Why Control Use Matters
In nearest-neighbor matching, a treated unit is paired with a control unit that is close on a chosen distance measure, often the difference in their estimated propensity scores. The choice of whether controls can be reused changes the assignment rule. With replacement, a control can serve as a match for more than one treated unit. Without replacement, a selected control is removed from the available pool; each control can be selected at most once.
That restriction makes every match affect the options available for the next treated unit. A control that is closest to one treated unit may also be the best option for another, but only one of them can use it. The matching order and the set of controls still available therefore matter. The procedure must specify an order for processing treated units, as well as a rule for resolving equal distances.
For example, let \(e_i\) denote unit \(i\)'s propensity score, the estimated probability of treatment given observed covariates. A common distance between treated unit \(i\) and control unit \(j\) is the absolute propensity-score difference:
Other distance choices are possible, but the rule must be fixed before assignments are made. In the examples below, matching uses absolute propensity-score distance, considers controls eligible unless already used, and processes treated units in the stated order. Assume ties are absent; in an application, a tie-breaking rule should be specified in advance.
The Assignment Rule Step by Step
Let \(C_k\) be the set of controls that remain available just before the \(k\)th treated unit is processed. If treated unit \(i_k\) is to be matched, the selected control is the unit in \(C_k\) with the smallest distance to \(i_k\). After selecting control \(j_k\), the next available set is \(C_{k+1}=C_k\setminus\{j_k\}\). Thus, a control selected earlier cannot be considered again.
This rule guarantees that each selected control is the nearest currently available control for the treated unit being processed. It does not guarantee that the resulting collection of pairs has the smallest possible total distance across all possible assignments. The procedure is sequential: each local choice changes the available set for later decisions.
Specify the score or other distance, treated-unit order, eligibility restrictions, and tie-breaking rule.
For the current treated unit, calculate its distance to each unused eligible control and select the smallest.
Remove it from the available pool before processing the next treated unit.
If no eligible control remains, do not create a pair; report the treated unit as unmatched.
Worked Assignments
Worked Example: Tracking Controls as They Are Consumed
Suppose three treated units have propensity scores \(0.70\), \(0.68\), and \(0.40\), in that processing order. Three controls have scores \(0.69\), \(0.66\), and \(0.42\). Label them \(C_1,C_2,C_3\), respectively. Match each treated unit to the nearest unused control.
For the first treated unit, \(T_1\), the distances are \(d(T_1,C_1)=|0.70-0.69|=0.01\), \(d(T_1,C_2)=|0.70-0.66|=0.04\), and \(d(T_1,C_3)=|0.70-0.42|=0.28\). The smallest distance is \(0.01\), so \(T_1\) is matched to \(C_1\). Control \(C_1\) is now unavailable.
For \(T_2\), compare only the unused controls: \(d(T_2,C_2)=|0.68-0.66|=0.02\) and \(d(T_2,C_3)=|0.68-0.42|=0.26\). Therefore \(T_2\) is matched to \(C_2\), and \(C_2\) is removed from the pool.
Only \(C_3\) remains for \(T_3\). Its distance is \(d(T_3,C_3)=|0.40-0.42|=0.02\), so the final pair is \(T_3\) with \(C_3\). All three treated units are matched, and every control is used once. Notice that each choice was made from the controls available at that step, not from the original control set.
Worked Example: When the Control Pool Is Exhausted
Now suppose treated scores, in order, are \(0.90\), \(0.70\), and \(0.50\), while the only controls have scores \(0.84\) and \(0.48\). There is no caliper or other eligibility restriction.
For \(T_1\), the distances are \(|0.90-0.84|=0.06\) and \(|0.90-0.48|=0.42\). Select the control with score \(0.84\); it is then unavailable.
For \(T_2\), the only unused control has score \(0.48\), giving distance \(|0.70-0.48|=0.22\). Select it. No controls remain for \(T_3\), so \(T_3\) is unmatched. The result is two pairs and one unmatched treated unit, not three pairs.
The matching algorithm itself does not decide what to do about the unmatched treated unit. An analysis that uses only the two matched treated units targets the treated units represented by those pairs, not automatically all treated units in the original sample. Reporting how many treated units were excluded, and comparing included and excluded treated units, is therefore important.
Worked Example: Calculating Pair Distances and Checking a Covariate
Consider two treated units with propensity scores \(0.72\) and \(0.58\), and three controls with scores \(0.70\), \(0.55\), and \(0.30\). Process the treated units in that order. Suppose the treated units' ages are 50 and 45 years, while the controls' ages, in score order, are 51, 42, and 60 years.
