Dynamic Mediation Index

• Dynamic Mediation Index is a class of time-resolved causal effect measures that quantify the proportion of treatment effects mediated over time.
• It integrates functional, Markov, VAR, and reinforcement learning frameworks to decompose immediate, delayed, and cumulative mediation effects.
• Applications span brain imaging, digital health, macroeconomics, and omics, employing methods such as spline expansion and latent factor models.

The Dynamic Mediation Index (DMI) is a class of time-resolved, pathway-specific causal effect measures designed to quantify the proportion or amount of a total treatment effect that is transmitted through intermediate variables—mediators—along a temporal dimension. Extensions of classical mediation analysis to dynamic settings have led to multiple formalizations of the DMI across longitudinal, functional, and high-frequency time series contexts, accommodating multivariate mediators, heterogeneous effects, and settings influenced by Markov/reinforcement learning structure. DMI enables the decomposition, quantification, and interpretation of evolving causal mechanisms in complex observational and experimental studies.

1. Formal Foundations of Dynamic Mediation Indices

Dynamic mediation analysis generalizes static mediation frameworks by permitting time-varying, potentially path-specific, or individualized effects. Canonical models include:

• Functional Mediation Models: Here, treatment $X(t)$, mediator $M(t)$, and outcome $Y(t)$ are processes over time $t$, and local coefficient functions (e.g., $\alpha(t)$, $\beta(t)$, $\gamma(t)$) parameterize time-specific relationships. The DMI at time $t$ is defined as

$$\mathrm{DMI}(t) = \frac{\mathrm{IE}(t)}{\mathrm{DE}(t)+\mathrm{IE}(t)}$$

where direct effect $\mathrm{DE}(t) = \gamma(t)$ and indirect effect $M(t)$0 in the concurrent model. Distributed-lag (historical) models further generalize this to allow for cumulative past-path effects (Zhao et al., 2018).

• Markov and Reinforcement Learning Models: Observed at discrete or infinite horizons, with treatment/action $M(t)$1, vector mediator $M(t)$2, and response $M(t)$3. DMI quantifies stagewise and cumulative mediation:

$M(t)$4

for mediator $M(t)$5, with further decompositions into immediate and delayed mediation effects (Luo et al., 2023, Ge et al., 2023).

• Impulse-Response Decomposition in VAR: For macroeconomic time series, DMI reflects the share of structural shock transmission via mediator variables by decomposing impulse-response functions (IRFs):

$M(t)$6

Here, $M(t)$7 is the IRF of $M(t)$8 from $M(t)$9 at lag $Y(t)$0, and $Y(t)$1 isolates the part due to the mediator $Y(t)$2 acting $Y(t)$3 periods after the shock (Dufour et al., 5 Sep 2025).

• Latent-Factor and Individualization Approaches: For high-dimensional and heterogeneous data, the DMI is individualized via latent factorization:

$Y(t)$4

where $Y(t)$5 and $Y(t)$6 are time- and individual-specific mediator and outcome coefficients. This enables extraction of both population and subject-level dynamic mediation profiles (Zhang et al., 2024).

2. Identification Assumptions and Causal Interpretation

Causal identification of DMI relies on time-dependent extensions of standard mediation conditions:

• Functional/Sequential Ignorability: No unmeasured confounding between treatment and mediator, or mediator and outcome, conditional on observed past and (if needed) latent factors (Zhao et al., 2018, Zhang et al., 2024).
• Consistency and No Interference: Observed outcomes align with the counterfactuals under the realized exposure trajectory; no spillover across units or time (Qian, 24 Jun 2025).
• Positivity: All relevant treatment–mediator–history combinations occur with nonzero probability.
• Orthogonality for Structural VAR: Correct identification of structural shocks is essential; omitted mediators or misspecified dynamics threaten identification (Dufour et al., 5 Sep 2025).

In reinforcement learning and Markov settings, sequential ignorability and Markovianity are needed for correct attribution of mediation effects along dynamic paths (Luo et al., 2023, Ge et al., 2023). Latent factor models necessitate a latent-variable ignorability assumption: adjustment for both observed and latent confounders suffices to block spurious causal paths (Zhang et al., 2024).

3. Estimation Procedures and Algorithmic Implementations

DMIs are typically estimated using a combination of regression, graphical model learning, penalization, and semiparametric methods:

• Spline/Basis Expansions: Time-varying coefficients are expressed in basis functions (e.g. B-splines, Fourier bases) and fit via penalized least squares with roughness penalties for smoothness (Zhao et al., 2018).
• Iterative Estimation for Multivariate Time Series: Estimation entails learning the dynamic structure (e.g., time-varying DAG among mediators), recursive regression for path coefficients, and aggregation over paths and time (Luo et al., 2023).
• Impulse-Response Decomposition and Bootstrapping: VAR coefficients are estimated by OLS, IRFs computed, the mediator’s contribution decomposed per the recursive formula, and inference via block bootstrap (Dufour et al., 5 Sep 2025).
• Multiply-Robust and Efficient Influence Function (EIF) Estimators: For high-dimensional and machine-learning-based settings (e.g., intensive longitudinal data), EIFs for natural excursion effects allow for doubly- and multiply-robust estimation, with cross-fitting for valid inference (Qian, 24 Jun 2025).
• Low-Rank and Sparse SEMs for Individualized Effects: Estimation uses nuclear-norm minimization, fused lasso penalties for smoothness, and SCAD or $Y(t)$7 penalties for sparsity in mediator selection. Singular value decomposition facilitates efficient global optimization (Zhang et al., 2024).

