crumble: A comprehensive framework for modern causal mediation analysis with intermediate confounding
Abstract
Causal mediation analysis is widely used to investigate how causal effects operate through specific pathways linking treatments or exposures to outcomes. Recently, \texttt{crumble} was developed to enable nonparametric estimation of several mediation parameters, even when mediators are continuous and/or multi-dimensional or when treatments are non-binary. But a practical and accessible guide to using \texttt{crumble} -- one that does not require deep familiarity with mediation analysis or semiparametric theory -- is currently lacking. This tutorial aims to an accessible introduction to \texttt{crumble} while minimizing technical complexity. We first review the mediation parameters implemented in \texttt{crumble} -- natural direct and indirect effects, randomized interventional effects, and recanting-twin effects. For each, we give the definition, interpretation, identification assumptions, and suitability in the presence or absence of intermediate confounding. Then, we demonstrate the usage of \texttt{crumble} by examining an example configuration. Next, we describe how \texttt{crumble} accommodates non-binary treatments through modified treatment policies. Finally, we illustrate the practical use of \texttt{crumble} through two case studies -- one with a binary treatment and one with a non-binary treatment -- based on the Job Search Intervention Study data.
Cite
@article{arxiv.2604.09902,
title = {crumble: A comprehensive framework for modern causal mediation analysis with intermediate confounding},
author = {Richard Liu and Nicholas T. Williams and Kara E. Rudolph and Ivan Diaz},
journal= {arXiv preprint arXiv:2604.09902},
year = {2026}
}