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?
More concretely, 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; as we will see, 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.
To keep the abstract formulas of this chapter tethered to a single concrete scenario, we adopt the depression intervention as a running example: \(T\) 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). We return to this example at each key estimand — TE, CDE, NDE/NIE, the Baron–Kenny decomposition — and note in Section 8.8 why the front-door criterion requires a different scenario.
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 naive 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. This makes 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
Throughout this chapter we work with the following prototype graph.
The graph encodes two causal pathways: the direct pathway \(T \to Y\), in which the treatment affects the outcome without passing through the mediator; and the indirect pathway \(T \to M \to Y\), in which the treatment first shifts the mediator, which in turn shifts the outcome.
8.2.2 Structural Equations
The prototype graph corresponds to the following nonparametric structural equation model: \[T = f_T(\mathbf{X},\, U,\, \varepsilon_T), \tag{8.1}\] \[M = f_M(T,\, \mathbf{X},\, \varepsilon_M), \tag{8.2}\] \[Y = f_Y(T,\, M,\, \mathbf{X},\, U,\, \varepsilon_Y), \tag{8.3}\] where each arrow corresponds to the presence of the parent in the child’s structural equation. In particular, \(U\) 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 \(U \to M\).
Disturbance structure. Writing a recursive system compatible with the graph is not yet a complete causal model: one must also say how the disturbances are jointly distributed. Unless stated otherwise, we adopt the NPSEM-IE reading of Equation 8.1–Equation 8.3: all shared causes represented in the model appear explicitly as variables (here \(\mathbf{X}\) and \(U\)), and the equation-specific disturbances \(\varepsilon_T\), \(\varepsilon_M\), \(\varepsilon_Y\) are mutually independent given those explicitly represented background variables. This independent-errors assumption is innocuous for the total-effect results of Section 8.3, but it becomes load-bearing when cross-world counterfactuals appear in Section 8.5–Section 8.6.
8.2.3 What Makes Mediation Harder Than Total Effect Estimation
The key difficulty is that \(M\) is a post-treatment variable. This creates two interrelated problems.
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, inducing spurious \(T\)–\(Y\) association through \(V\). This is precisely why naively regressing \(Y\) on \((T, M)\) does not isolate the direct effect.
Mediator–outcome confounding. Even when treatment is randomized — eliminating confounding on the \(T\) side — 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. This is the central challenge of mediation analysis.
8.2.4 A Working Graph for the Identification Sections
The prototype graph retains the unobserved confounder \(U\) on purpose: the chapter ultimately demonstrates, in Section 8.8, that the causal effect of \(T\) on \(Y\) can be identified even when \(U\) is hidden. In the intervening sections, however, we rely on back-door adjustment formulas that require every \(T\)–\(Y\) and \(T\)–\(M\) confounder to be captured by the observed covariate set \(\mathbf{X}\).
8.3 Total Causal Effect
Before introducing direct and indirect effect concepts, we fix the reference quantity: the total causal effect of \(T\) on \(Y\).
The total effect captures the combined impact of all causal pathways from \(T\) to \(Y\). It is identified by the back-door formula whenever \(\mathbf{X}\) satisfies the back-door criterion: \[\mathrm{TE} = \sum_{\mathbf{x}} \bigl[\E[Y \mid T{=}1, \mathbf{X}{=}\mathbf{x}] - \E[Y \mid T{=}0, \mathbf{X}{=}\mathbf{x}]\bigr] P(\mathbf{X}{=}\mathbf{x}).\] As throughout the book, the formula additionally presumes the intervention semantics and consistency established in Chapters 3–5, together with treatment positivity, so that both conditional means are well-defined wherever the outer average visits.
Running example. For the depression trial, 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 by which the intervention acts on depression — cognitive restructuring, behavioral activation, therapeutic alliance, and so on. Mediation analysis asks: of this total, how much is due to sleep?
The goal of mediation analysis is to define direct and indirect effect estimands that target the separate pathways within this total. The next two sections introduce two such families, each formalizing a different notion of what “direct” and “indirect” mean; of the two, only the second — the natural direct and indirect effects — is constructed to yield an additive decomposition of the total effect.
8.4 Controlled Direct Effect
8.4.1 Definition
The do-operator provides a direct definition of the controlled direct effect: intervene on both \(T\) and \(M\) simultaneously, fixing \(M\) at a specified level \(m\). Fixing \(M = m\) shuts down causal pathways that pass through \(M\); any remaining effect of \(T\) on \(Y\) operates through pathways that bypass \(M\) — in the prototype graph, precisely the direct edge \(T \to Y\).
Because Equation 8.4 involves two simultaneous interventions, it corresponds to the mutilated graph \(\Gcal_{\overline{T}\,\overline{M}}\) in which all edges into both \(T\) and \(M\) are deleted.
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, regardless of assignment. Any reduction in depression that survives this manipulation must come from non-sleep mechanisms. Whether sleep can be externally fixed in a clinical trial is a separate question — sleep is notoriously difficult to control — which is why the policy interpretation of the CDE in this scenario is strained even when the estimand is statistically identified.
8.4.2 Identification of the CDE
Since the CDE involves a joint intervention \(\doop(T, M)\), its identification in this chapter’s baseline-covariate setting reduces to a back-door-type adjustment applied to the pair \((T, M)\) jointly.
Theorem 8.1 (Baseline Adjustment for the CDE) Suppose \(\mathbf{Z}\) is a set of baseline (pre-treatment) covariates satisfying, for the joint intervention \(\doop(T, M)\) on \(Y\): (i) \(\mathbf{Z}\) contains no descendant of \(T\) or \(M\), and (ii) \(\mathbf{Z}\) d-separates the joint exposure set \(\{T, M\}\) from \(Y\) in \(\Gcal_{\underline{T}\,\underline{M}}\), the graph obtained by deleting all arrows out of \(T\) and out of \(M\). Suppose further that (iii) joint positivity holds at \((t, m)\): \[P(T{=}t,\, M{=}m \mid \mathbf{Z}{=}\mathbf{z}) > 0 \quad \text{for $P$-almost every $\mathbf{z}$ in the support of } \mathbf{Z}.\] Then \[\E[Y \mid \doop(T{=}t),\, \doop(M{=}m)] = \sum_{\mathbf{z}} \E[Y \mid t,\, m,\, \mathbf{z}]\, P(\mathbf{z}). \tag{8.5}\]
Condition (iii) ensures that the conditional expectation is defined at every \(\mathbf{z}\) that receives positive weight under \(P(\mathbf{z})\). Joint positivity is the CDE counterpart of the overlap conditions (P1)–(P2) introduced for the mediation formula in Section 8.6. Even when \(T\) is randomized, (iii) is not automatic: it additionally requires that every level \(m\) of the mediator has positive conditional probability given the treatment arm and \(\mathbf{Z}\). When some \((t, m, \mathbf{z})\) cells are never observed, \(\mathrm{CDE}(m)\) is not empirically identified at that \(m\).
