What Full Matching Describes
Matching creates groups of treated and control units to make comparisons more credible with respect to measured covariates. Full matching describes one possible structure for those groups: each matched set contains at least one treated unit and at least one control, and different sets may contain different numbers of each. The word “full” refers to the structure’s capacity to include units in mixed-treatment sets; it does not mean that the sets are all the same size or that the design automatically achieves good balance.
A matched set is a group defined by the matching design, not a claim that its members are identical. The set may contain one treated unit and several controls, several treated units and one control, or more than one unit from each group. Whether the covariates of the members are sufficiently similar depends on the matching criteria and the data—not merely on the fact that the grouping is a full match.
In this definition, a partition has two important properties. First, the sets do not overlap: a unit cannot be counted in two different sets. Second, their union is the full eligible sample: no eligible unit is left out. If a design intentionally excludes some units before defining the matched sample, the partition applies to the eligible units retained for matching. This tutorial uses \(\mathcal E\) to mean the units that the design intends to include.
For set \(S_k\), write \(n_{1k}\) for its number of treated units and \(n_{0k}\) for its number of controls. A set is valid precisely when \(n_{1k}\geq 1\) and \(n_{0k}\geq 1\). Its size is \(n_{1k}+n_{0k}\), and its treated-to-control ratio is \(n_{1k}:n_{0k}\). Stating which group is in the numerator matters: \(1:3\) means one treated unit for three controls, whereas \(3:1\) means three treated units for one control.
Reading the Set Structure
A full match can contain sets of several forms. A set with counts \((n_{1k},n_{0k})=(1,1)\) is a matched pair. A set with counts \((1,3)\) has one treated unit and three controls; a set with counts \((2,1)\) has two treated units and one control. These are all valid full-matching sets because each contains both treatment groups. The set ratio describes membership; it does not, by itself, specify an outcome-analysis method or an estimand.
The total counts across the sets must agree with the eligible sample. If there are \(N_1\) treated units and \(N_0\) controls in \(\mathcal E\), then a complete partition must satisfy:
These equalities are useful structural checks. They do not establish that matching was successful on covariates, but a failure of either equality shows that the proposed sets do not partition the intended eligible sample.
Worked Example: Checking a Small Full Match
Suppose a hypothetical eligible sample has three treated units, \(T_1,T_2,T_3\), and four controls, \(C_1,C_2,C_3,C_4\). A proposed grouping is \(S_1=\{T_1,C_1,C_2\}\), \(S_2=\{T_2,T_3,C_3\}\), and \(S_3=\{C_4,T_1\}\). Is it a full matching?
First check each set. \(S_1\) has counts \((n_{11},n_{01})=(1,2)\), and \(S_2\) has counts \((n_{12},n_{02})=(2,1)\); each includes both groups. \(S_3\) also includes both groups, but it repeats \(T_1\), which is already in \(S_1\). Thus the sets overlap, so the proposed grouping is not a partition.
A corrected grouping could be \(S_1=\{T_1,C_1,C_2\}\) and \(S_2=\{T_2,T_3,C_3,C_4\}\). Now the sets are disjoint, every eligible unit appears exactly once, and each set contains both groups. The set counts are \((1,2)\) and \((2,2)\), so the ratios are \(1:2\) and \(1:1\), respectively.
The totals verify the correction: \(1+2=3\) treated units and \(2+2=4\) controls. The grouping is structurally a full match. That conclusion does not tell us whether the members are close on relevant covariates.
Set membership also determines how many treated-control comparisons are available within a set. If every treated member is paired conceptually with every control member in that set, then set \(S_k\) contains \(n_{1k}n_{0k}\) cross-group pairings. This count follows because each of the \(n_{1k}\) treated units can be combined with each of the \(n_{0k}\) controls. Summing over sets gives:
This is a count of possible within-set comparisons, not a count of distinct matched sets and not necessarily the number of independent observations in a later analysis. For example, one treated unit in a set with four controls yields four possible treated-control pairings, but it is still one matched set.
Worked Example: Ratios and Within-Set Pairings
Consider a hypothetical full match with three sets. Set \(S_1\) contains one treated unit and two controls; \(S_2\) contains two treated units and one control; \(S_3\) contains two treated units and three controls. Find the total numbers of units in each group and the total number of within-set treated-control pairings.
The treated count is \(1+2+2=5\), and the control count is \(2+1+3=6\). Each set has both groups, so the set compositions satisfy the mixed-treatment requirement.
The pairings within the sets are \(1\times2=2\), \(2\times1=2\), and \(2\times3=6\). Therefore the total is \(2+2+6=10\) within-set treated-control pairings. This calculation does not turn the design into ten matched pairs: there are three matched sets, and units can contribute to multiple possible pairings within a set.
The ratios are \(1:2\), \(2:1\), and \(2:3\). Their variation is permitted by full matching. Reporting only a single overall ratio would conceal that different sets have different compositions.
