When Full Matching Cannot Place Every Unit
Full matching groups treated and control units into subclasses, with at least one unit from each treatment group in every subclass. Calipers can rule out some treated-control pairings, while restrictions on the treated-to-control ratio can rule out some otherwise possible subclasses. Consequently, a procedure that seeks a valid full match may be unable to retain every observed unit. “Full” describes the structure of the retained match; it does not guarantee that every unit in the original sample can be assigned.
The practical task is to determine which restriction prevents a unit from being placed, how many units are affected, and what options are defensible. These decisions matter because excluding units changes the matched sample and can narrow the population represented by the analysis.
For clarity, the examples use the pairwise caliper rule: every treated-control pair within a subclass must be eligible. Let a subclass contain \(m\) treated units and \(n\) controls. If the allowed treated-to-control ratio is from \(L\) to \(U\), feasibility requires \(L\leq m/n\leq U\), as well as \(m\geq1\) and \(n\geq1\). Other implementations may define caliper feasibility differently; the relevant eligibility rule should be stated and checked before interpreting a match.
A useful first diagnostic is to list each unit's eligible partners. A unit with none cannot enter any subclass under the stated caliper rule. But having at least one eligible partner is not enough: the available units may still be unable to form a subclass that meets the ratio limits. Thus eligibility is a necessary condition for inclusion, not always a sufficient one.
Diagnosing Infeasible Units
A convenient way to organize the diagnosis is as a bipartite graph: treated units appear on one side, controls on the other, and an edge joins two units when they satisfy the caliper. A vertex with no edges is isolated. It cannot be placed in a subclass containing an eligible opposite-treatment unit. The graph also shows why a simple count of treated and controls is insufficient: eligibility depends on which units can be grouped together, not just on the overall group sizes.
Ratio limits add a second check. For example, if each subclass must contain at least twice as many treated units as controls, then a one-treated, one-control pair is not feasible even when its distance satisfies the caliper. A unit with an eligible partner may still be unusable if no collection of mutually eligible units can meet the required ratio. Feasibility therefore depends on the combination of the eligibility pattern and the ratio rule.
Worked Example: Finding Units Isolated by a Caliper
Suppose three treated units have propensity scores \(0.18\), \(0.31\), and \(0.82\), and four controls have scores \(0.20\), \(0.37\), \(0.44\), and \(0.60\). Use an absolute propensity-score caliper \(c=0.08\), with subclass ratios between 1:2 and 2:1.
For the treated unit at \(0.18\), the closest control is at \(0.20\). The distance is \(|0.18-0.20|=0.02\), so this pair is eligible. For the treated unit at \(0.31\), the distances to the controls at \(0.20\) and \(0.37\) are \(0.11\) and \(0.06\), respectively; only the \(0.37\) control is eligible. The treated unit at \(0.82\) is at least \(0.82-0.60=0.22\) from every control, which exceeds \(0.08\), so it has no eligible partner.
The control at \(0.44\) is closest to the treated unit at \(0.31\), but that distance is \(|0.44-0.31|=0.13\). The control at \(0.60\) is also too far from every treated unit. Thus the eligible pairs are \(0.18\) with \(0.20\), and \(0.31\) with \(0.37\). Each pair has a 1:1 ratio, within the allowed range. The full match can retain those four units in two subclasses; the treated unit at \(0.82\) and controls at \(0.44\) and \(0.60\) cannot be retained under these restrictions.
Worked Example: Eligible Partners but No Feasible Ratio
Now suppose treated scores are \(0.20\) and \(0.30\), control scores are \(0.21\) and \(0.31\), and \(c=0.02\). Require at least two treated units per control in every subclass, so \(m/n\geq2\).
The eligible distances are \(|0.20-0.21|=0.01\) and \(|0.30-0.31|=0.01\). The other cross-group distances are \(|0.20-0.31|=0.11\) and \(|0.30-0.21|=0.09\), both greater than \(0.02\). The eligibility graph therefore consists of two separate one-treated, one-control pairs.
Each pair has \(m/n=1/1=1\), which fails the required inequality \(m/n\geq2\). Combining both pairs into one subclass does not solve the problem: that subclass would have two treated units and two controls, so \(m/n=2/2=1\) as well. No eligible subclass can include these units under this ratio restriction. The failure is not caused by a lack of caliper-eligible partners; it is caused by the ratio limit combined with the pattern of eligible pairings.
What Can Be Done?
There is no automatic correction that both retains every unit and preserves every restriction. The defensible choice depends on the study design and the purpose of the restrictions. First identify the reason for infeasibility; then consider whether that restriction can be changed without undermining the design. Possible responses include excluding units that cannot be placed, revising a caliper or ratio limit with a clear rationale, or using a different matching specification. Each response changes what the matched sample represents or what design constraint it enforces.
Exclusion is appropriate when the restrictions are substantively important and the units genuinely lack feasible comparison units. Report how many treated units and controls were excluded, and explain the reason. Do not describe an excluded unit as matched simply because it appears in the original data or was considered by the algorithm. If excluded units differ systematically in propensity scores or covariates, the retained sample may represent a narrower region of covariate support than the original study population.
