Causal Mediation Analysis

Motivating Example

You run a randomized experiment: a job training program for unemployed workers. The randomization is clean, and you find a significant positive effect on earnings. The program works.

But your program officer asks the most natural follow-up question: How does it work? Does the program boost earnings because it builds human capital (workers learn new skills)? Because it provides a signal to employers (a certificate on the resume)? Or because it expands the workers' professional networks (connections lead to job offers)?

These queries are mechanism questions, and they require mediation analysis. You want to decompose the total effect into the parts that flow through each pathway:

This decomposition turns out to be much harder than it looks.

A. Overview

The Baron and Kenny Legacy

For decades, management researchers followed the Baron and Kenny (1986) approach to mediation.

The approach involves three OLS regressions:

1. Regress the outcome on the treatment: $Y = c \cdot T + e_1$ (check that $c$ is significant)
2. Regress the mediator on the treatment: $M = a \cdot T + e_2$ (check that $a$ is significant)
3. Regress the outcome on both treatment and mediator: $Y = c' \cdot T + b \cdot M + e_3$ (check that $b$ is significant and that $c'$ is reduced compared to $c$)

The indirect effect is $a \times b$, and mediation is "established" if $c'$ is smaller than $c$ (partial mediation) or non-significant (full mediation).

This procedure is intuitive, easy to implement, and has been widely cited in the social sciences. It is also, in many settings, deeply problematic. Here is why.

What Baron and Kenny Gets Wrong

The Baron and Kenny approach implicitly assumes:

1. No treatment-mediator interaction. The effect of $M$ on $Y$ is the same regardless of treatment status. This assumption rules out, for example, the possibility that human capital matters more for program participants than for non-participants.

2. No unmeasured confounders of the mediator-outcome relationship. This requirement is the killer assumption. Even if $T$ is randomized, $M$ is not randomized. Workers who gain the most human capital from training might also be the most motivated, and motivation independently affects earnings.

3. Linear, additive effects. The three-regression approach assumes a specific parametric structure.

The modern causal mediation framework, developed by Imai et al. (2010) and by VanderWeele (2015), makes these assumptions explicit and provides tools for assessing whether they are plausible.

Common Confusions

"What is sequential ignorability?"

Sequential ignorability is the key assumption of the modern causal mediation framework. It has two parts:

1. Treatment is ignorable given pre-treatment covariates. Satisfied by randomization of $T$ or by assuming selection on observables.

2. The mediator is ignorable given treatment and pre-treatment covariates. Conditional on $T$ and observed covariates, $M$ is as good as randomly assigned. This condition is the much harder assumption.

The second part is demanding because even when $T$ is randomized, $M$ is typically endogenous. Workers who gain more human capital from training may differ from those who gain less, in ways that also affect earnings. Sequential ignorability requires that all such confounders are observed and controlled for.

"Can I do mediation with observational data?"

Technically yes, but the bar is very high. Doubly robust methods can help with model misspecification in the outcome and mediator equations, but they do not solve the fundamental confounding problem. You need both parts of sequential ignorability to hold. Without randomization of treatment, you need unconfoundedness of $T$. Without randomization of $M$ conditional on $T$, you need unconfoundedness of $M$ given $T$ and $X$. In practice, this requirement means you need an extremely rich set of covariates and a strong theoretical justification.

Even with experimental data (randomized $T$), the second part of sequential ignorability is a strong assumption because $M$ is not randomized.

"What about Zhao et al. (2010)?"

Zhao et al. (2010) provided an important critique of Baron and Kenny from within the management literature. They argued that the "step 1" requirement (significant total effect) is unnecessary and introduced a more sensible classification of mediation types (complementary, competitive, indirect-only, direct-only, no-effect). This classification was a significant step forward, but it still operates within the regression framework rather than the full causal framework.

B. Identification

Potential Outcomes Framework for Mediation

Let $M_i(t)$ be the mediator value under treatment $t$. Let $Y_i(t, m)$ be the outcome under treatment $t$ and mediator value $m$.

