8 Mediation and Front-Door Identification
8.1 Motivation: Mechanisms
The identification results of Chapters 5–7 all answer the same question: what is the total causal effect of \(T\) on \(Y\)? Mediation analysis asks a finer question: through what mechanism does that effect operate?
The total effect of \(T\) on \(Y\) may flow along multiple causal pathways. Some of this effect passes through an intermediate variable \(M\) — the mediator — along the path \(T \to M \to Y\). The remainder flows directly along \(T \to Y\), bypassing the mediator entirely. Mediation analysis aims to study mechanisms by defining direct and indirect effect concepts that target each pathway; only some of these concepts yield an additive decomposition of the total effect.
This mechanism question matters for scientific and policy reasons. In a clinical trial of a behavioral intervention (\(T\)) on depression (\(Y\)), a researcher may want to know how much of the benefit operates through improved sleep quality (\(M\)) versus other pathways — because if sleep is the main channel, targeting sleep directly may be a more efficient intervention. In an economics study of education (\(T\)) on wages (\(Y\)), how much operates through occupation (\(M\)) versus cognitive skills? The answer determines whether a policy should target educational attainment or occupational access.
Running example. Throughout this chapter we anchor the abstract formulas to a single concrete scenario: \(T\) is a randomized behavioral intervention, \(M\) is self-reported sleep quality measured mid-trial, and \(Y\) is a depression score (e.g., on the PHQ-9 scale).
The challenge is that mediators are post-treatment variables: they are affected by the treatment, and may themselves be confounded with the outcome. Conditioning on a post-treatment variable creates exactly the collider and selection-bias problems studied in Chapters 2 and 3. A naïve approach — simply including \(M\) as a covariate in a regression of \(Y\) on \(T\) — conflates adjustment with mediation and can introduce bias even in a randomized experiment.
This chapter also develops the front-door criterion, a distinct identification strategy that uses the mediation structure of the DAG to identify causal effects even when treatment and outcome are confounded by an unobserved variable, making mediation analysis relevant not only to mechanism research but also to the core identification problem of earlier chapters.
8.2 The Mediation DAG
8.2.1 The Prototype Graph
The graph encodes two causal pathways: the direct pathway \(T \to Y\) (treatment affects outcome without passing through the mediator) and the indirect pathway \(T \to M \to Y\) (treatment first shifts the mediator, which in turn shifts the outcome).
8.2.2 Structural Equations
The prototype graph corresponds to the nonparametric SEM: \[T = f_T(\mathbf{X},\, U,\, \varepsilon_T), \qquad M = f_M(T,\, \mathbf{X},\, \varepsilon_M), \qquad Y = f_Y(T,\, M,\, \mathbf{X},\, U,\, \varepsilon_Y), \tag{8.1}\] where each arrow corresponds to the presence of the parent in the child’s structural equation. \(U\) enters both \(T\)’s and \(Y\)’s equations, making the confounding paths explicit; \(U\) is absent from \(M\)’s equation, reflecting the absence of an arrow \(U \to M\) in the DAG.
8.2.3 What Makes Mediation Harder Than Total Effect Estimation
Collider bias. Conditioning on \(M\) can open collider paths. Suppose \(U \to T\) and \(V \to M\) and \(V \to Y\), with \(V\) unobserved. The path \(T \to M \leftarrow V \to Y\) is blocked when \(M\) is not conditioned on, but opens as soon as \(M\) is included as a covariate. This is precisely why naively regressing \(Y\) on \((T, M)\) does not isolate the direct effect.
Mediator–outcome confounding. Even when treatment is randomized, the mediator \(M\) is never randomized. An unobserved variable \(V\) with \(V \to M\) and \(V \to Y\) creates a back-door path from \(M\) to \(Y\) that randomization of \(T\) does not close.
8.2.4 A Working Graph for the Identification Sections
8.3 Total Causal Effect
Under the working assumption, TE is identified by the back-door formula: \[\mathrm{TE} = \sum_{\mathbf{x}} [\E[Y \mid T{=}1, \mathbf{X}{=}\mathbf{x}] - \E[Y \mid T{=}0, \mathbf{X}{=}\mathbf{x}]]\, P(\mathbf{X}{=}\mathbf{x}).\]
Running example. The TE is the expected change in depression score if the entire study population were assigned to the intervention versus control. It combines the effect operating through sleep improvement with every other pathway. Mediation analysis asks: of this total, how much is due to sleep?
