Causal Mediation Analysis

The Total Effect (TE) is the overall causal effect of a treatment (X) on an outcome (Y), encompassing all possible pathways. It is the difference in the expected outcome when the treatment is present versus absent, formally: TE = E[Y | do(X=1)] - E[Y | do(X=0)]. In mediation, the TE is the sum of the direct and indirect effects.

The Natural Direct Effect (NDE) quantifies the portion of the total effect that operates on the outcome through pathways not involving the specified mediator (M). It measures the change in Y when X changes, but the mediator is held at the value it would have naturally taken without the treatment. It answers: 'What is the effect of X on Y if we disable the path through M?'

The Natural Indirect Effect (NIE) quantifies the portion of the total effect that operates on the outcome through the specified mediator (M). It measures the change in Y when the treatment is fixed, but the mediator changes to the value it would have under treatment. Formally, it captures: TE = NDE + NIE. It answers: 'What is the effect of X on Y that is transmitted via M?'

The Controlled Direct Effect (CDE) is an alternative to the NDE. It measures the effect of the treatment on the outcome when the mediator is experimentally set to a specific, fixed value for the entire population (M = m). Unlike the NDE, it does not allow the mediator to vary naturally. The CDE is useful for policy questions about manipulating both X and M simultaneously.

Sequential Ignorability is the core set of assumptions required to identify natural direct and indirect effects from observational data. It consists of two main conditions:

• No Unmeasured Confounding of X->Y and X->M: All common causes of treatment and outcome/mediator are observed.
• No Unmeasured Confounding of M->Y: After conditioning on treatment and pre-treatment covariates, there are no unobserved common causes of the mediator and outcome. Violations of these assumptions, particularly the second, are a major source of bias.

The Mediation Formulas provide the mathematical expressions to estimate NDE and NIE from observed data under the sequential ignorability assumptions. For a binary treatment, they integrate over the distribution of covariates (C) and the mediator:

• NDE = Σ_c Σ_m [E(Y | X=1, M=m, C=c) - E(Y | X=0, M=m, C=c)] * P(M=m | X=0, C=c) * P(C=c)
• NIE = Σ_c Σ_m E(Y | X=1, M=m, C=c) * [P(M=m | X=1, C=c) - P(M=m | X=0, C=c)] * P(C=c) These formulas are implemented in software packages like mediation in R.

A Structural Causal Model (SCM) is the formal mathematical framework underpinning mediation analysis. It represents causal relationships as a system of structural equations, typically visualized as a causal graph. An SCM explicitly defines how each variable is generated from its direct causes and independent noise, enabling precise definitions of interventions (do-operator) and counterfactuals. Mediation analysis uses the SCM to formally define direct, indirect, and total effects.

A causal graph, or Directed Acyclic Graph (DAG), is the visual representation of an SCM. Nodes are variables, and directed edges represent assumed direct causal influences. For mediation analysis, the DAG explicitly shows the treatment (T), mediator (M), and outcome (Y), along with any confounders. Key paths include:

• Direct path: T → Y
• Indirect path: T → M → Y
• Backdoor paths: Non-causal associations, e.g., T ← C → Y, which must be blocked for identification. Understanding d-separation in the DAG is essential for identifying which covariates to control for.

The intervention, denoted by the do-operator (e.g., do(T=1)), is the mathematical representation of forcibly setting a variable to a value, simulating a randomized experiment. Do-calculus is a set of three inference rules that allow researchers to translate expressions involving interventions (P(Y | do(T))) into observable statistical quantities, provided the causal graph is known. Mediation formulas, such as the Pearl's mediation formula for natural direct and indirect effects, are derived using do-calculus to express these effects in terms of estimable conditional probabilities from observational data.

A counterfactual is a statement about what would have happened under a different, hypothetical condition. It represents the highest level on the causal hierarchy (ladder of causation). In mediation, effects are defined using counterfactual outcomes. For example, the natural indirect effect compares the outcome if the treatment were applied but the mediator took its value as if the treatment were not applied: Y(T=1, M(T=0)). This contrasts with the controlled direct effect, which holds the mediator fixed at a specific level. Counterfactual definitions provide the most rigorous grounding for mediation analysis.

Causal identifiability is the property that a causal quantity (like a mediation effect) can be uniquely computed from the available data and the assumed causal model. For mediation, specific identifiability assumptions must hold:

• No unmeasured confounding of the T-Y, T-M, and M-Y relationships.
• No mediator-outcome confounder affected by treatment.
• Consistency and positivity. If these assumptions are violated, the direct and indirect effects may not be nonparametrically identifiable from observational data, requiring alternative designs (e.g., experiments) or sensitivity analyses.

The frontdoor criterion is a graphical identification strategy used when a mediator variable provides a viable path to estimate a causal effect in the presence of unmeasured confounding between treatment and outcome. It is a complement to the backdoor criterion. The path must satisfy:

1. The treatment affects the mediator.
2. The mediator affects the outcome.
3. All effect of the treatment on the outcome passes through the mediator.
4. There is no unmeasured confounding of the mediator-outcome relationship. If these hold, the causal effect can be computed by summing over the mediator's values, even with confounding on the main T-Y path.