Sequential Transport for Causal Mediation Analysis

Agathe Fernandes Machado; Iryna Voitsitska; Arthur Charpentier; Ewen Gallic

stat.MEMar 16, 2026

Sequential Transport for Causal Mediation Analysis

Authors: Agathe Fernandes Machado, Iryna Voitsitska, Arthur Charpentier, Ewen Gallic

Organizations: Université du Québec à Montréal (UQAM), Canada · Ukrainian Catholic University, Ukraine · Kyoto University, Japan · Aix Marseille Université, CNRS, AMSE, Marseille, France ·

Abstract

We propose sequential transport (ST), a distributional framework for mediation analysis that combines optimal transport (OT) with a mediator directed acyclic graph (DAG). Instead of relying on cross-world counterfactual assumptions, ST constructs unit-level mediator counterfactuals by minimally transporting each mediator, either marginally or conditionally, toward its distribution under an alternative treatment while preserving the causal dependencies encoded by the DAG. For numerical mediators, ST uses monotone (conditional) OT maps based on conditional CDF/quantile estimators; for categorical mediators, it extends naturally via simplex-based transport. We establish consistency of the estimated transport maps and of the induced unit-level decompositions into mutatis mutandis direct and indirect effects under standard regularity and support conditions. When the treatment is randomized or ignorable (possibly conditional on covariates), these decompositions admit a causal interpretation; otherwise, they provide a principled distributional attribution of differences between groups aligned with the mediator structure. Gaussian examples show that ST recovers classical mediation formulas, while additional simulations confirm good performance in nonlinear and mixed-type settings. An application to the COMPAS dataset illustrates how ST yields deterministic, DAG-consistent counterfactual mediators and a fine-grained mediator-level attribution of disparities.

Figures & tables

Panel of Figure 1 — Figure 1: DAG for the simulated data.

Figure 2: Estimation (200 replicated samples,

n=600

) of average indirect effect

\bar{\delta}

, average direct effect

\bar{\zeta}

and average total causal effect

\bar{\tau}

Panel of Figure 3 — Figure 3: Simplified DAG for the COMPAS dataset.

Figure 4: Individual mutatis mutandis effects (distributional decomposition) for the COMPAS dataset estimated with counterfactual-based methods using Fairadapt (FPT-RF), optimal transport (OT), penalized OT (SKH), and sequential transport (ST) according to the causal graph of Figure 3 .

Metric	CM	FPT	NF	OT	SKH	ST
$\bar{\delta}$	-0.09	0.00	-0.06	-0.06	0.04	-0.06
$\bar{\zeta}$	0.00	-0.03	0.00	-0.03	-0.04	-0.02
$\bar{\tau}$	-0.09	-0.03	-0.06	-0.09	0.00	-0.08

Table 1: Estimated average causal effects for COMPAS , based on causal mediation analysis with LSEM using mediation package (CM), with counterfactual-transport-based-methods: optimal transport (OT), penalized OT (SKH), sequential transport (ST) and fairadapt (FPT) according to the DAG of Figure 3 , and using normalizing flows (NF).

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: Untreated units (green dots), treated Units (yellow dots), and transported units (blue triangles) under optimal transport (OT), entropic regularization with Sinkhorn algorithm (SKH) and sequential transport: ST(1) (transporting

X_{1}

first and then

X_{1}\to X_{2}

) and ST(2) (transporting

X_{2}

first and then

X_{2}\to X_{1}

). Lines indicate the transport mappings.

Figure 6: Estimated

\tau_{i}

for the two untreated units (black squares),

\boldsymbol{x}_{i_{1}}=(-1.5,-1.4)

(left) and

\boldsymbol{x}_{i_{2}}=(-2.4,-3.8)

(right), using different transport methods. Green dots: untreated; yellow dots: treated.

Metric	i	OT	OT-M	SKH	ST(1)	ST(2)
$\delta_{i}$	1	1.3	0.4	0.5	-0.4	1.4
$\delta_{i}$	2	4.2	4.1	1.0	1.1	3.9
$\zeta_{i}$	1	2.7	3.2	3.2	3.7	3.5
$\zeta_{i}$	2	4.5	7.7	3.4	3.3	7.6
$\tau_{i}$	1	4.0	3.5	3.7	3.3	4.9
$\tau_{i}$	2	8.8	11.8	4.4	4.4	11.9

Appendix

Table 2: Estimated individual causal effects for two units,

\boldsymbol{x}_{i_{1}}=(-1.5,-1.4)

and

\boldsymbol{x}_{i_{2}}=(-2.4,-3.8)

, based on counterfactuals build using optimal transport (OT), OT-Based matching (OT-M), penalized OT (SKH), and two sequential appoaches

X_{1}\rightarrow X_{2}

(ST(1)) and

X_{2}\rightarrow X_{1}

(ST(2)).

Figure 7: Individual causal effects for the Gaussian toy example estimated with counterfactual-based methods using optimal transport (OT), penalized OT (SKH), and two sequential approaches

X_{1}\rightarrow X_{2}

(ST(1)) and

X_{2}\rightarrow X_{1}

(ST(2)). The red dashed line depicts theoretical expectations.

Figure 8: Example of simulated data for the toy example of section D.1 for various distances between the means of the Gaussian distributions in groups 0 and 1.

Figure 9: Estimation error of the average causal effects ( $\bar{\delta}$ , $\bar{\zeta}$ , $\bar{\tau}$ ) across 200 replicated samples ( $n=500$ ), in the Gaussian case with two mediators (toy example of section D.1 ), for increasing values Of $\alpha$ .

Figure 10: Estimation error of the average causal effects (

\bar{\delta}

\bar{\zeta}

\bar{\tau}

) across 200 replicated samples (

n=500

), in the Gaussian case with two mediators (toy example of section D.1 ), for increasing proportion of untreated.

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