8.4 Controlled Direct Effect
Equation 8.2 involves two simultaneous interventions, corresponding to the mutilated graph \(\mathcal{G}_{\overline{TM}}\) (all edges into both \(T\) and \(M\) deleted). Fixing \(M = m\) for everyone shuts down the indirect pathway \(T \to M \to Y\): any remaining \(T\)-to-\(Y\) effect flows only through the direct edge.
Running example. \(\mathrm{CDE}(m)\) at $m = $ “poor sleep” is the expected change in depression score comparing intervention to control if every participant’s sleep quality were externally held at the poor-sleep level. Whether sleep can be externally fixed in a clinical trial is a separate question — which is why the policy interpretation of the CDE in this scenario is strained.
8.4.1 Identification of the CDE
Under the working assumption, \(\mathbf{Z} = \mathbf{X}\) is a valid adjustment set.
Graph surgery and back-door adjustment are distinct steps. Graph surgery defines the interventional target by deleting arrows into \(T\) and \(M\). Expressing that target as a functional of the observed distribution is a separate step requiring a valid adjustment set \(\mathbf{Z}\) in the original graph. If \(U\) is unobserved and \(\mathbf{X}\) alone cannot block the back-door path \(T \leftarrow U \to Y\), then Equation 8.3 with \(\mathbf{Z} = \mathbf{X}\) does not identify the CDE. Alternative strategies (front-door, IV) are required.
8.4.2 Physical Manipulability and the CDE Does Not Decompose
The CDE is only scientifically meaningful when an intervention to fix \(M\) at a specified level \(m\) is physically realizable. In many substantive settings, the mediator cannot be independently manipulated (education-occupation, behavioral intervention-sleep), making the CDE’s policy interpretation strained.
The CDE and some “controlled indirect effect” do not sum to the total effect in general. The residual \(\mathrm{TE} - \mathrm{CDE}(m)\) depends on \(m\) and has no clean do-calculus expression corresponding to “the indirect pathway.” The correct estimands for a pathway decomposition are the NDE and NIE, introduced next.
8.5 Natural Direct and Indirect Effects
8.5.1 Cross-World Counterfactuals
The CDE fixes the mediator by external intervention. A more scientifically natural question is: what is the effect of \(T\) on \(Y\) that bypasses \(M\) when \(M\) is held at the value it would naturally take under the reference treatment \(T = 0\)? This requires the nested potential outcomes notation \(Y(t, M(t'))\), which denotes the outcome observed if \(T\) were set to \(t\) and \(M\) were simultaneously set to the value it would naturally take if \(T\) were \(t'\). These are called cross-world counterfactuals because \(t\) and \(t'\) may differ.
Running example. The NDE is the part of the depression reduction that comes from cognitive, behavioral, or therapeutic-alliance pathways, not from sleep. The NIE is the complementary piece: how much of the depression reduction is due to the sleep improvement that the intervention itself causes.
The TE = NDE + NIE decomposition: \[\mathrm{TE} = \mathrm{NDE} + \mathrm{NIE}. \tag{8.6}\]
Proof. \(\mathrm{NDE} + \mathrm{NIE} = \E[Y(1, M(0)) - Y(0, M(0))] + \E[Y(1, M(1)) - Y(1, M(0))] = \E[Y(1, M(1)) - Y(0, M(0))] = \E[Y(1) - Y(0)] = \mathrm{TE}\). \(\square\)
8.6 Identification of Natural Effects
8.6.1 Sequential Ignorability
Assumptions 1–3 are the natural extensions of the Baron–Kenny conditions to the potential outcomes setting. Assumption 4 is the critical new requirement: it rules out a variable \(L\) that is caused by the treatment and confounds the mediator–outcome relationship.
What randomization of \(T\) does and does not provide. Randomizing \(T\) satisfies Assumptions 1 and 2 by design. It does not satisfy Assumptions 3 or 4. The mediator \(M\) is a post-treatment variable that is never randomized; any unobserved variable \(V\) with \(V \to M\) and \(V \to Y\) violates Assumption 3 regardless of how \(T\) was assigned. Assumption 4 is even more demanding: it can be violated by a variable \(L\) that is itself caused by the treatment.
