Matching With Replacement and Control Reuse

Why Reuse a Control?

Matching tries to compare treated units with untreated units that have similar covariates or propensity scores. If matching is done without replacement, each control can be used at most once. That restriction can leave some treated units with poor matches or no match at all. Matching with replacement relaxes the restriction: a control that is a good match for several treated units can be used several times.

Reuse changes how much each control contributes to the comparison. It does not, by itself, change which treated units are included. This distinction matters: the matched controls may be represented unevenly, while the treated-unit target can remain the average effect for the same matched treated units.

Definition: In one-to-one matching with replacement, each included treated unit \(i\) is assigned one untreated control \(M(i)\), and the same control may be assigned to more than one treated unit. Let \(m_j\) be the number of treated units assigned to control \(j\). Then \(m_j\) is that control’s reuse count, and \(\sum_{j:A_j=0}m_j=n_T^M\), where \(n_T^M\) is the number of included treated units.

What the Estimate Represents

Write \(A_i=1\) for a treated unit and \(A_i=0\) for a control unit. Let \(Y_i(1)\) and \(Y_i(0)\) denote unit \(i\)’s potential outcomes under treatment and control, and let \(Y_i\) be the observed outcome. The individual treatment effect is \(\tau_i=Y_i(1)-Y_i(0)\). For a matched set of treated units, the empirical treated-unit average effect is the mean of their \(\tau_i\) values; this is the finite-sample target corresponding to an ATT for those treated units.

In observational data, the treated units’ \(Y_i(0)\) values are not observed. A common one-to-one matched estimate substitutes each assigned control’s observed outcome for the missing control outcome. The same control outcome therefore appears once for every treated unit assigned to that control.

\widehat{\tau}_{M} =\frac{1}{n_T^M}\sum_{i:A_i=1,\ i\text{ included}}\left(Y_i-Y_{M(i)}\right) =\frac{1}{n_T^M}\sum_{i:A_i=1,\ i\text{ included}}Y_i -\frac{1}{n_T^M}\sum_{j:A_j=0}m_jY_j

The second expression makes the control representation explicit: the comparison mean is a weighted mean of the distinct matched controls, with reuse count \(m_j\) as the weight. The treated mean, by contrast, includes each matched treated unit once. Thus, if the included treated units stay the same, reusing controls changes their representation in the comparison mean, not the empirical treated-unit target.

Conditions for a causal interpretation: The matching estimate is not automatically a causal effect. A causal interpretation requires consistency and no interference, conditional exchangeability of treatment and potential outcomes given the matched covariates, and positivity for the treated units being targeted. The matching procedure must also produce sufficiently comparable controls for those treated units; in practice, covariate balance and match quality must be assessed. If treated units are discarded, the target is the included treated group rather than all treated units.

Under these assumptions, matched controls can stand in for the missing control outcomes of their treated matches. In finite data, however, matching is an approximation: a close propensity-score match does not guarantee balance on every covariate, and exchangeability is not established by matching alone.

A Short Derivation of the Reuse Weights

Start with the average of the matched treated-control differences. Regroup the control-outcome terms by distinct control unit. If control \(j\) is assigned to \(m_j\) treated units, its outcome occurs \(m_j\) times in the sum. Therefore the sum over matched pairs and the sum over distinct controls with reuse counts are exactly equal:

\sum_{i:A_i=1,\ i\text{ included}}Y_{M(i)} =\sum_{j:A_j=0}m_jY_j.

Dividing by \(n_T^M\) gives the weighted comparison mean in the estimator above. The weights sum to one after normalization because \(\sum_jm_j=n_T^M\). This is not the same as taking the unweighted mean of distinct controls: doing so would give a control used once the same contribution as a control used many times, contrary to the matched-pair estimator.

Worked Example: One Control Used Twice

Suppose three hypothetical treated patients are all matched, with replacement, to two hypothetical controls. Their observed outcomes, measured in points on a symptom scale, are:

Treated outcomes: \(14, 17, 15\). Their matched control outcomes are \(10, 10, 12\), respectively; the first control is reused for the first two treated patients. Assume the matching is adequate and the causal assumptions stated above are plausible for this illustration.

The three matched differences are \(14-10=4\), \(17-10=7\), and \(15-12=3\) points. Hence

\widehat{\tau}_{M} =\frac{4+7+3}{3} =\frac{14}{3} =4.667\text{ points}.

Equivalently, the treated mean is \((14+17+15)/3=15.333\) points. The control comparison mean gives the reused control two contributions: \[ \frac{2(10)+1(12)}{3}=\frac{32}{3}=10.667\text{ points}. \] The difference between these means is \(15.333-10.667=4.667\) points, as before.

The unweighted mean of the two distinct controls would be \((10+12)/2=11\) points, yielding a difference of \(15.333-11=4.333\) points. That is a different calculation: it does not give the reused control its two matched-pair contributions. Neither calculation is automatically causal; the interpretation of the matched estimate still depends on the assumptions and match quality.

