Why Put a Caliper on Full Matching?
Full matching can place one or more treated units and one or more control units in each subclass. Its flexibility can retain many units and accommodate unequal treated-control ratios. But without a distance restriction, an assignment may pair units that are too dissimilar on the matching scale to be credible counterparts. A caliper prevents that by declaring some treated-control links ineligible before subclasses are constructed.
The essential order is: define a distance and a caliper, remove links that exceed the caliper, then construct subclasses using only what remains. The restriction is on potential treated-control links, not on every pair of units in a subclass. A subclass can contain several units on either side; it need not be a complete set of mutually eligible pairs. Its members must, however, be connected through eligible links used in the assignment.
The distance scale and \(\delta\) are inseparable. For example, a caliper of 0.2 on an unstandardized logit propensity-score scale is not the same restriction as 0.2 standard deviations of that scale. State whether distance is an absolute difference, a standardized difference, or another prespecified metric. In propensity-score matching, a common choice is the absolute difference in estimated logit propensity scores, sometimes divided by their standard deviation. The caliper then has units appropriate to that choice.
From Eligible Links to Subclasses
A convenient representation is a bipartite graph: treated units form one set of vertices, controls form the other, and an edge joins \(i\) to \(j\) exactly when \(d_{ij}\leq\delta\). The caliper operation deletes edges; it does not alter the distances or turn an ineligible pair into an eligible one. Once the graph is built, an assignment can group units into subclasses only where eligible edges provide connections.
For a proposed subclass, write \(m_T\) for its number of treated units and \(m_C\) for its number of controls. Its treated-control ratio is \(m_T/m_C\). If the design requires, for example, each side to be no more than twice the size of the other, then \(1/2\leq m_T/m_C\leq2\) must hold separately in every subclass. A compliant overall ratio does not make a noncompliant subclass acceptable.
A graph component is a maximal group of vertices connected by paths of eligible edges. No eligible link crosses between distinct components. Therefore, a subclass cannot combine units from separate components while maintaining a connected assignment based only on eligible links. A component containing both treatment groups can sometimes be used as one subclass, provided its ratio satisfies the requirements. It may instead be divided into smaller connected subclasses when the assignment and ratio constraints permit.
Choose the matching scale and state the caliper \(\delta\) before examining outcomes. Make the inclusion rule explicit: a link is allowed when \(d_{ij}\leq\delta\).
Calculate distances for treated-control pairs and remove every edge with distance greater than \(\delta\). Keep the distance scale and threshold attached to the graph or its audit record.
Use only eligible edges to connect units. Check that each subclass contains both groups and meets any required treated-control ratio bounds.
Confirm that every included unit is assigned as intended, no selected link violates the caliper, and all subclass constraints hold. Report the caliper, distance definition, and any units that cannot be assigned.
Worked Constructions
Worked Example: Filtering Links at a Caliper
Suppose a hypothetical design uses standardized absolute logit propensity-score distance, with caliper \(\delta=0.30\). The distances for six treated-control pairs are listed below. Determine which links are eligible.
| Pair | Distance | Caliper check | Eligible? |
|---|---|---|---|
| T1–C1 | 0.12 | 0.12 ≤ 0.30 | Yes |
| T1–C2 | 0.38 | 0.38 > 0.30 | No |
| T2–C1 | 0.25 | 0.25 ≤ 0.30 | Yes |
| T2–C2 | 0.20 | 0.20 ≤ 0.30 | Yes |
| T3–C2 | 0.28 | 0.28 ≤ 0.30 | Yes |
| T3–C3 | 0.42 | 0.42 > 0.30 | No |
The eligible links include T1–C1, T2–C1, T2–C2, and T3–C2. The other two pairs cannot be used in any subclass connection under this design. If T1–C2 is appealing because it would create a convenient grouping, that convenience does not override the prespecified caliper.
Worked Example: Building Ratio-Compliant Subclasses
Continue with a hypothetical eligible graph containing treated units T1, T2, T3 and controls C1, C2, C3. Its eligible links and distances are T1–C1 (0.12), T2–C1 (0.25), T2–C2 (0.20), T3–C2 (0.28), and T3–C3 (0.18). All other treated-control distances exceed \(\delta=0.30\). Suppose each subclass must have a treated-control ratio between \(1/2\) and \(2\).
One feasible construction uses two subclasses. Put T1, T2, C1, and C2 in the first, connected by the eligible links T1–C1, T2–C1, and T2–C2. Its ratio is $$ \frac{m_T}{m_C}=\frac{2}{2}=1, $$ which lies between \(1/2\) and \(2\). Put T3 and C3 in the second subclass, connected by T3–C3. Its ratio is \(1/1=1\), also within bounds. Every included unit is assigned, and every link used is within the caliper.
The eligible link T3–C2 does not have to be selected in this particular assignment. Eligibility means a link may be used; it does not require every eligible link to be selected or every eligible pair in a subclass to be connected directly. This example demonstrates feasibility, not a claim that this construction is optimal for every matching objective.
Worked Example: A Caliper Can Make Ratio Constraints Infeasible
Consider three treated units, T1, T2, and T3, and two controls, C1 and C2. At caliper \(\delta=0.20\), suppose the only eligible links are T1–C1, T2–C1, T3–C1, and C2–T3. The graph is connected through T3–C1 and T3–C2. Now require each subclass to have at most two treated units per control, so \(m_T/m_C\leq2\).
