From a Distance Matrix to Matched Sets
A full-matching allocation is a collection of matched sets, also called subclasses. Each set contains at least one treated unit and at least one control. Unlike pair matching, a set need not contain exactly one unit from each group: it can have one treated unit and several controls, several treated units and one control, or multiple units of both types.
A distance matrix gives the treated-control distance for each possible pair. A smaller distance indicates greater closeness under the chosen distance measure. The matrix helps make and assess allocation choices, but it is not itself the allocation: the analyst must specify which units belong together. The examples below use hypothetical distances and illustrate feasible allocations; they do not claim that a particular allocation is uniquely optimal.
For set \(S_k\), let \(n_{Tk}\) be its treated count and \(n_{Ck}\) its control count. The set’s composition is reported as \((n_{Tk},n_{Ck})\), or equivalently as the treated-to-control count \(n_{Tk}:n_{Ck}\). This count describes the set’s membership; it is not a distance and does not by itself summarize the quality of the matches.
Worked Allocation With One Treated Unit per Set
Worked Example: Assigning Controls to Treated Units
Suppose a hypothetical distance matrix contains three treated units and four controls. Each entry is a distance, with smaller values indicating closer pairs.
| C1 | C2 | C3 | C4 | |
|---|---|---|---|---|
| T1 | 1 | 2 | 8 | 9 |
| T2 | 7 | 3 | 2 | 8 |
| T3 | 8 | 7 | 4 | 1 |
One feasible allocation groups \(T1\) with \(C1,C2\), groups \(T2\) with \(C3\), and groups \(T3\) with \(C4\). The selected distances are \(d_{11}=1\), \(d_{12}=2\), \(d_{23}=2\), and \(d_{34}=1\). These entries show why the proposed groups are plausible under the displayed distances; they do not establish that the allocation is optimal under every possible objective.
The set counts follow directly from the membership:
Set 1: \(S_1=\{T1,C1,C2\}\). There is one treated unit and two controls, so \(n_{T1}=1\), \(n_{C1}=2\), and the count is \(1:2\).
Set 2: \(S_2=\{T2,C3\}\). There is one treated unit and one control, so \(n_{T2}=1\), \(n_{C2}=1\), and the count is \(1:1\).
Set 3: \(S_3=\{T3,C4\}\). There is one treated unit and one control, so \(n_{T3}=1\), \(n_{C3}=1\), and the count is \(1:1\).
As a coverage check, the treated counts add to \(1+1+1=3\), the number of treated units in the matrix. The control counts add to \(2+1+1=4\), the number of controls. No unit appears in more than one set, and every set contains both treatment groups.
Allocations Can Also Group Treated Units
Full matching is not limited to putting one treated unit in every set. A set may instead contain multiple treated units and one control. The matrix still supplies pairwise distances, while the proposed membership determines which distances are relevant to checking that set.
Worked Example: Two Treated Units With One Control
Consider this hypothetical matrix with four treated units and three controls.
| C1 | C2 | C3 | |
|---|---|---|---|
| T1 | 1 | 6 | 7 |
| T2 | 2 | 5 | 8 |
| T3 | 7 | 1 | 5 |
| T4 | 8 | 2 | 4 |
A feasible allocation is \(S_1=\{T1,T2,C1\}\), \(S_2=\{T3,C2\}\), and \(S_3=\{T4,C3\}\). In \(S_1\), the two treated-control distances are \(d_{11}=1\) and \(d_{21}=2\). The other sets use \(d_{32}=1\) and \(d_{43}=4\), respectively.
For \(S_1\), count the membership rather than the number of matrix entries: it contains two treated units and one control, so \((n_{T1},n_{C1})=(2,1)\), a \(2:1\) set. For \(S_2\), the counts are \((1,1)\), and for \(S_3\), they are also \((1,1)\).
The total treated count is \(2+1+1=4\), and the total control count is \(1+1+1=3\). All seven units are used once. This confirms the allocation’s coverage and set feasibility. It does not show that these particular groups are best for a study: that judgment also depends on the distance definition, the allocation objective, and whether the resulting matches are substantively acceptable.
