Propensity Score Matching - Index

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
• Need classical PS matching framework, balancing theorem, ATT vs ATE? → Propensity Score Matching - Balancing Theorem and Failure Modes
• Need NN matching, caliper width, optimal/full matching algorithms? → Matching Algorithms and Caliper
• Need love plots, SMD thresholds, overlap plots, Rubin balance criteria? → Covariate Balance and Matching Diagnostics

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	$e(X)=P(T=1\mid X)$ is a balancing score; under strong ignorability, conditioning on scalar $e(X)$ 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
Balancing theorem, ATT estimand, PSM vs IPW, when matching fails	Propensity Score Matching - Balancing Theorem and Failure Modes	concept	The Selection Problem, Conditional Independence Assumption, Frequentist Causal Estimation	$e(X)$ is a sufficient statistic for covariate balance; PSM discards units, IPW reweights — complementary methods
NN greedy, caliper $\delta =0.2{\sigma }_{\text{logit}}$, optimal/full matching, subclassification	Matching Algorithms and Caliper	concept	Propensity Score Matching - Balancing Theorem and Failure Modes	Caliper prevents poor matches; optimal matching minimises total distance globally; full matching uses all data
SMD, love plot, overlap plot, variance ratio, Rubin (2001) criteria	Covariate Balance and Matching Diagnostics	concept	Propensity Score Matching - Balancing Theorem and Failure Modes, Matching Algorithms and Caliper	Balance assessed via effect sizes ($

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: $e_i(X_i)=P(T_i=1\mid X_i)$, strong ignorability (unconfoundedness + positivity), balancing-score theorem $X\perp T\mid e(X)$, 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.
• Propensity Score Matching - Balancing Theorem and Failure Modes — CONTAINS: Propensity score definition, balancing theorem (R&R 1983 Thm 1), strong ignorability definition, ATT vs ATE estimand comparison, PSM vs IPW table, the matching workflow, why PSM fails for activity bias
• Matching Algorithms and Caliper — CONTAINS: Logit PS distance metric, NN greedy matching (with/without replacement), caliper definition and $0.2{\sigma }_{\text{logit}}$ width rule, optimal matching, full matching, 1:k matching, subclassification, Mahalanobis distance, common support trimming, MatchIt R code
• Covariate Balance and Matching Diagnostics — CONTAINS: SMD formula and $|$SMD$|<0.1$ threshold, why $p$-values are wrong for balance, love plot (definition + cobalt R code), propensity score overlap plot, variance ratio, KS statistic, ESS for weighting, Rubin (2001) three-criteria theorem, outcome-blinding principle, ATT estimation after matching

External Connections

Potential Outcomes Framework · Conditional Independence Assumption · Frequentist Causal Estimation · The Selection Problem · Omitted Variables Bias · Synthetic Control · Bayesian Inverse Probability Weighting · Bayesian Inverse Probability Weighting · Activity Bias in Advertising

Sources

• Stuart 2010 - Matching Methods for Causal Inference - A Review — Stuart, E. A. (2010). Matching Methods for Causal Inference: A Review and a Look Forward. Statistical Science 25(1), 1-21.