Path-specific harm decomposition: A partial identification framework

Ruizi Yan; Dennis Frauen; Maresa Schröder; Stefan Feuerriegel

stat.MLSep 24, 2026

Path-specific harm decomposition: A partial identification framework

Authors: Ruizi Yan, Dennis Frauen, Maresa Schröder, Stefan Feuerriegel

Organizations: Department of Statistics, University of Oxford · LMU Munich · Munich Center for Machine Learning (MCML)

Abstract

A central goal when designing treatment policies is often to "do no harm", that is, to avoid interventions that improve average outcomes while worsening outcomes for some individuals. A widely used notion for harm is the fraction of negatively affected (FNA), defined as the probability that an intervention decreases an individual's outcome. However, in many applications, treatments operate through mediators, and a single "total" FNA can obscure whether harm arises primarily through direct pathways or indirect (mediator-induced) pathways. In this work, we introduce a path-specific analogue of the FNA. For this, we disentangle total harm into direct and indirect harm in causal mediation settings. However, these quantities depend on joint distributions of potential outcomes that are not point-identified even in randomised controlled trials. As a remedy, we develop a novel partial identification framework for direct and indirect FNA. In our framework, we (i) derive sharp Makarov bounds for the FNA, and (ii) propose a semiparametrically efficient estimator with valid confidence intervals for these bounds under mild margin conditions. We demonstrate our framework across various numerical experiments. To the best of our knowledge, we are the first to study path-specific decomposition of causal harm and to develop an orthogonal inference framework for its analysis.

Figures & tables

Work	Harm notion	Level	Path-specific	Identification	Orthogonal	Cov.-assisted
[ 3 ]	Actual causality	event	✗	—	✗	✗
[ 48 ]	Negative ATE	pop.	✗	point	✗	✗
[ 40 ]	Probability of necessity	pop.	✓	partial	✗	✗
[ 18 ]	Total FNA	pop.	✗	partial	✗	✗
[ 9 ]	Total FNA	pop.	✗	partial	✗	✓
[ 21 ]	Total FNA	pop.	✗	partial	✓	✓

Table 1 : Overview of related work on modelling harm.

Pathway	Average effect	True FNA
Direct	$\mathrm{NDE}=1.154$	$\text{FNA}_{\mathrm{dir}}=0.410$
Indirect	$\mathrm{NIE}=0.067$	$\text{FNA}_{\mathrm{ind}}=0.223$
Total	$\mathrm{ATE}=1.220$	$\text{FNA}_{\mathrm{tot}}=0.220$

Table 2 : Oracle average effects and true path-specific FNA.

$d$	Pathway	FNA	Oracle	Estimator	Mean est.	Bias	$\mathbf{Coverage}$	Width
$d=3$	Direct	$0.410$	$[0.220,\,0.450]$	Plug-in	$[0.124,\,0.492]$	$(-0.096,\,+0.042)$	${1.000}$	$0.368$
	Direct	$0.410$	$[0.220,\,0.450]$	Orthogonal	$\mathbf{[0.156,\,0.463]}$	$(\mathbf{-0.064},\,\mathbf{+0.013})$	$\mathbf{1.000}$	$\mathbf{0.307}$
	Indirect	$0.223$	$[0.177,\,0.797]$	Plug-in	$[0.037,\,0.954]$	$(-0.140,\,+0.157)$	$1.000$	$0.917$
	Indirect	$0.223$	$[0.177,\,0.797]$	Orthogonal	$\mathbf{[0.128,\,0.850]}$	$(\mathbf{-0.049},\,\mathbf{+0.053})$	$\mathbf{1.000}$	$\mathbf{0.722}$
	Total	$0.220$	$[0.082,\,0.333]$	Plug-in	$[0.104,\,0.475]$	$(+0.022,\,+0.142)$	$0.070$	$0.371$
	Total	$0.220$	$[0.082,\,0.333]$	Orthogonal	$\mathbf{[0.069,\,0.361]}$	$(\mathbf{-0.013},\,\mathbf{+0.028})$	$\mathbf{1.000}$	$\mathbf{0.292}$

Table 3 : Finite-sample (

n=5000

) performance for estimating covariate-assisted FNA bounds. Results: mean over 300 simulations. Bold indicates the better-performing estimator within each pathway.

Pathway	Estimator	Est. bounds	Conservative set
Direct	Plug-in	$[0.122\ (0.003),\,0.615\ (0.005)]$	$[0.116,\,0.625]$
Direct	Orth.	$\mathbf{[0.319\ (0.023),\,0.459\ (0.043)]}$	$\mathbf{[0.274,\,0.542]}$
Indirect	Plug-in	$[0.036\ (0.000),\,0.996\ (0.000)]$	$[0.035,\,0.997]$
Indirect	Orth.	$\mathbf{[0.042\ (0.039),\,0.935\ (0.018)]}$	$\mathbf{[0.000,\,0.969]}$
Total	Plug-in	$[0.133\ (0.004),\,0.639\ (0.005)]$	$[0.126,\,0.650]$
Total	Orth.	$\mathbf{[0.337\ (0.014),\,0.482\ (0.018)]}$	$\mathbf{[0.310,\,0.517]}$

Table 4 : Estimated FNA bounds on the MTO data. Standard errors are shown in parentheses. The conservative set is obtained by expanding the estimated lower and upper bounds by

1.96

standard errors. Plug-in standard errors ignore nuisance estimation and are reported for reference only.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Notation	Meaning
Abbreviations
FNA	Fraction negatively affected.
CDF	Cumulative distribution function.
EIF	Efficient influence function.
DML	Double/debiased machine learning.
RCT	randomised controlled trial.

Appendix

Table 5 : Summary of notations and abbreviations used throughout the paper.

