Why Allow the Number of Controls to Vary?
In fixed-ratio matching, each retained treated unit receives the same number of controls. That rule is simple, but it can be restrictive: some treated units may have several close eligible controls, while others have only one. Variable-ratio matching allows the number of controls to differ across treated units, within stated lower and upper limits. The goal is to use available controls without requiring every matched set to have an identical size.
More controls can contribute information to a treated unit’s comparison, but they are useful only when they are suitable matches. Requiring a treated unit to take several distant controls can weaken covariate comparability. Conversely, imposing a high minimum may leave treated units unmatched when eligible controls are scarce. Variable-ratio matching therefore makes the ratio limits and the rule for allocating controls part of the design, rather than treating them as incidental bookkeeping.
The ratio describes controls per treated unit, not treated units per control. When the design also uses a caliper, a control is eligible for a treated unit only if its distance is within the caliper; the selected controls must meet both the eligibility restriction and the ratio limits. The distance measure and its scale must be specified. For example, one might use absolute differences in propensity scores, \(d(i,j)=|e(X_i)-e(X_j)|\), or distances on the logit propensity score. The matching algorithm then uses eligible controls to construct the sets.
Feasibility Before Allocation
For \(m\) treated units that are all intended to be retained, the ratio limits require at least \(mL\) control assignments and allow at most \(mU\). With matching without replacement, if there are \(N_C\) available controls, then \(mL\leq N_C\) is necessary. It is not sufficient: a control may be eligible for only some treated units, so the required assignments may still be impossible. Feasibility must be checked against the actual eligible matches, not just the total sample size.
A practical design first identifies which treated-control pairs are eligible, then secures the minimum \(L\) for each retained treated unit, and only then considers adding controls up to \(U\). This priority helps prevent extra assignments to one set from consuming controls needed to satisfy another set’s minimum. The assignment order and tie-breaking rule should be stated, because a greedy procedure can produce different sets under different orders. If the minima cannot all be met, the design must specify whether to leave some treated units unmatched, revise a restriction, or use a different design. It should not silently label an undersized set as satisfying the original rule.
Worked Example: Assigning Two or Three Controls Per Treated Unit
Suppose three treated units have propensity scores \(0.20\), \(0.50\), and \(0.80\). Seven controls have scores \(0.18\), \(0.24\), \(0.45\), \(0.53\), \(0.76\), \(0.83\), and \(0.88\). Use absolute propensity-score distance, a caliper of \(0.10\), matching without replacement, and ratio limits \(L=1\), \(U=3\). For clarity, assign the closest eligible controls to each treated unit.
For the treated unit with score \(0.20\), the eligible control distances are \(|0.20-0.18|=0.02\) and \(|0.20-0.24|=0.04\). Both controls are within the caliper, so this set receives two controls.
For the treated unit with score \(0.50\), the distances to scores \(0.45\) and \(0.53\) are \(0.05\) and \(0.03\). Both are eligible, so it also receives two controls.
For the treated unit with score \(0.80\), the distances to \(0.76\), \(0.83\), and \(0.88\) are \(0.04\), \(0.03\), and \(0.08\). All three are eligible, so it receives three controls.
The resulting control counts are \(k_1=2\), \(k_2=2\), and \(k_3=3\). Each satisfies \(1\leq k_i\leq3\), and no control is reused. The unused-control count is \(7-(2+2+3)=0\). This allocation meets the stated limits, although in a larger problem the assignment choices would need to be coordinated to avoid using a control needed by another treated unit.
Construction and a Feasibility Check
A transparent variable-ratio procedure separates required matches from optional additional matches. The following outline assumes matching without replacement and a specified eligibility rule, such as a caliper. It describes a design strategy, not a guarantee that every allocation order will find the best possible assignment.
Specify \(L\), \(U\), the distance measure, any eligibility restriction, whether controls may be reused, and how ties or processing order will be handled.
For each treated unit, identify all controls that satisfy the stated restrictions. A control outside a caliper is not a candidate, no matter how useful it would be for reaching the minimum.
Allocate at least \(L\) eligible controls to each treated unit intended for retention, avoiding reuse if matching without replacement. If this cannot be done for every treated unit, identify the infeasible units rather than obscuring the failure.
Among remaining eligible controls, add the closest candidates according to the chosen rule, stopping when each set has \(U\) controls or no suitable control remains.
Check each count \(k_i\), confirm that every assigned pair is eligible, verify that controls are not duplicated when prohibited, and report treated units left unmatched and controls left unused.
The minimum-first step is important because optional additions can use up scarce controls. Even minimum-first allocation may depend on ordering: a treated unit with only one eligible control may need priority over a treated unit with several choices. A procedure should state how it handles that situation. Feasibility is a property of the entire assignment problem; checking only whether each treated unit has at least \(L\) possible controls can be misleading when those possible controls overlap.
Worked Example: Enough Controls Overall, but Not Enough for Every Minimum
Consider three treated units, ratio limits \(L=1\), \(U=2\), and matching without replacement. There are two controls, \(C_1\) and \(C_2\). Under the stated caliper, treated units \(T_1\) and \(T_2\) can each match only to \(C_1\), while \(T_3\) can match only to \(C_2\).