Reuse and the Treated-Unit Target Are Different Choices

For one-to-one matching, each included treated unit contributes one difference to the estimate. Reusing a control does not add another treated unit or give an included treated unit extra weight. If every treated unit eligible for the analysis is matched, the finite-sample target remains the average effect for those treated units, whether controls are reused or not.

A different target arises if a matching procedure leaves some treated units unmatched and excludes them. The target then refers to the included treated units. That change is caused by changing which treated units enter the analysis, not by reusing controls. A population ATT interpretation also requires assumptions about how the analyzed treated units relate to the population of treated units; keeping the same sample target does not by itself guarantee population generalizability.

Worked Example: Reuse Does Not Drop a Treated Unit

Consider three hypothetical treated units with observed outcomes of \(12\), \(16\), and \(20\) hours. For illustration, suppose their matched control outcomes are \(10\), \(10\), and \(10\) hours; the same control is reused for the first two treated units, and another control is used for the third. Suppose all three treated units are included and these matches are adequate under the stated assumptions.

Their differences are \(2\), \(6\), and \(10\) hours, so the estimate for these three treated units is

\widehat{\tau}_{M}=\frac{2+6+10}{3}=6\text{ hours}.

Each treated unit contributes once, including the third unit. The first control’s reuse affects its contribution to the control mean, but does not change the three treated units represented in the average.

Now suppose a no-replacement procedure matches only the first two treated units and leaves the third unmatched. The estimate for the included treated units is \((2+6)/2=4\) hours. This is a different treated-unit target. The change from \(6\) to \(4\) hours in this illustration comes from excluding the third treated unit, not from the mere act of reusing a control. In real data, the estimates would also depend on which matches each procedure produces.

Diagnostics and Failure Modes

Control reuse is useful when suitable controls are scarce, but it can concentrate the comparison on a small number of units. A large reuse count may reflect limited overlap or a region with few eligible controls. Inspect the matching assignments alongside covariate balance and the distances between matched units; a matched label alone does not establish that the units are comparable.

For example, a descriptive measure of how concentrated the control contributions are is

n_{\mathrm{eff,control}} =\frac{\left(\sum_jm_j\right)^2}{\sum_jm_j^2}.

This equals the number of distinct controls when all have equal reuse counts, and decreases as contributions become more concentrated. It is a diagnostic summary of the reuse weights, not a replacement for a valid standard error or a guarantee of independent information.

Repeated use also makes matched differences statistically dependent: two differences that use the same control share that control’s observed outcome. An uncertainty calculation that treats all matched differences as independent can therefore be inappropriate. Inference should account for the matching procedure and the dependence induced by reuse; the correct approach depends on the design and inferential target.

Choice or feature	What it changes	What to check
Reuse a control	How the matched comparison mean represents distinct controls	Reuse counts, match quality, balance, and dependence for inference
Exclude unmatched treated units	Which treated units define the matched-sample target	Which units were excluded and how the target population changes
Match without replacement	Control availability for later matches; it may also lead to unmatched treated units	Whether the resulting matches remain adequate and which treated units remain included

A practical alternative to reuse is matching without replacement, which prevents any control from appearing in more than one pair but can force less suitable matches or leave treated units unmatched. Excluding units that cannot be matched may improve comparability for the retained group while changing the target. These are design choices with different consequences; none should be selected solely because it produces a preferred estimate.

Common Mistakes

Counting a reused control as a new person. Repeated appearances are repeated contributions of one observed control, not additional independent units.
Taking the unweighted mean of distinct controls. That ignores how often each control was matched and does not reproduce the one-to-one matched-pair estimate.
Claiming reuse changes the treated target. Reuse changes control contributions. The treated target changes if the included treated units or their weights change.
Assuming matching proves comparability. Check covariate balance and match distances, and justify the causal assumptions separately.
Treating matched differences as independent despite shared controls. Reuse creates dependence that must be considered in uncertainty estimation.

Key takeaway: In one-to-one matching with replacement, each included treated unit contributes one matched difference, while a control contributes once for each time it is reused. This changes the control-side representation; changing the treated-unit target requires changing which treated units or treated-unit weights define the estimand.

Check Your Understanding

Answer each question, then check the solutions below.

Four treated units are matched to controls with reuse counts \(2,1,1\). What is the sum of the reuse counts, and what does it represent?
Three treated outcomes are \(8, 11, 13\), and their matched control outcomes are \(5, 5, 9\). The first control is reused. Calculate the matched estimate and state its units if outcomes are in milligrams per liter.
If a control is reused but all matched treated units remain included once, has the treated-unit target changed? Explain.
Name two diagnostics or inferential issues to examine when controls are reused.

Solutions. (1) The sum is \(2+1+1=4\), equal to the number of included treated units, since each has one match. (2) The differences are \(8-5=3\), \(11-5=6\), and \(13-9=4\) milligrams per liter. Thus \(\widehat{\tau}_M=(3+6+4)/3=13/3=4.333\) milligrams per liter. (3) No. Each included treated unit still contributes once, so reuse alone changes the control-side representation, not the treated-unit target. (4) Check reuse counts and covariate balance or match distances; also account for dependence between differences that share a control when estimating uncertainty.