8.6.2 The Mediation Formula
Interpretation. The mediation formula “mixes” the outcome regression under \(T = t\) with the mediator distribution under \(T = t'\). To compute the NDE, set \(t = 1\) and \(t' = 0\): take the conditional mean of \(Y\) at treatment 1, but weight the mediator by its distribution under treatment 0. This counterfactual reweighting is what makes the formula non-trivial. The presence of the second treatment index \(t'\) on the right-hand side is the visible trace of the cross-world step.
The NDE and NIE from the formula: \[\mathrm{NDE} = \sum_{m,\mathbf{x}} [\E[Y \mid 1, m, \mathbf{x}] - \E[Y \mid 0, m, \mathbf{x}]]\, P(M{=}m \mid T{=}0, \mathbf{x})\, P(\mathbf{x}),\] \[\mathrm{NIE} = \sum_{m,\mathbf{x}} \E[Y \mid 1, m, \mathbf{x}]\, [P(M{=}m \mid T{=}1, \mathbf{x}) - P(M{=}m \mid T{=}0, \mathbf{x})]\, P(\mathbf{x}).\]
8.7 The Linear Mediation Model: A Historical Special Case
8.7.1 The Baron–Kenny Three-Equation System
The regression-based approach of Baron and Kenny (1986) restricts the reduced prototype graph to a linear SEM: \[Y = \alpha_1 + \tau T + \boldsymbol{\gamma}_1^\top\mathbf{X} + \varepsilon_1, \tag{8.8}\] \[M = \alpha_2 + a T + \boldsymbol{\gamma}_2^\top\mathbf{X} + \varepsilon_2, \tag{8.9}\] \[Y = \alpha_3 + \tau' T + b M + \boldsymbol{\gamma}_3^\top\mathbf{X} + \varepsilon_3. \tag{8.10}\]
The four coefficients: \(\tau\) (total effect), \(a\) (first-stage effect of \(T\) on \(M\)), \(\tau'\) (direct effect of \(T\) on \(Y\) controlling for \(M\)), \(b\) (second-stage effect of \(M\) on \(Y\) controlling for \(T\)).
8.7.2 The Component Pathways
First stage (\(T \to M\)): Equation Equation 8.9 implements the back-door formula for \(T\) on \(M\). Under Condition 2, conditioning on \(\mathbf{X}\) blocks all back-door paths from \(T\) to \(M\), and \(a\) identifies \(\E[M(1)] - \E[M(0)]\).
Second stage (\(M \to Y\) given \(T\)): Equation Equation 8.10 implements the back-door formula for \(M\) on \(Y\) given \(T\). Conditioning on \((T, \mathbf{X})\) blocks every back-door path from \(M\) to \(Y\) in the reduced prototype graph, provided Assumption 3 (no unmeasured \(M\)–\(Y\) confounding given \(T\)) holds.
8.7.3 The Product and Difference Formulas
Proof. Substitute Equation 8.9 into Equation 8.10: \(Y = (\alpha_3 + b\alpha_2) + (\tau' + ab)T + (\boldsymbol{\gamma}_3 + b\boldsymbol{\gamma}_2)^\top\mathbf{X} + (b\varepsilon_2 + \varepsilon_3)\). Comparing with Equation 8.8 gives \(\tau = \tau' + ab\). \(\square\)
Running example. Suppose a randomized trial yields \(\hat\tau = 0.50\), \(\hat a = 0.40\), \(\hat b = 0.60\), \(\hat\tau' = 0.26\).
- Indirect effect (product method): \(\hat a \hat b = 0.40 \times 0.60 = 0.24\).
- Direct effect (difference method): \(\hat\tau - \hat a \hat b = 0.50 - 0.24 = 0.26 = \hat\tau'\).
- Proportion mediated: \(\hat a \hat b / \hat\tau = 0.24/0.50 = 0.48\) — roughly 48% of the total effect operates through sleep improvement.
