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Caliper matching permits a candidate matching pair only when a prespecified distance is at most a tolerance $c$. For propensity score $e(X)$, possible rules are $|e(X_i)-e(X_j)|\le c$ or $|\mathrm{logit}e(X_i)-\mathrm{logit}e(X_j)|\le c$. These are different distances; the numeric tolerance must be interpreted on the selected scale.^[ref]

For a treated unit with propensity score $0.40$, a raw-score caliper $c=0.03$ admits a control with score $0.42$ and excludes one with score $0.46$. 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.

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 $n_T$ 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.

Using 50 controls per UTC can reduce sampling variance by averaging over more control outcomes. The gain has sharply diminishing returns because controls matched to the same UTC are not 50 independent treated-control contrasts.

The disadvantage is poorer covariate balance: the 50th-nearest school will usually be much less comparable than the fifth-nearest school, increasing residual confounding and changing the target population. A caliper or variable matching ratio should therefore prevent distant matches even if many controls are available.

Peter C. Austin develops methods for observational studies, including propensity score matching. In his simulation study of caliper matching, treated and untreated people are paired only when their scores are sufficiently close.^[ref]

The caliper creates a practical tradeoff: a wider tolerance can retain more pairs but allow larger differences in measured baseline risk; a tighter tolerance can discard people and change the population represented by the comparison. Austin compared resulting bias, mean squared error and confidence interval coverage under specified data-generating scenarios. His numerical recommendations concern those estimands and scenarios, rather than a guarantee that matching removes unmeasured confounding.^[ref]