Formula Equation 8.5 is the standard back-door adjustment formula applied to the pair \((T, M)\) as a joint treatment. Under the working assumption of Section 8.2.4, \(\mathbf{Z} = \mathbf{X}\) satisfies condition (ii): deleting the arrows out of \(T\) and \(M\) removes \(T \to M\), \(T \to Y\), and \(M \to Y\), and the only remaining paths from \(\{T, M\}\) to \(Y\), namely \(T \leftarrow \mathbf{X} \to Y\) and \(M \leftarrow \mathbf{X} \to Y\), are blocked by \(\mathbf{X}\).
Graph surgery and back-door adjustment are distinct steps. It is worth pausing on what the theorem does and does not say, because the two steps of identification can be conflated. Graph surgery defines the interventional target by deleting arrows into \(T\) and \(M\); this specifies what we want to compute. Expressing that target as a functional of the observed distribution is a separate step that requires a valid adjustment set \(\mathbf{Z}\) in the original graph. The mutilated graph does not by itself justify the right-hand side of Equation 8.5. Consider what happens if we set aside the working assumption and return to the full prototype with \(U\) unobserved: the back-door path \(T \leftarrow U \to Y\) is present in the original graph and is not blocked by any observed covariate, so \(\mathbf{X}\) alone is not a valid adjustment set. Adjustment identifies the CDE in that setting only when \(\mathbf{Z}\) actually blocks the \(U\)-path. When \(U\) remains unobserved, alternative strategies are required (e.g. the front-door formula of Section 8.8, or an instrumental variable as in Chapter 7).
8.4.3 Physical Manipulability and the Meaning of the CDE
The CDE answers the question: what is the effect of \(T\) on \(Y\) when the mediator is prevented from changing? This question has a clean causal answer precisely because \(\doop(M{=}m)\) is a genuine intervention: it severs all edges into \(M\), placing \(M\) at \(m\) regardless of what \(T\) does.
8.4.4 The CDE Does Not Have a Complementary Indirect Effect
A common misconception is that the CDE and some complementary “controlled indirect effect” sum to the total effect, analogous to the NDE \(+\) NIE \(=\) TE decomposition. This is not generally true.
Fixing \(M = m\) by intervention removes treatment-induced variation in the mediator, but the residual quantity \(\mathrm{TE} - \mathrm{CDE}(m)\) is not a controlled indirect effect. It is a well-defined causal contrast — every term below is an interventional mean — but it depends on the selected mediator level \(m\), it does not correspond to any intervention that isolates only the indirect pathway, and it cannot be interpreted as the effect transmitted through \(M\) in a complementary pathway decomposition. Formally: \[\begin{aligned} \mathrm{TE} - \mathrm{CDE}(m) ={}& \Bigl(\E\!\left[Y \mid \doop(T{=}1)\right] - \E\!\left[Y \mid \doop(T{=}1),\,\doop(M{=}m)\right]\Bigr) \\ &- \Bigl(\E\!\left[Y \mid \doop(T{=}0)\right] - \E\!\left[Y \mid \doop(T{=}0),\,\doop(M{=}m)\right]\Bigr), \end{aligned} \tag{8.6}\] which mixes the effect of intervening on \(M\) under \(T=1\) and \(T=0\) in a way that has no simple pathway interpretation. The decomposition TE \(=\) NDE \(+\) NIE holds because the natural effects involve cross-world counterfactuals that are constructed precisely to be complementary; the CDE has no analogous cross-world partner.
The practical implication is that if the research goal is to decompose the total effect into direct and indirect components, the correct estimands are the NDE and NIE, not \(\mathrm{CDE}(m)\) and \(\mathrm{TE} - \mathrm{CDE}(m)\). The CDE is the right estimand when the question is specifically about the effect of \(T\) with \(M\) controlled at a given level — a different and more limited question.
8.5 Natural Direct and Indirect Effects
8.5.1 Motivation: 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 \(T = 0\)?
Answering this requires a comparison across two intervention worlds: the world where \(T = 1\) but \(M\) is held at the value it would naturally take under \(T = 0\), versus the world where \(T = 0\). The first of these cannot be realized by any single experiment on the same unit — \(T\) would have to be set to 1 (to determine \(Y\)) and to 0 (to fix \(M\) at its natural \(T=0\) value) simultaneously. It requires the nested potential outcomes notation \(Y(t, M(t'))\), which denotes the outcome that would be 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.
8.5.2 Definitions
Running example. The NDE is the expected reduction in depression score when the intervention is delivered but each participant’s sleep is held at the value it would have had without the intervention — the part of the benefit that comes from cognitive, behavioral, or therapeutic-alliance pathways, not from sleep. The NIE is the complementary piece. By construction \(\mathrm{NDE} + \mathrm{NIE} = \mathrm{TE}\). One caveat carries over from the CDE discussion: the nested counterfactual \(Y(t, m)\) presupposes that the mediator value is a well-defined causal input. If two different ways of arriving at the same sleep score would have different effects on depression, then “holding sleep at the value it would have had without the intervention” is ambiguous, and the NDE inherits that ambiguity. Throughout, we read \(Y(t, m)\) structurally, as the output of the equation \(f_Y\) in Equation 8.3 evaluated at \((t, m)\).
Decomposition. The total effect decomposes as \[\mathrm{TE} = \mathrm{NDE} + \mathrm{NIE}. \tag{8.9}\]
Proof. \[\begin{aligned} \mathrm{NDE} + \mathrm{NIE} &= \E[Y(1, M(0)) - Y(0, M(0))] + \E[Y(1, M(1)) - Y(1, M(0))] \\ &= \E[Y(1, M(1)) - Y(0, M(0))] = \E[Y(1) - Y(0)] = \mathrm{TE}. \qquad\square \end{aligned}\] The final equality is not pure algebra: it uses composition, \(Y(t, M(t)) = Y(t)\), which holds under the recursive structural interpretation adopted in Section 8.2 — setting \(T = t\) and simultaneously setting \(M\) to the very value it would take under \(T = t\) reproduces the single intervention \(\doop(T{=}t)\).
8.5.3 Interpreting the Cross-World Nature
The NDE involves the counterfactual \(Y(1, M(0))\): the outcome when \(T\) is set to 1 but \(M\) is held at the value it would naturally take under \(T = 0\). This is a cross-world quantity because the two coordinates of the argument refer to different intervention worlds. It cannot be observed for any individual, and in general it cannot be written as a do-expression.
This is not merely a philosophical subtlety. The cross-world nature has direct implications for identification: as shown in the next section, identifying the NDE and NIE requires assumptions that are strictly stronger than those needed for the CDE or the total effect.
8.6 Identification of Natural Effects
8.6.1 Sequential Ignorability
Identification of the NDE and NIE requires the conditions known collectively as sequential ignorability (Imai et al. 2010), together with consistency, composition, and positivity. We state sequential ignorability as three assumptions of two logically different kinds: Assumptions 1 and 2 below are observed-world statements — conditional independencies of the type met in Chapters 4–6 — while Assumption 3 is a cross-world statement relating counterfactuals from two different intervention worlds.
