Suppose there are two groups in a study: treatment and control. There are two potential outcomes for an individual, \(i\): outcome under treatment, \(Y_i(1)\), and outcome under control, \(Y_i(0)\). Only one of the two potential outcomes can be realised and observed as \(Y_i\).

The **treatment effect** for an individual is defined as the difference in potential outcomes for that individual:

\(\mathit{TE}_i = Y_i(1) – Y_i(0)\).

Since we cannot observe both potential outcomes for any individual, we usually we make do with a **sample** or **population ***average* treatment effect (SATE and PATE). Although these are unobservable (they are the averages of unobservable differences in potential outcomes), they can be estimated. For example, with random treatment assignment, the difference in observed sample mean outcomes for the treatment and control is an unbiased estimator of SATE. If we also have a random sample from the population of interest, then this difference in sample means gives us an unbiased estimate of PATE.

Okay, so what happens if we add a mediator? The potential outcome is expanded to depend on both treatment group and mediator value.

Let \(Y_i(t, m)\) denote the potential outcome for \(i\) under treatment \(t\) and with mediator value \(m\).

Let \(M_i(t)\) denote the potential value of the mediator under treatment \(t\).

The **(total) treatment effect** is now:

\(\mathit{TE}_i = Y_i(1, M_i(1)) – Y_i(0, M_i(0))\).

Informally, the idea here is that we calculate the potential outcome under treatment, with the mediator value as it is under treatment, and subtract from that the potential outcome under control with the mediator value as it is under control.

The **causal mediation effect (CME)** is what we get when we hold the treatment assignment constant, but work out the difference in potential outcomes when the mediators are set to values they have under treatment and control:

\(\mathit{CME}_i(t) = Y_i(t, M_i(1)) – Y_i(t, M_i(0))\)

The **direct effect (DE)** holds the mediator constant and varies treatment:

\(\mathit{DE}_i(t) = Y_i(1, M_i(t)) – Y_i(0, M_i(t))\)

Note how both CME and DE depend on the treatment group. If there is no interaction between treatment and mediator, then

\(\mathit{CME}_i(0) = \mathit{CME}_i(1) = \mathit{CME}\)

and

\(\mathit{DE}_i(0) = \mathit{DE}_i(1) = \mathit{DE}\).

ACME and ADE are the averages of these effects. Again, since they are defined in terms of potential values (of outcome and mediator), they cannot be directly observed, but – given some assumptions – there are estimators.

Baron and Kenny (1986) provide an estimator in terms of regression equations. I’ll focus on two of their steps and assume there is no need to adjust for any covariates. I’ll also assume that there is no interaction between treatment and moderator.

First, regress the mediator (\(m\)) on the binary treatment indicator (\(t\)):

\(m = \alpha_1 + \beta_1 t\).

The slope \(\beta_1\) tells us how much the mediator changes between the two treatment conditions on average.

Second, regress the outcome (\(y\)) on both mediator and treatment indicator:

\(y = \alpha_2 + \beta_2 t + \beta_3 m\).

The slope \(\beta_2\) provides the average direct effect (ADE), since this model holds the mediator constant (note how this mirrors the definition of DE in terms of potential outcomes).

Now to work out the average causal mediation effect (ACME), we need to wiggle the outcome by however much the mediator moves between treat and control, whilst holding the treatment group constant. Slope \(\beta_1\) tells us how much the treatment shifts the mediator. Slope \(\beta_3\) tells us how much the outcome increases for every unit increase in the mediator, holding treatment constant. So \(\beta_1 \beta_3\) is ACME.

For more, especially on complicating the Baron and Kenny approach, see Imai et al. (2010).

### References

Baron, R. M., & 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.

Imai, K., Keele, L., & Yamamoto, T. (2010). Identification, Inference and Sensitivity Analysis for Causal Mediation Effects. *Statistical Science*, *25*, 51–71.