The overall control count is \(N_C=2\), whereas satisfying all three minima would require \(mL=3(1)=3\) control assignments. Thus \(mL\leq N_C\) fails, and retaining all three treated units is impossible. The overlap makes the shortage explicit: only one of \(T_1\) and \(T_2\) can receive \(C_1\), and only \(T_3\) can receive \(C_2\).
For example, assigning \(C_1\) to \(T_1\) and \(C_2\) to \(T_3\) gives \(k_1=1\), \(k_2=0\), and \(k_3=1\). The second treated unit is unmatched; its set does not meet the minimum. Assigning \(C_1\) to \(T_2\) instead leaves \(T_1\) unmatched. Reporting the actual unmatched unit or units is essential. The fact that each treated unit has at least one eligible control considered in isolation does not make a simultaneous no-replacement assignment feasible.
If instead three distinct controls were available overall, the simple count condition would hold, but the assignment could still fail if all three treated units depended on the same small subset of controls. The eligible-pair structure must therefore be checked in addition to the total count.
What the Ratio Changes in an Outcome Comparison
Variable-ratio matching also changes how matched sets contribute to a treatment-effect summary. For a continuous outcome \(Y\), let \(Y_i\) be the observed outcome for treated unit \(i\), and let \(Y_{ij}\) be the outcome for control \(j\) assigned to its set \(C_i\). A common treated-unit-weighted matched contrast is
where \(M\) is the number of retained treated units. Each treated unit contributes one set-level treated-minus-control difference, regardless of whether its set has one control or several. This matches an average across retained treated units, rather than allowing treated units with more controls to count more heavily merely because their sets are larger. The contrast is not automatically a causal effect: a causal interpretation requires appropriate design and identification assumptions, including no unmeasured confounding conditional on the covariates used for matching, as well as adequate overlap for the retained units.
Worked Example: Averaging Set-Level Differences
Suppose three retained treated units have outcomes \(10\), \(12\), and \(9\) units. Their matched controls have outcomes \((8,9)\), \((10,11,13)\), and \((8)\), respectively. The control counts are \(2\), \(3\), and \(1\), so the sets have variable ratios within limits \(1\) to \(3\).
The first set’s difference is \(\displaystyle 10-\frac{8+9}{2}=10-8.5=1.5\) units. The second is \(\displaystyle 12-\frac{10+11+13}{3}=12-\frac{34}{3}=0.6667\) units. The third is \(9-8=1\) unit.
The treated-unit-weighted estimate is \(\displaystyle \widehat{\tau}_{\mathrm{ATT}}=\frac{1.5+0.6667+1}{3}=1.056\) units. This calculation gives equal weight to each of the three treated units, while averaging the controls within each set. Pooling all six controls into one overall control mean would answer a different weighting question; it would not reproduce this set-level estimator.
Diagnostics, Trade-Offs, and Common Mistakes
After matching, check the design rather than assuming that the ratio limits guarantee good matches. Useful diagnostics include the number of controls per treated unit, the distribution of within-set distances, the number of unused controls, and the number and characteristics of unmatched treated units. Also assess covariate balance in the matched sample. A design can satisfy every ratio constraint yet have poor covariate balance, especially if the upper ratio encourages adding controls that are eligible but relatively distant.
The main contrast with fixed-ratio matching is flexibility: fixed-ratio matching uses the same \(k_i\) for every retained treated unit, whereas variable-ratio matching permits \(k_i\) to vary between \(L\) and \(U\). A fixed 1:1 rule may discard useful additional controls; a fixed 1:3 rule may be infeasible for treated units with few eligible controls. Variable ratios can accommodate such differences, but they require a clear allocation rule and careful reporting. They do not remove the need to justify the matching distance, eligibility restriction, or choice of retained treated units.
- Choosing a large maximum and assuming it must be better. Additional controls help only if they are suitable. A larger \(U\) may add distant controls, worsen balance, or use controls needed elsewhere.
- Checking only the total number of controls. \(mL\leq N_C\) is necessary for an all-treated, no-replacement design, but overlapping eligibility can still make the assignment infeasible.
- Calling a set complete when it misses its minimum. If \(k_i<L\), that treated unit has not met the specified rule. Report it as unmatched or explain a prespecified change in design.
- Pooling controls without considering the target weighting. Pooling can give more influence to treated units with larger sets. State whether the analysis weights sets equally or uses another justified weighting scheme.
Check Your Understanding
Answer each question, then compare your reasoning with the brief solutions.
- Four treated units are intended for retention under a 1:1-to-1:2 design without replacement. What is the minimum number of control assignments required, and what simple necessary condition does the available control count have to meet?
- Why is it not enough to show that every treated unit has at least one eligible control when using matching without replacement?
- For a set with a treated outcome of 15 and control outcomes 12 and 14, calculate the set-level treated-minus-control difference.
- Name two diagnostics that can reveal a variable-ratio design is not performing well.
Solutions. (1) The minimum is \(4(1)=4\) assignments. At least four controls must be available, so the necessary count condition is \(mL\leq N_C\); it does not guarantee feasibility under the eligibility restrictions. (2) Different treated units may depend on the same single control, which cannot be assigned to both sets without replacement. (3) The control mean is \((12+14)/2=13\), so the difference is \(15-13=2\) outcome units. (4) Examples include poor covariate balance, many treated units left unmatched, large within-set distances, or an unexpected pattern of unused controls.