Introduction

Identifying a causal effect is hard. Decomposing it into the portion that operates through a specific pathway (a mediator) is harder still. Mediation analysis has been a standard tool in psychology and public health for decades and a source of methodological controversy in causal inference for just as long.

The central question of mediation analysis is: does treatment D affect outcome Y directly, or indirectly through a mediator M? For example: does education raise earnings because it builds cognitive skills ((D → M → Y, the indirect pathway), because it signals ability to employers ((D → Y the direct pathway), or both? Or: does a conditional cash transfer improve children's long-run earnings because it keeps them in school ($M$ = years of schooling) or for other reasons (nutrition, reduced child labour)?

These questions are scientifically important. Answers guide policy: if the education effect operates entirely through signalling, subsidising schooling has different implications than if it operates through skill formation. Yet the causal inference community is deeply divided on whether natural direct and indirect effects can be identified, and at what cost.

1 The Framework: Natural Direct and Indirect Effects

The potential outcomes notation for mediation involves the mediator's potential value: M_i(d) is the mediator under treatment D = d, and Y_i(d,m) is the outcome under treatment D=d and mediator M=m. The Natural Direct Effect (NDE) and Natural Indirect Effect (NIE) are (Pearl, 2001; Robins and Greenland, 1992):

The total effect decomposes as TE = NDE+NIE

Equation (1) is the effect of D on Y when the mediator is held at its natural value under no treatment: M(0). This is a cross-world quantity: Y(1,M(0)) involves simultaneously setting D=1 (in one counterfactual world) and M to its value when $D=0$ (in another).

Equation (2) is the change in Y due to the change in M induced by D, with D set to its treated value throughout.

Condition (3) involves potential outcomes from two different treatment regimes simultaneously: Y(d, m) (under D = d, M = m) and M(d′) (under D = d′). No experiment can simultaneously assign the same unit to two treatment conditions; no data—experimental or observational—can test this assumption. It is empirically unfalsifiable.  

Mediator-outcome confounding is fatal and unresolvable. Even with a randomised treatment D, the mediator M is not randomised. Any unmeasured variable U that affects both M and Y (a mediator-outcome confounder) violates (3) and biases the NDE and NIE. Crucially, unlike treatment-outcome confounding (which can be addressed by randomising D) or instrument-outcome confounding (which can be addressed by choosing better instruments), there is no design-based way to eliminate mediator-outcome confounding. Robins (2003) showed that in non-linear models, the NDE is generally non-identified even with a randomised treatment unless strong parametric assumptions are imposed.  

The classic Baron-Kenny approach is worse. The widely used Baron-Kenny (Baron and Kenny, 1986) sequential regression approach—regress M on D, then regress Y on D and M—can be shown to estimate NDE and NIE only under the most restrictive conditions (no treatment-mediator interaction, no mediator-outcome confounding, linear models). In the presence of treatment-mediator interaction or non-linearity, it estimates the wrong quantities and can yield the wrong sign (Imai et al., 2010).  

3 The Case For: Identifiability Under Weaker Assumptions

Sensitivity analysis saves the day. Imai et al. (2010) develop a comprehensive framework for causal mediation analysis that makes the key assumption—no unmeasured mediator-outcome confounding—explicit and proposes a formal sensitivity analysis. Researchers report: "the NIE would be zero if the residual correlation between M and Y were ρ* = 0.3"—turning an untestable assumption into a quantitative statement that domain experts can evaluate. The R package mediation implements this framework.  

Design-based identification without cross-world assumptions. Imai et al. (2010) also show that by using two separate randomised encouragements—an instrument Z_D for D and an instrument Z_M for M in a factorial encouragement design—natural indirect effects can be identified without the cross-world assumption. The design manipulates both treatment and mediator independently, giving separate first-stage variation for each. While this design is logistically demanding, it is feasible in some experiments (e.g., randomising both whether a job training programme is offered and whether a specific skill module is delivered).  

Principal stratification avoids cross-world quantities. Frangakis and Rubin (2002) propose defining direct effects in terms of the principal strata: subpopulations defined by the joint value of (M(0), M(1)). The principal stratum direct effect (PSDE) for the stratum M(0) = M(1) = m is:   

This is the direct effect in the subpopulation where the mediator would be m regardless of treatment—it avoids cross-world potential outcomes entirely. The PSDE requires only no unmeasured confounding of the treatment-outcome relationship among units with the same (M(0), M(1))—a much weaker condition than (3).  

Controlled direct effects are identified under weaker conditions. VanderWeele (2015) emphasises the controlled direct effect:   

the effect of D when M is externally set to m (not at its natural value). This is a do-interventional quantity in Pearl's language and requires only no mediator-outcome confounding conditional on X—no cross-world assumption. The CDE answers a different question (what happens if we could control M to equal m?) but is often more policy-relevant and is always better identified.

4 The Synthesis Position

The debate reveals a genuine scientific tension: the most interesting mediation estimands (NDE, NIE) require the most controversial assumptions; the best-identified estimands (CDE, principal stratum effects) answer slightly different questions.

The way forward is not to abandon mediation analysis, but to:

1. Report sensitivity analyses for the cross-world assumption and mediator-outcome confounding as a default.
2. Consider principal stratification or CDE as the primary estimand when the identifying assumptions for NDE/NIE are implausible.
3. Invest in design-based solutions (factorial encouragement designs) where logistically feasible.
4. Be explicit about what quantity is being estimated and under what assumptions.

5 What Would Help Resolve the Debate?

1. More empirical studies using factorial encouragement designs, which can identify NIE without cross-world assumptions and provide a benchmark for sensitivity analysis.
2. Consensus on reporting standards: all mediation analyses should report sensitivity analyses for mediator-outcome confounding.
3. Theoretical advances on partial identification of NDE/NIE under weaker assumptions (e.g., using proxy mediators or multiple instruments for the mediator).

Conclusion

Mediation analysis is neither impossible nor straightforward. Natural direct and indirect effects require untestable cross-world assumptions that no experiment can satisfy and no data can falsify. Yet they capture scientifically important questions that principal stratum effects and controlled direct effects partially, but not fully, address. The responsible approach is to combine sensitivity analysis for identification failures, design-based solutions where feasible, and transparency about which estimand is being targeted and what assumptions it requires.

References

1. Baron, R. M. and Kenny, D. A. (1986). The moderator-mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations. Journal of Personality and Social Psychology, 51(6):1173-1182.
2. Frangakis, C. E. and Rubin, D. B. (2002). Principal stratification in causal inference. Biometrics, 58(1):21-29.
3. Imai, K., Keele, L., Tingley, D., and Yamamoto, T. (2011). Unpacking the black box of causality: Learning about causal mechanisms from experimental and observational studies. American Political Science Review, 105(4):765-789.
4. Pearl, J. (2001). Direct and indirect effects. In Proceedings of the 17th Conference on Uncertainty in Artificial Intelligence, pages 411-420.
5. Robins, J. M. and Greenland, S. (1992). Identifiability and exchangeability for direct and indirect effects. Epidemiology, 3(2):143-155.
6. Robins, J. M. (2003). Semantics of causal DAG models and the identification of direct and indirect effects. In Highly Structured Stochastic Systems, pages 70-81. Oxford UP.
7. VanderWeele, T. J. (2015). Explanation in Causal Inference: Methods for Mediation and Interaction. Oxford University Press.