26-data-wise-scholar-skills-mathematical-identification-theory

brycewang-stanford/Auto-Empirical-Research-SkillsResearchAudit pendingSnapshot 01b705fd84fa

Summary

This source did not publish a separate summary. Review SKILL.md before using the skill.

SKILL.md

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

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)

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!)

# 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))
}

# 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)
  )
}

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

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

    U
   / \
  ↓   ↓
  A → M → Y

  U
  ↓
Z → A → Y

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

\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}

A → C ← Y

A → M → Y

U1 → X ← U2
↓         ↓
A ——————→ Y

26-data-wise-scholar-skills-mathematical-identification-theory

Summary

Identification Theory

Core Concepts

What is Identification?

Why Identification Matters

Two Frameworks

1. Potential Outcomes (Rubin/Neyman)

2. Structural Causal Models (Pearl)

DAG Framework

Directed Acyclic Graphs (DAGs)

Key DAG Terminology

D-Separation

The Core Concept

Path Blocking Rules

D-separation Formula

Backdoor Criterion

Definition

Backdoor Adjustment Formula

Front-Door Criterion

Framework Equivalence

Key Identification Assumptions

For Treatment Effects

For Mediation Effects

Standard Identification Results

1. Average Treatment Effect (ATE)

2. Average Treatment Effect on Treated (ATT)

3. Natural Direct and Indirect Effects (Mediation)

4. Controlled Direct Effect (CDE)

DAG-Based Identification

The Back-Door Criterion

The Front-Door Criterion

Instrumental Variables

Sequential Identification (Multiple Mediators)

Sequential Mediation (A → M1 → M2 → Y)

Identification Formula (No Intermediate Confounding)

Partial Identification

Manski Bounds (No Assumptions)

Sensitivity Analysis

E-Value

Identification Strategies by Design

Randomized Controlled Trials (RCTs)

Observational Studies

Natural Experiments

Identification in the MediationVerse

medfit: Foundation

probmed: Effect Size

RMediation: Confidence Intervals

medrobust: Sensitivity

medsim: Validation

Identification Proof Template

Common Identification Pitfalls

1. Conditioning on Colliders

2. Conditioning on Mediators

3. Overcontrol Bias

4. M-Bias

5. Table 2 Fallacy

Verification Questions

Integration with Other Skills

Key References

Related skills