For the treated unit with score \(0.72\), the three score distances are \(0.02\), \(0.17\), and \(0.42\); it is matched to the control with score \(0.70\). That control's age is 51 years, one year from the treated unit's age of 50. For the treated unit with score \(0.58\), the remaining controls have score distances \(0.03\) and \(0.28\); it is matched to the control with score \(0.55\), whose age is 42 years, three years from the treated unit's age of 45.
The matched pairs' absolute score distances are \(0.02\) and \(0.03\), so their mean absolute score distance is \((0.02+0.03)/2=0.025\). Their absolute age differences are 1 and 3 years, with mean \((1+3)/2=2\) years. These calculations describe the matched pairs, but do not establish that age—or any other covariate—is adequately balanced overall. Balance requires examining the relevant covariates in the matched sample, not just confirming that the propensity scores are close.
Assumptions and Interpretation
The assignment rule is a deterministic algorithm once scores, distance, order, eligibility rules, and tie-breaking are specified. It does not require a causal assumption to produce pairs. A causal interpretation of an outcome comparison using those pairs requires additional conditions. In potential-outcomes terms, common conditions include consistency (the observed outcome equals the potential outcome under the treatment actually received), no interference between units, conditional exchangeability given the measured pretreatment covariates, and adequate overlap in treatment probabilities for the population being compared. These conditions are substantive, not consequences of matching.
When every treated unit is matched and the target is the average treatment effect among the treated, a simple paired outcome contrast is the average of the treated outcome minus its matched control outcome. If only some treated units are matched, the same calculation pertains to the matched treated subset unless further assumptions justify generalization. Matching can improve comparability on measured covariates, but it cannot remove confounding by unmeasured variables, and a small score distance alone does not prove that the causal assumptions hold.
Here \(M\) is the number of matched pairs, \(Y_{i_k}\) is the observed outcome for treated unit \(i_k\), and \(Y_{j_k}\) is the observed outcome for its control. The expression is an estimator based on the matched sample; its causal interpretation depends on the target population and the assumptions just stated.
Diagnostics, Alternatives, and Limitations
After matching, inspect the number and characteristics of matched and unmatched units, the distribution of pair distances, and balance in the pretreatment covariates that matter for treatment assignment and outcome. A small average distance can conceal a few very poor pairs, so examine the full distribution, including its largest values. Also check whether matched treated units differ systematically from unmatched treated units. Serious remaining imbalance or a concentration of large distances signals that the matching design may not provide the intended comparability.
Without replacement, each control contributes to at most one pair. This avoids repeatedly relying on a small number of controls, but can leave later treated units with poor matches or no match at all. Matching with replacement instead allows a control to be reused when it is close to several treated units; this can improve individual pair proximity, while making some controls contribute repeatedly and requiring the analysis to account for that reuse. Neither rule guarantees covariate balance or resolves unmeasured confounding.
The greedy no-replacement rule is also sensitive to design choices such as treated-unit order and tie handling. The matching order should be explicit and reproducible; changing it can change which controls remain available later. This is a reason to document the order, not a reason to treat one order as universally correct. Other approaches, including optimal pair matching, formulate the assignment across pairs rather than making each decision only from the current available pool. Their objectives and trade-offs differ from the simple sequential rule taught here.
A common mistake is to calculate a treated unit's closest control from the full control list every time, even after that control has already been selected. That repeats a control and silently changes the method into matching with replacement. The correction is to remove each selected control immediately and recalculate later choices using only unused controls. Another mistake is to interpret successful matching as proof of causal validity; matching is a design procedure, not a substitute for defensible assumptions or post-match balance assessment.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- Treated scores are \(0.60\) and \(0.50\), processed in that order. Control scores are \(0.58\) and \(0.49\). Which pairs are formed without replacement?
- There are four treated units and three controls, with no eligibility restrictions. What is the greatest possible number of one-to-one pairs, and what must happen to at least one treated unit?
- Why is a control selected for the first treated unit not considered again for the second treated unit?
- Does a mean absolute propensity-score distance of \(0.01\) establish that all covariates are balanced? Explain.
- If some treated units are unmatched, what population does a simple matched-pair outcome contrast directly describe?
Solutions. (1) The first treated unit has distances \(|0.60-0.58|=0.02\) and \(|0.60-0.49|=0.11\), so it takes the control with score \(0.58\). The remaining control has score \(0.49\), which is then paired with the second treated unit at distance \(|0.50-0.49|=0.01\). (2) At most three pairs can be formed because only three controls are available and each can be used once; at least one treated unit must be unmatched. (3) The defining rule removes each selected control from the available pool, so no control is used more than once. (4) No. A small mean score distance summarizes score proximity but does not show balance on every covariate; covariate balance must be checked directly, and the mean can also hide individual large distances. (5) It directly describes the treated units represented in the matched pairs. Generalizing to unmatched treated units requires additional justification.