The Natural Direct Effect (NDE) is:

The NDE captures all paths from $T$ to $Y$ that do not go through $M$.

The Natural Indirect Effect (NIE) is:

The NIE captures the path from $T$ to $Y$ that goes through $M$.

The total effect decomposes cleanly:

Sequential Ignorability (Formal)

Assumption 1: $\{Y(t', m), M(t)\} \perp\!\!\!\perp T \mid X = x$ for all $t, t', m$

Assumption 2: $Y(t', m) \perp\!\!\!\perp M \mid T = t, X = x$ for all $t, t', m$

C. Visual Intuition

The classic mediation DAG:

T --> M --> Y
T ---------> Y

The problem arises from unmeasured confounders:

T --> M --> Y
T ---------> Y
      U ----> M
      U ----> Y

If $U$ exists and is unobserved, controlling for $M$ does not isolate the indirect effect — it introduces bias through $U$.

D. Mathematical Derivation

E. Implementation

1library(mediation)
2
3# Step 1: Mediator model
4model.m <- lm(mediator ~ treatment + x1 + x2, data = df)
5
6# Step 2: Outcome model
7model.y <- lm(outcome ~ treatment + mediator + x1 + x2, data = df)
8
9# Step 3: Causal mediation analysis
10med_out <- mediate(model.m, model.y,
11                 treat = "treatment",
12                 mediator = "mediator",
13                 boot = TRUE, sims = 1000)
14summary(med_out)
15plot(med_out)
16
17# Step 4: Sensitivity analysis
18sens_out <- medsens(med_out, rho.by = 0.05, effect.type = "indirect")
19summary(sens_out)
20plot(sens_out)
21# Report: at what value of rho does the indirect effect cross zero?

F. Diagnostics

1. Sensitivity analysis for sequential ignorability. Report the Imai-Keele-Tingley sensitivity parameter $\rho$. How large must the mediator-outcome confounding be to nullify the indirect effect? If the breakpoint $\rho$ is small (e.g., 0.05-0.10), your finding is fragile.

2. Treatment-mediator interaction. Test for $T \times M$ interactions in the outcome model. If significant, the simple $a \times b$ decomposition is biased.

3. Multiple mediators. If you have several mediators, check whether they are causally ordered. If $M_1$ causes $M_2$, you cannot simply run separate mediation analyses for each.

4. Pre-treatment covariates. Check that all covariates you condition on are truly pre-treatment. Including post-treatment variables introduces bias.

5. Comparison with Baron and Kenny. Report both the $a \times b$ product and the modern causal mediation estimates. If they agree, your results are robust to the choice of framework.

6. Placebo mediators. Test mediators that should not be affected by the treatment. If you find "mediation" through a placebo mediator, something is wrong.

Interpreting Your Results

Significant indirect effect, robust to sensitivity analysis: You have credible evidence for mediation. Report the proportion mediated and the sensitivity parameter at which the indirect effect becomes zero.

Significant indirect effect, sensitive to confounding: The indirect effect is suggestive but not robust. Be honest. Report that even moderate confounding of the mediator-outcome relationship could explain the result.

Non-significant indirect effect: Be careful about concluding "no mediation." You might lack statistical power, or the mediator might be poorly measured. Measurement error in the mediator attenuates the indirect effect.

G. What Can Go Wrong

H. Practice

I. Swap-In: When to Use Something Else

• Total-effect estimation (DiD, IV, RDD): When the total effect is sufficient and the mediator is not well-measured or the mechanism is not the primary question.
• Structural equation modeling (SEM): When the causal model involves multiple mediators with known structure and the goal is simultaneous estimation of all pathways.
• Instrumental mediation: When the sequential ignorability assumption is implausible and an instrument for the mediator is available to identify the indirect effect.
• Controlled direct effects: When the question is about blocking a specific pathway (what would happen if the mediator were held fixed) rather than decomposing the total effect into indirect and direct components.

J. Reviewer Checklist