Counting Rules and a Basic Verification
The count checks can be written generally. Let \(I(i,k)=1\) if treated unit \(T_i\) belongs to set \(S_k\), and \(I(i,k)=0\) otherwise. Define \(J(j,k)\) in the same way for control unit \(C_j\). If there are \(N_T\) treated units and \(N_C\) controls, then
For a complete allocation, every treated unit belongs to exactly one set, so \(\sum_{k=1}^{K} I(i,k)=1\) for every \(i\). Summing the set counts and changing the order of summation gives
The same argument gives \(\sum_{k=1}^{K} n_{Ck}=N_C\). Thus the two total-count identities are necessary checks for a complete allocation. They are not sufficient on their own: an allocation could have the right totals but still place two units in the same set twice while omitting other units. Verify membership as well as totals, and check that each set has positive counts on both sides.
When Both Groups Appear Multiple Times in a Set
Worked Example: A Set With Multiple Units on Each Side
Suppose a hypothetical matrix has three treated units and four controls:
| C1 | C2 | C3 | C4 | |
|---|---|---|---|---|
| T1 | 1 | 2 | 9 | 8 |
| T2 | 7 | 8 | 2 | 3 |
| T3 | 8 | 7 | 3 | 1 |
Consider \(S_1=\{T1,C1,C2\}\) and \(S_2=\{T2,T3,C3,C4\}\). The distances associated with \(S_1\) are \(d_{11}=1\) and \(d_{12}=2\). Those associated with \(S_2\) are \(d_{23}=2\), \(d_{24}=3\), \(d_{33}=3\), and \(d_{34}=1\).
For \(S_1\), there is one treated unit and two controls, so its count is \((1,2)\). For \(S_2\), there are two treated units and two controls, so its count is \((2,2)\). The count \(2:2\) describes membership; it does not mean that each treated unit has been separately paired with exactly one control.
The treated total is \(1+2=3\), and the control total is \(2+2=4\). Each unit appears exactly once, and both sets include at least one treated unit and one control. The arithmetic establishes that this is a valid full-matching allocation under the stated membership rules.
Diagnostics, Comparisons, and Limitations
A practical allocation check has three parts: confirm the membership of every set, count each treatment group within each set, and verify the overall totals. Then inspect the selected distances in the matrix. A set can be formally feasible yet contain a treated-control relationship that is not acceptable for the study. A count table alone cannot reveal that problem.
- Missing unit: If the set counts sum to fewer than the numbers of units available, inspect the membership list for an omitted unit.
- Repeated unit: A correct total can conceal reuse or duplication. Check that every unit appears in exactly one set for the allocations described here.
- One-sided set: A set containing treated units only or controls only is not a matched set under the definition used here.
- Confusing counts and distances: A \(1:2\) set has one treated unit and two controls. The count says nothing by itself about how close their covariates are.
- Overreading small distances: Distances are meaningful only relative to the variables, scaling, and distance measure used to construct them. A small numerical value is not automatically a guarantee of good covariate balance.
Pair matching imposes the simpler structure of one treated unit with one control, while variable-ratio matching permits a treated unit to be matched with differing numbers of controls. Full matching allows either treatment group to have multiple members within a set. This flexibility can use units that a pair-only structure would leave out, but it also means the allocation and its set counts must be reported clearly. It does not, by itself, guarantee good balance, remove bias from unmeasured confounding, or establish a causal effect.
The examples use allocations with no additional ratio restrictions. If some selected distances are unacceptable, or if a design requires limits on set composition, the allocation must be reconsidered under those requirements; simply calculating counts does not enforce them. Keep the distinction clear: feasibility asks whether the sets satisfy the stated membership rules, while optimality and substantive adequacy require further criteria and assessment.
Check Your Understanding
Work through the questions, then compare your reasoning with the solutions.
- A proposed set contains \(T1,T2,C1\). What are its treated and control counts?
- An allocation has three sets with treated counts \(1,2,1\) and control counts \(2,1,2\). How many treated units and controls does it include in total?
- Why does a total control count equal to the number of controls not, by itself, prove that every control was assigned exactly once?
- Can a set containing two treated units and two controls be a full-matching set under the definition used here? State its count.
Solutions. (1) There are two treated units and one control, so the count is \(2:1\). (2) The treated total is \(1+2+1=4\); the control total is \(2+1+2=5\). These totals check coverage only if the membership lists also show no omissions or duplicates. (3) One control could be repeated while another is omitted, leaving the total unchanged; check each unit’s membership directly. (4) Yes. Both groups are represented, and the count is \(2:2\).