• Scientific Understanding: Moves beyond “black box” descriptions to explanatory theories. How does the world work?
• Refining Theory: Tests specific theoretical propositions about causal processes
• Designing Better Interventions: If we know why something works, we might find more efficient or potent ways to achieve the same outcome (e.g., Vitamin C tablets instead of limes)
  • Mediators themselves can have mediators!
• Generalisability: Understanding the mechanism helps predict if an effect will hold in different contexts where the mediator might operate differently
• Audiences and policymakers almost always ask about mechanisms!

Source: Righetti (2023)

Source: Righetti (2023)

• Remember from heterogeneous effects: causal effects often vary!
• If \(a_i\) and \(b_i\) vary across individuals, then the average indirect effect is \(E[a_i b_i]\)
• \(E[a_i b_i] = E[a_i] E[b_i] + Cov(a_i, b_i)\)
• Simply multiplying the average effect of \(Z\) on \(M\) (\(\hat{a} \approx E[a_i]\)) by the average effect of \(M\) on \(Y\) (\(\hat{b} \approx E[b_i]\)) only gives the average indirect effect if \([Cov(a_i, b_i) = 0]\).
• This covariance term is generally not zero and not identifiable.
• So, the simple \(c = d + ab\) decomposition breaks down with heterogeneous effects.

• Even if we ignore heterogeneous effects for a moment… Equation (3) is problematic: \(Y_i = \alpha_3 + d Z_i + b M_i + e_{3i}\)
• \(Z\) is random, but \(M\) is an outcome variable. It’s very likely correlated with the error term \(e_{3i}\).
• Why? Unobserved confounders might affect both M and Y.
• Example (Bhavnani): Unmeasured local ‘egalitarianism’ (\(e_{3i}\)) might encourage women to run for office (\(M\)) and make voters more likely to elect women (\(Y\)), independently of the reservation policy (\(Z\)).
• Including \(M\) in the regression is like including a post-treatment variable that is correlated with the error term!

• Relies on strong, often implausible assumptions:
  • Constant treatment effects (for \(c = d + ab\))
  • No unobserved confounding between \(M\) and \(Y\) (for unbiased \(\hat{b}\) and \(\hat{d}\))
• These assumptions are not guaranteed by random assignment of \(Z\) alone.
• Prone to bias that exaggerates the mediator’s role and downplays the treatment’s direct effect.
• Use with extreme caution, if at all, for causal mediation claims based solely on \(Z\) randomisation.

• Let’s apply our familiar framework. For each individual \(i\):
  • \(Z_i\): Randomly assigned treatment (0 or 1)
  • \(M_i(z)\): Potential value of the mediator if assigned to treatment \(z\)
  • \(Y_i(z)\): Potential value of the outcome if assigned to treatment \(z\)
• Observed variables:
  • \(M_i = M_i(Z_i)\)
  • \(Y_i = Y_i(Z_i)\)
• Average Treatment Effect (Total Effect): \(ATE = E[Y_i(1) - Y_i(0)]\)
• Effect of \(Z\) on \(M\): \(E[M_i(1) - M_i(0)]\)
• These are estimable from a standard experiment randomising \(Z\).

• To formally define direct and indirect effects, we need a more complex notation (Imai, Keele, Yamamoto 2010; Robins & Greenland 1992):
• \(Y_i(m, z)\): Potential outcome for individual \(i\) if their mediator were set to value \(m\) AND they were assigned treatment \(z\).
• This allows us to think about hypothetical scenarios:
  • What would \(Y\) be if we gave the treatment (\(z=1\)) but somehow forced the mediator to the level it would have taken under control (\(m=M_i(0)\))? This would be \(Y_i(M_i(0), 1)\).
  • These are complex potential outcomes because they involve situations that can never happen in reality (more on that soon!).
• Linking back: \(Y_i(1) = Y_i(M_i(1), 1)\) and \(Y_i(0) = Y_i(M_i(0), 0)\).

