using sensitivity functions to index selected cross-world dependence departures.Dependence within a common assignment class does not itself create a recanting district; dependence across nodes assigned different treatment histories is represented by the sensitivity functions.Source: S3.4 Sensitivity analysis examples · pp. 65–68
  • For each fixed sensitivity-function choice, consistency requires correct intermediate-variable laws and either a correct outcome regression or treatment mechanism.Source: S3.4 Sensitivity analysis examples · p. 69

S4.1 Simulation 1: Confirming theoretical properties

S4.2 Simulation 2: Misspecified parametric models versus flexible estimation

  • Super Learner substantially reduced persistent bias versus misspecified GLMs, with n-scaled variances relatively similar; misspecification primarily shifted estimator centering, not empirical dispersion.Figure 3 compares root-n scaled bias and n-scaled variance for both density-based and regression-based estimators.Source: S4.2 Simulation 2: Misspecified… · p. 76, Figure 3 · p. 76

S4.3 Simulation 3: Evaluating robustness of estimators against misspecification

  • Simulation 3 was designed to evaluate estimator robustness conditions using data generated from specified models, but the supplied passages do not report its results.Source: S4.3 Simulation 3: Evaluating… · p. 77
  • Component regressions for treatment-history regressions must be correctly specified whenever consistency of the corresponding joint regression is required.The component regressions are those for each treatment Aj entering the joint treatment-history regression.Source: S4.3 Simulation 3: Evaluating… · p. 80
  • Six regimes spanning two endpoints and four cut points, plus all-correct and all-misspecified configurations, assess consistency regimes without implying efficiency under partial nuisance correctness.The regimes correspond to the consistency results in Corollaries 1 and 2.Source: S4.3 Simulation 3: Evaluating… · p. 80

S5. Additional details on the real-data application

  • The regression-based one-step estimator used GLMs or Super Learner for nuisance functions, with 95% Wald intervals from the empirical variance of the estimated effect influence function and no probability truncation.The effect influence function was obtained as the difference between those for the two target means.Source: S5. Additional details on the… · p. 81
  • The analytic cohort contained 2,850 participants after exclusions and limited carry-forward of isolated missing measurements; it included 1,649 smokers, 1,023 nonsmokers, and 178 quitters.Continuous intermediate variables were retained on their original scales rather than discretized.Source: S5. Additional details on the… · p. 81
  • Baseline SBP was lowest among smokers, while nonsmokers were more often female and quitters tended to be older and more highly educated than current smokers.The cohort also differed across tobacco-use groups in occupational and marital-status distributions.Source: S5. Additional details on the… · pp. 81–82
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