Despite the neutral-sounding name, these conditions are substantially stronger than the ignorability assumptions used for the total effect or the CDE. Assumptions 2 and 3 govern the mediator–outcome relationship, which is never directly controlled by the investigator; Assumption 3, moreover, is untestable in principle: no experiment, however elaborate, can place one unit in two intervention worlds at once. Students who have internalized the back-door criterion should resist the temptation to read sequential ignorability as a mild extension of it: the assumptions address different sub-problems, and the hardest one has nothing to do with how \(T\) was assigned.
Assumptions 1–2 are the natural extensions of the Baron–Kenny conditions of Section 8.7 to the potential outcomes setting: observed-world exchangeability for the treatment and for the mediator. Assumption 3 is the genuinely new requirement, and the second box explains its structural pedigree: it holds under the NPSEM-IE when, in addition to the pre-treatment confounding control already demanded by Assumptions 1–2, there is no treatment-induced mediator–outcome confounder \(L\).
What randomization of \(T\) does and does not provide. Randomizing the treatment \(T\) satisfies Assumption 1 (both its \(T\)–\(Y\) and \(T\)–\(M\) components) by design. It does not satisfy Assumption 2 or Assumption 3. 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 2 regardless of how \(T\) was assigned. The structural condition behind Assumption 3 is even more demanding: it can be defeated by a variable \(L\) that is itself caused by the treatment, so randomizing \(T\) actually creates the conditions under which such violations can arise. The upshot is that a randomized experiment identifies the total effect and the first-stage \(T \to M\) effect cleanly, but it does not by itself identify the second-stage \(M \to Y\) effect, and therefore it does not by itself identify the NDE or NIE.
8.6.2 The Mediation Formula
Under the sequential ignorability assumptions, consistency and composition, and the positivity conditions (P1)–(P2), the cross-world expectation \(\E[Y(t, M(t'))]\) is identified from observational data.
Theorem 8.2 (Mediation Formula (Pearl 2001)) Under sequential ignorability (Assumptions 1–3), consistency and composition, positivity (P1)–(P2), and with discrete \(M\) and \(\mathbf{X}\): \[\E\!\left[Y(t, M(t'))\right] = \sum_{m}\sum_{\mathbf{x}} \E\!\left[Y \mid T{=}t,\, M{=}m,\, \mathbf{X}{=}\mathbf{x}\right] P(M{=}m \mid T{=}t',\, \mathbf{X}{=}\mathbf{x})\, P(\mathbf{X}{=}\mathbf{x}). \tag{8.10}\]
Proof sketch. We derive the formula in five labeled steps, each invoking a specific identification assumption.
Step 1 (law of total expectation and composition). Condition on \(\mathbf{X}\) and marginalize over \(M(t')\): \[\E[Y(t, M(t'))] = \sum_{m,\mathbf{x}} \E[Y(t, m) \mid M(t'){=}m,\, \mathbf{X}{=}\mathbf{x}]\, P(M(t'){=}m \mid \mathbf{X}{=}\mathbf{x})\, P(\mathbf{X}{=}\mathbf{x}),\] using the composition axiom \(Y(t, M(t')) = Y(t, m)\) on the event \(\{M(t') = m\}\).
Step 2 (cross-world independence supplied by the causal model). Invoke Assumption 3: \(Y(t, m) \indep M(t') \mid \mathbf{X}\). This is the step where the structural model earns its keep. Under an NPSEM-IE compatible with the working graph, the counterfactuals \(M(t')\) and \(Y(t, m)\) are functions of disjoint disturbances conditional on \(\mathbf{X}\). It is a cross-world restriction: strictly stronger than anything randomization of \(T\) provides, and not testable from the observed-data distribution. With the cross-world independence in hand, the conditioning on \(M(t')\) in Step 1 can be dropped: \[\E[Y(t, m) \mid M(t'){=}m,\, \mathbf{X}{=}\mathbf{x}] = \E[Y(t, m) \mid \mathbf{X}{=}\mathbf{x}].\] This is the move that distinguishes the mediation formula from an ordinary back-door adjustment: the two counterfactual worlds must be decoupled before the observed-data substitutions that follow.
Step 3 (Assumptions 1 and 2). By \(Y(t, m) \indep T \mid \mathbf{X}\) (the \(Y(t,m)\) marginal of Assumption 1) and \(Y(t, m) \indep M \mid T, \mathbf{X}\) (Assumption 2), conditioning on the observables \(T\) and \(M\) leaves the potential-outcome mean unchanged: \[\E[Y(t, m) \mid \mathbf{X}{=}\mathbf{x}] = \E[Y(t, m) \mid T{=}t,\, M{=}m,\, \mathbf{X}{=}\mathbf{x}].\]
Step 4 (consistency for \(Y\)). On the event \(\{T = t,\, M = m\}\), \(Y(t, m) = Y\), and so the left side equals \(\E[Y \mid T{=}t, M{=}m, \mathbf{X}{=}\mathbf{x}]\).
Step 5 (treatment–mediator component of Assumption 1, and consistency for \(M\)). Rewrite the mediator distribution using \(M(t') \indep T \mid \mathbf{X}\) and consistency (\(M(t') = M\) when \(T = t'\)): \[P(M(t'){=}m \mid \mathbf{X}{=}\mathbf{x}) = P(M{=}m \mid T{=}t',\, \mathbf{X}{=}\mathbf{x}).\]
Substituting Steps 2–5 into Step 1 yields Equation 8.10. The positivity conditions ensure that every conditional expectation and every conditional distribution on the right-hand side is well-defined at each cell that receives positive weight. \(\square\)
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 \(Y\) evaluated at treatment 1, but weight the mediator by its distribution under treatment 0. This counterfactual reweighting is what makes the formula non-trivial. Unlike an ordinary back-door adjustment, which is a functional of a single treatment arm, the mediation formula combines two arms, and the bridge between them is the cross-world independence used in Step 2. Whenever a student is tempted to read Equation 8.10 as a routine standardization formula, the presence of the second treatment index \(t'\) on the right-hand side is the visible trace of that cross-world step.