• Using this notation, we can define effects more precisely:
• Total Effect (ATE): \(E[Y_i(1) - Y_i(0)] = E[Y_i(M_i(1), 1) - Y_i(M_i(0), 0)]\)
• Controlled Direct Effect (CDE): Effect of \(Z\) on \(Y\), holding \(M\) fixed at some level \(m\).
  • \(CDE(m) = E[Y_i(m, 1) - Y_i(m, 0)]\)
• Natural Direct Effect (NDE): Effect of \(Z\) on \(Y\) if \(M\) were held at the level it would have taken under control.
  • \(NDE = E[Y_i(M_i(0), 1) - Y_i(M_i(0), 0)]\)
• Natural Indirect Effect (NIE): Effect of \(M\) changing from \(M_i(0)\) to \(M_i(1)\), holding \(Z\) fixed (usually at \(Z=1\) for policy relevance).
  • \(NIE = E[Y_i(M_i(1), 1) - Y_i(M_i(0), 1)]\)
• Under certain assumptions, \(ATE = NDE + NIE\).

• Look closely at the definitions of NDE and NIE:
  • NDE involves \(Y_i(M_i(0), 1)\)
  • NIE involves \(Y_i(M_i(0), 1)\)
• These are “complex” or “counterfactual” potential outcomes.
  • \(Y_i(M_i(0), 1)\) represents the outcome under treatment (\(Z=1\)), but with the mediator at the level it would take under control (\(M=M(0)\)).
• This state can never happen in reality! If \(Z_i=1\), we observe \(M_i(1)\), not \(M_i(0)\).
• These potential outcomes are fundamentally unobservable from any single experiment just randomising \(Z\).
• Now you see why mediation analysis is so tricky and why we don’t do it often!

• Estimating the theoretically precise Natural Direct Effect (NDE) and Natural Indirect Effect (NIE) requires knowing the values of complex potential outcomes like \(Y_i(M_i(0), 1)\).
• Since these are unobservable, we cannot estimate NDE and NIE without making strong assumptions.
• Common assumptions include versions of “sequential ignorability” (Imai et al.), essentially assuming M is “as-if randomised” conditional on Z and pre-treatment covariates – very similar to the assumption needed for the regression approach to be unbiased.
• Just randomising Z is not sufficient to identify mediation pathways without these extra assumptions.

• What if we can show that \(Z\) has no effect on \(M\)?
• If \(M_i(1) = M_i(0)\) for all individuals \(i\) (the sharp null hypothesis), then \(M\) cannot possibly mediate the effect of Z.
• In this case, the complex potential outcomes simplify: \(Y_i(M_i(0), 1) = Y_i(M_i(1), 1) = Y_i(1)\). The NIE becomes zero.
• How to test this?
  • Estimate the average effect: \(E[M_i(1) - M_i(0)]\). If it’s precisely zero…
  • Test for heterogeneity: Check if \(Var(M_i(1))\) differs from \(Var(M_i(0))\). (As discussed in Lecture 18). If variances are similar and ATE is zero, it lends support to the sharp null.
• Ruling out mediators is easier than quantifying mediation. Useful for eliminating hypotheses.

• What if we design an experiment that randomises both \(Z\) and \(M\)? (A factorial design!)
• Example: 2x2 design
  • Group 1: \(Z=0\), \(M=0\) (Control)
  • Group 2: \(Z=0\), \(M=1\) (Manipulate \(M\) only)
  • Group 3: \(Z=1\), \(M=0\) (Manipulate \(Z\), block \(M\)?)
  • Group 4: \(Z=1\), \(M=1\) (Manipulate \(Z\) and \(M\))
• This helps estimate Controlled Direct Effects (CDEs). E.g., Effect of \(Z\) holding \(M\) at 0 is (Group 3 - Group 1). Effect of \(M\) holding \(Z\) at 1 is (Group 4 - Group 3).
• BUT: This still doesn’t directly estimate the NDE or NIE, because \(M\) is set experimentally, not allowed to take its “natural” value \(M_i(z)\). But why is this important?
• Manipulating \(M\) might be hard or artificial (e.g., giving Vitamin C tablets vs. limes might have different effects beyond Vitamin C).

• Often, we can’t directly set \(M\). Instead, we use an “encouragement” \(Z\) to try and influence \(M\).
• Think back to non-compliance and Instrumental Variables (IV):
  • \(Z\) = Assignment/Encouragement (e.g., offer tutoring)
  • \(M\) = Treatment Received (e.g., actually attend tutoring)
  • \(Y\) = Outcome (e.g., test score)
• We can use \(Z\) as an instrument for \(M\) to estimate the effect of \(M\) on \(Y\) for Compliers (CACE/LATE).
• This is sometimes framed as a mediation analysis: \(Z\) affects \(Y\) through its effect on \(M\).
• Formula: \(CACE_{M \to Y} = \frac{ITT_{Y}}{ITT_{M}} = \frac{E[Y|Z=1] - E[Y|Z=0]}{E[M|Z=1] - E[M|Z=0]}\)

