What is Positivity Assumption?
Positivity Assumption is the causal-identification condition that every treatment being compared has nonzero probability for every covariate profile in the target population where an effect is to be learned.
Quick Facts
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How It Works
Separate structural impossibility from sparse evidence
A structural violation means a treatment cannot occur for some eligible profile, so the missing counterfactual contrast is not identified there without additional extrapolation assumptions. A practical violation means both treatments are possible in principle but one arm is rare in the available sample. Zhu and colleagues explain why the distinction changes whether collecting data, redesigning eligibility, or redefining the target population is appropriate.
Diagnose support in the space that matters
Inspect treatment counts in substantively important strata, propensity distributions by arm, covariate balance, maximum weights, tail-weight concentration, and Effective Sample Size. Diagnostics should be repeated for deployment slices. Estimated propensity overlap can hide mismatched covariate combinations, while flexible high-dimensional adjustment can create thin local neighborhoods even when marginal plots look healthy.
Every remedy changes assumptions or the estimand
Trimming and calipers remove unsupported units; overlap weights emphasize the equipoise population; coarsening covariates may leave residual Confounding; outcome regression extrapolates into unsupported regions. None silently restores the original ATE. Report the retained population, thresholds, weight diagnostics, sensitivity across defensible rules, and which claims rely on observed comparisons versus model extrapolation.
Key Characteristics
- Requires each compared treatment to be possible in relevant covariate strata
- Is also described through overlap or common support
- Depends on the estimand, target population, and adjustment set
- Distinguishes structural violations from finite-sample practical violations
- Weak overlap creates extrapolation, unstable weights, and low effective sample size
- Cannot be repaired without changing data, assumptions, or the target population
Common Use Cases
- Auditing support before Propensity Score weighting or matching
- Checking whether a CATE is estimable for an intended user segment
- Detecting unsupported actions in Off-Policy Evaluation logs
- Choosing between trimming, overlap weighting, and target-population revision
- Defining deployment abstention rules for unsupported covariate profiles
Example
Loading code...Frequently Asked Questions
Are Positivity and overlap exactly the same?
They are often used interchangeably, but Positivity is a population-level probability condition and observed overlap is empirical evidence about support in a finite sample. Good-looking sample overlap cannot prove the population condition, and poor sample overlap may reflect either sparse data or structural impossibility.
Can a larger sample fix a Positivity violation?
It can reduce a practical violation when both treatments are genuinely possible but one is rarely observed. It cannot fix a structural violation such as a treatment contraindication. That requires changing the question, target population, intervention, or explicit extrapolation assumptions.
Do Propensity Scores between 0.1 and 0.9 prove adequate overlap?
No universal cutoff proves overlap. Thresholds are context-dependent, estimated scores have error, and similar scores can represent different covariate combinations. Examine treatment counts, joint covariate support, weights, balance, Effective Sample Size, and sensitivity to the chosen rule.
Does trimming extreme Propensity Scores recover the original ATE?
Usually not. Trimming removes units and changes the population over which effects are averaged. The resulting estimand should be named and the excluded population described; otherwise a stable estimate can be misreported as evidence for people the data no longer represent.
Can outcome regression solve nonoverlap?
It can produce predictions outside observed support, but those predictions are extrapolations driven by model structure rather than treated-versus-control evidence. Report that dependency, test plausible models, and avoid presenting unsupported precision as identified causal information.