Strong Ignorability

Definition

An assumption combining Ignorability and Positivity

$$\begin{align} \text{(1) Ignorability)} \quad & W \perp\!\!\!\perp (Y(0), Y(1)) \mid X \\ \text{(2) Positivity)} \quad & 0 < P(W=1 \mid X=x) < 1, \quad \forall x \end{align}$$

This is a concept defined by Rosenbaum & Rubin (1983), and it is a sufficient condition for identifying causal effects in observational studies.

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Intuitive Understanding

The Role of the Two Conditions

Assumption	Role
Ignorability	Treatment selection is fully explained by $X$ (no hidden confounders)
Positivity	Both treatment and control are observable at every $X$ (no extrapolation)

Both are needed:

• Ignorability only: estimation is impossible in certain regions
• Positivity only: selection bias is not resolved

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Why Is It Important?

The Key Theorem of the Propensity Score

Under Strong Ignorability:

$$W \perp\!\!\!\perp (Y(0), Y(1)) \mid X \implies W \perp\!\!\!\perp (Y(0), Y(1)) \mid e(X)$$

The high-dimensional $X$ can be reduced to the scalar propensity score $e(X)$.

Causal Effect Identification

When Strong Ignorability holds:

$$E[Y(w) \mid X=x] = E[Y \mid W=w, X=x]$$

Therefore:

$$\text{ATE} = E_X[E[Y \mid W=1, X] - E[Y \mid W=0, X]]$$

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Testability

Component	Testability
Ignorability	Impossible (unobserved confounders cannot be confirmed)
Positivity	Possible (check the propensity score distribution)
Strong Ignorability	Partially possible

Indirect Diagnostics

1. Covariate Balance Check: Check covariate balance after reweighting
2. Placebo Tests: Check for effects on pre-treatment outcomes
3. Sensitivity Analysis: Assess the impact of assumption violations

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Related Methodologies

Methods that assume Strong Ignorability:

• IPW - Inverse Propensity Weighting
• Propensity Score Matching - Matching of similar individuals
• Doubly Robust Estimator - Doubly robust estimation
• CBPS - Covariate Balancing Propensity Score
• Meta-learners - S/T/X/R-learner
• Causal Forest - Heterogeneous effect estimation

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When the Assumption Is Relaxed

When Strong Ignorability is not satisfied:

Ignorability Violation (Hidden Confounders)

• Sensitivity Analysis - Sensitivity analysis
• Deconfounder - Inferring latent confounders
• Deep IV - Using instrumental variables

Positivity Violation (Lack of Overlap)

• Trimming - Removing extreme PS
• Overlap Weighting - Robust weighting
• Bounds estimation

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Related Concepts

• Causal Assumptions Overview - Integrated overview of the three core assumptions
• Ignorability - Conditional independence assumption
• Positivity - Overlap assumption
• Propensity Score - The key tool of Strong Ignorability
• Balancing Score - Theoretical foundation

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References

• Rosenbaum, P. R., & Rubin, D. B. (1983). The central role of the propensity score in observational studies for causal effects
• yaoSurveyCausalInference2021 - Section 2.3