Propensity-score matching (causal inference)

What is Propensity-score matching (causal inference)?

The propensity score is the probability of being treated given the covariates: e(X) = P(T = 1 | X). Rosenbaum & Rubin's key result: if you match treated and control units on e(X), the matched groups are balanced on X in expectation. Subsequent comparison of outcomes between matched groups estimates the ATT under no-unmeasured-confounding.

Matching methods: nearest-neighbour (fast, greedy, may discard treated units if controls are exhausted), optimal (global minimisation of total distance, slower), full matching (every unit used, variable cluster sizes — most efficient but harder to interpret). Caliper width (e.g. 0.2 SD of logit-PS) restricts matches to similar PS — improves balance at the cost of dropping unmatchable treated units.

Standardised mean differences (SMDs) on covariates before vs after matching: |SMD| < 0.1 is the conventional balance threshold. Achieving |SMD| < 0.1 on all key covariates is the headline check that the matched groups are comparable.

When should I use Propensity-score matching (causal inference)?

• Observational data where treatment was not randomly assigned and you have a sufficient set of measured confounders.
• Reducing dependence of the ATT on the outcome model (matching is non-parametric on the outcome).
• Pre-step before running a Cox or linear model on the matched data.

What data does it need?

Treatment (binary) + outcome (numeric) + covariates (multi-numeric) + matching method + caliper + ratio + replacement toggle.

What does it report?

SMD before vs after (|SMD| < 0.1 highlighted) + ATT + 95% CI + p (paired t on 1:1 nearest matches; weighted lm otherwise).

What does it assume?

• No unmeasured confounding (the key assumption; not testable from data).
• Overlap in propensity scores between treated and controls.
• Correctly specified propensity model.

How do I interpret the result?

Matching can only adjust for measured confounders — unmeasured confounding remains. Use Rosenbaum sensitivity analysis to quantify how strong unmeasured confounding would have to be to overturn the conclusion.

See also

• Logistic regression (binary)
• GEE (generalized estimating equations)

References

• Austin (2011). An introduction to propensity score methods for reducing the effects of confounding. MBR 46(3).