The NDE and NIE from the formula. Applying Equation 8.10 with \((t, t') = (1, 0)\) and \((0, 0)\): \[\begin{aligned} \mathrm{NDE} &= \sum_{m,\mathbf{x}} \bigl[\E[Y \mid T{=}1, M{=}m, \mathbf{X}{=}\mathbf{x}] - \E[Y \mid T{=}0, M{=}m, \mathbf{X}{=}\mathbf{x}]\bigr] \\ &\qquad\quad \times P(M{=}m \mid T{=}0, \mathbf{X}{=}\mathbf{x})\, P(\mathbf{X}{=}\mathbf{x}), \\[4pt] \mathrm{NIE} &= \sum_{m,\mathbf{x}} \E[Y \mid T{=}1, M{=}m, \mathbf{X}{=}\mathbf{x}] \\ &\qquad\quad \times \bigl[P(M{=}m \mid T{=}1, \mathbf{X}{=}\mathbf{x}) - P(M{=}m \mid T{=}0, \mathbf{X}{=}\mathbf{x})\bigr] P(\mathbf{X}{=}\mathbf{x}). \end{aligned}\]
Identification versus estimation. The last three sections have been entirely about causal estimands and their identification: what quantities are we trying to learn, and under what structural assumptions are they expressible as functionals of the observed data distribution? No specific model for \(P(Y \mid T, M, \mathbf{X})\) or \(P(M \mid T, \mathbf{X})\) was required. Section 8.7 turns to a parametric linear model that delivers familiar closed-form estimators under additional modeling assumptions — linearity, additivity, and no \(T \times M\) interaction — that go beyond the identification conditions above. Fitting the regression system does not solve the identification problem; it assumes it has already been solved.
8.7 The Linear Mediation Model: A Historical Special Case
The framework developed above — potential outcomes, cross-world counterfactuals, sequential ignorability — is the modern conceptual foundation of mediation analysis. Before that framework existed, practitioners used a simpler regression-based approach that works cleanly under linearity and no interaction. That approach, due to Baron and Kenny (1986), was enormously influential and remains widely cited. We study it here for three reasons: it builds intuition for the two-stage structure of mediation, its coefficients \(a\), \(b\), \(\tau'\) reappear as special cases of the NDE and NIE under linearity, and it is the dominant approach in many applied literatures that students will encounter.
It is not, however, the general framework. The algebraic decomposition \(\tau = \tau' + ab\) is a consequence of linearity, not a causal identity. Outside linear, no-interaction models it fails, and the product and difference methods it produces are not estimates of the NDE or NIE.
8.7.1 The Baron–Kenny Three-Equation System
The regression-based approach restricts the reduced prototype graph to a linear SEM, conditioning throughout on \(\mathbf{X}\): \[Y = \alpha_1 + \tau T + \boldsymbol{\gamma}_1^{\top}\mathbf{X} + \varepsilon_1, \tag{8.12}\] \[M = \alpha_2 + a T + \boldsymbol{\gamma}_2^{\top}\mathbf{X} + \varepsilon_2, \tag{8.13}\] \[Y = \alpha_3 + \tau' T + b M + \boldsymbol{\gamma}_3^{\top}\mathbf{X} + \varepsilon_3. \tag{8.14}\]
The four coefficients of interest are: \(\tau\), the total effect of \(T\) on \(Y\); \(a\), the first-stage effect of \(T\) on \(M\); \(\tau'\), the direct effect of \(T\) on \(Y\) controlling for \(M\); and \(b\), the second-stage effect of \(M\) on \(Y\) controlling for \(T\).
8.7.2 The Component Pathways
Equations Equation 8.12–Equation 8.14 identify three distinct causal sub-problems, each by the back-door formula.
First stage (\(T \to M\)). Equation Equation 8.13 is the parametric implementation of the back-door formula for the effect of \(T\) on \(M\): under linearity and Condition 2 below, conditioning on \(\mathbf{X}\) blocks all back-door paths from \(T\) to \(M\), and the OLS coefficient \(a\) identifies \(\E[M(1)] - \E[M(0)]\). In a randomized experiment, \(a\) is identified without conditioning on anything.
Second stage (\(M \to Y\) given \(T\)). Equation Equation 8.14 implements the back-door formula for the effect of \(M\) on \(Y\) given \(T\). In the reduced prototype graph, \(M\) has parents \(T\) and \(\mathbf{X}\), so every back-door path from \(M\) begins with \(M \leftarrow T\) or \(M \leftarrow \mathbf{X}\). The four such paths are \(M \leftarrow T \to Y\) (blocked by \(T\)); \(M \leftarrow T \leftarrow \mathbf{X} \to Y\) (blocked by \(T\) or \(\mathbf{X}\)); \(M \leftarrow \mathbf{X} \to Y\) (blocked by \(\mathbf{X}\)); and \(M \leftarrow \mathbf{X} \to T \to Y\) (blocked by \(T\) or \(\mathbf{X}\)). Conditioning on \((T, \mathbf{X})\) therefore blocks every back-door path from \(M\) to \(Y\) in this graph, provided Assumption 2 of Section 8.6 holds.
8.7.3 The Product and Difference Formulas
Proposition 8.1 (Mediation Decomposition in the Linear Model) Under Equation 8.12–Equation 8.14, \[\tau \;=\; \tau' \;+\; a b. \tag{8.15}\] The indirect and direct effects are therefore identified by \(\tau_{\mathrm{ind}} = ab\) (product method) and \(\tau_{\mathrm{dir}} = \tau - ab = \tau'\) (difference method).
Proof. Substitute Equation 8.13 into Equation 8.14: \[\begin{aligned} Y &= \alpha_3 + \tau' T + b(\alpha_2 + aT + \boldsymbol{\gamma}_2^{\top}\mathbf{X} + \varepsilon_2) + \boldsymbol{\gamma}_3^{\top}\mathbf{X} + \varepsilon_3 \\ &= (\alpha_3 + b\alpha_2) + (\tau' + ab)\,T + (\boldsymbol{\gamma}_3 + b\boldsymbol{\gamma}_2)^{\top}\mathbf{X} + (b\varepsilon_2 + \varepsilon_3). \end{aligned}\] Comparing with Equation 8.12 gives \(\tau = \tau' + ab\). \(\square\)
| 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 \(M\) |
Running example: a numerical walk-through. Suppose a randomized trial of the behavioral intervention (\(T\)) on depression (\(Y\)), with sleep quality (\(M\)) as the mediator and baseline covariates \(\mathbf{X}\), yields \[\hat\tau = 0.50, \qquad \hat a = 0.40, \qquad \hat b = 0.60, \qquad \hat\tau' = 0.26.\] For concreteness, orient the outcome so that larger \(Y\) means greater improvement (e.g. \(Y\) is the reduction in PHQ-9 score from baseline); a beneficial intervention then has positive coefficients throughout. Then:
- Indirect effect via sleep (product method): \(\hat a \hat b = 0.40 \times 0.60 = 0.24\).
- Direct effect (difference method): \(\hat\tau - \hat a \hat b = 0.50 - 0.24 = 0.26 = \hat\tau'\), which matches the fitted \(\hat\tau'\), illustrating the algebraic identity.
- Proportion mediated: \(0.24/0.50 = 0.48\), so roughly 48% of the total effect operates through sleep improvement in this sample.
These numbers serve the illustration only; Problem 4 asks students to recompute them and to construct a Sobel confidence interval.
