Control-Unit Matching and the ATC Target

Why Matching Controls Changes the Question

Matching can be organized around treated units or around control units. The choice matters because the units that remain after matching determine whose treatment effect the comparison is intended to describe. When controls are selected for their similarity to treated units, the resulting target is generally the effect among those retained controls, not automatically the effect among every control in the original population.

This distinction is important when treatment effects vary across people or settings. If controls who can be matched differ systematically from controls who cannot, the average effect for retained controls may differ from the average effect for all controls—even if the matching procedure makes the retained treated and control groups look similar on measured covariates.

Definition: Let $D=1$ denote treatment and $D=0$ denote control, and let $Y(1)$ and $Y(0)$ be a unit’s potential outcomes under treatment and control. The average treatment effect on the controls is $$ \operatorname{ATC}=E[Y(1)-Y(0)\mid D=0]. $$ Let $M=1$ indicate that a control unit is retained by the matching procedure. The retained-control effect is $$ \tau_{\mathrm{ret}}=E[Y(1)-Y(0)\mid D=0,M=1]. $$ The ATC averages effects over all controls; $\tau_{\mathrm{ret}}$ averages them over controls retained for the matched analysis.

Here, $M$ is a retention indicator for controls; it does not mean treatment assignment. Matching is a design step, not a treatment intervention. A control unit’s $Y(0)$ is observed, while its $Y(1)$ is counterfactual. The matched treated units provide information about that missing outcome only under appropriate assumptions.

Assumptions and the Target in Covariate Terms

Write $X$ for measured pretreatment covariates. To express the two targets as covariate averages, define the control-specific conditional treatment effect $\tau_0(x)=E[Y(1)-Y(0)\mid D=0,X=x]$. This definition conditions on control status as well as $X$. It is therefore the relevant conditional effect for averaging over controls.

\operatorname{ATC} =E[\tau_0(X)\mid D=0] =\int \tau_0(x)f(x\mid D=0)\,dx, \qquad \tau_{\mathrm{ret}} =E[\tau_0(X)\mid D=0,M=1] =\int \tau_0(x)f(x\mid D=0,M=1)\,dx.

The second equality for the retained-control target requires that, conditional on $D=0$ and $X$, retention does not select units according to their potential outcomes or their treatment effects. For example, it is appropriate when retention is determined by measured baseline covariates and the matching design, with no additional outcome-related selection after conditioning on those covariates. If retention is related to potential effects even among controls with the same $X$, the integral instead needs the conditional effect $E[Y(1)-Y(0)\mid D=0,X=x,M=1]$.

The covariate-average expressions clarify the difference between the targets: the conditional effect function can be the same, while the covariate distributions over which it is averaged differ. The targets coincide if the average conditional effect is the same under those two distributions. They also coincide under a constant treatment effect. Neither condition follows merely from achieving covariate balance between the matched treated and control groups.

Assumptions for a causal matched contrast: Potential outcomes are well defined and consistent with observed outcomes, and there is no interference between units. Conditional exchangeability, $(Y(1),Y(0))\perp D\mid X$, makes treated and control units comparable in potential outcomes at the same $X$. For estimating effects among retained controls, treated units must be available at covariate values represented among those controls: $P(D=1\mid X=x)>0$ for the relevant $x$. The matching must also achieve adequate covariate comparability, and retention must not select on potential outcomes after conditioning on the covariates used. These are substantive assumptions; matching alone does not guarantee them.

Full conditional exchangeability is a convenient sufficient condition. For the missing treated outcome among controls, the essential comparison is that, given $X$, treated outcomes represent the treated potential outcomes of controls. Positivity is target-specific: an effect among retained controls may be estimable even when treated units are unavailable for some controls who were not retained. That does not make the ATC for all controls estimable.

Worked Examples of the Target

Worked Example: The ATC and the Retained-Control Effect Differ

Consider a hypothetical population of four controls. Two have covariate profile $X=L$, and two have profile $X=H$. Suppose their treatment effects are 2 outcome units for each $L$ control and 8 outcome units for each $H$ control. A matching design retains the two $L$ controls and no $H$ controls.

The ATC averages the effects of all four controls: $$ \operatorname{ATC}=\frac{2+2+8+8}{4} =\frac{20}{4}=5\text{ outcome units}. $$ The retained-control effect averages only the two retained controls: $$ \tau_{\mathrm{ret}}=\frac{2+2}{2} =\frac{4}{2}=2\text{ outcome units}. $$

Thus the control-matched analysis targets an effect of 2 outcome units for the retained controls, not the ATC of 5 outcome units. The difference arises because the effect varies by covariate profile and the retained controls have a different profile distribution. In an actual study, the effects for controls are not all observed; this example stipulates them to make the estimands and their distinction explicit.

Worked Example: Averaging Conditional Effects Over Different Control Distributions

Suppose, hypothetically, that among controls the conditional effect is 1 unit for $X=L$ and 5 units for $X=H$. In the full control group, 10 of 20 controls have each profile. Matching retains 8 controls with $X=L$ and 2 with $X=H$.

