Why Some Units Cannot Be Matched
A caliper is a feasibility rule: a treated unit and a control unit may be matched only if their distance is no greater than a prespecified limit. That rule can protect against pairing units that are too dissimilar on the matching scale. It also means that a unit may have no eligible counterpart at all. Such a unit cannot be included in a match satisfying the caliper, regardless of how the algorithm orders candidates or chooses among them.
This is different from a unit that has eligible partners but is left unmatched because those partners were assigned elsewhere, for example under a no-replacement procedure. The distinction matters for diagnosing the matching process and for explaining why the final matched sample may be smaller than the original sample.
The distance must be interpreted on the scale actually used. For example, if matching uses the logit of the estimated propensity score, \(d(i,j)\) may be the absolute difference between the two units' logit scores, and \(c\) is measured on that same scale. If matching uses another score or distance, the definition is unchanged, but both the distances and caliper must refer to that choice.
The matching design should specify the distance, the caliper, and whether matching is with or without replacement before outcomes are used to assess effects. A standard causal interpretation additionally relies on the relevant causal assumptions, including consistency, conditional exchangeability given measured covariates, and adequate overlap for the target population. A caliper can restrict comparisons to closer units, but it does not establish these assumptions or guarantee balance on every covariate.
What the Matching Algorithm Can—and Cannot—Do
It is useful to view the data as two groups of vertices, treated units and control units. Draw an edge between a treated unit and a control unit exactly when their pair is admissible. A unit with no admissible partners has no edges. Any matching procedure that respects the caliper can only use edges in this graph. Consequently, a unit with no edges cannot appear in a matched pair unless the caliper rule is changed or violated.
A common procedure first identifies eligible pairs and then applies its matching rule—such as nearest-neighbor matching within the caliper. A unit with no eligible partner is skipped and remains unmatched. The algorithm should not substitute the nearest unit if that distance exceeds \(c\): doing so would no longer be caliper-constrained matching. The unit remains in the original data, but it is not part of the matched analysis that requires an admissible counterpart.
Here \(\mathcal{C}(i)\) is the set of admissible control partners for treated unit \(i\). If \(\mathcal{C}(i)=\varnothing\), no caliper-respecting matched pair can contain \(i\). The same definition applies to a control by considering its admissible treated partners. This is a direct consequence of the pairwise constraint: every matched pair must satisfy \(d(i,j)\leq c\), and an empty candidate set contains no pair that can satisfy it.
The algorithm's handling of isolated units does not by itself determine what happens to units that do have candidates. Under matching without replacement, two treated units may compete for one eligible control, leaving one treated unit unmatched even though it had an admissible partner before assignments were made. Matching with replacement can permit a control to serve multiple treated units, subject to the design, but it cannot create an admissible partner where none exists. Thus, report both the number with zero eligible partners and the number left unmatched after allocation.
Worked Example: Finding Treated Units With No Eligible Controls
Suppose matching uses absolute differences in logit propensity scores, with caliper \(c=0.20\). Three treated units have scores \(0.10,\ 0.80,\ 1.20\), and three controls have scores \(0.05,\ 0.30,\ 0.95\). Identify the candidate controls for each treated unit.
For the treated unit with score \(0.10\), the distances to the controls are \(|0.10-0.05|=0.05\), \(|0.10-0.30|=0.20\), and \(|0.10-0.95|=0.85\). The first two controls are admissible because their distances are at most \(0.20\).
For the treated unit with score \(0.80\), the distances are \(0.75,\ 0.50,\) and \(0.15\). Only the control with score \(0.95\) is admissible.
For the treated unit with score \(1.20\), the distances are \(1.15,\ 0.90,\) and \(0.25\). All exceed \(0.20\), so this treated unit has no admissible control partner and must remain unmatched under this caliper.
The fact that the score \(1.20\) is closest to the control score \(0.95\) does not make that pair eligible: its distance is \(0.25>0.20\). The caliper is a hard limit, not a preference for which the algorithm can make an exception.
Worked Examples of Exclusion and Competition
Worked Example: Counting the Matched Treated Sample
Consider four treated units, \(A,B,C,D\), and three controls, \(1,2,3\). With caliper \(c=0.15\), suppose the pairwise distances are:
| Treated unit | Control 1 | Control 2 | Control 3 | Admissible controls |
|---|---|---|---|---|
| A | 0.04 | 0.31 | 0.40 | 1 |
| B | 0.22 | 0.18 | 0.35 | None |
| C | 0.10 | 0.09 | 0.28 | 1 and 2 |
| D | 0.42 | 0.33 | 0.07 | 3 |
The admissibility check is \(d\leq 0.15\). Thus \(A\) can match control 1, \(C\) can match control 1 or 2, and \(D\) can match control 3. Unit \(B\) has no admissible partner: even its nearest control is at distance \(0.18\), which exceeds the caliper.
Under a no-replacement procedure, one possible allocation is \(A\)-1, \(C\)-2, and \(D\)-3, retaining three of the four treated units. The allocation is feasible because every listed distance is at most \(0.15\). Unit \(B\) is excluded regardless of allocation. If the outcome is measured in days of hospitalization, for example, an average of treated-control differences over these matched treated units would describe the matched treated sample; it would not automatically describe all four treated units.
Worked Example: An Eligible Unit Left Unmatched by Competition
Now consider two treated units with scores \(0.50\) and \(0.62\), and two controls with scores \(0.55\) and \(0.90\). Use absolute score differences and caliper \(c=0.15\).
