Worked Binary-Mediator Mediation Analysis

From a Joint Distribution to Natural Effects

When treatment, mediator, and outcome are binary, the observed data can be summarized by the joint distribution of mediator and outcome within each treatment group. The challenge is to turn those observed probabilities into counterfactual risks that answer questions such as: what would the outcome risk be under treatment if the mediator instead followed the distribution it would have under no treatment?

This tutorial works through that calculation step by step. The outcome is binary, so its expected value is its risk. Effects will be expressed as differences in risk, measured in percentage points when converted to percentages. These are population quantities; the hypothetical distributions below are treated as known exactly, so the examples do not calculate sampling uncertainty.

For a chosen reference value \(a'\), the natural direct effect compares \(E[Y_{1,M_{a'}}]\) with \(E[Y_{0,M_{a'}}]\): treatment changes while the mediator follows its natural value under \(a'\). The corresponding natural indirect effect at treatment value \(a\) compares \(E[Y_{a,M_1}]\) with \(E[Y_{a,M_0}]\): the treatment setting is held fixed while the mediator’s natural distribution changes. A common additive decomposition is

Here \(\text{NDE}(0)\) is the natural direct effect using the untreated mediator as the reference, and \(\text{NIE}(1)\) is the natural indirect effect with treatment set to 1. The total effect (TE) is the contrast between the two natural treatment regimes. These quantities add on the risk-difference scale; a different effect scale need not have the same additive decomposition.

Identification and the Standardization Formula

The observed joint distribution identifies these counterfactual risks only under causal assumptions. In the following formulas there are no baseline covariates; in applications with covariates, the relevant conditional risks and mediator probabilities must also be standardized over the covariate distribution.

Write \(q_a(m)=P(M=m\mid A=a)\) for the observed mediator distribution under treatment \(a\), and \(r_{am}=P(Y=1\mid A=a,M=m)\) for the observed outcome risk in treatment-mediator group \((a,m)\). Under the assumptions above, the mediation formula gives

The standardization is over the mediator distribution for \(a'\), while the outcome risks are taken from treatment group \(a\). That distinction is essential: changing which distribution is used for the weights changes the counterfactual risk being calculated.

A Short Justification

Because \(M_{a'}\) is binary, the natural counterfactual outcome can be partitioned according to whether \(M_{a'}=0\) or \(M_{a'}=1\). Taking expectations gives

where \(I(\cdot)\) equals 1 when its condition holds and 0 otherwise. The cross-world condition lets each term factor as \(E[Y_{a,m}]P(M_{a'}=m)\). Treatment exchangeability and consistency identify \(P(M_{a'}=m)\) as \(P(M=m\mid A=a')\). Mediator exchangeability and consistency identify \(E[Y_{a,m}]\) as \(E[Y\mid A=a,M=m]\). Since \(Y\) is binary, this conditional expectation is \(r_{am}\), yielding the standardization formula.

The additive decomposition follows by adding and subtracting the same intermediate counterfactual risk:

The first bracket is \(\text{NIE}(1)\), and the second is \(\text{NDE}(0)\). The equality is arithmetic on the risk-difference scale, not a claim that effects must add on every scale.

Checks Before Interpreting the Results

A joint table is enough to calculate the standardized risks, but not enough to establish a causal interpretation. Before reporting natural effects, check that the data support all treatment and mediator groups used in the formula, and assess whether the no-unmeasured-confounding and cross-world assumptions are scientifically defensible. Positivity can be checked in observed data; the other causal assumptions generally require design knowledge and substantive argument.

Keep the two ingredients of each standardized risk separate. The outcome risk \(r_{am}\) comes from treatment level \(a\); the mediator weight \(q_{a'}(m)\) comes from the reference level \(a'\). For example, \(E[Y_{1,M_0}]\) uses treated outcome risks and untreated mediator probabilities. Substituting \(q_1(m)\) instead would calculate \(E[Y_{1,M_1}]\), not \(E[Y_{1,M_0}]\).

• Using an observed group risk as a natural risk. The observed risk \(P(Y=1\mid A=a)\) averages over the mediator distribution observed at \(a\). It equals the corresponding natural risk \(E[Y_{a,M_a}]\) under consistency and treatment exchangeability, but it does not by itself provide a cross-treatment standardized risk such as \(E[Y_{1,M_0}]\).
• Reversing the weights. In \(E[Y_{a,M_{a'}}]\), use outcome risks from \(A=a\) and mediator probabilities from \(A=a'\).
• Forgetting the effect scale. The displayed additive decomposition holds for risk differences. Do not assume the direct and indirect risk differences add to a risk ratio or odds ratio.
• Treating assumptions as verified by the table. A complete joint distribution permits arithmetic, but it cannot rule out unmeasured confounding or establish the cross-world condition.