Restricting Treated-Control Ratios in Full Matching

Why Restrict Ratios Within Matched Sets?

A full-matching design groups treated and control units into subclasses, with at least one unit from each group in every subclass. The numbers of treated and control units can vary from set to set. That flexibility is useful, but it also means one subclass might contain many treated units and very few controls, while another contains the opposite configuration. A ratio restriction places an explicit bound on this variation.

The restriction is a rule about counts within each subclass. It is not a requirement that the entire matched sample have a particular ratio, and it does not by itself say whether units within a set are close in propensity score or covariates. Those are distinct design considerations.

Definition: For a full-matching subclass \(s\), let \(n_{Ts}\) be its number of treated units and \(n_{Cs}\) its number of control units. Its treated-to-control ratio is \(r_s=n_{Ts}/n_{Cs}\). Given lower and upper limits \(L\) and \(U\), where \(0<L\le U\), the subclass satisfies the ratio restriction if and only if \(L\le r_s\le U\). The limits are inclusive unless the design specification explicitly says otherwise.

This tutorial uses the order treated-to-control: a subclass with two treated units and four controls has ratio \(2/4=0.5\), or \(1:2\). Because ratio notation is sometimes written in either order, always check which group appears first before judging a set. A ratio interval such as \([0.5,2]\) permits treated units to be half as numerous as controls, equally numerous, or up to twice as numerous, subject to the bounds.

Turn Ratio Bounds Into Count Checks

A subclass must have positive counts from both groups, so \(n_{Ts}\ge1\) and \(n_{Cs}\ge1\). Once the treated count is fixed, the ratio limits can be translated into a range of integer control counts. Starting from \(L\le n_{Ts}/n_{Cs}\le U\), multiply by the positive control count and rearrange:

L\le \frac{n_{Ts}}{n_{Cs}}\le U \quad\Longleftrightarrow\quad \frac{n_{Ts}}{U}\le n_{Cs}\le \frac{n_{Ts}}{L}.

The control count must be an integer. Therefore, for a fixed treated count, the allowable values are

\max\left(1,\left\lceil\frac{n_{Ts}}{U}\right\rceil\right) \le n_{Cs}\le \left\lfloor\frac{n_{Ts}}{L}\right\rfloor.

The ceiling rounds up to the next integer, and the floor rounds down. If the lower integer bound exceeds the upper bound, no subclass with that treated count can meet the specified ratio limits. Equivalently, with the control count fixed, the treated count must satisfy \(\lceil L n_{Cs}\rceil\le n_{Ts}\le\lfloor U n_{Cs}\rfloor\), with at least one treated unit.

Key rule: Evaluate the ratio separately for every subclass. A full-matching allocation meets a subclass-level ratio restriction only when every subclass contains both groups and every individual ratio lies within \([L,U]\).

Worked Checks at the Boundaries

Worked Example: Checking Four Subclasses

Suppose a hypothetical allocation has four subclasses with counts \((n_T,n_C)\) equal to \((2,3)\), \((3,1)\), \((1,2)\), and \((4,2)\). The design requires \(0.5\le n_T/n_C\le2\). Check each subclass.

For \((2,3)\), the ratio is \(2/3=0.667\), which is between 0.5 and 2, so the set passes.

For \((3,1)\), the ratio is \(3/1=3\), which exceeds 2, so the set fails the upper limit.

For \((1,2)\), the ratio is \(1/2=0.5\), exactly the lower limit, so the set passes. For \((4,2)\), the ratio is \(4/2=2\), exactly the upper limit, so it also passes.

The allocation therefore violates the restriction because one of its four subclasses fails. The fact that the other three pass does not compensate for that violation.

Worked Example: Finding Allowable Integer Counts

A hypothetical subclass is to contain exactly three treated units. The required ratio limits are \(0.75\le n_T/n_C\le1.5\). Find every allowable control count.

Substitute \(n_T=3\), \(L=0.75\), and \(U=1.5\) into the integer bounds:

\left\lceil\frac{3}{1.5}\right\rceil \le n_C\le \left\lfloor\frac{3}{0.75}\right\rfloor \quad\Longrightarrow\quad 2\le n_C\le4.

Thus the allowable control counts are 2, 3, and 4. Checking the endpoint values gives \(3/2=1.5\) and \(3/4=0.75\); both meet the inclusive limits. A count of one control would give \(3/1=3\), above the upper limit, while five controls would give \(3/5=0.6\), below the lower limit.

The range is determined by the integer calculation, not by rounding a ratio to a convenient label. For example, \(n_C=5\) does not pass merely because the counts might be described informally as “about three treated for five controls.”

Why the Overall Ratio Is Not Enough

Subclasses can have opposing imbalances that cancel when their counts are added. The overall ratio is a weighted summary of subclass counts; it does not certify that each subclass meets its own limits. This distinction matters whenever the design specification constrains the composition of each matched set.

Worked Example: An Acceptable Overall Ratio Hides Violations

Consider a hypothetical allocation of six treated and eight control units in two subclasses. The ratio limits are \(0.5\le n_T/n_C\le2\). In subclass A there are five treated units and two controls; in subclass B there is one treated unit and six controls.

