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Nonparametric efficient estimation of the longitudinal front-door functional

Marie S. Breum, Helene C. W. Rytgaard, Torben Martinussen, Erin E. Gabriel

Abstract

The front-door criterion is an identification strategy for the intervention-specific mean outcome in settings where the standard back-door criterion fails due to unmeasured exposure-outcome confounders, but an intermediate variable exists that completely mediates the effect of exposure on the outcome and is not affected by unmeasured confounding. The front-door criterion has been extended to the longitudinal setting, where exposure and mediator vary over time. However, with the exception of a simple plug-in estimator, no suitable estimation techniques have been proposed. In this work, we derive nonparametric efficient estimators of the longitudinal front-door functional. The estimators accommodate high-dimensional mediators, are multiply robust, and allow for the use of data-adaptive methods for estimating nuisance functions while still providing valid inference. The theoretical properties of the estimators are illustrated in a simulation study, and we apply the estimators to a trial of peanut allergy in infants.

Nonparametric efficient estimation of the longitudinal front-door functional

Abstract

The front-door criterion is an identification strategy for the intervention-specific mean outcome in settings where the standard back-door criterion fails due to unmeasured exposure-outcome confounders, but an intermediate variable exists that completely mediates the effect of exposure on the outcome and is not affected by unmeasured confounding. The front-door criterion has been extended to the longitudinal setting, where exposure and mediator vary over time. However, with the exception of a simple plug-in estimator, no suitable estimation techniques have been proposed. In this work, we derive nonparametric efficient estimators of the longitudinal front-door functional. The estimators accommodate high-dimensional mediators, are multiply robust, and allow for the use of data-adaptive methods for estimating nuisance functions while still providing valid inference. The theoretical properties of the estimators are illustrated in a simulation study, and we apply the estimators to a trial of peanut allergy in infants.

Paper Structure

This paper contains 39 sections, 15 theorems, 50 equations, 15 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Set-up and notation
  4. Front-door identification of the intervention-specific mean outcome
  5. Alternative interpretations of the longitudinal front-door functional
  6. Representations of the F-functional
  7. Nested expectation representations
  8. IPW representations
  9. Estimation
  10. Efficient influence function
  11. Estimation of the mediator density ratio
  12. Proposed estimators
  13. One-step estimator
  14. TMLE
  15. Large sample properties
  16. ...and 24 more sections

Key Result

Theorem 2.1

Under Assumptions ass:consistency-ass:positivity, $\Psi^{\bar{a}_T}(P)$ is identified via the longitudinal front-door functional where $\pi_t(a_t' \mid w, \bar{a}_{t-1}', \bar{m}_{t-1})$ and $g_t(m_t \mid w, \bar{a}_t, \bar{m}_{t-1})$ are the conditional densities of $A_t$ and $M_t$, $Q_Y( w, \bar{a}_T, \bar{m}_T) \equiv \mathbb{E}\left(Y \mid w , \bar{a}_T, \bar{m}_T\right)$ is the conditional

Figures (15)

  • Figure 1: (a) Example DAG in a setting where the Assumption 2 holds for one time-point. (b) Example DAG in a setting where the Assumption 2 holds for three time-points. The arrows from $W$ are suppressed, but arrows may exist to all measured variables and between $W$ and $U$ in either direction.
  • Figure 2: Example DAG with three time-points.
  • Figure 3: Simulation results for DGM (1). Absolute bias scaled by $\sqrt{n}$.
  • Figure 4: Simulation results for DGM (1). $\hbox{MSE}$ scaled by $n$.
  • Figure 5: Simulation results for DMG (1). Coverage probability.
  • ...and 10 more figures

Theorems & Definitions (24)

  • Theorem 2.1: Identification
  • proof
  • Theorem 4.1: EIF
  • proof
  • Corollary 1
  • Lemma 4.1: Multiple robustness
  • Corollary 2
  • Theorem 4.2: Asymptotic linearity
  • proof
  • Lemma S1
  • ...and 14 more