Matching With Replacement

Why Allow a Control to Be Matched More Than Once?

Nearest-neighbor matching pairs each treated unit with a control unit that has a similar propensity score. With replacement, a control remains available after it has been selected, so it can serve as the match for several treated units. This can help when only a few controls are close to the treated units in score, but it also means the matched data may contain repeated use of the same control.

The choice is about the matching allocation, not about changing the propensity score or the causal assumptions. Matching with replacement does not make a poorly supported treated unit comparable to a control, and it does not by itself establish that an observed outcome difference is causal. The earlier material on propensity-score distances and hand-calculating nearest-neighbor matches provides the distance framework used here.

Definition: Let \(A_i=1\) indicate that unit \(i\) is treated and \(A_i=0\) indicate that it is a control. Let \(e(X_i)=P(A_i=1\mid X_i)\) be the propensity score for covariates \(X_i\). In one-to-one nearest-neighbor matching with replacement, each treated unit \(i\) is assigned a control \(j(i)\) whose score minimizes the absolute distance \(|e(X_i)-e(X_j)|\) among eligible controls. A selected control remains eligible for later treated units and may therefore be selected more than once.

Unless a different distance is specified, this tutorial uses absolute differences between propensity scores. The matching rule can be written as:

j(i)=\operatorname*{arg\,min}_{j:A_j=0} \left|e(X_i)-e(X_j)\right|,\qquad A_i=1.

If two controls tie for the minimum distance, an additional tie-breaking rule is needed; ties are not resolved by the definition alone. Here, examples use unique nearest controls. The replacement rule means that the minimization is performed separately for each treated unit without removing previously selected controls from the eligible set.

Assumptions and the Matched Contrast

For a causal interpretation, define \(Y_i(1)\) and \(Y_i(0)\) as unit \(i\)'s potential outcomes under treatment and control. The treated-unit target is the average treatment effect among treated units (ATT), \(E[Y(1)-Y(0)\mid A=1]\). Matching estimates this target by comparing treated outcomes with outcomes from selected controls. The design requires more than a close numerical distance.

Consistency and a well-defined treatment: the observed outcome equals the potential outcome under the treatment actually received, and treatment versions are sufficiently well defined for the question.
No interference: one unit's treatment does not change another unit's outcome, or any interference is handled in the definition of treatment and outcome.
Conditional exchangeability: after conditioning on measured covariates \(X\), treatment assignment is independent of the potential outcomes. Matching observed covariates through a propensity score cannot remove bias from unmeasured confounders.
Positivity for the treated: treated units in the target population have a nonzero probability of being untreated at their covariate values. In practice, useful controls must also lie sufficiently close in the chosen score distance.
Meaningful score and distance: scores must be estimated or specified in a way that supports relevant covariate balance, and distance on the score must be a defensible basis for forming matches.

The propensity score is a balancing score under the standard conditions, but a finite sample need not have exact balance just because scores were estimated. After matching, assess covariate balance and the distances actually obtained. A close score match is not a substitute for checking balance in important covariates.

Matched ATT estimate: For \(n_T\) treated units, let \(j(i)\) denote the matched control for treated unit \(i\). The one-to-one matched estimate is

\widehat{\tau}_{\mathrm{ATT}} =\frac{1}{n_T}\sum_{i:A_i=1} \left(Y_i-Y_{j(i)}\right).

A control selected \(m_j\) times contributes its outcome \(m_j\) times to the matched control total. Equivalently, the estimate subtracts the average outcome among treated units from a control mean that weights each control by its match count:

\widehat{\tau}_{\mathrm{ATT}} =\frac{1}{n_T}\sum_{i:A_i=1}Y_i -\frac{1}{n_T}\sum_{j:A_j=0}m_jY_j, \qquad \sum_{j:A_j=0}m_j=n_T.

This identity follows by expanding the pairwise differences: every treated outcome appears once, and each control outcome appears once for every treated unit assigned to it. Thus replacement changes the weights in the matched control mean; it does not increase the number of distinct controls.