8.7.4 Inference: The Sobel Test and Bootstrap
The delta method gives an approximate variance for the product \(\hat a \hat b\): \[\widehat{\mathrm{Var}}(\hat a \hat b) \approx \hat b^2\, \widehat{\mathrm{Var}}(\hat a) + \hat a^2\, \widehat{\mathrm{Var}}(\hat b) + 2\, \hat a\, \hat b\, \widehat{\mathrm{Cov}}(\hat a, \hat b). \tag{8.16}\] The Sobel test (Sobel 1982) uses the simplified approximation that drops the cross-covariance: \[\widehat{\mathrm{Var}}_{\mathrm{Sobel}}(\hat a \hat b) = \hat b^2\, \widehat{\mathrm{Var}}(\hat a) + \hat a^2\, \widehat{\mathrm{Var}}(\hat b), \tag{8.17}\] yielding the \(z\)-statistic \(z = \hat a \hat b / \sqrt{\widehat{\mathrm{Var}}_{\mathrm{Sobel}}(\hat a \hat b)}\). Because \(\hat a\) and \(\hat b\) come from separate regressions, their cross-covariance is often treated as negligible, but it is not zero in general and dropping it is a convention rather than an exact simplification.
Statistical validity versus causal interpretation. The Sobel test is a statistical inference procedure for the product of two fitted regression coefficients. Under standard regression conditions, Equation 8.17 is an approximation to the sampling variance of \(\hat a \hat b\), and the resulting \(z\)-statistic tests whether the regression product \(ab\) is zero. That is a property of the fitted model. Interpreting the tested product as an indirect causal effect is a separate claim that additionally requires the causal identification conditions and the structural restrictions below. When those assumptions fail, the Sobel approximation may still be a reasonable statistical test of whether the regression product is zero, but that product no longer corresponds to the indirect effect of \(T\) on \(Y\) through \(M\).
In practice, bootstrap confidence intervals for \(ab\) are preferred over the Sobel test for inference, because the distribution of a product of estimates is skewed in finite samples — especially when either \(\hat a\) or \(\hat b\) is small.
8.7.5 The Baron–Kenny Assumptions: Two Distinct Categories
The conditions required for the Baron–Kenny decomposition fall into two fundamentally different categories that should not be conflated.
In the reduced prototype graph, Conditions 1 and 2 hold because \(\mathbf{X}\) blocks every \(T\)–\(Y\) and \(T\)–\(M\) back-door path by construction. Condition 3 is different: it requires the absence of any variable \(V \to M\) and \(V \to Y\) not captured by \((T, \mathbf{X})\), and it is never guaranteed by randomization of \(T\) because \(M\) is a post-treatment variable that is not itself randomized. Restrictions 4 and 5 have no graphical counterpart.
Baron–Kenny as a special case of the modern framework. Under Conditions 1–3 and Restrictions 4–5, the Baron–Kenny decomposition delivers valid identification formulas for the NDE and NIE: linearity and no interaction guarantee that the population coefficients satisfy \(\mathrm{NDE} = \tau'\) and \(\mathrm{NIE} = ab\). Translating these identification formulas into consistent estimators requires a separate set of conditions. Outside Conditions 1–3 and Restrictions 4–5, \(\tau'\) and \(ab\) need not identify the CDE or the natural effects: the listed conditions are sufficient rather than necessary, and special structures can restore particular equalities, but in general the coefficients of a misspecified structural system carry no causal interpretation.
Chapter pivot: from decomposing an identified total effect to identifying a confounded one. Everything up to this point has taken identification of the total effect for granted — the working assumption of Section 8.2.4 absorbed all \(T\)–\(Y\) confounding into observed covariates — and asked how that effect decomposes across pathways. The remainder of the chapter reverses the direction of travel. In Section 8.8, the mediator stops being the object of study and becomes an identification device: the total effect itself is unidentified by adjustment, and the mediation structure of the graph is what rescues it.
8.8 Front-Door Identification
8.8.1 The Front-Door DAG
Every identification strategy above assumed that an observed covariate set \(\mathbf{X}\) exists that blocks the back-door paths from \(T\) to \(Y\) through \(U\). What if \(U\) is wholly unobserved and no such adjustment set exists? In the prototype mediation DAG, the back-door criterion then fails for the total effect, and the methods of earlier sections are unavailable. Chapter 3 previewed the front-door strategy for exactly this situation; this section gives the full development — the identification theorem with its support conditions, the do-calculus proof, the failure modes, and a numerical illustration — and connects it to mediation.
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: remove the direct \(T \to Y\) edge, so that \(M\) fully mediates the effect of \(T\) on \(Y\); and require that \(U\) has no arrow into \(M\), so that 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 on depression plausibly operates through several non-sleep channels — cognitive restructuring, behavioral activation, therapeutic alliance — so the direct edge \(T \to Y\) is present. The front-door formula is therefore unavailable for that setting. The canonical example in which the three front-door conditions are all plausible is Pearl’s smoking–tar–cancer graph: \(T\) denotes smoking, \(M\) the amount of tar deposited in the lungs, \(Y\) a lung-cancer outcome, and \(U\) an unobserved genetic susceptibility. If all of smoking’s carcinogenic effect flows through tar, and the genetic susceptibility does not act on tar directly, the front-door formula identifies the causal effect of smoking on cancer without ever observing \(U\). This is a stylized identification example, not a claim that measured tar burden satisfies the front-door conditions in actual epidemiologic data: a genetic susceptibility could plausibly affect inhalation behavior, tar metabolism, or the measurement of tar itself.
8.8.2 The Three Front-Door Conditions
In the front-door graph: Condition 1 holds because there is no \(T \to Y\) edge. Condition 2 holds, but for a subtler reason than “no back-door path exists”: there is a back-door path from \(T\) to \(M\), namely \(T \leftarrow U \to Y \leftarrow M\), but it is blocked by the unconditioned collider \(Y\). Because \(U\) has no arrow into \(M\), no open back-door path remains — and the criterion requires only the absence of open paths. Condition 3 holds because the only back-door path from \(M\) to \(Y\) is \(M \leftarrow T \leftarrow U \to Y\), which is blocked by conditioning on \(T\).
Example 8.1 (When the Front-Door Criterion Fails) Each of the three front-door conditions is load-bearing. We illustrate what breaks by modifying the front-door graph in two separate ways. As Case (a) shows, a single added edge can violate more than one condition at once.
Case (a): Conditions 2 and 3 fail — \(U\) also causes \(M\). Start with the front-door graph and add the edge \(U \to M\), so the unobserved confounder directly affects the mediator as well as the treatment and the outcome.
Condition 1 still holds: every directed path from \(T\) to \(Y\) passes through \(M\). Condition 2, however, now fails: the path \(T \leftarrow U \to M\) is an unblocked back-door path from \(T\) to \(M\). Because \(U\) is unobserved, there is no observed variable that can block this path. Condition 3 also fails: the path \(M \leftarrow U \to Y\) is an open back-door path from \(M\) to \(Y\), and conditioning on \(T\) does not block it, since \(T\) does not lie on the path.
