Why Restrict Matches to Shared Covariate Values?
A small propensity-score distance does not guarantee that a pair is comparable on every covariate that matters. In particular, a treated unit and a control unit can have similar estimated propensity scores while belonging to different hospitals, regions, or clinically important categories. Exact matching on key covariates prevents such pairs from being considered: a match is eligible only when the units share the specified categorical values. A distance rule then chooses among eligible candidates.
This is a design restriction, not a treatment-effect estimator by itself. It changes which pairs are possible and which treated units can be retained. The distance used to choose among candidates may be the propensity score or its logit; as in earlier material on distances for propensity score matching, the distance and any caliper should be specified before matching. Exact agreement on selected categories does not ensure balance on the remaining covariates, nor does it by itself establish a causal interpretation.
The joint category matters. If the designated covariates are hospital and sex, “Hospital A, female” is one exact-match group, while “Hospital A, male” is another. Agreement on hospital alone is not enough. Every variable declared exact must agree at the level actually recorded. Exact matching is not the same as matching on a numerical covariate that happens to take the same value: the method is most commonly used for categorical characteristics selected in advance as essential to share.
Conditions and the Matching Rule
The mechanics of forming pairs require correctly recorded categories, treatment labels, and a defined distance for eligible candidates. If a propensity score is used, it must be estimated or otherwise specified for the units being matched. A propensity-score distance can help rank eligible controls, but it cannot make an ineligible control eligible. If a caliper is also imposed, a candidate must meet both rules: exact category agreement and the caliper threshold.
For a causal interpretation of a matched treatment-effect estimate, design choices are not enough. The usual causal conditions include consistency (the observed outcome equals the potential outcome under the treatment actually received), no interference between units, conditional exchangeability given the measured covariates, and positivity for the target population. Positivity requires that both treatment states be possible at relevant covariate values. Exact matching can make a practical lack of overlap visible: a category containing treated units but no controls cannot supply an exact control match. These assumptions are substantive and are not verified merely because a matching algorithm returned pairs.
A common implementation is nearest-neighbor matching within exact-match groups. For each treated unit, consider only controls in its group, compute the specified distance to each eligible control, and select the closest one. Matching without replacement removes a selected control from later consideration; with replacement, it remains available. A caliper, if used, discards candidates whose distance exceeds the prespecified maximum. If no eligible candidate remains, that treated unit is unmatched.
Define the categories to be shared, including how multiple categorical covariates form joint groups. Decide how missing or ambiguous category values will be handled before matching.
For each treated unit, retain only controls with the same joint category. Do not compare distances across different exact-match groups.
Rank eligible controls by the chosen distance, apply any caliper, and select matches using the prespecified replacement and tie rules.
Report unmatched treated units and assess covariate balance and overlap in the retained sample, including covariates not designated for exact matching.
Worked Matching Examples
Worked Example: Nearest Neighbors Within Exact Groups
Hypothetical data. Suppose the exact-match variables are hospital and sex, and matching is without replacement using absolute differences in estimated propensity scores. In the “Hospital A, female” group, two treated units have scores \(0.30\) and \(0.46\); three controls have scores \(0.28\), \(0.40\), and \(0.51\). All other units are in different exact-match groups.
For the treated unit with score \(0.30\), the eligible distances are \( |0.30-0.28|=0.02\), \( |0.30-0.40|=0.10\), and \( |0.30-0.51|=0.21\). The nearest control is the one with score \(0.28\), at distance \(0.02\).
After that control is used, the treated unit with score \(0.46\) can choose between the remaining controls. Its distances are \( |0.46-0.40|=0.06\) and \( |0.46-0.51|=0.05\), so it is matched to the control with score \(0.51\). The control with score \(0.40\) remains unused.
These are matches only because both pairs agree on hospital and sex. A control in Hospital B with score \(0.31\) would have a smaller score distance for the first treated unit, but it is not a candidate. The result also depends on the specified treated-unit order when matching without replacement; this example follows the stated order rather than claiming a globally best allocation.
Worked Example: No Exact Candidate Means No Match
Hypothetical data. Exact matching is required on treatment site. A treated unit at Site C has an estimated propensity score of \(0.42\). The available controls are at Sites A and B, with scores \(0.41\) and \(0.43\), respectively. Suppose the distance is the absolute difference in propensity scores, with a caliper of \(0.05\).
The score distances are \( |0.42-0.41|=0.01\) and \( |0.42-0.43|=0.01\). Both would satisfy the caliper if distance were the only restriction. But neither control is at Site C, so neither is an exact-match candidate. The treated unit has no match and is excluded from the matched analysis under this design.
This conclusion is not a failure of the distance calculation; it is the direct consequence of the exact site requirement. If there were a Site C control at score \(0.48\), it would be eligible by category but fail the caliper because \( |0.42-0.48|=0.06>0.05\). Thus exact agreement and the distance threshold are separate requirements.
Worked Example: A Matched-Pair ATT Summary
Hypothetical data. Suppose two treated units are matched to controls within their exact groups, and the outcome is a follow-up score measured in points. The treated and control outcomes are \(18\) and \(15\) points for the first pair, and \(12\) and \(14\) points for the second pair. If the target is the average treatment effect for these matched treated units, compute the mean paired outcome difference.
