Why a Caliper Is a Feasibility Rule
A distance-based matching procedure may find a control unit that is closer to a treated unit than the other available controls, yet still regard that pair as too far apart to use. A caliper makes this restriction explicit: it sets a maximum permitted distance for a treated-control pair. The caliper does not, by itself, decide which admissible pair to select. It first determines which pairs are allowed to be considered.
This distinction matters because nearest-neighbor matching and caliper matching answer different questions. Nearest-neighbor matching identifies a closest available candidate under the chosen distance. A caliper asks whether a particular pair is sufficiently close to pass a fixed limit. The closest candidate can fail that limit; in that case, being the closest does not make it admissible.
For propensity score matching on the propensity-score scale, a common distance is the absolute difference between the two estimated scores. If \(\hat e_i\) is the estimated propensity score for treated unit \(i\) and \(\hat e_j\) is the estimated score for control unit \(j\), then \(d_{ij}=|\hat e_i-\hat e_j|\). A threshold of \(c=0.05\), for example, permits score differences no larger than 0.05. The numerical threshold has meaning only together with the distance definition and scale.
The boundary is inclusive: a pair with distance exactly \(c\) is admissible. A pair with distance greater than \(c\) is not. If distances are rounded for display, use the underlying values when they are available; rounding can otherwise make a pair appear to lie on the boundary when it is slightly above or below it.
Applying the Rule Pair by Pair
The rule is direct once the distance and threshold are specified. For each possible treated-control pair, compare its distance with \(c\). No ranking is needed for this comparison, and the distance to one control does not change whether another pair passes the threshold. A convenient representation is a matrix of distances in which entries at or below \(c\) are marked admissible.
Worked Example: Classifying Propensity-Score Pairs
Suppose two treated units have estimated scores 0.20 and 0.44, and three controls have scores 0.18, 0.31, and 0.49. Use absolute score difference as the distance and apply a caliper of \(c=0.12\). Find every admissible pair.
For the treated unit with score 0.20, the distances to the three controls are
Since \(0.02\leq 0.12\) and \(0.11\leq 0.12\), the controls with scores 0.18 and 0.31 are admissible for this treated unit. The control with score 0.49 is not, because \(0.29>0.12\).
For the treated unit with score 0.44, the distances are
Only \(0.05\leq 0.12\), so the control with score 0.49 is the only admissible control for this treated unit. The complete feasibility set therefore contains three pairs: (0.20, 0.18), (0.20, 0.31), and (0.44, 0.49), where each ordered pair lists a treated score followed by a control score. This enumeration establishes eligibility; it does not specify a general selection algorithm.
In graph terms, treated units and control units form two separate sets of vertices. Draw an edge between a treated unit and a control unit exactly when their distance is at most \(c\). The resulting bipartite graph displays the full set of admissible pairs. This representation is useful because it prevents a common mix-up: an edge means that the pair is allowed, not that the pair has been selected.
Admissible Pairs Are Not the Same as a Matching
A matching assignment must satisfy the design's additional constraints. In one-to-one matching without replacement, for example, each treated unit can be paired with at most one control, and each control can be used at most once. A set of individually admissible pairs may fail to produce an assignment that matches every treated unit under those constraints. With replacement, reuse of controls changes that feasibility question. The caliper defines eligible edges; the matching design determines which collections of edges can be used together.
Worked Example: Individual Eligibility Versus Joint Feasibility
Suppose two treated units have scores 0.30 and 0.34. Two controls have scores 0.31 and 0.45. Use absolute score difference and set \(c=0.04\). Determine the admissible pairs, then check whether both treated units can be matched in a one-to-one assignment without replacement.
For the treated unit at 0.30, the distances are
Thus its only admissible control is the one at 0.31. For the treated unit at 0.34, the distances are
This treated unit also has only one admissible control: the one at 0.31. Both treated units have at least one eligible partner, but those eligible partners are the same control. Without replacement, that control cannot be assigned to both treated units, so no one-to-one assignment can match both. With replacement, reuse of that control would be permitted, subject to the chosen design. This example shows why counting eligible pairs or checking each treated unit separately does not establish joint feasibility.
