Why Matching Structure Matters
A matched data set is organized into pairs or sets of treated and control units. The structure determines how many units each set can contain and whether a unit can appear in more than one set. Those rules constrain which allocations are possible: a ratio that looks suitable on paper may be impossible with the available numbers of treated and control units, or once unacceptable matches are ruled out.
The examples below use \(Z=1\) for a treated unit and \(Z=0\) for a control unit. A matched set contains units assigned to the same set for comparison. For the common hypothetical example, an allowed link means that the treated and control units are close enough under a stipulated caliper. The caliper is a maximum permitted distance; a link outside it cannot be used. The particular distances are illustrative, not data from a real study.
Four Structures and Their Unit-Use Rules
In one-to-one matching, every matched set is a pair: one treated unit and one control unit. Under the usual without-replacement rule, each retained unit appears in exactly one pair, although some original units may be left unmatched. One-to-one matching is the \(1:1\) fixed-ratio case.
In fixed-ratio matching, each set has the same prespecified numbers of treated and control units. For example, a \(1:2\) set contains one treated unit and two controls; a \(2:1\) set contains two treated units and one control. When matching is without replacement, every retained unit is assigned to at most one set. A fixed ratio does not by itself guarantee that all units can be used: both the available counts and the allowed links matter.
In variable-ratio matching, the number of units in a set may vary within specified limits. A common form gives each retained treated unit one or more controls, with the number of controls varying across treated units. The sets still obey the chosen unit-use rule; in the examples here, each control is used at most once. “Variable ratio” means set sizes can differ, not that units are automatically reused or that any set size is permitted.
In full matching, each matched set contains at least one treated unit and at least one control unit. A set may contain one treated unit and several controls, several treated units and one control, or multiple units from both groups. Among units included in the full match, each unit belongs to one set. “Full” describes the flexible set structure; it does not guarantee that every unit in the original pool must be retained if constraints or the absence of feasible partners require exclusions.
| Structure | Typical set composition | Unit use in the examples | Key restriction |
|---|---|---|---|
| One-to-one | 1 treated : 1 control | At most one set per unit | Every retained unit has a distinct partner |
| Fixed ratio | The same \(a:b\) treated-to-control ratio in every set | At most one set per unit | Counts must support the ratio in every set |
| Variable ratio | Set ratios vary within specified bounds | At most one set per unit | Set sizes may differ but must meet the bounds |
| Full matching | Any set with at least one unit from each group | Each included unit belongs to one set | Every set must contain both groups |
A Shared Hypothetical Pool
Suppose a hypothetical pool contains four treated units, \(T_1,T_2,T_3,T_4\), and six controls, \(C_1,\ldots,C_6\). The table gives a distance for each treated-control pair. Only distances no greater than \(0.10\) are allowed by the illustrative caliper. A dash means that the distance is greater than \(0.10\), so that link is unavailable.
| Treated unit | \(C_1\) | \(C_2\) | \(C_3\) | \(C_4\) | \(C_5\) | \(C_6\) |
|---|---|---|---|---|---|---|
| \(T_1\) | 0.02 | 0.05 | — | — | — | — |
| \(T_2\) | 0.03 | 0.04 | 0.08 | 0.09 | — | — |
| \(T_3\) | — | — | 0.02 | 0.03 | 0.06 | — |
| \(T_4\) | — | — | — | 0.02 | 0.04 | 0.07 |
For example, \(T_2\) can be linked to \(C_4\) because the distance is \(0.09\), but \(T_1\) cannot be linked to \(C_3\) under this caliper. Feasibility is therefore more than a matter of counting: a proposed allocation must use distinct units when required and every assigned link must be allowed.
Worked Example: One-to-One Matching
Construct a one-to-one allocation that retains all four treated units, uses each control at most once, and respects the \(0.10\) caliper.
One feasible allocation is \((T_1,C_1)\), \((T_2,C_2)\), \((T_3,C_3)\), and \((T_4,C_4)\). The four distances are \(0.02\), \(0.04\), \(0.02\), and \(0.02\), respectively, and each is no greater than \(0.10\). No control appears twice.
The sum of the displayed pair distances is \(0.02+0.04+0.02+0.02=0.10\), and their mean is \(0.10/4=0.025\). These calculations describe this allocation’s distances; they do not establish that it is the unique or best possible allocation. Controls \(C_5\) and \(C_6\) are unused. One-to-one matching can retain all treated units here without retaining every control.
Count Feasibility and Fixed Ratios
A simple count gives an initial feasibility check for a fixed ratio. Suppose each set contains one treated unit and \(r\) controls, with matching without replacement. If \(q\) sets are formed, they use \(q\) treated units and \(rq\) controls. With \(n_1\) treated units and \(n_0\) controls available, the counts must satisfy \(q\leq n_1\) and \(rq\leq n_0\).
This is a necessary count bound, not a guarantee of feasible matching. Even when enough controls exist, the caliper-compatible links may not allow the required allocation. For a general fixed ratio of \(a\) treated units to \(b\) controls per set, \(q\) sets use \(aq\) treated units and \(bq\) controls; both groups must have enough units, and feasible cross-group links must also exist.
Worked Example: Fixed One-to-Two Matching
Can all four treated units in the shared pool be placed into \(1:2\) sets without replacement? If not, give a feasible allocation with as many treated units as the counts permit.
Four such sets would require \(2(4)=8\) distinct controls, but only six are available. Thus, even before checking the caliper, all four treated units cannot be retained in \(1:2\) sets without replacement. The count bound is \(q\leq\min(4,\lfloor6/2\rfloor)=3\).
