A propensity score is the estimated probability that a given unit received treatment, conditional on its observed covariates. Propensity score matching (PSM) uses that single number to pair treated and untreated units who looked equally likely to receive treatment before it happened, then compares outcomes across the matched pairs. It is a way to approximate the balance a randomized controlled trial gets for free — but only for the covariates that were actually measured and included in the score.
That last clause is the part worth reading twice before anything else on this page. Matching does not “control for confounding” in the way that phrase is often used. It controls for measured confounding — the variables the researcher collected, chose, and put into the propensity-score model. Any confounder that was not measured, or that the researcher chose not to include, is untouched by the match. A well-executed PSM analysis can produce two groups that look statistically indistinguishable on every covariate in the model and still be confounded by something outside it. This is the single most common misreading of a PSM result: treating post-matching covariate balance as evidence that the comparison is now “as good as randomized.” It is evidence that the observed covariates are balanced. Nothing more.
How propensity score matching works
PSM is typically run in three stages.
- Estimate the propensity score. Fit a model — almost always logistic regression, though other classifiers are used — predicting treatment assignment (1/0) from the pre-treatment covariates believed to influence both treatment assignment and the outcome. The fitted probability from that model is each unit’s propensity score, a single number between 0 and 1.
- Match on the score. Pair each treated unit with one or more untreated units with a similar propensity score. Common approaches: nearest-neighbor matching (with or without replacement), caliper matching (only match within a fixed maximum distance, e.g. 0.2 of a standard deviation of the logit of the propensity score — a convention from Austin, 2011, not a fixed rule), and full or optimal matching, which minimizes total distance across all pairs rather than matching greedily. Unmatched units — typically those with no comparable counterpart, often at the extremes of the score distribution — are dropped from the analysis.
- Check balance, then estimate the effect. Before trusting any outcome comparison, check whether matching actually balanced the covariates. Then estimate the treatment effect on the matched sample, usually with a simple comparison of means, a paired test, or a regression run on the matched data (matching plus regression adjustment together is a common and defensible combination, sometimes called “doubly robust” when done carefully).
Reading a covariate balance table
The standard artefact for judging whether a match worked is a balance table reporting the standardized mean difference (SMD) for each covariate, before and after matching. SMD expresses the difference in covariate means between treated and untreated groups in units of pooled standard deviation — the same logic as Cohen’s d, applied to a baseline characteristic instead of an outcome. Unlike a p-value from a t-test, SMD does not shrink automatically as sample size grows, which is exactly why it, not a significance test, is the recommended way to assess balance (a t-test on baseline covariates conflates “the groups differ” with “the sample is large enough to detect a small difference” — see the note on p-value misreadings below).
A worked example. Suppose a study is comparing outcomes for patients who did versus did not receive a new post-surgical protocol, using PSM to account for the fact that assignment wasn’t random:
| Covariate | SMD, unmatched | SMD, matched |
|---|---|---|
| Age | 0.41 | 0.06 |
| Baseline severity score | 0.55 | 0.09 |
| Sex (proportion female) | 0.12 | 0.03 |
| Comorbidity count | 0.38 | 0.11 |
| Prior hospitalization (yes/no) | 0.29 | 0.05 |
The sentence a researcher would write from that table: “Before matching, treated and untreated patients differed meaningfully on baseline severity and comorbidity burden (SMD 0.55 and 0.38); after 1:1 nearest-neighbor matching on the propensity score, all standardized mean differences fell below the conventional 0.1 threshold, indicating adequate covariate balance in the matched sample.” That 0.1 (sometimes stated as 0.25) threshold is a convention proposed in the methodological literature — most commonly cited to Austin (2009, 2011) — not a statistical law; some fields treat anything under 0.1 as well-balanced and under 0.25 as acceptable, but there is no single universally enforced cutoff, and a paper should state which convention it is using and why.
Note what the table does not show: whether the outcome comparison that follows is unconfounded by anything not on this list. A perfectly balanced table on five measured covariates says nothing about a sixth, unmeasured one.
The assumptions PSM depends on — and what happens when they fail
- Unconfoundedness / conditional ignorability. Treatment assignment is assumed to be “as good as random” once conditioned on the observed covariates — formally, the potential outcomes are independent of treatment given the covariates in the propensity model. This is an assumption, not something the data can verify. It fails whenever a real confounder was not measured or not included, and no amount of matching, balance-checking, or post-hoc statistical polish can detect that failure from the matched data alone. This is what “PSM cannot fix unmeasured confounding” means concretely: the method is only as good as the covariate list that went into the score.
