Recanting witness and natural direct effects: Violations of assumptions or definitions?

There have been numerous publications on the advantages and disadvantages of estimating natural (pure) effects compared to controlled effects. One of the main criticisms of natural effects is that it requires an additional assumption for identifiability, namely that the exposure does not cause a confounder of the mediator-outcome relationship. However, every analysis in every study should begin with a research question expressed in ordinary language. Researchers then develop/use mathematical expressions or estimators to best answer these ordinary language questions. When a recanting witness is present, the paper illustrates that there are no violations of assumptions. Rather, using directed acyclic graphs, the typical estimators for natural effects are simply no longer answering any meaningful question. Although some might view this as semantics, the proposed approach illustrates why the more recent methods of path-specific effects and separable effects are more valid and transparent compared to previous methods for decomposition analysis.