Three sets can be formed as \(\{T_1,C_1,C_2\}\), \(\{T_3,C_3,C_5\}\), and \(\{T_4,C_4,C_6\}\). The required distances are \(0.02,0.05,0.02,0.06,0.02,\) and \(0.07\); all are within the caliper. The allocation uses six distinct controls and three treated units, leaving \(T_2\) unmatched. It reaches the count bound, so no \(1:2\) allocation without replacement can retain more than three treated units in this pool.
Variable-Ratio and Full-Matching Allocations
Allowing set sizes to vary can make it possible to retain units that a fixed ratio would leave unused. The permitted range still matters: if every treated unit must have at least one control, each treated unit needs a distinct available control under matching without replacement. A lower bound on controls per treated unit, an upper bound on controls per set, and any caliper links all constrain feasibility.
Full matching permits a broader range of set compositions than the common variable-ratio form in which each set has exactly one treated unit. In particular, a full-matching set may include more than one treated unit, provided it also includes at least one control. The total ratio across the whole matched sample does not need to equal the ratio in each individual set.
Worked Example: Variable-Ratio Sets
Form variable-ratio sets that retain all four treated units and use all six controls once, with each treated unit receiving at least one control and no set receiving more than two.
One feasible allocation is \(\{T_1,C_1\}\), \(\{T_2,C_2,C_3\}\), \(\{T_3,C_4\}\), and \(\{T_4,C_5,C_6\}\). The respective treated-to-control ratios are \(1:1\), \(1:2\), \(1:1\), and \(1:2\). The distances used are \(0.02\), \(0.04\), \(0.08\), \(0.03\), \(0.04\), and \(0.07\). Each is at most \(0.10\), and every unit appears in exactly one set.
The total is four treated units and six controls, but the sets do not all have the same ratio. This illustrates the structural difference from fixed \(1:2\) matching: that fixed rule could form at most three sets here, whereas this variable-ratio allocation retains all four treated units by giving some of them one control rather than two.
Worked Example: A Full-Matching Allocation
Show how full matching can use a set containing multiple treated units while retaining every unit in the shared pool.
Consider the sets \(\{T_1,C_1,C_2\}\), \(\{T_2,T_3,C_3,C_4\}\), and \(\{T_4,C_5,C_6\}\). The middle set contains two treated units and two controls; it is allowed as a full-matching set because it contains at least one unit from each group.
Every treated-control link within these proposed sets satisfies the caliper: for the first set the distances are \(0.02\) and \(0.05\); for the middle set they are \(0.08,0.09,0.02,\) and \(0.03\); and for the last set they are \(0.04\) and \(0.07\). Thus all eight links are at most \(0.10\). The sets use \(T_1,T_2,T_3,T_4\) and \(C_1,\ldots,C_6\) exactly once. Their treated-to-control ratios are \(1:2\), \(2:2\), and \(1:2\), so the ratio varies across sets.
What the Structures Do—and Do Not—Guarantee
Matching structures organize units; they do not, by themselves, establish that treated and control units are comparable on all relevant characteristics or that a causal effect is identified. Those conclusions depend on the assumptions and design features of the study, not just on whether sets are pairs or larger subclasses. The introductory material on full matching covers its matched-set construction; the comparison here concerns how its set rules differ from the other structures.
The unit-use rule matters for describing the matched sample. Under without-replacement matching, each retained unit contributes to one set. If a procedure instead matches with replacement, a control may be assigned to more than one treated unit. That changes the allocation: the same control is reused, rather than several distinct controls being added. A list of matched pairs can then contain multiple appearances of one control, so counting pair records is not the same as counting distinct people.
Excluding unmatched units also changes which units the allocation contains. For instance, the fixed \(1:2\) example retained three of four treated units, while the variable-ratio example retained all four. The structural difference alone does not show that one retained sample is more appropriate for a particular research question. It does show why the matching rule and the actual retained units should be reported clearly.
Diagnostics and Common Pitfalls
Before treating an allocation as feasible, check its structure directly. Count the treated and control units in every set, confirm the required ratio or allowed range, verify that no unit is reused when matching is without replacement, and check every assigned link against the caliper. Then count distinct retained units by treatment group and list those left unmatched. A global count can look acceptable while one set violates the fixed ratio or contains a prohibited link.
- Confusing a feasible count with a feasible allocation. Having six controls is enough by count for three \(1:2\) sets, but only an assignment with permitted links and distinct controls is a valid allocation.
- Treating “full” as “every original unit must be retained.” Full matching allows flexible mixed-treatment sets; it does not override calipers or guarantee a partner for every unit.
- Assuming variable-ratio sets reuse controls. Variable set sizes do not imply replacement. State separately whether units may appear in multiple sets.
- Reading the overall ratio as each set’s ratio. Full matching can have different ratios in different sets even when the overall treated-to-control count is fixed.
Check Your Understanding
Answer each question using the definitions and hypothetical pool in this tutorial. Solutions follow the questions.
- With five treated units and eleven controls, what is the maximum number of \(1:2\) sets possible without replacement by counts alone?
- In one sentence, distinguish variable-ratio matching from full matching in terms of set composition.
- In the shared pool, does the fixed \(1:2\) allocation in the worked example use all four treated units? Give the count reason.
- Why is a set containing two treated units and two controls allowed under full matching?
Solutions. (1) The count bound is \(\min(5,\lfloor11/2\rfloor)=\min(5,5)=5\) sets; this establishes count feasibility only, not feasible links. (2) Variable-ratio matching permits set sizes to vary within its specified rules, while full matching permits any set composition with at least one treated and one control unit. (3) No: four \(1:2\) sets need eight distinct controls, and only six are available, so at most \(\lfloor6/2\rfloor=3\) treated units can be retained. (4) Full matching requires at least one unit from each group in every set; it does not require exactly one treated unit or a fixed ratio in each set.