Propensity Score Matching
Classical Rosenbaum-Rubin propensity-score matching framework and its diagnostics, ingested from Stuart (2010), “Matching Methods for Causal Inference: A Review and a Look Forward,” Statistical Science 25(1).
Routing Summary
- Need the big picture, the four-step workflow, or the four uses of the propensity score? → Propensity Score Matching - Overview
- Need the definition of the propensity score, the balancing-score theorem, or strong ignorability? → Propensity Score and the Balancing Property
- Need a distance measure (exact / Mahalanobis / caliper) or a matching algorithm (nearest-neighbor, optimal, full, subclassification, IPTW)? → Matching Methods and Distance Measures
- Need to check match quality (standardized mean differences, variance ratios, QQ plots) or know why not to use a balance hypothesis test? → Covariate Balance Diagnostics
- Need the overlap / common-support requirement, trimming, or its effect on the estimand? → Common Support and Overlap
Concept Map
| Concept | Note | Type | Depends On | Key Result |
|---|---|---|---|---|
| Workflow & four uses of PS | Propensity Score Matching - Overview | overview | Potential Outcomes, Balancing Property | Separate design (no outcomes) from analysis; PS used for matching, subclassification, weighting, adjustment |
| PS definition + Rosenbaum-Rubin theorem | Propensity Score and the Balancing Property | theorem | Potential Outcomes, Conditional Independence | is a balancing score; under strong ignorability, conditioning on scalar removes observed-covariate bias |
| Distances & matching structures | Matching Methods and Distance Measures | concept | Balancing Property | Exact/Mahalanobis/caliper distances; k:1, greedy vs. optimal, with/without replacement, full matching, IPTW |
| Balance diagnostics | Covariate Balance Diagnostics | concept | Balancing Property, Distance Measures | Use standardized mean diffs & variance ratios (<0.25, 0.5-2), QQ plots; never use balance p-values |
| Overlap / common support | Common Support and Overlap | concept | Balancing Property | Estimate effects only on the region of common support; trim outside it; overlap constrains ATE vs. ATT |
Notes
- Propensity Score Matching - Overview — CONTAINS: definition of matching, two settings, outcome-free design vs. analysis, the four implementation steps, ATT/ATE estimands, the four uses of the propensity score, Chapin curse-of-dimensionality example, matching-vs-regression complementarity.
- Propensity Score and the Balancing Property — CONTAINS: , strong ignorability (unconfoundedness + positivity), balancing-score theorem , ignorability given the PS, liberal variable-selection rule (balance not c-statistic), linear/logit PS, caliper bias-reduction figures (0.2 SD → 98%).
- Matching Methods and Distance Measures — CONTAINS: four affinely-invariant distances with formulas, Mahalanobis-within-caliper formula, k:1 nearest neighbor & ratio matching, greedy vs. optimal, with/without replacement & frequency weights, subclassification (5-10 → 90% bias), full matching (Hansen SAT example), IPTW & weighting-by-odds formulas, weight trimming, doubly-robust.
- Covariate Balance Diagnostics — CONTAINS: standardized difference in means formula, Rubin (2001) three balance measures, thresholds (SMD<0.25, var ratio 0.5-2), the balance-test caution (in-sample property; tests conflate balance with power), QQ plots and before/after standardized-difference plots.
- Common Support and Overlap — CONTAINS: region of common support / positivity, propensity-score trimming and convex-hull (King-Zeng), how calipers vs. weighting handle overlap, estimand implications (ATE vs. ATT), why matching surfaces non-overlap that regression hides.
External Connections
Potential Outcomes Framework · Conditional Independence Assumption · Frequentist Causal Estimation · The Selection Problem · Omitted Variables Bias · Synthetic Control · Bayesian Inverse Probability Weighting · Bayesian Propensity Score Weighting · Activity Bias in Advertising
Sources
- Stuart 2010 - Matching Methods for Causal Inference - A Review.pdf — Stuart, E. A. (2010). Matching Methods for Causal Inference: A Review and a Look Forward. Statistical Science 25(1), 1-21.