Matching constructs comparisons between units receiving different treatments but having similar pretreatment covariates. Similarity can be measured directly on selected covariates or through a propensity score. The design specifies the distance, allowed discrepancies, number of controls per treated unit and whether a control can be reused.
With one distinct control matched to each of treated units, a simple estimator isThis averages differences in the represented treated population. Dropping unmatched treated units changes that population, and reusing controls induces dependence between pair differences. Caliper matching explicitly limits allowed discrepancies.
Matching seeks covariate balance; it does not create a randomized controlled trial. Identifying a causal effect still requires conditional exchangeability given the matched pretreatment covariates, adequate positivity, and consistency in causal inference. Unmeasured determinants of assignment and outcome can remain different even when measured covariates are well balanced. Conditioning on a causal mediator or inducing collider bias is not repaired by calling the procedure matching.
Covariate balance means that the compared treatment groups have similar distributions of specified pretreatment covariates. Exact distributional balance in a weighted population isThis is a property of the represented covariates and chosen weights. Matching and inverse probability weighting can aim to approximate it.
Equal means are weaker than equal probability distributions. For example, a covariate taking values equally often in one group and equally often in the other has mean zero in both, but variances one and four. A zero standardized mean difference would miss that discrepancy. Inspect relevant spreads, tails, categories and combinations as well as means.
A nonsignificant difference test does not demonstrate covariate balance: small samples may have low statistical power, while large samples can detect negligible discrepancies. Covariate balance on measured pretreatment variables also cannot establish conditional exchangeability for unmeasured potential outcomes.
This gives a dimensionless effect size, so mean differences can be compared across measurement scales.
Caliper matching permits a candidate matching pair only when a prespecified distance is at most a tolerance . For propensity score , possible rules are or . These are different distances; the numeric tolerance must be interpreted on the selected scale.[ref]
For a treated unit with propensity score , a raw-score caliper admits a control with score and excludes one with score . The matching algorithm still has to choose among admissible controls and decide whether controls may be reused. A tighter caliper excludes distant pairs but can leave more treated units unmatched. Excluding those units changes the population represented by the matched estimate; a caliper is not a guarantee of an unchanged average treatment effect on the treated. Check covariate balance after matching, and account for the paired or reused-control structure in statistical inference.
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In statistics, "matching" refers to a technique used in observational studies and experiments to control for confounding variables when estimating causal effects. The main goal of matching is to create comparable groups that differ only in the treatment or intervention of interest, thus reducing bias in the estimation of treatment effects. There are several common forms of matching: 1. **Propensity Score Matching (PSM):** This is one of the most widely used methods.