How a Caliper Changes Nearest-Neighbor Matching
Nearest-neighbor matching ranks possible controls by their distance from a treated unit. A caliper adds a restriction: a control may be selected only if its distance is no greater than a specified threshold. Thus, the rule is not simply “choose the nearest control,” but “choose the nearest control that is admissible under the caliper.” This distinction matters when controls are used up in a no-replacement procedure, when several treated units compete for the same control, and when the closest eligible control differs from the closest control in the original pool.
Here the matching distance is the absolute difference in estimated logit propensity scores, as in calipers on the logit propensity score. The caliper width is fixed for the examples; it is a distance threshold, not a probability and not a significance level. The matching calculations are mechanical once the scores, width, replacement rule, treated-unit order, and tie rule are specified.
The inequality includes the boundary: a control exactly \(c\) units away is admissible. Because the distance is measured on the logit scale, \(c\) is expressed in log-odds units. The caliper choice itself is a separate design decision; the earlier material on choosing a caliper width and calipers on the logit propensity score concerns how to specify that threshold.
The Matching Rule, Stated Precisely
For each treated unit \(i\), let \(C_i\) be the set of controls that are admissible and currently available. The matching rule can be written as a constrained minimization:
If \(C_i\) is empty, the minimization has no solution and no control is assigned to that treated unit. With replacement, the available-control set remains the same after a match: a control can be selected again. Without replacement, a selected control is removed from the available pool before the next treated unit is processed. Therefore, for matching without replacement, the assignment also depends on the order in which treated units are processed.
A short derivation clarifies why the threshold and ranking must be applied together. The caliper condition defines the feasible set \(C_i=\{j:Z_j=0,\ |\hat s_i-\hat s_j|\leq c,\ j\text{ available}\}\). A nearest-neighbor choice is the minimum distance over that set, not over all controls. Hence a control outside the threshold cannot be selected merely because its distance ranks first in a list that ignores eligibility. If the feasible set is empty, there is no minimum over eligible controls to assign.
Record the score scale, caliper width, replacement rule, treated-unit order, and tie rule before making assignments.
For the treated unit being processed, compute the absolute score difference to each control currently available.
Keep only controls whose distance is at most \(c\). Rank this eligible set from smallest to largest distance.
Select the nearest eligible control. If matching without replacement, remove it from the pool before processing the next treated unit.
Worked Example: Making Sequential Matches Without Replacement
Suppose three treated units have logit scores \(0.10\), \(0.32\), and \(0.70\), and four controls have scores \(0.04\), \(0.24\), \(0.53\), and \(0.91\). Use caliper \(c=0.20\), process treated units in the order listed, and match without replacement. Call the treated units \(T_1,T_2,T_3\) and the controls \(C_1,C_2,C_3,C_4\) in the order listed.
For \(T_1\), the distances to \(C_1,C_2,C_3,C_4\) are \(0.06,0.14,0.43,0.81\), respectively. The first two controls are within \(0.20\), and \(C_1\) is closer. Assign \(T_1\) to \(C_1\), then remove \(C_1\) from the pool.
For \(T_2\), compare only the remaining controls. The distances to \(C_2,C_3,C_4\) are \(0.08,0.21,0.59\). Only \(C_2\) is within the caliper, so assign \(T_2\) to \(C_2\) and remove it.
For \(T_3\), the remaining-control distances to \(C_3,C_4\) are \(0.17,0.21\). The former is admissible and the latter is not. Assign \(T_3\) to \(C_3\). The resulting pairs are \((T_1,C_1)\), \((T_2,C_2)\), and \((T_3,C_3)\). In particular, \(C_4\) is not selected: being the next control in the score ordering does not make it admissible when its distance is \(0.21>0.20\).
Replacement and Order Affect the Available Set
The caliper determines which controls may be considered; replacement determines which controls remain available after each assignment. With replacement, a control can be the nearest admissible control for more than one treated unit. Without replacement, a control used earlier cannot be assigned again. The distance calculations may be identical under the two rules, but the candidate set for later assignments differs.
Worked Example: A Previously Used Control and the Caliper
Two treated units have scores \(0.50\) and \(0.52\). Two controls have scores \(0.50\) and \(0.62\). Use \(c=0.15\), match without replacement, and process the treated units in the order listed.
For the first treated unit, the distances are \(|0.50-0.50|=0.00\) and \(|0.50-0.62|=0.12\). Both controls meet the caliper, and the score-\(0.50\) control is nearer. Assign it to the first treated unit and remove it.
For the second treated unit, the score-\(0.50\) control is no longer available. The remaining control has distance \(|0.52-0.62|=0.10\), which is at most \(0.15\). Assign the second treated unit to the score-\(0.62\) control. This is the nearest admissible choice among the remaining controls. A calculation that continues to treat the already-used control as available would not follow the specified no-replacement design.