| Effect | Formula | Path(s) |
|---|---|---|
| Total | \(\tau\) | \(T \to Y\) and \(T \to M \to Y\) combined |
| Direct | \(\tau' = \tau - ab\) | \(T \to Y\) only |
| Indirect | \(ab\) | \(T \to M \to Y\) only |
| Proportion mediated | \(ab/\tau\) | Share of total effect via \(M\) |
8.7.4 Inference: The Sobel Test and Bootstrap
The delta method gives an approximate variance for the product \(\hat a \hat b\): \[\widehat{\mathrm{Var}}(\hat a \hat b) \approx \hat b^2\, \widehat{\mathrm{Var}}(\hat a) + \hat a^2\, \widehat{\mathrm{Var}}(\hat b) + 2\, \hat a\, \hat b\, \widehat{\mathrm{Cov}}(\hat a, \hat b). \tag{8.12}\]
The Sobel test (Sobel 1982) drops the cross-covariance: \[\widehat{\mathrm{Var}}_{\mathrm{Sobel}}(\hat a \hat b) = \hat b^2\, \widehat{\mathrm{Var}}(\hat a) + \hat a^2\, \widehat{\mathrm{Var}}(\hat b), \tag{8.13}\] yielding \(z = \hat a \hat b / \sqrt{\widehat{\mathrm{Var}}_{\mathrm{Sobel}}(\hat a \hat b)}\). In practice, bootstrap confidence intervals for \(ab\) are preferred over the Sobel test because the distribution of a product of estimates is skewed in finite samples.
8.7.5 The Baron–Kenny Assumptions: Two Distinct Categories
Causal ignorability conditions (conditions 1–3): these are causal identification assumptions about unmeasured confounding. Violating them introduces bias that no amount of additional data can remove.
- No unmeasured \(T\)–\(Y\) confounding: \(\varepsilon_1 \indep T \mid \mathbf{X}\).
- No unmeasured \(T\)–\(M\) confounding: \(\varepsilon_2 \indep T \mid \mathbf{X}\).
- No unmeasured \(M\)–\(Y\) confounding given \(T\): \(\varepsilon_3 \indep M \mid T, \mathbf{X}\).
Structural modeling restrictions (conditions 4–5): these are parametric assumptions about functional form. Violating them does not introduce identification bias in the causal sense, but \(ab\) and \(\tau'\) no longer equal the NDE and NIE.
- Linearity and additivity: equations Equation 8.8–Equation 8.10 are correctly specified as linear and additive.
- No \(T \times M\) interaction: the coefficient \(b\) is the same for all values of \(T\).
Conditions 1–3 cannot be tested from observed data at all. Conditions 4–5 can be partially probed by residual diagnostics and interaction terms.
8.8 Front-Door Identification
8.8.1 The Front-Door DAG
Every identification strategy in Section 8.4–Section 8.6 assumed an observed covariate set \(\mathbf{X}\) that blocks the back-door paths from \(T\) to \(Y\) through \(U\). What if \(U\) is wholly unobserved and no such adjustment set exists? The front-door criterion turns this obstacle into an opportunity: under two additional restrictions on the prototype mediation graph, the mediation structure itself provides identification of the total effect without conditioning on \(U\).
The two restrictions are: (i) remove the direct \(T \to Y\) edge, so \(M\) fully mediates; and (ii) require \(U\) has no arrow into \(M\), so the \(T \to M\) sub-effect is unconfounded.
Why the running example does not apply here. The depression/sleep scenario fails Condition 1 (full mediation): a behavioral intervention plausibly operates through several non-sleep channels. The canonical front-door example is Pearl’s smoking–tar–cancer graph: \(T\) = smoking, \(M\) = tar deposits, \(Y\) = lung cancer, \(U\) = genetic susceptibility. If all of smoking’s carcinogenic effect flows through tar, and genetic susceptibility does not act on tar directly, the front-door formula identifies the causal effect without observing \(U\).
8.8.2 The Three Front-Door Conditions
In the front-door DAG: Condition 1 holds (no \(T \to Y\) edge). Condition 2 holds (\(U\) has no arrow into \(M\)). Condition 3 holds: the only back-door path from \(M\) to \(Y\) is \(M \leftarrow T \leftarrow U \to Y\), which is blocked by conditioning on \(T\).