The distances from treated unit \(A\) at \(0.50\) are \(|0.50-0.55|=0.05\) and \(|0.50-0.90|=0.40\). Its only candidate is the control at \(0.55\). For treated unit \(B\) at \(0.62\), the distances are \(|0.62-0.55|=0.07\) and \(|0.62-0.90|=0.28\). Its only candidate is also the control at \(0.55\).
Neither treated unit is isolated: each has one admissible partner. But if matching is without replacement and \(A\) is processed first, \(A\) may take the control at \(0.55\); then \(B\) cannot use it and is left unmatched. The reason is competition for a limited control, not absence of an admissible partner in the original candidate set. With replacement, both treated units could be paired to that control, because both distances satisfy the caliper. This comparison changes the allocation rule, not the caliper or the candidate distances.
A report that says only “one treated unit had no match” would miss this distinction. Here the correct account is that no treated unit initially lacked a candidate, but one was left unmatched under the specified no-replacement allocation.
What Exclusion Means for Interpretation
Removing units without admissible partners changes which units contribute to the matched estimate. If treated units outside the control-score range are excluded, an estimate based on the retained treated units pertains most directly to the treated units who could be matched under the design—not automatically to every treated unit in the original sample. If controls are excluded for lack of treated partners, they likewise do not contribute to the matched comparisons.
This is not necessarily a flaw. Restricting comparisons can improve comparability for the included units, while acknowledging that the data do not support the same comparison for everyone. But exclusion can limit generalizability and can alter the population represented by the estimate. The matching target and the exclusions should therefore be described alongside the effect estimate. The caliper alone does not prove that the resulting matched sample is balanced or that a causal effect is identified.
The distinction between no admissible partner and allocation-related unmatched status also matters when comparing algorithms. With the same distance and caliper, changing order or replacement rules can change which eligible units are paired. It cannot rescue a unit whose candidate set is empty. A wider caliper may add candidates, but it can also admit more distant pairs; that is a design change, not a harmless repair. A narrower caliper may improve closeness among permitted pairs while excluding more units. These trade-offs should be assessed before examining outcomes.
Diagnostics and Common Mistakes
A useful diagnostic begins with the candidate sets, not only the final match list. For each treated unit, count admissible controls; for each control, count admissible treated units. Report how many units have zero candidates on each side, and separately report the number left unmatched after allocation. Summaries or plots of the matching scores by treatment group can reveal regions where one group has little or no counterpart support.
Then examine the composition of retained and excluded units. Compare their observed covariates and score ranges, and assess covariate balance in the matched sample using appropriate balance summaries. Good balance among matched units does not establish balance for excluded units, and score overlap does not guarantee balance on every measured covariate. If one part of the sample is repeatedly excluded, the resulting target population may be substantially narrower than the original one.
- Forcing the nearest pair. The wrong reasoning is, “There is a nearest control, so it can be used.” Every treated unit has a nearest control when controls exist, but that pair may violate the caliper. The correction is to check \(d\leq c\); if no control passes, leave the unit unmatched.
- Calling every unmatched unit isolated. A unit can have an admissible partner and still lose it to another unit under matching without replacement. The correction is to distinguish zero candidates before allocation from unmatched status after allocation.
- Assuming exclusions are inconsequential. Dropped units may differ systematically from retained units. The correction is to report who was excluded and state which population the matched comparisons represent.
- Treating caliper compliance as proof of causal validity. The correction is to assess design assumptions and covariate balance separately; admissible score distances alone do not establish exchangeability or remove unmeasured confounding.
| Approach or design choice | What it can change | What it cannot do |
|---|---|---|
| Nearest-neighbor matching within a fixed caliper | Chooses the closest eligible partner under the matching rule | Cannot match a unit with an empty candidate set |
| Matching with replacement | Can allow an eligible control to serve more than one treated unit | Cannot make an inadmissible pair admissible |
| Widening the caliper | May add candidate pairs and reduce zero-candidate counts | Cannot preserve the original restriction; newly allowed pairs may be less similar |
| Matching without a caliper | Can avoid exclusions caused specifically by that caliper | May force pairs across larger distances and does not guarantee good comparability |
These alternatives address different constraints. If many units are isolated, first verify that the score, distance scale, and caliper were implemented as intended. Then consider whether the chosen restriction is appropriate for the scientific target and design. Do not widen the caliper merely to increase the number matched without considering the resulting pair quality and the change in the population represented.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A treated unit has distances \(0.12,\ 0.19,\ 0.31\) to three controls, with caliper \(c=0.18\). Does it have an admissible partner? What if the nearest control is already used under matching without replacement?
- Two treated units each have exactly one admissible control, and it is the same control. Explain how one treated unit could be left unmatched despite having an admissible partner.
- Why does increasing the caliper potentially reduce the number of isolated units, and what trade-off does that create?
- A matched analysis excludes treated units with no admissible controls. Can its estimate automatically be interpreted as the effect for all treated units? Explain.
Solutions. (1) Yes. The distances \(0.12\) and \(0.19\) are not both admissible: only \(0.12\leq 0.18\), so there is one eligible control. If that control has already been used without replacement, the treated unit may remain unmatched after allocation, but it was not isolated before allocation. (2) Under no replacement, the first treated unit processed may take the sole eligible control, leaving the other without an available control. The latter had an admissible partner in the initial candidate set, so this is competition, not isolation. (3) A wider caliper can bring previously too-distant pairs within the limit, creating additional candidate edges. The trade-off is that the design permits less similar pairs, which may weaken comparability. (4) No. The estimate directly reflects the matched treated units under the design. Extending it to all treated units requires additional justification because excluded treated units may differ and had no eligible controls under the chosen caliper.