Why the formula breaks. Both stages collapse. Stage 1 uses \(P(M{=}m \mid T{=}t)\) as if the \(T \to M\) link were unconfounded. But with \(U \to M\), the association between \(T\) and \(M\) is partly due to their shared cause \(U\): the quantity \(P(m \mid t)\) mixes cause and confounding and no longer identifies \(P(m \mid \doop(T{=}t))\). Likewise, because the open path \(M \leftarrow U \to Y\) survives conditioning on \(T\), the Stage 2 functional does not generally equal \(P(y \mid \doop(M{=}m))\). Failure of either condition alone would already suffice to invalidate the formula.
Case (b): Condition 3 fails — the \(M \to Y\) link has an extra unobserved confounder. Return to the prototypical graph and add a second unobserved variable \(V\) that affects both \(M\) and \(Y\).
Conditions 1 and 2 still hold. But Condition 3 now fails: the path \(M \leftarrow V \to Y\) is a back-door path from \(M\) to \(Y\) that is not blocked by conditioning on \(T\). The path does not pass through \(T\), so fixing \(T\) does nothing to close it.
Why the formula breaks. Stage 2 uses \(\sum_{t'} P(y \mid t', m)\,P(t')\) as the back-door adjustment formula for \(P(y \mid \doop(M{=}m))\), with \(T\) as the adjustment variable. But \(T\) only satisfies the back-door criterion for \(M \to Y\) when it blocks all back-door paths from \(M\) to \(Y\). The new path \(M \leftarrow V \to Y\) is not blocked by \(T\), so \(T\) is no longer a valid adjustment set for Stage 2.
Summary.
| Modification | Condition(s) violated | Consequence |
|---|---|---|
| Add \(T \to Y\) | Cond. 1 | A causal path bypasses \(M\) |
| Add \(U \to M\) | Conds. 2 and 3 | Stages 1 and 2 both fail |
| Add \(V \to M\) and \(V \to Y\) | Cond. 3 | Stage 2 fails |
In each case the graphical check detects the failure before any formula is written down. The first row’s modification — adding a direct \(T \to Y\) edge — shows that the absence of such an edge in the front-door graph is not an independent requirement: it is simply one way of satisfying Condition 1, which demands that all directed paths from \(T\) to \(Y\) pass through \(M\).
8.8.3 Derivation of the Front-Door Formula
Theorem 8.3 (Front-Door Formula (Pearl 1995)) Suppose \(M\) satisfies the front-door criterion for the effect of \(T\) on \(Y\). Assume further the positivity conditions:
(F1) Treatment positivity. \(P(T = t) > 0\) at the evaluation level \(t\), so that \(P(m \mid t)\) is well-defined; every \(t'\) entering the inner sum lies in the support of \(T\).
(F2) Mediator overlap across treatment arms. For every mediator value \(m\) with \(P(M{=}m \mid T{=}t) > 0\) and every \(t'\) with \(P(T{=}t') > 0\), \(P(M{=}m \mid T{=}t') > 0\).
Then \[P\!\left(y \mid \doop(T{=}t)\right) = \sum_m P(m \mid t) \sum_{t'} P(y \mid m, t')\, P(t'). \tag{8.18}\]
Condition (F2) is the front-door analogue of the mediator overlap (P2): it ensures that \(P(y \mid m, t')\) is empirically defined at every \((m, t')\) cell that receives positive weight in Equation 8.18.
Proof. We apply the do-calculus in three steps, each exploiting one of the front-door conditions.
Step 1: Identify the effect of \(T\) on \(M\). Apply Rule 2 with \(X = \varnothing\), \(Z = T\), \(W = \varnothing\); the required graph is \(\Gcal_{\underline{T}}\), obtained by deleting \(T\)’s outgoing arrow. The only remaining path between \(T\) and \(M\) is \(T \leftarrow U \to Y \leftarrow M\), blocked at the unconditioned collider \(Y\); Condition 2 is what guarantees that no open path survives (an edge \(U \to M\) would leave \(T \leftarrow U \to M\) open). Hence \((M \indep T)_{\Gcal_{\underline{T}}}\), and Rule 2 gives \[P\!\left(m \mid \doop(T{=}t)\right) = P(m \mid t).\]
Step 2: Identify the effect of \(M\) on \(Y\). By Condition 3, conditioning on \(T\) blocks all back-door paths from \(M\) to \(Y\), so \(\{T\}\) is a valid back-door adjustment set: \[P\!\left(y \mid \doop(M{=}m)\right) = \sum_{t'} P(y \mid m, t')\, P(t').\]
Step 3: Combine via full mediation. The law of total probability applied under the intervention \(\doop(T{=}t)\) gives \[P\!\left(y \mid \doop(T{=}t)\right) = \sum_m P\!\left(m \mid \doop(T{=}t)\right)\, P\!\left(y \mid \doop(T{=}t),\, m\right), \tag{8.19}\] where the second factor conditions on \(M{=}m\). Converting that conditioning into the intervention \(\doop(M{=}m)\) is the delicate step, and it is here that Conditions 3 and 1 do their work, in that order.
First, apply Rule 2 with \(X = T\), \(Z = M\), \(W = \varnothing\); the required graph is \(\Gcal_{\overline{T}\,\underline{M}}\). Deleting the arrows into \(T\) and out of \(M\) leaves only the edges \(T \to M\) and \(U \to Y\), so no path connects \(M\) to \(Y\) and \((Y \indep M \mid T)_{\Gcal_{\overline{T}\,\underline{M}}}\) holds — Condition 3 secures this in general, since any surviving path would be a back-door path from \(M\) to \(Y\) not blocked by \(T\). Rule 2 gives \[P\!\left(y \mid \doop(T{=}t),\, m\right) = P\!\left(y \mid \doop(T{=}t),\, \doop(M{=}m)\right).\] Second, apply Rule 3 with \(X = M\), \(Z = T\), \(W = \varnothing\); the required graph is \(\Gcal_{\overline{M}\,\overline{T}}\). Deleting the arrows into both \(M\) and \(T\) isolates \(T\), so the now-redundant \(\doop(T{=}t)\) can be dropped: \[P\!\left(y \mid \doop(T{=}t),\, \doop(M{=}m)\right) = P\!\left(y \mid \doop(M{=}m)\right).\] Condition 1 secures this second step in general: every directed path from \(T\) to \(Y\) passes through \(M\), whose incoming arrows have been deleted, so no route from \(T\) to \(Y\) survives. This is the formal content of “the total effect of \(T\) factors through \(M\).” Substituting Steps 1 and 2 into Equation 8.19 yields Equation 8.18. \(\square\)
8.8.4 Intuition: Routing Around Confounding
The front-door formula achieves identification in two steps that each use only unconfounded variation.
- \(T\) to \(M\): There is no confounding on the \(T \to M\) edge (\(U\) does not affect \(M\)), so \(P(m \mid t)\) is the causal effect of \(T\) on \(M\).
- \(M\) to \(Y\): There is back-door confounding on \(M \to Y\) through the path \(M \leftarrow T \leftarrow U \to Y\), but \(T\) is a non-collider on this path, so conditioning on \(T\) closes it. The resulting conditional \(P(y \mid m, t')\) is then averaged over the observed marginal distribution of \(T\).
