Identification Theory

Comprehensive framework for causal identification in statistical methodology

Use this skill when working on: causal identification, mediation analysis identification, DAG-based reasoning, potential outcomes, identification assumptions, partial identification, sensitivity analysis, or deriving identification formulas.

Core Concepts

What is Identification?

A causal parameter $\psi$ is identified if it can be uniquely determined from the observed data distribution $P(O)$.

Formally: $\psi$ is identified if $P_1(O) = P_2(O) \Rightarrow \psi_1 = \psi_2$.

Why Identification Matters

Causal Question → Target Estimand → Identification → Estimation → Inference ↓ ↓ ↓ ↓ ↓ "Does A E[Y(1)-Y(0)] Express in Statistical Confidence cause Y?" terms of P(O) methods intervals

Without identification, no amount of data can answer causal questions.

Two Frameworks

1. Potential Outcomes (Rubin/Neyman)

Primitives:

• $Y(a)$ = potential outcome under treatment $a$

• Only $Y = Y(A)$ is observed (consistency)

• Fundamental problem: never observe both $Y(0)$ and $Y(1)$ for same unit

Advantages:

• Clear definition of causal effects

• Natural for experimental reasoning

• Connects to missing data theory

2. Structural Causal Models (Pearl)

Primitives:

• Directed Acyclic Graph (DAG) encoding causal structure

• Structural equations: $Y := f_Y(PA_Y, U_Y)$

• Interventions via do-operator: $P(Y | do(A=a))$

Advantages:

• Visual representation of assumptions

• Systematic identification algorithms

• Clear separation of statistical and causal assumptions

DAG Framework

Directed Acyclic Graphs (DAGs)

A DAG $\mathcal{G} = (V, E)$ consists of:

• Vertices $V$: Random variables

• Directed edges $E$: Direct causal relationships

• Acyclic: No directed cycles

Key DAG Terminology

Term Definition Notation

Parents Direct causes $PA_Y$

Children Direct effects $CH_Y$

Ancestors All causes $AN_Y$

Descendants All effects $DE_Y$

Collider Node with two incoming arrows $A \to C \leftarrow B$

Mediator Node on causal path $A \to M \to Y$

Confounder Common cause $A \leftarrow C \to Y$

DAG specification and visualization using dagitty

library(dagitty)

Define mediation DAG

mediation_dag <- dagitty(' dag { A [exposure] M [mediator] Y [outcome] X [confounder]

X -> A
X -> M
X -> Y
A -> M
A -> Y
M -> Y

} ')

Visualize

plot(mediation_dag)

Find adjustment sets

adjustmentSets(mediation_dag, exposure = "A", outcome = "Y")

Check implied conditional independencies

impliedConditionalIndependencies(mediation_dag)

D-Separation

The Core Concept

Two nodes $A$ and $B$ are d-separated by set $Z$ if every path between them is blocked.

Path Blocking Rules

Path Type Blocked by conditioning on...

Chain: $A \to M \to B$ $M$ (blocks)

Fork: $A \leftarrow C \to B$ $C$ (blocks)

Collider: $A \to C \leftarrow B$ NOT $C$ (conditioning opens!)

D-separation Formula

$$A \perp!!!\perp_{\mathcal{G}} B \mid Z \iff \text{every path } A \text{---} B \text{ is blocked by } Z$$

Check d-separation using dagitty

check_dseparation <- function(dag, x, y, z = NULL) { if (is.null(z)) { dseparated(dag, x, y) } else { dseparated(dag, x, y, z) } }

Find all d-separating sets

find_dsep_sets <- function(dag, x, y) {

All adjustment sets that d-separate x and y

adjustmentSets(dag, exposure = x, outcome = y, effect = "total") }

Verify conditional independence implications

verify_ci_implications <- function(dag, data) { implied_ci <- impliedConditionalIndependencies(dag)

results <- lapply(implied_ci, function(ci) { # Parse the CI statement vars <- strsplit(as.character(ci), " \|\| | \| ")[[1]] x <- vars[1] y <- vars[2] z <- if (length(vars) > 2) vars[3:length(vars)] else NULL

