Mediator

Definition

A mediator is an intermediate variable lying on the causal pathway through which a treatment (X) affects an outcome (Y). In the structure X → M → Y, M is the mediator.

DAG representation:

  Treatment (X) → Mediator (M) → Outcome (Y)

Key characteristics:

• Part or all of X’s effect is transmitted to Y through M
• Controlling for M blocks the genuine causal effect

Total, Direct, and Indirect Effects

        X → M → Y     (Indirect effect: through M)
        X ----→ Y     (Direct effect: not through M)

Total Effect (TE)

$TE = E[Y|do(X=1)] - E[Y|do(X=0)]$

• The total effect of X on Y
• Direct + Indirect effect

Direct Effect (DE)

$DE = E[Y|do(X=1, M=m)] - E[Y|do(X=0, M=m)]$

• The X → Y effect when M is held fixed
• The pathway that does not pass through M

Indirect Effect (IE)

$IE = TE - DE$

• The X → Y effect through M
• “Mediated effect”

Why NOT Control for Mediators?

Problem 1: Underestimation of Total Effect

When you control for M:

• The indirect effect (X → M → Y) is blocked
• The total effect is underestimated

Original: X ---(direct)--> Y
          X ---(via M)---> Y   ← this part disappears

Problem 2: Endogenous Selection Bias

When the mediator is connected to a confounder:

      U (unmeasured)
     ↙        ↘
    M          Y
    ↑
    X

When you control for M:

• A spurious association is created between X and U (M becomes a collider)
• Bias is introduced

Example: Education → Adult Intelligence → Income

                    Childhood Intelligence
                        ↙        ↘
Education → Adult Intelligence → Income
     ↘_________________________↗

• Without control: the total effect of Education on Income
• With control for Adult Intelligence:
  1. The indirect effect is blocked (underestimation)
  2. Childhood Intelligence plays the role of U → bias

When to Control for Mediator?

Goal: Estimate Total Effect

→ Do NOT control for mediator

Goal: Estimate Direct Effect Only

→ Control for mediator (caution required)

However:

• Estimating the direct effect requires additional assumptions
• Bias is possible if M is not randomized
• Even in an experimental study, M must be randomized to be valid

Mediation Analysis

Traditional mediation (Baron & Kenny, 1986):

• Has many limitations
• Difficult to interpret causally

Modern causal mediation (Imai et al., 2010; Pearl):

• Sequential ignorability assumption
• Stricter identification conditions

Mediator vs. Confounder vs. Collider

Variable Type	DAG Structure	Effect of Control
Confounder	X ← Z → Y	Removes spurious association ✓
Collider	X → C ← Y	Creates spurious association ✗
Mediator	X → M → Y	Blocks the causal effect ✗

Commonality (Collider & Mediator): both are post-treatment variables

Identifying Mediators

Temporal Criterion

• Occurs after the treatment and before the outcome
• Is affected by the treatment

Causal Pathway

• X → M: X affects M
• M → Y: M affects Y
• M lies on the X→Y pathway

Example Mediators

Treatment	Mediator	Outcome
Education	Skills/Knowledge	Income
Therapy	Coping mechanisms	Depression
Drug	Biological pathway	Health outcome
Exercise	Fitness level	Weight loss

Partial vs. Full Mediation

Full Mediation

X → M → Y   (only through M)

• All of X’s effect goes through M
• Direct effect = 0

Partial Mediation

X → M → Y
X ---→ Y   (also direct effect)

• Part of X’s effect goes through M
• Direct effect ≠ 0

Post-treatment Variable Rule

Rule of Thumb:

Do not control for variables that occur after the treatment

Reasons:

1. Mediator: blocks the causal effect
2. Collider: creates spurious association
3. Descendant of collider: indirect collider bias

Exceptions:

• When estimating the direct effect is the explicit goal
• When using appropriate causal mediation analysis

Related Concepts

• DAG - visualizing causal structure
• Confounder - common cause (must be controlled)
• Collider - common effect (must not be controlled)
• CATE - treatment effect estimation
• Causal Mediation Analysis - analyzing mediation effects

References

• rohrerThinkingClearlyCorrelations - the dangers of controlling for mediators
• Baron, R. M., & Kenny, D. A. (1986). The moderator-mediator variable distinction
• Pearl, J. (2014). Interpretation and identification of causal mediation
• Imai, K., et al. (2010). A general approach to causal mediation analysis