A tie in distance does not change whether a control is admissible, but it does mean that the minimizer is not unique. A reproducible analysis must state a tie rule, such as a fixed unit identifier order or a prespecified random rule. In a no-replacement procedure, different tie resolutions can affect which controls remain for later treated units. The same is true of the treated-unit processing order.
Worked Example: How Processing Order Can Change Matches
Let treated units \(A\) and \(B\) have scores \(0.10\) and \(0.24\), and controls \(C_1\) and \(C_2\) have scores \(0.12\) and \(-0.02\). Use caliper \(c=0.15\), match without replacement, and use the stated order to process treated units.
If \(A\) is processed first, its distances are \(0.02\) to \(C_1\) and \(0.12\) to \(C_2\). Both are admissible, so \(A\) receives \(C_1\), the nearer control. Then \(B\)'s only remaining control is \(C_2\), at distance \(|0.24-(-0.02)|=0.26\), which exceeds the caliper. Thus \(B\) receives no match in this order.
Now process \(B\) first. Its distances are \(0.12\) to \(C_1\) and \(0.26\) to \(C_2\). Only \(C_1\) is admissible, so \(B\) receives \(C_1\). Unit \(A\) then has only \(C_2\) available, at distance \(|0.10-(-0.02)|=0.12\), so \(A\) receives \(C_2\). Both treated units are matched in this order. Every assignment obeys the same caliper and nearest-eligible rule; the difference comes from the greedy order and no-replacement constraint.
Checking the Procedure and Its Limits
Useful checks focus on whether the implementation respects both parts of the rule. For every assigned pair, verify that the control was available at that step, that the absolute score difference is no greater than \(c\), and that no closer available admissible control was passed over. For no-replacement matching, also verify that no control appears in more than one pair. Keep a record of the score difference for each pair; a table of assignments alone can conceal a caliper violation.
- Caliper boundary: A distance equal to \(c\) is eligible under the \(\leq c\) rule. Rounding scores before checking the boundary can incorrectly change eligibility; use unrounded values for assignment when available.
- Wrong candidate pool: Under no replacement, rank controls that are still available, not controls already assigned. Under replacement, do not remove a selected control.
- Order sensitivity: If units are processed sequentially without replacement, report the order or use a prespecified order. Trying several orders and reporting only a preferred result makes the matching procedure difficult to reproduce.
- Ties: A tie rule should be specified in advance and applied consistently. Do not silently choose whichever tied control produces a preferred downstream result.
- Score-model problems: Extreme estimated scores or separation can make scores unstable and can leave poor overlap. A matching algorithm cannot repair a poorly estimated score model or create comparable units where the data provide little support.
This procedure is a local, greedy choice: it selects the nearest admissible control for the treated unit currently being processed. It does not generally guarantee the best overall allocation across all treated units. A different global assignment method can optimize a total matching criterion subject to constraints, but that is a different design choice. Other alternatives include nearest-neighbor matching without a caliper, which may permit more distant pairs, and matching with replacement, which permits reuse of a close control. These alternatives trade off restrictions, reuse, and the resulting set of matched units; none is automatically preferable in every application.
A caliper matching rule by itself does not establish a causal effect. Causal interpretation still depends on the design and identification assumptions, including consistency, conditional exchangeability given the measured covariates, and positivity in the target population. A small score distance is not proof that all relevant covariates are balanced, and it cannot address unmeasured confounding. Matching can also change which treated units contribute to the comparison when some have no eligible control; conclusions should describe the population represented by the matched analysis rather than implying that every original unit was compared.
Check Your Understanding
Work each question using the stated caliper rule. Solutions follow the questions.
- A treated unit has logit score \(0.30\). Available controls have scores \(0.18\), \(0.42\), and \(0.50\), with \(c=0.12\). Which control is selected?
- A treated unit has score \(0.60\), and the available controls have scores \(0.40\) and \(0.51\). With \(c=0.08\), is either control admissible?
- Under no replacement, a selected control has distance \(0.03\) from a later treated unit, while the nearest remaining control has distance \(0.09\). If \(c=0.10\), which control can be selected?
- Why can reversing the treated-unit order change matches in greedy matching without replacement, even when scores and the caliper are unchanged?
Solutions. (1) The distances are \(|0.30-0.18|=0.12\), \(|0.30-0.42|=0.12\), and \(|0.30-0.50|=0.20\). The first two controls are tied at the caliper boundary, so either may be selected according to the prespecified tie rule. (2) The distances are \(0.20\) and \(0.09\); both exceed \(0.08\), so neither is admissible. (3) The control at distance \(0.03\) is unavailable because it was already selected. The remaining control at distance \(0.09\) is within \(0.10\), so it is selected. (4) An early treated unit may use a control that would have been the best admissible option for a later unit. Removing that control changes the later unit's available set, so reversing the order can change assignments or whether a unit receives a match.