• For the IV estimate of the effect of \(M\) on \(Y\) to be valid, we need the exclusion restriction.
• In the mediation context, this means \(Z\) must affect \(Y\) only through \(M\). There can be no direct path from \(Z\) to \(Y\) (\(d=0\)).
• This is a very strong assumption! Often, the encouragement (\(Z\)) might affect the outcome (\(Y\)) through other channels besides the intended mediator (\(M\)).
• Example (Bhavnani): Seat reservations (\(Z\)) might affect future elections (\(Y\)) not just by creating incumbents (\(M1\)), but also by changing voter attitudes (\(M2\)) or mobilising different voters (\(M3\)).
• If \(Z\) has a direct effect or affects multiple mediators, the simple IV approach doesn’t isolate the effect through one specific \(M\).
• Identifying effects through multiple mediators requires multiple encouragements (instruments) with different effects on the mediators – very complex!

• Given the challenges of formally estimating direct/indirect effects or using IV with strong assumptions…
• An alternative: Implicit Mediation Analysis.
• Instead of trying to measure M and model the Z → M → Y pathway explicitly…
• Focus on the treatment Z itself. Many treatments are “bundles” of different components.
• Design experiments that add or subtract specific components of the treatment bundle.
• Compare the effects of these different treatment variations.
• This implicitly tests the importance of the components (and the mediators they likely affect) without needing to measure M or make strong assumptions about unobservables.

• CCT programmes give cash to poor families if they meet certain conditions (e.g., school attendance, health check-ups).
• Potential mediators: Increased income (cash effect), Changed behaviour due to rules (conditionality effect).
• Implicit Mediation Design: (e.g., Baird et al. 2009)
  • Group 1: Control (no programme)
  • Group 2: Unconditional Cash Transfer (UCT - gets cash, no rules)
  • Group 3: Conditional Cash Transfer (CCT - gets cash + rules)
• Comparisons:
  • (Group 2 - Group 1): Effect of cash alone.
  • (Group 3 - Group 1): Effect of cash + conditions.
  • (Group 3 - Group 2): Effect of conditions (holding cash constant).
• This tells us about the importance of the conditions (likely mediator: behaviour change) vs. the cash (likely mediator: income) without directly measuring parental behaviour or income changes and modelling them.

• Famous experiment testing social pressure effects on voting.
• Treatment “ingredients” gradually added:
  • Group 1: Control (no mail)
  • Group 2: “Civic Duty” mailer (basic encouragement)
  • Group 3: “Hawthorne” mailer (Civic Duty + told they are being studied)
  • Group 4: “Self” mailer (Hawthorne + shown own household’s past voting record)
  • Group 5: “Neighbors” mailer (Self + shown neighbours’ past voting records)
• Implicit mediators: Sense of duty, being watched, accountability for own record, comparison to neighbours.

Illustrative Turnout Rates:

(Based on Gerber, Green, and Larimer 2008)

• Clear pattern: Adding social pressure components significantly increases turnout.
• Comparing Group 5 vs 4 suggests disclosing neighbours’ records adds ~3.3% effect.
• Comparing Group 4 vs 3 suggests disclosing own record adds ~2.3% effect.
• Tells us which ingredients matter without modelling psychological states.

• This approach identifies which aspects of a complex intervention drive the effect.
• It suggests the likely importance of different mediating processes (e.g., social comparison is powerful for turnout) without needing to directly measure or model the mediator variable (e.g., feelings of shame/pride).
• It’s inherently design-based and stays within the clean framework of comparing randomly assigned groups.
• Very useful for programme evaluation and refinement. What are the active ingredients?

• Avoids bias inherent in regression-based mediation with non-randomised mediators.
• Stays within the unbiased statistical framework of comparing randomised groups.
• Lends itself to exploration and discovery of effective treatment variations.
• Relies on weaker, design-based assumptions rather than statistical assumptions about unobservables.
• Often more practical and feasible than trying to perfectly manipulate or measure mediators in field settings.

• Be critical of causal mediation claims, especially those based solely on standard regression methods after randomising only Z.
• Ask about the assumptions being made (constant effects? no confounding? exclusion restriction?).
• Favour design-based approaches where possible.
• Implicit mediation offers a robust way to gain insights into mechanisms by comparing different versions of a treatment.
• Understanding mechanisms is crucial, but requires careful thought about identification strategies!