The key insight is that \(U\) confounds \(T\) and \(Y\) but not the \(T \to M\) edge. The front-door formula exploits this asymmetry to identify the total causal effect without ever observing or conditioning on \(U\).
8.8.5 Front-Door vs. Prototype Mediation
| Feature | Prototype mediation DAG | Front-door DAG |
|---|---|---|
| Direct \(T \to Y\) edge | Present | Absent |
| \(U\) confounds \(T\)–\(Y\) | Yes | Yes |
| Goal | Mechanism analysis (decomposition under natural effects; level-specific direct effect under CDE) | Identify total effect despite \(T\)–\(Y\) confounding |
| Requires Assumption 2 (\(M\)–\(Y\) no confounding given \(T\)) | Yes | Satisfied by front-door Condition 3, not by randomization or by default |
| Identified by front-door formula | No (direct \(T \to Y\) breaks Condition 1) | Yes |
8.9 Mediation vs. Instrumental Variables
Mediation analysis and instrumental variables both involve a third variable connected to the treatment–outcome relationship, but the causal role of that variable is fundamentally different in the two frameworks.
8.9.1 Structural Comparison
| 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 under confounding | Mechanism analysis of \(T \to Y\) effect |
| Key assumption | Exclusion: \(Z\) affects \(Y\) only through \(T\) | Sequential ignorability: mediator–outcome ignorability plus the cross-world independence |
| Estimand | LATE (Wald, under monotonicity); under a homogeneous linear effect, the common structural coefficient (Ch. 7) | NDE, NIE, or CDE |
| Unobserved \(T\)–\(Y\) confounders | Permitted (IV routes around them) | Must be addressed separately (Assumption 1) |
| Testability | Relevance testable; exclusion untestable | Sequential ignorability untestable |
8.9.2 The Conceptual Contrast
The contrast is sharpest in terms of what the third variable does in each framework.
In IV, \(Z\) is a handle: it generates exogenous variation in \(T\) that is free of the back-door confounding path \(T \leftarrow U \rightarrow Y\). The instrument is valuable precisely because it is not on the causal path from \(T\) to \(Y\) — the exclusion restriction says that \(Z\) cannot directly affect \(Y\).
In mediation analysis, \(M\) is a pathway: it transmits part of the causal effect of \(T\) to \(Y\). The mediator is valuable precisely because it is on the causal path. The analysis goal depends on the estimand: under natural effects, the question is how much of the total effect flows through this pathway; under the CDE, the question is what part of \(T\)’s effect on \(Y\) remains when \(M\) is held fixed at a chosen level.
8.9.3 Can the Same Variable Be Both?
It is worth asking whether the same variable \(M\) could serve as both a mediator and an instrument. The answer is: 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 — that \(M\) affects \(Y\) only through \(T\). These are mutually incompatible structural assumptions about the same intermediate variable.
The qualifier for the same treatment–outcome relation matters. A variable that mediates one causal question may, in a differently oriented question, serve as an instrument-like exogenous source of variation. For example, occupation might mediate the effect of education on wages while functioning as an exogenous source of variation in some unrelated downstream analysis where neither education nor wages is the target. The incompatibility is between the two roles within one fixed \((T, Y)\) pair, not across different scientific questions.
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 — a goal that IV also pursues. But, as the warning box in Section 8.8 stresses, the front-door \(M\) is not an instrument: it lies on the causal path, satisfies a graphical front-door condition, and substitutes structural restrictions on the graph for the exclusion restriction that defines an IV.
8.10 Summary
Mediation studies mechanisms. Mediation analysis aims to study the mechanisms by which \(T\) affects \(Y\) by defining direct and indirect effect concepts that target the pathways \(T \to Y\) and \(T \to M \to Y\). Of the two direct-effect families developed in this chapter, only the natural effects are constructed to yield an additive decomposition of the total effect; the controlled direct effect is not. The challenge throughout is that the mediator is a post-treatment variable that may be confounded.
The total effect is the baseline estimand. The TE \(= \E[Y(1) - Y(0)]\) captures all pathways combined. It may be identified by randomization, by back-door adjustment on a sufficient covariate set, by front-door identification, or by other identifying structures; which strategy applies depends on the available causal structure.
The CDE uses do-calculus. The controlled direct effect fixes \(M = m\) by joint intervention \(\doop(T, M)\). Under consistency, a well-defined joint intervention, an appropriate adjustment condition for the pair \((T, M)\), and joint positivity, it is identified by standardization (Theorem 8.1). The CDE depends on the fixed level \(m\) and does not have a 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 decompose the total effect as TE \(=\) NDE \(+\) NIE, and are identified by the mediation formula under sequential ignorability. The critical cross-world independence (Assumption 3) is not secured by randomization of \(T\). Under the NPSEM-IE used in these notes, it follows from the independent-error structural model together with adequate control of pre-treatment common causes and the absence of a treatment-induced mediator–outcome confounder.
The linear model simplifies but restricts. The Baron–Kenny three-equation system gives the clean formulas \(\tau_{\mathrm{ind}} = ab\) and \(\tau_{\mathrm{dir}} = \tau'\), with \(\tau = \tau' + ab\). This decomposition is purely algebraic and holds only under linearity and no interaction. In nonlinear or interaction settings, the product and difference methods disagree and NDE \(\neq\) CDE.
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. It achieves this 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 baseline adjustment for the joint intervention \((T, M)\) as in Theorem 8.1.
- Add an unobserved \(U\) with \(U \to T\) and \(U \to Y\). Is the CDE identified by any adjustment set consisting of observed variables? In particular, does standardization over \(X\) remain valid? Explain which condition of Theorem 8.1 fails.
- Instead add \(U\) with \(U \to M\) and \(U \to Y\). Answer the same questions.
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 the \(\mathrm{CDE}(m)\) using the back-door formula.
- Give a sufficient graphical condition under which fixing \(M\) removes no causal pathway from \(T\) to \(Y\) — for example, that no directed path from \(T\) to \(Y\) passes through \(M\). Interpret this condition.
- (second pass) Show by example that the graphical condition of part (b) alone does not make \(\mathrm{CDE}(m)\) equal to the total effect for all \(m\): exhibit a structural equation for \(Y\) with a \(T \times M\) interaction in which the two quantities differ even though \(M\) is not affected by \(T\). What additional functional condition closes the gap?
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, writing the noise terms explicitly: \(Y(t, M(t')) = \beta t + \gamma \alpha t' + \gamma\eta + \varepsilon\). Note that the same \(\eta\) appears for every value of \(t'\) — the structural construction shares the mediator disturbance across intervention worlds, and this is exactly what makes the nested counterfactual meaningful.
- Derive \(\mathrm{NDE} = \beta\) and \(\mathrm{NIE} = \alpha\gamma\) from Equation 8.7–Equation 8.8.