Component	Parameter	Value
Treatment model	$\alpha_{A}$ intercept	$0.20$
Mediator model	$\alpha_{M}$ intercept	$-0.90$
Mediator model	$\beta_{M}$ treatment effect on mediator	$2.00$
Outcome model	$d_{L},\,d_{R}$ direct effect limits	$3.00,\ -0.75$
	$q_{L},\,q_{R}$ mediator effect limits	$-1.10,\ 1.20$
	$k$ transition sharpness	$6.0$

Appendix

Table 6 : Parameter values common to all designs.

	3 covariates			5 covariates			10 covariates
	$\eta_{A}$	$\eta_{M}$	$\eta_{Y}$	$\eta_{A}$	$\eta_{M}$	$\eta_{Y}$	$\eta_{A}$	$\eta_{M}$	$\eta_{Y}$
$X_{1}$	$0.35$	$0.15$	$0.10$	$0.35$	$0.15$	$0.10$	$0.35$	$0.15$	$0.10$
$X_{2}$	$-0.25$	$0.40$	$0.30$	$-0.25$	$0.40$	$0.30$	$-0.20$	$0.30$	$0.25$
$X_{3}$	$0.50$	$-0.30$	$-0.20$	$0.50$	$-0.30$	$-0.20$	$0.40$	$-0.25$	$-0.15$
$X_{4}$	–	–	–	$-0.15$	$0.20$	$0.25$	$-0.15$	$0.20$	$0.20$
$X_{5}$	–	–	–	$0.30$	$-0.10$	$-0.15$	$0.25$	$-0.10$	$-0.10$

Appendix

Table 7 : Index-function coefficients for the designs of Section 6 . Entries are the coefficients of

\eta_{A}

\eta_{M}

, and

\eta_{Y}

in Eqs. ( 59 ) and ( 60 ). A dash marks a covariate not present in the design.

Design	Index functions
Polynomial	$\eta_{A}(x)=0.35x_{1}-0.25x_{2}+0.50x_{3}+0.15x_{1}^{2}-0.10x_{2}^{2}+0.20x_{1}x_{2}$
	$\eta_{M}(x)=0.15x_{1}+0.40x_{2}-0.30x_{3}-0.10x_{1}^{2}+0.20x_{1}x_{2}$
	$\eta_{Y}(x)=0.10x_{1}+0.30x_{2}-0.20x_{3}+0.15x_{1}^{2}-0.10x_{1}x_{2}$
Sinusoidal	$\eta_{A}(x)=0.55\sin x_{1}-0.40\cos x_{2}+0.50x_{3}$
	$\eta_{M}(x)=0.35\sin x_{1}+0.25\cos x_{2}-0.30x_{3}$
	$\eta_{Y}(x)=0.25\sin x_{1}+0.35\cos x_{2}-0.20x_{3}$

Appendix

Table 8 : Index functions for the nonlinear designs. The covariate law is that of the three-covariate design, and all parameters in Table 6 are unchanged.

DGP	Pathway	FNA	Oracle	Estimator	Mean est.	Bias	$\mathbf{Coverage}$	Width
Sinusoidal	Direct	$0.410$	$[0.205,\,0.450]$	Plug-in	$[0.102,\,0.507]$	$(-0.103,\,+0.057)$	$1.000$	$0.406$
	Direct	$0.410$	$[0.205,\,0.450]$	Orthogonal	$\mathbf{[0.137,\,0.469]}$	$\mathbf{(-0.068,\,+0.019)}$	$\mathbf{1.000}$	$\mathbf{0.332}$
	Indirect	$0.223$	$[0.183,\,0.794]$	Plug-in	$[0.024,\,0.972]$	$(-0.158,\,+0.179)$	$1.000$	$0.948$
	Indirect	$0.223$	$[0.183,\,0.794]$	Orthogonal	$\mathbf{[0.145,\,0.836]}$	$\mathbf{(-0.037,\,+0.043)}$	$\mathbf{1.000}$	$\mathbf{0.691}$
	Total	$0.220$	$[0.070,\,0.332]$	Plug-in	$[0.092,\,0.499]$	$(+0.022,\,+0.168)$	$0.123$	$0.407$
	Total	$0.220$	$[0.070,\,0.332]$	Orthogonal	$\mathbf{[0.055,\,0.356]}$	$\mathbf{(-0.015,\,+0.025)}$	$\mathbf{1.000}$	$\mathbf{0.301}$

Appendix

Table 9 : Finite-sample (

n=5000

) performance for estimating covariate-assisted FNA bounds under nonlinear DGPs. Results: mean over 300 simulations. Bold indicates the better-performing estimator within each pathway.

\widehat{F}_{a,a^{\prime}}^{(-k)}(t\mid x)=\widehat{\mathbb{E}}_{-k}\{\widehat{\Gamma}_{a,a^{\prime}}^{t}(O)\mid X=x\}.

Appendix

Algorithm 1 Two-stage orthogonal estimation of path-specific FNA bounds

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This paper studies the problem of identifying the treatment that maximizes the expected natural direct potential outcome (NDPO), which captures the potential outcome of an intervention while excluding the pathway transmitted through a mediator that researchers may wish to remove from evaluation. We first establish population-level identification of the expected NDPO in a causal bandit setting using observable interventional distributions. We then develop a fixed-confidence best-arm identification (BAI) algorithm based on the Track-and-Stop (TaS) framework, employing a cutting-set method to solve the resulting semi-infinite optimization problem. The proposed algorithm achieves sample-efficient identification with a high-probability correctness guarantee. We prove that it satisfies

δ

-correctness and asymptotic optimality. Finally, we validate the approach through empirical evaluations on a large-scale real-world advertising dataset (IPinYou).

Abstract

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