The table below summarizes major DMI estimation strategies and their contexts:

Model Class	Estimation Approach	Key Reference
Functional mediation	Spline basis, penalized least squares	(Zhao et al., 2018)
Multivariate time series (RL/Markov)	Recursive regression, DAG learning, bootstrapping	(Luo et al., 2023)
Structural VAR (macroeconomics)	IRF decomposition, local projections, VAR bootstrap	(Dufour et al., 5 Sep 2025)
Intensive longitudinal (MRTs)	EIFs, multiply-robust, ML for nuisances	(Qian, 24 Jun 2025)
Latent factor, individualized	Nuclear-norm/fused lasso, SVD, sparse regression	(Zhang et al., 2024)

4. Path-Specific Interpretation and Decomposition

DMIs allow intricate decomposition of dynamic causal effects:

• Time-Point Specific Proportion: In functional models, $Y(t)$8 indicates at each instant $Y(t)$9 the fraction of the total effect that passes through the mediator. If $t$0, mediation dominates; if $t$1, direct effects dominate (Zhao et al., 2018).
• Immediate and Delayed Effects: RL and Markov settings admit partition into instantaneous mediation (IIME) and delayed/carryover mediation (DIME), identifying both acute and persistent transmission mechanisms (Luo et al., 2023, Ge et al., 2023).
• Distributed Temporal Mediation: In VAR analysis, the DMI at lag $t$2 quantifies the role of the mediator in propagating shocks at specific time horizons, thereby characterizing front-loaded versus delayed mediation channels (Dufour et al., 5 Sep 2025).
• Heterogeneity and Personalization: Latent factor approaches yield individual- and time-specific mediation indices $t$3, supporting subgroup analysis and dynamic clustering for highly heterogeneous systems (Zhang et al., 2024).

5. Applications and Empirical Results

Dynamic Mediation Indices have been applied in diverse domains:

• Functional brain imaging: The DMI(t) recovers temporal “mediation windows” in neural response, uncovering the phases when brain regions act as principal conduits for stimulus-to-response transmission (Zhao et al., 2018).
• Mobile health and digital behavioral intervention: In digital trial data, DMIs decompose direct prompt effects and mediation via physiological signals, revealing, e.g., negative mood effects mediated via resting heart rate and sleep quality (Luo et al., 2023, Qian, 24 Jun 2025).
• Macroeconomics: DMI quantifies, across horizons post-policy shock, the importance of investor sentiment versus credit risk in explaining output changes. In one empirical example, investor sentiment accounted for 60% of GDP response three months after a monetary tightening, while default risk mediated negligible effect (Dufour et al., 5 Sep 2025).
• Omics and personalized medicine: Latent factor DMIs expose subject-level, time-resolved mediation signatures in DNA methylation–mediated mechanisms, supporting variable selection and profiling in high-dimensional systems (Zhang et al., 2024).

6. Methodological Limitations and Extensions

Key limitations across DMI frameworks include:

• Parametric Assumptions: Many estimation methods assume linearity and additive SEM structures, which may be violated in practice.
• Confounder Control: Identification critically depends on adequately modeling observed, time-varying, and possibly latent confounders.
• Learning Complex Structures: For multivariate or high-dimensional mediators, accurate DAG learning and penalization are challenging and computationally expensive.
• Stationarity and Temporal Alignment: Infinite-horizon or stationary approximations may not capture nonstationary or irregularly sampled systems.

Extensions proposed in the literature include nonlinear and generalized linear models, adaptation to irregular and heterogeneous sampling, and the incorporation of latent/unmeasured mediators or time-varying confounding (Luo et al., 2023, Zhang et al., 2024, Dufour et al., 5 Sep 2025).

7. Connections to Granger and Sims Causality, and Unified Causal Mediation

A key contribution of the DMI concept is the formal linkage of time-series (Granger, Sims) and structural causal mediation perspectives. The DMI operationalizes how a mediator’s role can be mined from standard VAR or SEM output, translating IRFs (Sims causality, total effect) into mediation mechanisms (Granger causality as mediation path). Absence of Granger causality (across lags or mediators) is sufficient for null mediation, unifying statistical predictability and causal pathway attribution in dynamic systems (Dufour et al., 5 Sep 2025).

This connection underlines the broad utility of the DMI, rendering it a theoretically principled and practically implementable tool for characterizing, quantifying, and visualizing temporal mediation across domains.