- Confirm \(\mathrm{NDE} + \mathrm{NIE} = \beta + \alpha\gamma = \mathrm{TE}\).
- Now suppose a \(T \times M\) interaction is added: \(Y = \beta T + \gamma M + \delta (T \cdot M) + \varepsilon\). Show that \(\mathrm{CDE}(m) = \beta + \delta m\) and \(\mathrm{NDE} = \beta + \delta\, \E[M(0)]\). Conclude that the two generally differ when \(\delta \neq 0\), and identify the one mediator level at which they coincide.
4. The Baron–Kenny three-equation system. In the reduced prototype graph with the linear SEM Equation 8.12–Equation 8.14:
- 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 estimated coefficients are \(\hat\tau = 0.50\), \(\hat a = 0.40\), \(\hat b = 0.60\), \(\hat\tau' = 0.26\). Compute the indirect effect by both the product and difference methods. Do they agree? Compute the proportion mediated, and state the conditions — direct and indirect effects of the same sign, and a total effect bounded away from zero — under which calling this ratio a “proportion” is meaningful. What happens to \(ab/\tau\) when \(\tau' = -ab\)?
- (optional computational practice) An analyst reports \(\hat a = 0.40\) with \(\widehat{\mathrm{SE}}(\hat a) = 0.08\), and \(\hat b = 0.60\) with \(\widehat{\mathrm{SE}}(\hat b) = 0.10\). Compute the Sobel standard error for \(\hat a \hat b\) using Equation 8.17 and construct an approximate 95% confidence interval.
5. The critical role of Assumption 2 (mediator–outcome ignorability). Consider the graph in which 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 8.14 ignoring \(V\) and obtains \(\hat b = 0.80\). Under a linear additive structural model in which \(V\) has positive effects on both \(M\) and \(Y\), in which direction is \(\hat b\) biased? Explain why the DAG alone, without the linearity and sign restrictions, does not determine the direction.
- State two additional data structures or design features that would permit identification of the second-stage effect — for example: (i) measuring a covariate set sufficient for the \(M\)–\(Y\) back-door paths, or (ii) an experiment that directly assigns the mediator. For each, say which assumption of Section 8.6 it restores.
6. Front-door identification. (Emphasis: the do-calculus derivation and diagnosing failure.) Consider the front-door graph.
- Verify that the three front-door conditions hold.
- Walk through the three-step proof of Theorem 8.3 for this graph: 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 Equation 8.18 still hold?
- Explain how front-door Condition 3 corresponds to the Baron–Kenny requirement that the \(M \to Y\) relation be unconfounded after conditioning on \(T\) (equivalently, Assumption 2 of Section 8.6). The front-door graph does not dispense with this requirement — it satisfies it by construction of the graph. Why does the condition hold in the front-door graph but fail in the graph with an \(M\)–\(Y\) confounder \(V\)?
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 an unobserved ability variable \(U\). State the complete set of identification assumptions from Section 8.6 needed to identify the NIE through \(M_A\), and explain why randomization of \(T\) secures only Assumption 1.
- 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 \(M_A\) and \(Z\) are both “intermediate” variables and that the analyses are interchangeable. Write a one-paragraph critique of this argument.
- Can the front-door formula be applied if motivation \(M_A\) fully mediates the effect of \(T\) on \(Y\) and \(U\) does not directly affect \(M_A\)? State the three conditions and assess whether they hold.
8. Front-door identification with baseline covariates. (Second pass; builds on Problem 6.) Consider the DAG: \(T \to M \to Y\), \(\mathbf{X} \to T\), \(\mathbf{X} \to M\), \(\mathbf{X} \to Y\), \(U \to T\), \(U \to Y\), with \(\mathbf{X}\) observed and pre-treatment, \(U\) unobserved, and no direct \(T \to Y\) edge.
- Show that the three front-door conditions hold conditionally on \(\mathbf{X}\): all directed paths from \(T\) to \(Y\) pass through \(M\); every back-door path from \(T\) to \(M\) is blocked given \(\mathbf{X}\) (enumerate the paths through \(\mathbf{X}\) and through \(U\), and say which node blocks each); and \((T, \mathbf{X})\) blocks every back-door path from \(M\) to \(Y\).
- Derive the covariate-adjusted front-door formula \[P\!\left(y \mid \doop(T{=}t)\right) = \sum_{\mathbf{x}} P(\mathbf{x}) \sum_m P(m \mid t, \mathbf{x}) \sum_{t'} P(y \mid m, t', \mathbf{x})\, P(t' \mid \mathbf{x})\] by repeating the three-step do-calculus argument of Theorem 8.3 with \(W = \{\mathbf{X}\}\) carried through each rule application. (You will also need \(P(\mathbf{x} \mid \doop(t)) = P(\mathbf{x})\); which rule justifies it, and why does it require \(\mathbf{X}\) to be pre-treatment?)
- Explain why the inner average uses the covariate-conditional distribution \(P(t' \mid \mathbf{x})\) rather than the marginal \(P(t')\) used in Equation 8.18.
- Now add the edge \(U \to M\). Which conditional front-door condition fails, and which stage of the two-stage argument collapses?
9. Nonidentifiability by construction (advanced / second pass). This problem practices the technique of the bow-graph example in Chapter 3: prove nonidentifiability by exhibiting two observationally equivalent models with different causal effects.
Consider the graph from Case (a) of Example 8.1: \(T \to M \to Y\), \(U \to T\), \(U \to M\), \(U \to Y\) (front-door Conditions 2 and 3 are violated). Let \(T, M, Y, U \in \{0,1\}\) with \(U \sim \mathrm{Bern}(1/2)\).
- Construct two SEMs \(\mathcal{M}_1\) and \(\mathcal{M}_2\), each compatible with the graph, that induce the same observed distribution of \((T, M, Y)\) but different interventional distributions: specifically, \(P_{\mathcal{M}_1}(Y{=}1 \mid \doop(T{=}1)) \ne P_{\mathcal{M}_2}(Y{=}1 \mid \doop(T{=}1))\). (Hint: set \(T = U\) in both models. In \(\mathcal{M}_1\) set \(M = T \wedge U\) and \(Y = M \vee U\); in \(\mathcal{M}_2\) set \(M = T \vee U\) and \(Y = M \vee U\), where \(\wedge\) and \(\vee\) denote Boolean AND and OR. Observationally \(T = M = Y = U\) in both models, so both produce \(P(T{=}0, M{=}0, Y{=}0) = P(T{=}1, M{=}1, Y{=}1) = \tfrac{1}{2}\). Then compute \(P(Y{=}1 \mid \doop(T{=}1))\) and \(P(Y{=}1 \mid \doop(T{=}0))\) in each model, and take their difference.)
- Explain precisely which identification strategy fails in this graph and why. Your explanation should reference (i) the back-door criterion (is there any observed variable that blocks \(T \leftarrow U \to Y\)?), (ii) the front-door criterion (which conditions fail and which stages collapse), and (iii) the completeness theorem (what does the existence of your two models imply about any identification method?).