8.8.3 Derivation of the Front-Door Formula
8.9 Mediation vs. Instrumental Variables
| Feature | Instrumental Variables | Mediation Analysis |
|---|---|---|
| Position of third variable | Pre-treatment (\(Z\) precedes \(T\)) | Post-treatment (\(M\) follows \(T\)) |
| Causal role | Exogenous source of variation in \(T\) | Pathway through which \(T\) affects \(Y\) |
| Primary goal | Identification of \(T \to Y\) effect | Mechanism analysis (decomposition) |
| Key assumption | Exclusion: \(Z\) affects \(Y\) only through \(T\) | Sequential ignorability: no unmeasured \(M\)–\(Y\) confounding |
| Estimand | LATE (Wald) or ATE (homogeneity) | NDE, NIE, or CDE |
| Unobserved \(T\)–\(Y\) confounders | Permitted | Must be addressed separately |
| Testability | Relevance testable; exclusion untestable | Sequential ignorability untestable |
Can the same variable be both? Not for the same treatment–outcome relation. Within a single causal question of how \(T\) affects \(Y\), the mediator role places \(M\) on the causal path (inclusion required), whereas the IV role demands the exclusion restriction. These are mutually incompatible structural assumptions. The front-door identification formula is the closest bridge between the two within a single \((T, Y)\) analysis: it uses the mediator \(M\) to identify the total effect of \(T\) on \(Y\) even when \(T\) is confounded — but the front-door \(M\) is not an instrument.
8.10 Chapter Summary
| Symbol | Meaning |
|---|---|
| TE | Total effect \(\E[Y(1) - Y(0)]\) |
| \(\mathrm{CDE}(m)\) | Controlled direct effect at mediator level \(m\) Equation 8.2 |
| NDE | Natural direct effect Equation 8.4 |
| NIE | Natural indirect effect Equation 8.5 |
| \(Y(t, M(t'))\) | Cross-world counterfactual (nested potential outcome) |
| \(\tau = \tau' + ab\) | Baron–Kenny decomposition (linear model only) |
| Equation 8.7 | Mediation formula (nonparametric) |
| Equation 8.14 | Front-door formula |
- Mediation studies mechanisms. Mediation analysis defines direct and indirect effect concepts that target the pathways \(T \to Y\) and \(T \to M \to Y\). Only natural effects yield an additive TE decomposition; the CDE does not.
- The total effect is the baseline estimand. TE \(= \E[Y(1) - Y(0)]\) captures all pathways. It may be identified by randomization, back-door adjustment, front-door identification, or IV.
- The CDE uses do-calculus. The CDE fixes \(M = m\) by joint intervention \(\doop(T, M)\) and is identified by the back-door formula applied to the pair \((T, M)\). The CDE depends on the fixed level \(m\) and has no natural “indirect” complement.
- Natural effects require potential outcomes. The NDE and NIE involve cross-world counterfactuals \(Y(t, M(t'))\) that cannot be expressed with the do-operator alone. They are identified by the mediation formula under sequential ignorability. The critical Assumption 4 (no treatment-induced mediator–outcome confounder) is not secured by randomization of \(T\) and must be defended on subject-matter grounds.
- The linear model simplifies but restricts. The Baron–Kenny system gives \(\tau_{\mathrm{ind}} = ab\) and \(\tau_{\mathrm{dir}} = \tau'\), with \(\tau = \tau' + ab\). This decomposition is purely algebraic and holds only under linearity and no interaction.
- Front-door identification uses mediation structure. When \(M\) fully mediates \(T \to Y\), no \(T \to M\) confounding exists, and \(T\) blocks the back-door paths from \(M\) to \(Y\), the front-door formula identifies the total effect despite unobserved \(T\)–\(Y\) confounding, by composing two unconfounded sub-effects.
- Mediation and IV are complementary, not equivalent. IV uses a pre-treatment variable to generate exogenous variation in \(T\); mediation uses a post-treatment variable to study how the causal effect operates. The same variable cannot simultaneously serve as a mediator and a valid IV for the same treatment–outcome relation.
8.11 Problems
1. Identifying the CDE. Consider the DAG: \(T \to M\), \(T \to Y\), \(M \to Y\), \(X \to T\), \(X \to M\), \(X \to Y\), with all variables observed.
- Write the identification formula for \(\E[Y \mid \doop(T{=}1), \doop(M{=}m)]\) using the back-door criterion for the joint intervention \((T, M)\).
- Add an unobserved \(U\) with \(U \to T\) and \(U \to Y\). Does the back-door formula still identify the CDE? Explain which condition fails.
- Instead add \(U\) with \(U \to M\) and \(U \to Y\). Does the back-door formula still identify the CDE? Explain.
2. CDE vs. total effect. In the reduced prototype graph, let \(\mathbf{Z}\) satisfy the back-door criterion for both the total effect and the joint intervention \((T, M)\).
- Write expressions for the total effect and \(\mathrm{CDE}(m)\) using the back-door formula.