- Positivity / common support. Every treated unit needs a realistic chance of having been untreated (and vice versa) given its covariates — in practice, the propensity-score distributions of the two groups need to overlap. Where they don’t (e.g. a covariate combination that predicts treatment with near-certainty), those units cannot be matched validly and should be dropped or reported as excluded, not forced into a match with a poor counterpart.
- Correct specification of the propensity model. If the logistic regression estimating the propensity score omits a relevant covariate, includes the wrong functional form, or misses an interaction, the resulting score — and every match built on it — inherits that error.
- SUTVA (no interference between units). One unit’s treatment status is assumed not to affect another unit’s outcome. This can be violated in settings with spillover, contagion, or shared resources.
What propensity score matching cannot fix
This is the caveat worth stating plainly rather than folding into a methods-limitations paragraph at the end: PSM balances measured covariates between groups; it does nothing to, and cannot, address confounding from variables that were never measured. A published PSM analysis with a beautifully balanced table can still be badly confounded if an important driver of both treatment choice and outcome — socioeconomic status, disease severity captured imperfectly by the available proxy, clinician judgment that isn’t recorded anywhere in the dataset — was left out of the model. This is not a hypothetical: it is the standard critique leveled at observational comparisons that lean on matching to claim causal-strength conclusions, and it is why PSM results are generally reported as associational evidence strengthened by design, not as equivalent to a randomized trial. Where the underlying data support it, a sensitivity analysis for unmeasured confounding (e.g. an E-value or Rosenbaum bounds) is the appropriate way to communicate how strong an unmeasured confounder would have to be to change the conclusion — matching alone cannot answer that question.
When PSM is (and isn’t) the right tool
PSM is a reasonable choice when: the research question is inherently observational (a randomized trial is unethical, impractical, or already happened without one), the covariates believed to drive treatment selection are actually measured and available, and the sample is large enough that dropping unmatched units doesn’t gut statistical power. It is a poor choice, or at least needs strong caveats, when: the confounders are only partially measured or measured with error, the treated and control groups have poor overlap in propensity scores, or the study is really trying to answer a causal question that regression adjustment, instrumental variables, difference-in-differences, or a randomized controlled trial would answer more convincingly. For a broader comparison of when observational designs versus randomized designs are appropriate, see RCT vs. observational study.
Reporting standards
A methods section using PSM should state, at minimum: which covariates went into the propensity-score model and why they were chosen, the matching algorithm and any caliper used, the balance diagnostic and threshold applied (with the SMD values, not just “balance was achieved”), how many units were unmatched and dropped, and whether a sensitivity analysis for unmeasured confounding was performed. Reviewers increasingly expect the balance table itself, not just a summary sentence — reporting “groups were well balanced after matching (all p > 0.05)” without SMDs is a known weak spot, since a non-significant t-test on a small matched sample can mask a covariate difference that is still practically meaningful.
Frequently asked questions
Does propensity score matching prove causation?
No. It strengthens the causal interpretability of an observational comparison relative to an unadjusted one, but it remains dependent on the unconfoundedness assumption, which cannot be verified from the data. It is best described as reducing confounding from measured covariates, not establishing causation the way a well-run randomized trial does.
What’s the difference between propensity score matching and regression adjustment?
Regression adjustment models the outcome directly, including covariates as predictors alongside the treatment indicator, using the full sample. PSM instead models treatment assignment, then restricts or reweights the sample to comparable units before looking at outcomes at all. The two rely on similar assumptions about measured confounding and are sometimes combined (matching followed by regression on the matched sample), which some (imperfectly) call doubly robust because it protects against misspecification of either model alone.
How large an SMD is “too large” to call two groups balanced?
There is no universal cutoff. The most commonly cited conventions are an SMD below 0.1 for good balance and below 0.25 as a looser acceptable bound, both traced to methodological work by Peter Austin. Treat these as reporting conventions to state explicitly, not as fixed statistical thresholds — the same caution that applies to Cohen’s d benchmarks for effect size.
Can propensity score matching fix a small sample size?
No — matching typically reduces the usable sample further, since unmatched units are dropped. It trades some statistical power for improved covariate balance between the groups that remain. If the sample is already small, matching can leave too few pairs for a stable estimate.
Is propensity score matching still considered current practice?
It remains widely used and taught, but methodologists have increasingly favored propensity score weighting (e.g. inverse probability of treatment weighting) over matching in some settings, since weighting retains the full sample rather than discarding unmatched units. Both approaches share the same core limitation: neither can address confounding from unmeasured variables.