Worked Example: Allocating Matches and Correctly Computing Distances

Suppose three hypothetical treated units have scores \(0.51\), \(0.54\), and \(0.57\). Two hypothetical controls have scores \(0.20\) and \(0.49\). Match each treated unit to its nearest control with replacement.

Treated score	Distance to control at 0.20	Distance to control at 0.49	Selected control
0.51	\|0.51 − 0.20\| = 0.31	\|0.51 − 0.49\| = 0.02	0.49
0.54	\|0.54 − 0.20\| = 0.34	\|0.54 − 0.49\| = 0.05	0.49
0.57	\|0.57 − 0.20\| = 0.37	\|0.57 − 0.49\| = 0.08	0.49

For example, the first distance in the table is \(0.51-0.20=0.31\), not \(0.34\). The latter is the distance from the second treated score, \(0.54\), to \(0.20\). The control with score \(0.49\) is nearest in all three comparisons, so it is reused three times. The control at \(0.20\) is not selected. This is a valid allocation under replacement, although the last selected match has a score distance of \(0.08\); whether that is close enough depends on the substantive setting and any prespecified restrictions.

From Repeated Matches to an Outcome Estimate

The allocation determines how often each control outcome enters the estimate. The next example shows both the pairwise calculation and its equivalent weighted-control calculation. The outcomes are hypothetical and measured in points.

Worked Example: Calculating the Matched Contrast

Use the scores and assignments from the preceding example. Suppose the treated outcomes corresponding to scores \(0.51\), \(0.54\), and \(0.57\) are 10, 14, and 12 points. Suppose the control outcome at score \(0.49\) is 9 points and the control outcome at score \(0.20\) is 8 points.

All three treated units are matched to the control with outcome 9. Their individual matched differences are \(10-9=1\), \(14-9=5\), and \(12-9=3\) points. Therefore:

\widehat{\tau}_{\mathrm{ATT}} =\frac{1+5+3}{3} =\frac{9}{3} =3\text{ points}.

The treated mean is \((10+14+12)/3=12\) points. The matched control mean counts the outcome 9 three times, so it is \((9+9+9)/3=9\) points. Their difference is also \(12-9=3\) points. The estimate describes a three-point average treated-minus-matched-control outcome difference for these matched units. It is not automatically a causal effect unless the identifying assumptions are credible.

A reused control can have substantial influence because its outcome is replicated in the matched comparison. This is not an arithmetic error: it is exactly what the stated matching design specifies. It does, however, make it important to report how many distinct controls were used and how often each was reused.

Worked Example: Match Counts and the Weighted Control Mean

Consider four hypothetical treated units with scores \(0.25\), \(0.46\), \(0.53\), and \(0.72\), and three controls with scores \(0.22\), \(0.48\), and \(0.75\). Their nearest-neighbor assignments are:

For \(0.25\), the distances are \(0.03\), \(0.23\), and \(0.50\), so select \(0.22\).
For \(0.46\), the distances are \(0.24\), \(0.02\), and \(0.29\), so select \(0.48\).
For \(0.53\), the distances are \(0.31\), \(0.05\), and \(0.22\), so select \(0.48\).
For \(0.72\), the distances are \(0.50\), \(0.24\), and \(0.03\), so select \(0.75\).

Thus the controls at \(0.22\), \(0.48\), and \(0.75\) have match counts 1, 2, and 1. Suppose their outcomes are 6, 8, and 10 units, while the four treated outcomes are 9, 11, 12, and 13 units. The treated mean is \((9+11+12+13)/4=11.25\) units. The weighted matched control mean is:

\frac{1(6)+2(8)+1(10)}{4} =\frac{32}{4} =8\text{ units}.