- Under what graphical condition does \(\mathrm{CDE}(m)\) equal the total effect for all \(m\)? Interpret this condition.
3. Natural direct and indirect effects. Verify the NDE + NIE = TE decomposition algebraically for the linear SEM \(M = \alpha T + \eta\), \(Y = \beta T + \gamma M + \varepsilon\) (no interaction).
- Compute \(Y(t, M(t'))\) in the linear model. Show that \(Y(t, M(t')) = \beta t + \gamma(\alpha t') + \text{noise}\).
- Derive \(\mathrm{NDE} = \beta\) and \(\mathrm{NIE} = \alpha\gamma\) from definitions Equation 8.4–Equation 8.5.
- Confirm \(\mathrm{NDE} + \mathrm{NIE} = \beta + \alpha\gamma = \mathrm{TE}\).
- Now suppose a \(T \times M\) interaction is added: \(Y = \beta T + \gamma M + \delta (T \cdot M) + \varepsilon\). Compute \(\mathrm{CDE}(m)\) and \(\mathrm{NDE}\). Show that \(\mathrm{NDE} \neq \mathrm{CDE}(m)\) when \(\delta \neq 0\).
4. The Baron–Kenny three-equation system. In the reduced prototype graph with the linear SEM Equation 8.8–Equation 8.10:
- State the three identification assumptions. For each, give the graphical condition in terms of back-door paths.
- Derive the equality \(\tau = \tau' + ab\) algebraically.
- Suppose \(\hat\tau = 0.50\), \(\hat a = 0.40\), \(\hat b = 0.60\), \(\hat\tau' = 0.26\). Compute the indirect effect by both the product and difference methods. Do they agree? Compute the proportion mediated.
- With \(\widehat{\mathrm{SE}}(\hat a) = 0.08\) and \(\widehat{\mathrm{SE}}(\hat b) = 0.10\), compute the Sobel standard error for \(\hat a \hat b\) using Equation 8.13 and construct an approximate 95% confidence interval.
5. The critical role of Assumption 3. Consider the graph where an unobserved \(V\) has \(V \to M\) and \(V \to Y\), with \(T\) randomized.
- Identify all back-door paths from \(M\) to \(Y\) in this graph.
- Can any combination of observed variables \((T, \mathbf{X})\) block all of these paths? Explain using d-separation.
- Suppose an analyst fits Equation 8.10 ignoring \(V\) and obtains \(\hat b = 0.80\). In which direction is \(\hat b\) biased if \(V\) has positive effects on both \(M\) and \(Y\)?
- State the additional data structure that would be needed to identify the second-stage effect nonparametrically.
6. Front-door identification. Consider the front-door graph.
- Verify that the three front-door conditions hold.
- Walk through the three-step proof of the Front-Door Formula (Equation 8.14): identify which do-calculus rule justifies each step.
- Add a direct edge \(T \to Y\) to the graph. Which front-door condition is violated? Does formula Equation 8.14 still hold?
- Explain why the front-door graph does not require no unmeasured \(M\)–\(Y\) confounding given \(T\) as an identifying assumption.
7. Mediation vs. instrumental variables. A researcher studies the effect of a job training program (\(T\)) on wages (\(Y\)). She proposes two intermediate variables: (A) motivation (\(M_A\)), measured after the program starts; (B) a lottery that randomly selects applicants for admission (\(Z\)), measured before the program.
- For variable (A): draw the mediation DAG including \(M_A\), \(T\), \(Y\), and unobserved ability \(U\). State the sequential ignorability assumption needed to identify the NIE through \(M_A\). Explain why randomization of \(T\) does not automatically satisfy this assumption.
- For variable (B): draw the IV DAG with \(Z\), \(T\), \(Y\), and \(U\). State the three IV assumptions. Explain why the exclusion restriction and the “mediator inclusion” of mediation analysis are mutually incompatible conditions for the same intermediate variable.
- The researcher argues that \(M_A\) (motivation) and \(Z\) (lottery) are both “intermediate” variables and that the analyses are interchangeable. Write a one-paragraph critique of this argument, using the distinctions from Section 8.9.
- Can the front-door formula be applied if motivation \(M_A\) fully mediates the effect of \(T\) on \(Y\) and \(U\) (unobserved ability) does not directly affect \(M_A\)? State the three conditions and assess whether they hold.