# Test with partial correlation or conditional independence test
test_result <- test_conditional_independence(data, x, y, z)

list(statement = as.character(ci), p_value = test_result$p.value)

})

do.call(rbind, lapply(results, as.data.frame)) }

Backdoor Criterion

Definition

A set $Z$ satisfies the backdoor criterion relative to $(A, Y)$ if:

• No node in $Z$ is a descendant of $A$

• $Z$ blocks every path between $A$ and $Y$ that contains an arrow into $A$

Backdoor Adjustment Formula

If $Z$ satisfies the backdoor criterion: $$P(Y | do(A = a)) = \sum_z P(Y | A = a, Z = z) P(Z = z)$$

or equivalently: $$E[Y(a)] = E_Z[E[Y | A = a, Z]]$$

Front-Door Criterion

When backdoor fails but mediator is unconfounded: $$P(Y | do(A)) = \sum_m P(M = m | A) \sum_{a'} P(Y | M = m, A = a') P(A = a')$$

Check backdoor criterion

check_backdoor <- function(dag, exposure, outcome, adjustment_set) {

Using dagitty

valid_sets <- adjustmentSets(dag, exposure = exposure, outcome = outcome, type = "minimal")

Check if proposed set is valid

is_valid <- any(sapply(valid_sets, function(s) { setequal(s, adjustment_set) }))

list( is_valid = is_valid, minimal_sets = valid_sets, proposed = adjustment_set ) }

Compute backdoor-adjusted estimate

backdoor_adjustment <- function(data, outcome, exposure, adjustment) { formula_str <- paste(outcome, "~", exposure, "+", paste(adjustment, collapse = " + ")) model <- lm(as.formula(formula_str), data = data)

Standardization

predictions_a1 <- predict(model, newdata = transform(data, setNames(list(1), exposure))) predictions_a0 <- predict(model, newdata = transform(data, setNames(list(0), exposure)))

list( ate = mean(predictions_a1 - predictions_a0), se = sqrt(var(predictions_a1 - predictions_a0) / nrow(data)) ) }

Full identification analysis

analyze_identification <- function(dag, exposure, outcome) { list( adjustment_sets = adjustmentSets(dag, exposure, outcome), instrumental_sets = instrumentalVariables(dag, exposure, outcome), direct_effects = adjustmentSets(dag, exposure, outcome, effect = "direct"), implied_independencies = impliedConditionalIndependencies(dag) ) }

Framework Equivalence

For most problems, both frameworks give equivalent results: $$E[Y(a)] = E[Y | do(A=a)]$$

Choose based on context and audience.

Key Identification Assumptions

For Treatment Effects

Assumption Formal Statement Interpretation

Consistency $Y = Y(A)$ Observed outcome equals potential outcome for received treatment

Positivity $P(A=a \mid X=x) > 0$ for all $x$ with $P(X=x) > 0$ Every covariate stratum has both treated and untreated

Exchangeability $Y(a) \perp!!!\perp A \mid X$ No unmeasured confounding given $X$

SUTVA No interference, single version of treatment Units don't affect each other

For Mediation Effects

Additional assumptions required:

Assumption Formal Statement Interpretation

Cross-world exchangeability $Y(a,m) \perp!!!\perp M(a^*) \mid X$ Counterfactual mediator independent of counterfactual outcome

No $A$-$M$ interaction (optional) $Y(a,m) - Y(a',m)$ constant in $m$ Simplifies identification

Compositional $Y(a) = Y(a, M(a))$ Potential outcome composition

Standard Identification Results

1. Average Treatment Effect (ATE)

Target: $\psi = E[Y(1) - Y(0)]$

Under exchangeability (A1), consistency (A2), positivity (A3):

$$\psi = E\left[E[Y | A=1, X] - E[Y | A=0, X]\right]$$

Proof sketch: \begin{align} E[Y(a)] &= E[E[Y(a) | X]] && \text{(iterated expectations)}
&= E[E[Y(a) | A=a, X]] && \text{(A1: exchangeability)}
&= E[E[Y | A=a, X]] && \text{(A2: consistency)} \end{align}