A short way to state the logic is to define \(A_{ij}=1\) when \(d_{ij}\leq c\), and \(A_{ij}=0\) otherwise. A proposed assignment can use only pairs with \(A_{ij}=1\). For one-to-one matching without replacement, assignment indicators \(m_{ij}\) must also satisfy \(m_{ij}\leq A_{ij}\), with at most one selected pair for each treated unit and at most one for each control. The caliper supplies the first constraint; it does not guarantee that the other constraints can all be satisfied.
A Boundary Check and a Short Justification
The caliper condition is a threshold test, not a statistical test. There is no p-value attached to a pair passing the rule. The reason for the inclusive inequality is that the caliper is defined as the largest permitted distance: distances equal to that maximum satisfy the stated limit, while larger distances violate it.
Worked Example: Handling a Pair on the Boundary
A treated unit has an estimated score of 0.35. Three controls have estimated scores 0.25, 0.26, and 0.47. Use absolute score difference and \(c=0.10\). Which pairs are admissible?
The distances are
The first pair is exactly at the threshold and is admissible because \(0.10\leq 0.10\). The second is admissible because \(0.09\leq 0.10\). The third is inadmissible because \(0.12>0.10\). Excluding the first control on the grounds that a distance must be strictly less than the caliper would apply a different rule from the inclusive condition defined here.
Diagnostics, Comparisons, and Limitations
Before using the resulting admissible-pair set, check the inputs and implementation. The distance scale, threshold, and inequality should be stated unambiguously. Confirm that treated and control scores were computed on a comparable basis, that each pair's distance was calculated using the specified definition, and that the software or hand calculation includes the boundary as intended. A simple distance table or graph can expose a scale mismatch, an accidental use of a different distance, or a rounding error.
A caliper is a restriction on pairwise distance, not a guarantee of good covariate balance. Nor does passing the rule establish that treatment assignment is unconfounded or that a causal effect is identified. Those conclusions require the relevant causal assumptions and diagnostics; a distance threshold alone cannot supply them.
Compared with unrestricted nearest-neighbor matching, a caliper can reject the nearest available control if even that closest pair exceeds the limit. Compared with exact matching, it permits pairs whose values differ, provided their specified distance is within the threshold; exact matching requires agreement on the variable or category being matched exactly. These are different feasibility rules. The rule's practical consequences depend on the chosen distance, threshold, and matching constraints.
A caliper can also leave some treated units without any admissible control, or create competition for a small number of eligible controls. Such patterns are diagnostic signals that the restrictions and available data may not support the desired assignment. They do not, by themselves, determine what alternative design should be used. The key limitation is that admissibility is local to the specified pairwise distance: it does not evaluate the full set of covariates, guarantee a complete assignment, or settle the scientific appropriateness of the threshold.
Check Your Understanding
Answer each question using the stated threshold rule. Brief solutions follow the questions.
- A pair has distance 0.07 and the caliper is 0.07. Is the pair admissible? Explain.
- One treated unit has distances 0.03, 0.08, and 0.11 to three controls. If \(c=0.09\), which pairs are admissible?
- Two treated units each have exactly one admissible control, and it is the same control. Can both be matched in a one-to-one design without replacement?
- Does the fact that a pair passes a propensity-score caliper establish covariate balance or causal validity? Explain briefly.
Solutions. (1) Yes. The rule is inclusive, and \(0.07\leq 0.07\). (2) The pairs at distances 0.03 and 0.08 are admissible; the pair at 0.11 is not, because \(0.11>0.09\). (3) No. The single control cannot be used twice without replacement, so both treated units cannot be assigned distinct controls. (4) No. Passing the caliper shows only that the specified pairwise distance is within the threshold; it does not guarantee balance on all covariates or establish the assumptions needed for causal interpretation.