The estimated matched contrast is \(11.25-8=3.25\) units. Checking by pairs gives \(9-6=3\), \(11-8=3\), \(12-8=4\), and \(13-10=3\), whose mean is \((3+3+4+3)/4=3.25\) units. The two calculations agree because the control at \(0.48\) contributes twice.

Diagnostics, Alternatives, and Limitations

Before interpreting an estimate, inspect the design that produced it. Useful diagnostics include the distribution of matched score distances, the number of distinct controls used, the maximum reuse count, and covariate balance before and after matching. A small average distance can conceal a few poor matches, so examine the full distribution or individual distances as well as a summary. Check whether high-reuse controls are unusual in important covariates; if they are, the matched estimate may rely heavily on a narrow part of the control sample.

If treated units have no nearby controls, the score distributions may have limited overlap. Replacement allows the available nearby control to be used repeatedly, but it cannot create information about treated units whose covariate profiles are not represented among controls. The result may therefore depend on a small number of controls and may not support a credible estimate for the intended treated population. Reconsider the target population or matching design rather than treating repeated use as a repair for absent support.

The main matching alternative in the next topic is matching without replacement: once selected, a control is no longer available. That rule can spread matches across more distinct controls, but it can also force later treated units to accept less-near controls. Replacement avoids that particular constraint at the cost of possible concentration and repeated-control dependence. Neither rule is universally superior; their consequences depend on score overlap, sample sizes, match quality, and the inferential procedure.

Other approaches, such as regression adjustment or weighting, use the data differently and can be useful comparisons, but they do not eliminate the need to assess confounding, overlap, and model adequacy. Inference after matching with replacement must account for the fact that matched observations are not simply independent pairs: the same control outcome can enter several differences. Treating all pair differences as independent can understate uncertainty. Use an inferential method appropriate to the matching design, and state clearly whether uncertainty from score estimation is included.

Common Mistakes

Removing a control after its first selection. That implements a no-replacement rule, not matching with replacement. Under replacement, every treated unit is matched from the full eligible control set.
Counting repeated appearances as new controls. Three matches to one control still represent one distinct control, even though that control contributes three times to a one-to-one matched estimate.
Calling the estimate causal because scores are close. Close scores do not establish conditional exchangeability, positivity, or adequate covariate balance. Those conditions require substantive and empirical assessment.
Using the wrong distance arithmetic. A distance is the absolute difference between a particular treated score and a particular control score. Keep the two scores identified, then subtract and take the absolute value.
Using an ordinary independent-pairs standard error without checking its assumptions. Reused controls make some pair differences share the same outcome, so the apparent pairs are dependent.

Key takeaway: With replacement, each treated unit receives its own nearest eligible control, and any control may be selected repeatedly. Match counts determine the implicit control weights. Assess distance quality, balance, overlap, reuse concentration, and dependence before drawing conclusions.

Check Your Understanding

Answer each question, then check the brief solutions below.

A treated unit has score 0.42. Eligible control scores are 0.37 and 0.46. Which control is nearest, and what is the distance?
Three treated units are matched to one control with outcome 7. Their outcomes are 8, 10, and 9. Calculate the matched contrast.
In one-to-one matching with replacement, four treated units use controls with match counts 1, 2, and 1. How many distinct controls are used, and how many control appearances enter the matched mean?
Why does matching with replacement not solve a lack of overlap?
What feature of the matched data makes a naive independent-pairs standard error potentially misleading?

Solutions. 1. The distances are \(|0.42-0.37|=0.05\) and \(|0.42-0.46|=0.04\), so select the control at 0.46, at distance 0.04. 2. The differences are \(8-7=1\), \(10-7=3\), and \(9-7=2\), giving \((1+3+2)/3=2\) outcome units. 3. Three distinct controls are used, and four control appearances enter the matched mean because the match counts sum to four. 4. Reuse can repeat an available control, but it cannot supply a control with covariates or a score near a treated unit when no such control exists. 5. Reused control outcomes appear in multiple pair differences, making those differences dependent.

What You'll Learn