2. Average Treatment Effect on Treated (ATT)

Target: $\psi_{ATT} = E[Y(1) - Y(0) | A=1]$

Under weaker exchangeability $Y(0) \perp!!!\perp A \mid X$:

$$\psi_{ATT} = E\left[E[Y | A=1, X] - E[Y | A=0, X] \mid A=1\right]$$

3. Natural Direct and Indirect Effects (Mediation)

Target:

• NDE: $E[Y(1, M(0)) - Y(0, M(0))]$

• NIE: $E[Y(1, M(1)) - Y(1, M(0))]$

Under mediation assumptions (see VanderWeele, 2015):

$$NDE = \int\int {E[Y|A=1,M=m,X=x] - E[Y|A=0,M=m,X=x]} , dP(m|A=0,X=x) , dP(x)$$

$$NIE = \int\int E[Y|A=1,M=m,X=x] {dP(m|A=1,X=x) - dP(m|A=0,X=x)} , dP(x)$$

4. Controlled Direct Effect (CDE)

Target: $CDE(m) = E[Y(1,m) - Y(0,m)]$

Simpler identification (no cross-world assumption):

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

DAG-Based Identification

The Back-Door Criterion

A set $X$ satisfies the back-door criterion relative to $(A, Y)$ if:

• No node in $X$ is a descendant of $A$

• $X$ blocks every path between $A$ and $Y$ that contains an arrow into $A$

If satisfied: $$P(Y | do(A=a)) = \sum_x P(Y | A=a, X=x) P(X=x)$$

The Front-Door Criterion

When there's an unmeasured confounder $U$ between $A$ and $Y$, but $M$ mediates all of $A$'s effect:

U

/
↓ ↓ A → M → Y

Identification: $$P(Y | do(A=a)) = \sum_m P(M=m | A=a) \sum_{a'} P(Y | M=m, A=a') P(A=a')$$

Instrumental Variables

When $Z$ affects $Y$ only through $A$:

U ↓ Z → A → Y

Local ATE identification (with monotonicity): $$LATE = \frac{E[Y | Z=1] - E[Y | Z=0]}{E[A | Z=1] - E[A | Z=0]}$$

Sequential Identification (Multiple Mediators)

Sequential Mediation (A → M1 → M2 → Y)

Product of three path identification requires:

• Standard confounding control for each arrow

• No intermediate confounders affected by treatment

• Sequential ignorability assumptions

Path-specific effects:

• Direct: $A \to Y$

• Through $M_1$ only: $A \to M_1 \to Y$

• Through $M_2$ only: $A \to M_2 \to Y$

• Through both: $A \to M_1 \to M_2 \to Y$

Identification Formula (No Intermediate Confounding)

$$\text{Effect through } M_1 \to M_2 = \int E\left[\frac{\partial^3}{\partial a \partial m_1 \partial m_2} E[Y|A,M_1,M_2,X]\right]$$

Expressed as product of coefficients: $\hat{\alpha}_1 \cdot \hat{\beta}_1 \cdot \hat{\gamma}_2$

Partial Identification

When point identification fails, we can still bound the parameter.

Manski Bounds (No Assumptions)

For ATE with missing outcomes: $$E[Y(1)] \in [E[Y \cdot A]/P(A=1) + y_{min}P(A=0), E[Y \cdot A]/P(A=1) + y_{max}P(A=0)]$$

Sensitivity Analysis

When exchangeability is uncertain, parameterize violation:

Unmeasured confounding parameter $\Gamma$: $$\frac{1}{\Gamma} \leq \frac{P(A=1|X,U=1)/P(A=0|X,U=1)}{P(A=1|X,U=0)/P(A=0|X,U=0)} \leq \Gamma$$

Compute bounds as function of $\Gamma$ (Rosenbaum bounds).

