Back-door Criterion

Definition

The Back-door Criterion (Pearl, 1993) is a graphical criterion for identifying a causal effect from observational data. It determines whether a set of variables $Z$ is sufficient to identify the causal effect of $X \rightarrow Y$.

Formal Definition:

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

1. No variable in $Z$ is a descendant of $X$
2. $Z$ blocks every back-door path connecting $X$ and $Y$

Back-door Path

Back-door path: a path from $X$ to $Y$ that begins with an arrow pointing into $X$

X ← ... → Y   (back-door path)
X → ... → Y   (front-door path, causal path)

Example:

    Z
   ↙ ↘
  X   Y

• Back-door path: $X \leftarrow Z \rightarrow Y$
• This path transmits non-causal association

Intuitive Understanding

Core idea:

Causal effect = Total association - Spurious association (via back-door)

Total association between X and Y:
  1. Causal path: X → ... → Y
  2. Back-door paths: X ← ... → Y (spurious)

Back-door criterion: block 2 to leave only 1

Analogy:

• Front door: the path through which X affects Y (causal)
• Back door: the path through which X and Y are connected via a common cause (non-causal)
• Close all the back doors → the causal effect becomes identifiable

Back-door Adjustment Formula

If $Z$ satisfies the back-door criterion:

$P(Y|do(X=x)) = \sum_z P(Y|X=x, Z=z) \cdot P(Z=z)$

Or, for the continuous case:

$E[Y|do(X=x)] = \int E[Y|X=x, Z=z] \cdot p(z) \, dz$

Meaning:

• Conditioning on $Z$ and averaging yields the causal effect
• The interventional distribution can be computed from observational data

Algorithm: Finding Adjustment Sets

Step 1: Enumerate All Back-door Paths

Starting from $X$, follow arrows pointing into $X$ and trace all paths that reach $Y$

Step 2: Determine How to Block Each Path

• Fork (X ← Z → Y): conditioning on Z blocks it
• Chain (… → Z → …): conditioning on Z blocks it
• Collider (…→ Z ←…): not conditioning on Z blocks it (already blocked)

Step 3: Select the Adjustment Set

• A set of variables that blocks all back-door paths
• Does not include any descendant of $X$

Examples

Example 1: Simple Confounding

    Z
   ↙ ↘
  X → Y

Back-door path: $X \leftarrow Z \rightarrow Y$ Adjustment set: $\{Z\}$ Formula: $E[Y|do(X)] = \sum_z E[Y|X,Z=z] \cdot P(Z=z)$

Example 2: Multiple Confounders

  Z1    Z2
   ↘  ↙  ↘
    X  →  Y

Back-door paths:

1. $X \leftarrow Z1 \rightarrow Y$ (none, if there is no direct Z1→Y)
2. …

Adjustment set: $\{Z1\}$, $\{Z2\}$, or $\{Z1, Z2\}$ depending on the situation

Example 3: Collider on Back-door Path

  Z1 → C ← Z2
   ↓       ↓
   X   →   Y

Back-door path: $X \leftarrow Z1 \rightarrow C \leftarrow Z2 \rightarrow Y$

• C is a collider → the path is already blocked!
• Adjustment set: $\emptyset$ (no control is needed)

Caution: controlling for C opens the path → bias arises

Example 4: Mediator

  Z
  ↓
  X → M → Y

Causal path: $X \rightarrow M \rightarrow Y$ Back-door path: none (Z affects only X)

Adjusting for $\{Z\}$ is optional (since there is no back-door)

Caution: do NOT adjust for M (it would block the front-door path)

Sufficient vs Minimal Adjustment Sets

Sufficient Adjustment Set

• Any set that satisfies the back-door criterion

Minimal Adjustment Set

• The smallest among the sufficient sets
• Contains no unnecessary variables

Trade-off:

• More variables: more robust (guards against omitted confounding)
• Fewer variables: more efficient (lower variance)

Limitations

1. Dependence on DAG correctness: if the DAG is wrong, the conclusion is wrong
2. Unmeasured confounders: if unmeasured variables exist, they cannot be blocked
3. Sufficient but not necessary: the back-door criterion is a sufficient, not a necessary, condition

Related Criteria

Front-door Criterion

X → M → Y
    ↑
    U (unobserved confounder)

• An alternative when back-door paths cannot be blocked
• Identification that exploits a mediator

Instrumental Variables

• An alternative when back-door paths cannot be blocked
• Exploits the Instrument → X → Y structure

Related Concepts

• DAG - Visualizing causal structure
• Confounder - Creates back-door paths
• Collider - Blocks back-door paths
• d-separation - Conditional independence in a DAG
• Propensity Score - Implements back-door adjustment
• Unconfoundedness - No unmeasured confounders

References

• Pearl, J. (1993). Comment: Graphical models, causality and intervention
• Pearl, J. (2009). Causality: Models, Reasoning, and Inference
• rohrerThinkingClearlyCorrelations - Introduces the back-door criterion