Putting all five units in one connected subclass gives \(m_T/m_C=3/2=1.5\), which satisfies the bound. Thus this graph does allow a full assignment under the stated rule, even though the control C1 is linked to all three treated units. The ratio applies to the subclass as a whole; it is not a limit on the number of eligible edges incident to a control.
If the caliper were tightened to \(0.10\) and the only remaining links were T1–C1, T2–C1, and T3–C1, the single connected group would have ratio \(3/1=3\), above the limit. It cannot be split into ratio-compliant connected subclasses: there is only one control, and each subclass must contain a control. This is a design infeasibility under the combined caliper and ratio constraints, not a reason to add a forbidden link. The assignment must be reviewed rather than silently violating either restriction.
Why the Construction Works—and Where It Can Fail
The basic graph logic gives a short feasibility check. If an included unit has no eligible opposite-group neighbor, then no selected eligible edge can connect that unit into a subclass containing both treatment groups. If the graph separates into components, no eligible path joins units across those components. These statements follow directly from the edge definition: every permitted connection must be an edge in \(E_\delta\), and paths can use only such edges. Ratio limits impose a further check on each proposed connected subclass.
This check is necessary, not always sufficient for a particular constrained assignment. A component can contain both groups yet have no partition into connected subclasses satisfying all ratio limits or other design constraints. An optimizer can search over feasible assignments, but it cannot create a solution when the restrictions are contradictory. Keep feasibility separate from optimality: a feasible assignment satisfies the rules; an optimal assignment is best according to the specified objective among feasible assignments.
The caliper restriction is also distinct from the objective used to choose among eligible assignments. A matching procedure may minimize a distance-based cost, subject to the permitted links and structural constraints. The caliper defines which links are allowed; the objective ranks allowable assignments. If all eligible edges have identical status under a feasibility check, that does not mean they have identical distances or costs.
After constructing subclasses, inspect the resulting design rather than relying on the threshold alone. Useful diagnostics include the number of eligible links per unit, the distribution of selected-link distances, component sizes and treatment composition, and whether each subclass meets the ratio rules. Assess covariate balance after matching using the planned covariates and appropriate summaries. A caliper constrains the matching distance; it does not guarantee balance on every covariate, nor does it guarantee that the propensity-score model or causal assumptions are adequate.
A tighter caliper removes more edges, which may improve closeness among the links that remain but can also fragment the graph and make assignment less feasible. A looser caliper admits more options but permits more distant links. Compare plausible thresholds as a design sensitivity analysis, using prespecified criteria and without choosing a threshold to obtain a preferred outcome estimate. Record the distance definition, threshold, ratio constraints, assignment procedure, and any exclusions so the design can be reproduced.
Calipers Compared With Other Design Choices
| Design choice | What it restricts | What it does not ensure |
|---|---|---|
| Full matching without a caliper | Subclass structure and any stated ratio or assignment constraints | That every treated-control link is sufficiently close on the chosen distance scale |
| Caliper-constrained full matching | Which treated-control links may connect units, in addition to subclass constraints | That a feasible assignment exists or that all covariates are balanced |
| Pair matching with a caliper | Eligible links and a one-to-one pair structure | The flexibility of full matching to form subclasses with unequal group sizes |
| Exact matching | Whether units agree exactly on specified variables or categories | Closeness on other variables not included in the exact restriction |
These methods answer different design needs. Exact matching restricts agreement on specified variables; a caliper restricts a defined distance; pair matching imposes a pair structure; full matching allows larger subclasses. A caliper can be used with a full-matching design, but it should not be described as if it converted full matching into pair matching.
Common Mistakes
- Applying the caliper after assignment. The incorrect reasoning is to build subclasses first and then declare distant links unacceptable. The correction is to remove those links before construction and form subclasses from the eligible graph.
- Allowing a link exactly at the threshold to be ambiguous. State the rule explicitly. Under \(d_{ij}\leq\delta\), a distance equal to the caliper is eligible.
- Assuming every pair within a subclass must pass the caliper. A subclass can include multiple units linked through eligible paths; the restriction applies to the links used to connect the subclass, not automatically to every cross-pair.
- Checking only the overall treatment-control ratio. Ratio restrictions apply to each subclass when specified. Verify each one individually.
- Treating a caliper as proof of balance or causal validity. It limits a chosen matching distance. Balance and the assumptions needed for causal interpretation require separate assessment.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- With \(\delta=0.15\), is a treated-control link at distance 0.15 eligible under the rule in this tutorial?
- A subclass contains three treated units and two controls. What is its treated-control ratio, and does it satisfy bounds \(1/2\leq m_T/m_C\leq2\)?
- Two graph components have no eligible edge or path between them. Can a subclass use units from both components while connecting them only with eligible links?
- Explain why an eligible link need not be selected in a particular full-matching assignment.
- A caliper-constrained assignment has good balance on the propensity-score distance but poor balance on one important covariate. Does the caliper establish adequate balance on that covariate?
Solutions. (1) Yes. The rule is \(d_{ij}\leq\delta\), so equality is allowed. (2) The ratio is \(3/2=1.5\), which is between \(0.5\) and \(2\), so it satisfies the bounds. (3) No. Without an eligible path, the assignment cannot connect those units using permitted links. (4) Eligibility makes a link allowable, not mandatory; the assignment selects links needed for its feasible subclasses and objective. (5) No. The caliper restricts the chosen distance, not every covariate; balance on the important covariate must be assessed separately.