Why Set Size Can Vary
Fixed pair matching requires each set to contain one treated unit and one control. Variable-ratio matching permits more than one control per treated unit under its specified structure. Full matching is more general in the composition it permits: a set may have multiple treated units, multiple controls, or both, while still requiring representation from each group. The distinction is about the allowable grouping structure, not a ranking in which one method is always preferable.
| Structure | Allowed set composition | What to check |
|---|---|---|
| Pair matching | One treated unit and one control per set | Every set has counts \((1,1)\) |
| Variable-ratio matching | Set sizes can vary according to the design, commonly with one treated unit and multiple controls | Each set meets the design’s specified ratio rules |
| Full matching | At least one treated unit and one control per set; counts may vary in either group | Each set is mixed, the sets are disjoint, and all intended units are included |
A variable set structure can accommodate samples in which the numbers of treated and control units differ, and it can represent more than one treated unit in a set. But flexibility is not a guarantee of balance. A set may be structurally valid and still bring together units that differ substantially on covariates. In applications, a distance measure and restrictions may guide which groupings are acceptable; the definition of full matching does not prescribe a specific distance measure or allocation algorithm.
The causal interpretation of any later comparison also depends on design and identification assumptions, not on the set format alone. In particular, measured-covariate similarity does not ensure that treated and control units are comparable on unmeasured factors. The earlier material on causal inference supplies the broader assumptions needed for causal claims. Here, the structural claim is narrower: a full matching is a particular way to group eligible treated and control units.
Worked Example: Separating Structural Validity From Similarity
Suppose a hypothetical eligible sample contains treated units \(T_1,T_2\) and controls \(C_1,C_2,C_3\). A proposed full match is \(S_1=\{T_1,C_1,C_2\}\) and \(S_2=\{T_2,C_3\}\). A measured covariate \(X\) has values \(X(T_1)=10\), \(X(C_1)=11\), \(X(C_2)=12\), \(X(T_2)=40\), and \(X(C_3)=41\). Verify the structure and describe what the covariate values show.
The first set has counts \((1,2)\), and the second has counts \((1,1)\). The sets do not overlap, and together they contain both treated units and all three controls exactly once. Thus the proposed grouping is a full match.
Within \(S_1\), the control values differ from the treated value by \(11-10=1\) and \(12-10=2\) units of \(X\). Within \(S_2\), the difference is \(41-40=1\) unit. These values are close within each set on this covariate, but this single-covariate check is not a general proof of balance: other measured covariates may differ, and unmeasured covariates are not assessed by these numbers. The example illustrates why structural validity and covariate similarity are separate questions.
Diagnostics, Assumptions, and Common Errors
Before interpreting a full-matching design, check its structure directly. List the treated and control counts for every set, confirm that neither count is zero, and verify that each intended unit appears once and only once. Then compare the total counts with the eligible sample. Separately examine the covariate balance and distances relevant to the design. A correct partition can still be a poor match if the members of a set are not sufficiently comparable for the intended analysis.
The definition itself assumes that treatment status is known for each unit and that the eligible sample has been specified. If the intended sample contains at least one treated unit and one control, a mixed set can be formed when there are no additional restrictions on which units may be grouped together. If the design imposes compatibility rules, however, the existence of a valid partition is not automatic; the rules may prevent some units from sharing sets. A full-matching structure is not a license to ignore such restrictions.
- Putting only one treatment group in a set. A group containing only controls is not a full-matching set, even if other sets contain treated units.
- Allowing a unit to appear more than once. Reusing a unit in multiple sets violates the partition definition unless a different design explicitly allows reuse.
- Calling every valid partition a good match. Mixed-treatment sets satisfy the structural definition, but balance and the suitability of comparisons still need to be assessed.
- Reversing the ratio. Always report treated count first when using the treated-to-control convention: \(2:1\) means two treated units for each control in that set.
- Equating possible pairings with sets. A set with two treated units and three controls has six possible cross-group pairings but remains one matched set.
The main limitation of the definition is that it says how units are grouped, not how the grouping should be chosen or how outcomes should be analyzed. It does not select an estimand, supply weights, determine uncertainty estimates, or establish causal identification. Those choices must be justified separately. The practical value of understanding the structure is that it makes the matched sample auditable: one can see who belongs to each comparison group and whether the design’s bookkeeping is correct.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A set contains three treated units and no controls. Can it be one set in a full matching? Why?
- A proposed match has sets with counts \((1,2)\), \((2,1)\), and \((1,1)\). How many treated units and controls are included, and how many within-set treated-control pairings are possible?
- In a set with two treated units and five controls, what is the treated-to-control ratio, and how many possible cross-group pairings are there?
- Why does verifying that each set has both treatment groups not establish that the design has good covariate balance?
Solutions. (1) No. A full-matching set must contain at least one treated unit and at least one control. (2) There are \(1+2+1=4\) treated units and \(2+1+1=4\) controls. The possible within-set pairings are \((1\times2)+(2\times1)+(1\times1)=2+2+1=5\). (3) The ratio is \(2:5\); the number of possible cross-group pairings is \(2\times5=10\). (4) Mixed membership is a structural condition only. Covariate similarity must be assessed separately, and even good measured-covariate balance does not by itself address unmeasured differences.