35.2 Key Assumptions

Matching relies on the standard set of assumptions underpinning selection on observables—also known as the back-door criterion (see Assumptions for Identifying Treatment Effects). When these assumptions hold, matching can yield valid estimates of causal effects by constructing treated and control groups that are comparable on observed covariates.

1. Strong Conditional Ignorability Assumption (Unconfoundedness)

Also known as the no hidden bias or ignorability assumption:

$$(Y(0),\, Y(1)) \,\perp\, T \,\big|\, X$$

This implies that, conditional on covariates $X$, treatment assignment is independent of the potential outcomes. In other words, there are no unobserved confounders once we adjust for $X$.

• This assumption is not testable, but it is more plausible when all relevant confounders are observed and included in $X$.
• It is often satisfied approximately when unobserved covariates are highly correlated with the observed ones.
• If unobserved variables are unrelated to $X$, you can:
  • Conduct sensitivity analysis to test the robustness of your estimates.
  • Apply design sensitivity techniques: If unobserved confounding is suspected, methods such as (Heller, Rosenbaum, and Small 2009)’s design sensitivity approaches or bounding approaches (e.g., the rbounds R package) can be used to test how robust findings are to hidden bias.

This is the cornerstone assumption of matching: without it, causal inference from observational data is generally invalid.

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2. Overlap (Positivity) Assumption (Common Support)

$$0 < P(T=1 \mid X) < 1 \quad \forall X$$

This condition ensures that, for every value of the covariates $X$, there is a positive probability of receiving both treatment and control.

• If this assumption fails, there are regions of covariate space where either treatment or control units are absent, making comparison impossible.
• Matching enforces this assumption by discarding units outside of the region of common support.

This pruning step is both a strength and limitation of matching—it improves internal validity at the cost of generalizability.

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3. Stable Unit Treatment Value Assumption (SUTVA)

SUTVA requires that:

• The potential outcomes for any individual unit do not depend on the treatment assignment of other units.

That is, there are no interference or spillover effects between units.

• Mathematically, $Y_i(T_i)$ depends only on $T_i$, not on $T_j$ for any $j \neq i$.
• Violations can occur in settings like:
  • Education (peer effects)
  • Epidemiology (disease transmission)
  • Marketing (network influence)

In cases with known spillover, efforts should be made to reduce interactions or explicitly model interference.

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Summary of Assumptions for Matching

Assumption	Description	Notation
Conditional Ignorability	No hidden confounding after conditioning on covariates	$(Y(0), Y(1)) \perp T \mid X$
Overlap (Positivity)	Each unit has a non-zero probability of treatment and control assignment	$0 < P(T=1 \mid X) < 1$
SUTVA	No interference between units; one unit’s outcome unaffected by another’s treatment	$Y_i(T_i)$ unaffected by $T_j$ for $j \ne i$

These three assumptions form the foundation for valid causal inference using matching methods.

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References

Heller, Ruth, Paul R Rosenbaum, and Dylan S Small. 2009. “Split Samples and Design Sensitivity in Observational Studies.” Journal of the American Statistical Association 104 (487): 1090–1101.