The pair differences, treated minus control, are \(18-15=3\) points and \(12-14=-2\) points. Their average is \[ \frac{3+(-2)}{2}=\frac{1}{2}=0.5000\text{ points}. \]
The matched-sample estimate is therefore \(0.5000\) points for the two retained treated units. It describes this matched treated sample; extending it to all treated units requires additional assumptions, especially when some treated units were excluded. Pairing on exact categories does not itself remove confounding from unmeasured covariates or guarantee causal identification.
Why the Restriction Changes the Analysis
The eligibility rule can be written as a restriction on the set of possible pairs. If \(M\) is the set of selected treated-control pairs, then every pair must satisfy \[ (i,j)\in M \quad\Longrightarrow\quad Z_i=1,\ Z_j=0,\ C_i=C_j. \] A distance-based rule selects among this restricted set; it cannot relax the equality condition. This simple implication explains two practical consequences. First, exact matching guarantees agreement on the designated categories for every retained pair. Second, if a treated unit’s category has no available controls, the eligible set is empty and no distance calculation can create a match.
The first consequence is a design guarantee only for the selected categories. Covariates omitted from \(C_i\) can still differ substantially, even when the propensity scores are close. Therefore, the balance assessment must include all measured confounders relevant to the design, not just the exact-match variables or the score. Standardized differences and distributional comparisons are useful summaries; a small propensity-score distance alone is not a balance diagnostic.
The second consequence is a change in the target population. If some treated units cannot be matched, a matched-pair estimate generally concerns the retained treated units rather than the original treated population. Report how many treated units were retained and excluded, and describe their exact-match groups and covariates. If exclusions are concentrated in particular groups, the resulting estimate may not generalize to those groups.
A short justification for checking overlap within groups follows directly from the eligibility rule. For a treated unit \(i\), define its eligible control set as \(A_i=\{j:Z_j=0,\ C_j=C_i\}\). If \(A_i\) is empty, no candidate exists. If it is nonempty but every candidate fails the caliper, no feasible match exists under the combined rule. Thus the number and characteristics of controls in each exact group, as well as distances within that group, determine empirical matchability. A favorable overall propensity-score range can conceal empty or poorly supported individual groups.
Diagnostics, Alternatives, and Limitations
After matching, check that each selected pair satisfies exact agreement and any distance or caliper restriction. Then examine the number of treated units matched, the number of controls used, repeated control use if replacement is allowed, and the distribution of distances. Assess balance on the remaining measured covariates before and after matching. If a category has many treated units but few controls, matching without replacement may leave treated units unmatched; matching with replacement can reuse a close control but concentrates representation on that control and requires an analysis that accounts for reuse.
Exact matching is more restrictive than ordinary nearest-neighbor matching, where the nearest control can come from any category unless other restrictions are imposed. The restriction is valuable when category agreement is substantively essential, but it can reduce match rates and precision. Coarsened exact matching, the next topic, relaxes equality on original values by grouping values into chosen coarser bins; it is a distinct design choice, not a rule to silently apply within exact matching. Mahalanobis distance matching is another alternative for comparing multivariate covariate profiles, but it does not replace the defining exact-category restriction unless explicitly combined with it.
Key limitations are that exact matching cannot balance variables that were not measured, estimates may apply only to retained units, and sparse categories can make the design infeasible. Categories must also be defined carefully: inconsistent labels, overly granular categories, or missing values treated as if they were ordinary shared categories can produce misleading matches. Do not decide after seeing outcomes to merge groups or change the exact variables; that makes the design outcome-dependent and weakens the interpretation of the analysis.
Common Mistake
Correction: A small distance ranks eligible candidates; it does not establish eligibility. Under exact matching on a category \(C\), a pair is allowed only when \(C_i=C_j\). If no control in the treated unit’s group meets the distance rule, report that unit as unmatched rather than borrowing a control from another group.
Check Your Understanding
Answer each question, then compare with the solutions below.
- Exact matching is required on clinic and sex. A treated woman at Clinic 1 has a control candidate at Clinic 1 who is male and another at Clinic 2 who is female. Are either eligible?
- A treated unit has propensity score \(0.35\). Two controls in its exact group have scores \(0.32\) and \(0.41\). Which is nearest, and what is the distance?
- Why can exact matching leave a treated unit unmatched even if a control elsewhere has an almost identical propensity score?
- Two matched pairs have outcome differences, treated minus control, of \(4\) and \(-1\) minutes. What is their mean difference, and to whom does it directly apply?
- Name one diagnostic beyond checking exact-category agreement, and explain what it assesses.
Solutions. (1) Neither is eligible: the first differs in sex and the second differs in clinic, so neither shares the joint category. (2) The distances are \( |0.35-0.32|=0.03\) and \( |0.35-0.41|=0.06\); the score-\(0.32\) control is nearest, at distance \(0.03\). (3) Exact matching restricts candidates to controls with the same designated category, so a control outside that group is ineligible regardless of distance. (4) The mean is \((4+(-1))/2=1.5\) minutes; it directly summarizes the two retained treated units represented by these pairs. (5) For example, assess balance on other measured covariates using standardized differences; this checks whether the groups are comparable beyond the categories forced to agree.