E-Value

Minimum strength of unmeasured confounding (on risk ratio scale) needed to explain away observed effect:

$$E\text{-value} = RR + \sqrt{RR \times (RR-1)}$$

Identification Strategies by Design

Randomized Controlled Trials (RCTs)

• Treatment assignment random → exchangeability holds by design

• Still need SUTVA, consistency

• For mediation: randomize $M$ as well, or use sequential ignorability

Observational Studies

Strategy Key Assumption Best For

Regression adjustment All confounders measured Rich covariate data

Propensity score Correct PS model High-dimensional confounders

Instrumental variables Valid instrument exists Unmeasured confounding

Regression discontinuity Continuity at threshold Sharp treatment rules

Difference-in-differences Parallel trends Panel data

Natural Experiments

• Exploit exogenous variation (policy changes, geographic variation)

• Requires careful argument for why variation is "as-if random"

Identification in the MediationVerse

medfit: Foundation

• Implements standard mediation identification

• VanderWeele regression-based approach

• Supports binary/continuous treatments and mediators

probmed: Effect Size

• $P_M$ identification requires identified NDE/NIE

• Handles case when NDE and NIE have opposite signs

RMediation: Confidence Intervals

• Takes identified effects as input

• Distribution of product of coefficients (PRODCLIN)

• Monte Carlo intervals

medrobust: Sensitivity

• When identification assumptions are uncertain

• Bounds on effects under confounding

• E-values for unmeasured confounding

medsim: Validation

• Simulate data where truth is known

• Verify identification formulas recover true effects

• Test estimator properties

Identification Proof Template

\begin{theorem}[Identification of $\psi$] Under Assumptions: \begin{enumerate}[label=A\arabic*.] \item (Consistency) $Y = Y(A)$, $M = M(A)$ \item (Positivity) $P(A=a|X) > \epsilon > 0$ for all $a \in \mathcal{A}$ \item (Exchangeability) $Y(a) \perp!!!\perp A \mid X$ \end{enumerate} the causal estimand $\psi = E[g(Y(a))]$ is identified by [ \psi = E_X\left[E[g(Y) \mid A=a, X]\right]. ] \end{theorem}

\begin{proof} \begin{align} E[g(Y(a))] &= E\left[E[g(Y(a)) \mid X]\right] && \text{(law of total expectation)} \ &= E\left[E[g(Y(a)) \mid A=a, X]\right] && \text{(by A3: exchangeability)} \ &= E\left[E[g(Y) \mid A=a, X]\right] && \text{(by A1: consistency)} \end{align} The RHS depends only on the observed data distribution $P(Y,A,X)$. \end{proof}

Common Identification Pitfalls

1. Conditioning on Colliders

A → C ← Y

Conditioning on $C$ opens a path between $A$ and $Y$.

2. Conditioning on Mediators

A → M → Y

Conditioning on $M$ blocks the indirect effect, doesn't control confounding.

3. Overcontrol Bias

Conditioning on descendants of treatment can bias estimates.

4. M-Bias

U1 → X ← U2 ↓ ↓ A ——————→ Y

Conditioning on $X$ opens path $A \leftarrow U_1 \rightarrow X \leftarrow U_2 \rightarrow Y$.

5. Table 2 Fallacy

Interpreting coefficients causally when model includes intermediate variables.

Verification Questions

When reviewing identification arguments, ask:

• Is the target estimand clearly defined?

• Are all assumptions explicitly stated?

• Is each step in the derivation justified?

• Are the assumptions plausible in this context?

• What if an assumption is violated?

• Is there a DAG that encodes the assumptions?

• Are there alternative identification strategies?

Integration with Other Skills

This skill works with:

• proof-architect - For writing identification proofs

• asymptotic-theory - For inference after identification

• methods-paper-writer - For presenting identification in manuscripts

• simulation-architect - For validating identification

Key References

Imai

Hernan

Pearl, J. (2009). Causality: Models, Reasoning, and Inference (2nd ed.)

VanderWeele, T.J. (2015). Explanation in Causal Inference

Hernán, M.A. & Robins, J.M. (2020). Causal Inference: What If

Imbens, G.W. & Rubin, D.B. (2015). Causal Inference for Statistics

Version: 1.0 Created: 2025-12-08 Domain: Causal Inference, Mediation Analysis