The overall ratio is

\frac{5+1}{2+6}=\frac{6}{8}=0.75,

which lies within the specified interval. But subclass A has ratio \(5/2=2.5\), above the upper limit, and subclass B has ratio \(1/6\approx0.167\), below the lower limit. The allocation fails the subclass-level restriction twice, despite its acceptable overall ratio.

For comparison, another allocation of the same six treated and eight control units could use subclasses \((2,4)\) and \((4,4)\). Their ratios are \(2/4=0.5\) and \(4/4=1\); both pass. The same overall numbers can therefore be arranged in a way that either satisfies or violates the set-level rule.

Applying the Restriction in a Full-Matching Design

A ratio restriction can be treated as an admissibility condition while constructing or reviewing a full match. For a proposed subclass, its group counts are checked against the limits; a violating subclass is not an admissible set under that design specification. When checking a complete allocation, verify both that the sets are mixed and that every unit appears in exactly one subclass, as required for a partition-based full matching, then check the ratio condition set by set.

The following procedure separates the counting rule from other design choices:

Record the convention and limits.
Write the ratio in the order treated divided by control, and state the lower bound \(L\) and upper bound \(U\).

Count both groups in every subclass.
Confirm that \(n_T\ge1\) and \(n_C\ge1\). A set containing only treated units or only controls is not a mixed full-matching subclass.

Evaluate each ratio or its integer bounds.
Compute \(n_T/n_C\) and compare it with \([L,U]\), or use the ceiling-and-floor bounds when one group’s count is fixed.

Report violations by set.
Identify the subclass and whether it violates the lower or upper limit. Do not replace this check with an overall ratio.

Ratio restrictions may affect which groupings are available and can make a proposed allocation inadmissible. They do not alone determine which admissible allocation is preferable: a design may also be assessed using its matching objective and the relevant covariate or propensity-score diagnostics. A ratio-compliant allocation is not automatically well matched on those features.

Diagnostics, Alternatives, and Limitations

A reliable audit uses unrounded counts. Compute \(n_T/n_C\) from the integer counts, and compare it directly to the stated bounds. If a displayed ratio is rounded, a value printed as 0.50 might lie just below a lower bound of 0.5 before rounding. Counts avoid that ambiguity. For example, if the lower limit is \(0.5\), the condition is exactly \(n_T\ge n_C/2\); for integer counts, equivalently \(n_T\ge\lceil n_C/2\rceil\).

Common warning signs include reversed ratio order, applying a restriction only to the aggregate, and rounding a noninteger bound in the wrong direction. In particular, solving for an integer count requires a ceiling on the lower limit and a floor on the upper limit. If a proposed subclass is impossible under those bounds, changing a rounding convention cannot make it admissible.

A ratio restriction is one design choice among alternatives. An unrestricted full matching permits any positive treated and control counts within a mixed subclass, whereas a restricted design rules out subclasses outside the selected interval. A fixed-ratio design is more restrictive still, allowing only a specific count pattern. Restrictions can prevent extreme set compositions, but tighter limits also reduce the set of possible groupings. The choice should be justified by the design goals and available units rather than treated as a guarantee of causal validity.

In particular, satisfying the ratio condition does not establish covariate balance, sufficient overlap, absence of unmeasured confounding, or the validity of an effect estimate. It is a structural constraint on matched-set counts. It also does not specify a distance threshold: a subclass can satisfy its ratio limits and still contain units that are poorly matched on a chosen distance. Those questions require their own design checks.

Common mistake: “The overall treated-to-control ratio is between the limits, so every matched set satisfies the rule.” This reasoning is incorrect because subclass imbalances can cancel in the totals. Compute and inspect \(n_{Ts}/n_{Cs}\) for every subclass; the overall ratio is not a substitute.

Check Your Understanding

Use the treated-to-control convention and assume all stated bounds are inclusive.

With limits \(0.5\le n_T/n_C\le1.5\), a subclass has four treated units. What integer control counts are allowed?
Does a subclass with two treated units and five controls meet limits \(0.5\le n_T/n_C\le2\)? Show the ratio and state the violated bound, if any.
Under limits \(0.5\le n_T/n_C\le2\), check two subclasses with counts \((2,1)\) and \((1,2)\). Does each pass, and does their combined ratio settle the set-level question?
Explain why rounding a computed ratio before checking a boundary can be misleading, and name a safer check.

Solutions. (1) The integer bounds are \(\lceil4/1.5\rceil=3\) and \(\lfloor4/0.5\rfloor=8\), so \(n_C\) may be any integer from 3 through 8. (2) The ratio is \(2/5=0.4\), below 0.5, so the subclass fails the lower limit. (3) The ratios are \(2/1=2\) and \(1/2=0.5\), so both pass at the upper and lower boundaries, respectively. Their combined ratio is \(3/3=1\), but the set-level conclusions come from checking both subclasses individually. (4) Rounding can make a value appear to meet a boundary when its unrounded value does not. Use the exact integer counts and compare \(n_T/n_C\) with the limits, or use the corresponding ceiling-and-floor count bounds.