Relaxing a caliper can create new eligible edges, but it also permits more distant treated-control pairs. Recalculate the eligibility pattern and assess whether the revised matches remain credible; a larger retained sample alone does not establish that the match is better. Relaxing ratio limits may allow subclasses with different numbers of treated units and controls, but it does not create an eligible pair where the caliper excludes all opposite-treatment units. Both changes should be planned and reported as design choices, not made solely to obtain a desired number of retained units.
A different propensity-score or distance specification may also change which pairs are eligible. This is not a guarantee that the problem disappears: the alternative must still be justified, and its resulting overlap and covariate balance should be examined. The earlier material on exact matching, Mahalanobis distance matching, and constrained matching provides relevant design alternatives; choosing among them requires attention to the study's covariates and design goals.
Worked Example: Comparing Two Caliper Choices
Suppose one treated unit has propensity score \(0.50\), and three controls have scores \(0.47\), \(0.52\), and \(0.60\). Require the treated-to-control ratio in a subclass to be between 1:3 and 1:1, so \(1/3\leq m/n\leq1\).
With \(c=0.04\), the distances are \(|0.50-0.47|=0.03\), \(|0.50-0.52|=0.02\), and \(|0.50-0.60|=0.10\). The first two controls are eligible; the third is not. Grouping the treated unit with the two eligible controls gives \(m/n=1/2=0.5\), which satisfies \(1/3\leq0.5\leq1\). The subclass retains two controls.
If the caliper is changed to \(c=0.10\), the third control becomes eligible because its distance is exactly \(0.10\). The resulting subclass has \(m/n=1/3\), which still satisfies the lower bound. All four units can now be placed in one subclass. This increases retention, but it does so by allowing a pair separated by \(0.10\) on the propensity-score scale. The numerical feasibility check is not evidence that the wider caliper is substantively acceptable; the increased distance should be considered alongside covariate balance and the design rationale.
Justifying the Feasibility Checks
The basic exclusion rule follows directly from the subclass requirements. If a unit has no eligible unit in the opposite treatment group, then every subclass containing it would have to include at least one opposite-treatment unit that violates the caliper rule. Such a subclass is invalid. Therefore the unit cannot belong to any feasible subclass under that rule.
Ratio feasibility can be checked separately once a candidate subclass is specified. For example, under \(L\leq m/n\leq U\), the lower limit is equivalent to \(m\geq Ln\), and the upper limit to \(m\leq Un\). In the second worked example, the only possible edge-based groups had \(m=n=1\), so \(m\geq2n\) would require \(1\geq2\), which is false. This arithmetic establishes infeasibility for those candidate groups; it does not imply that every problem with eligible edges is infeasible. Larger compatible groups may meet the ratio bounds, so the full eligibility pattern must be considered.
List eligible opposite-treatment partners for each unit using the exact distance scale and caliper threshold.
These units cannot enter a valid subclass under the current caliper rule.
For candidate subclasses, calculate \(m/n\) and compare it with both ratio limits.
Record excluded units and reconsider calipers or ratio limits only with a design-based justification; then reassess balance and overlap.
Diagnostics, Alternatives, and Common Pitfalls
A useful report does more than give a total number matched. It states the number of units excluded in each treatment group, distinguishes caliper-isolated units from units blocked by ratio constraints, and summarizes the propensity-score and covariate patterns of retained and excluded units. Examining the eligible-pair structure can reveal whether exclusions are concentrated in a region where one treatment group has little or no support. After any design change, reassess covariate balance and whether the revised constraints were met.
Exclusion, constraint relaxation, and a different matching specification are not interchangeable. Exclusion respects the original restrictions but reduces the matched sample. Relaxing a caliper permits more distant pairs; changing ratio limits permits different subclass compositions. A different distance specification changes which units are considered close. The appropriate comparison depends on the design objective, not simply on which option retains the most observations.
- Pitfall: assuming “full” means every original unit is retained. Full matching requires both treatment groups in each retained subclass; units with no feasible placement may be omitted.
- Pitfall: treating any eligible edge as a feasible match. The edge may not be part of a subclass satisfying the ratio limits, and other caliper restrictions may also matter.
- Pitfall: widening a caliper only to reduce exclusions. A wider threshold changes which pairs are permitted. Check the resulting distances and covariate balance, and explain the choice.
- Pitfall: reporting only the overall matched sample size. Treatment-specific exclusions and their locations in the propensity-score and covariate distributions affect how the retained sample should be interpreted.
Check Your Understanding
Answer each question, then compare your reasoning with the brief solutions.
- A treated unit has propensity score \(0.40\); the closest control has score \(0.47\). With an absolute-score caliper of \(0.05\), can the treated unit enter a subclass under the pairwise caliper rule?
- A subclass has 2 treated units and 4 controls. Does it satisfy ratio limits \(1/2\leq m/n\leq1\)? Show the ratio.
- Why does a unit with one eligible opposite-treatment partner not necessarily have a feasible full-matching placement?
- Name two diagnostics to report when a caliper or ratio restriction excludes units.
Solutions. (1) No. The distance is \(|0.40-0.47|=0.07>0.05\); if no other control is closer and within the caliper, the unit has no eligible partner. (2) Yes. \(m/n=2/4=0.5\), and \(1/2\leq0.5\leq1\). (3) The available partner may not allow a subclass to meet the ratio limits, or the eligibility pattern may not support a larger feasible subclass. (4) Report exclusions separately by treatment group and distinguish caliper-ineligible units from ratio-infeasible placements; also examine where excluded units lie in the propensity-score or covariate distributions.