The control-specific conditional effects are $\tau_0(L)=1$ and $\tau_0(H)=5$. The full-control distribution gives $$ \operatorname{ATC} =\frac{10}{20}(1)+\frac{10}{20}(5) =0.5+2.5=3\text{ outcome units}. $$ The retained-control distribution gives $$ \tau_{\mathrm{ret}} =\frac{8}{10}(1)+\frac{2}{10}(5) =0.8+1.0=1.8\text{ outcome units}. $$

This calculation uses $\tau_0(x)$, the conditional effect among controls. It does not substitute an effect averaged over treated and control units at $X=x$. That distinction matters unless an additional assumption justifies equating those conditional effects.

The size and direction of the shift depend on how retention changes the mix of effect-modifying covariates. For binary $M$, the difference can also be written as a covariance:

E[\tau_0(X)\mid D=0,M=1]-E[\tau_0(X)\mid D=0] = \frac{\operatorname{Cov}(\tau_0(X),M\mid D=0)} {P(M=1\mid D=0)}.

To verify the identity, within the control group write $p=P(M=1\mid D=0)$. Then $E[\tau_0(X)\mid D=0,M=1]=E[\tau_0(X)M\mid D=0]/p$. Subtracting $E[\tau_0(X)\mid D=0]$ yields $[E(\tau_0(X)M\mid D=0)-E(\tau_0(X)\mid D=0)E(M\mid D=0)]/p$, which is the displayed covariance divided by $p$. If retention is more likely for profiles with larger effects, the covariance is positive and the retained-control effect exceeds the ATC; the reverse association yields a smaller retained-control effect.

Estimating the Retained-Control Effect

For a simple one-to-one matched design without replacement, let $n_m$ be the number of retained controls, with one treated match for each. Let $j(i)$ denote the treated unit matched to retained control $i$. An unadjusted matched contrast is

\widehat{\tau}_{\mathrm{ret}} =\frac{1}{n_m}\sum_{i:D_i=0,\,M_i=1} \{Y_{j(i)}-Y_i\}.

Each retained control receives equal weight in this expression. Under exact matching on $X$, conditional exchangeability, and an appropriate design that supplies treated matches in the same covariate profiles, the treated outcomes estimate the controls’ mean $Y(1)$ while the controls’ observed outcomes provide $Y(0)$. With sufficiently close rather than exact matches, the contrast additionally depends on how well the matching controls residual covariate differences; matching does not make those differences disappear by definition.

Worked Example: Calculating a Matched-Control Contrast

In a hypothetical study, three retained controls are each matched to a treated unit with the same measured covariate profile. The observed outcomes, in symptom-score points, are:

Matched pair	Treated outcome	Control outcome	Treated minus control
1	14	10	4
2	11	9	2
3	16	10	6

The estimate is the mean of the three treated-minus-control differences: $$ \widehat{\tau}_{\mathrm{ret}} =\frac{(14-10)+(11-9)+(16-10)}{3} =\frac{4+2+6}{3} =4.000\text{ symptom-score points}. $$

This arithmetic gives the matched contrast, not proof that the causal effect is 4 points. A causal interpretation requires the assumptions stated above. In particular, exact agreement on the measured profiles does not rule out unmeasured confounding, and the estimate concerns the three retained controls rather than every control in the source population.

Diagnostics, Alternatives, and Limitations

A control-matching analysis should make the target visible and then check whether the design supports it. Report how many controls were retained and describe how their covariate distribution differs from that of all controls. Assess covariate balance between the retained controls and their treated matches, especially for covariates that may predict outcomes or modify treatment effects. Inspect whether some retained controls have no credible treated comparison. A small matched sample or visibly poor covariate comparability is a warning that the matched contrast may be unstable or confounded.

A particularly important diagnostic question is whether unmatched controls occupy covariate profiles absent from the treated group. If so, those controls cannot be recovered into the ATC simply by interpreting the matched estimate more broadly. An estimate for all controls would require additional support and assumptions, such as a defensible outcome model extrapolating to those profiles. Reporting the matched estimate as an ATC without such justification misstates the target.

The main alternative matching orientation is treated-unit matching, which aims to retain treated units and commonly targets the ATT. Control-unit matching instead retains controls and naturally targets their effect when each retained control is represented equally. An analysis that keeps or weights all controls may target the ATC, but it relies on adequate treated comparisons across the covariate distribution of all controls and on the assumptions of its adjustment method. These alternatives answer different target-population questions; none is universally preferable.

A common mistake is to reason, “The matched treated and control groups are balanced, so the estimate is the ATC.” Balance describes comparability of the groups used in the matched analysis; it does not restore controls that were excluded or make the matched controls representative of all controls. The correction is to name the estimand from the retained population first, then justify any claim that it also represents the all-control target.

Control-unit matching is most useful when the effect for controls is the substantive question and appropriate treated comparisons can be found for the relevant controls. Its limitations are consequential: it may discard many controls, change the covariate mix and effect average, and leave limited information for assessing generalizability. Even excellent measured balance cannot establish conditional exchangeability for unmeasured factors. Always report the retained-control target and describe the assumptions required for a causal interpretation.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

In one sentence, distinguish the ATC from the retained-control effect.
Among controls, suppose 75% have $X=L$ with $\tau_0(L)=2$, and 25% have $X=H$ with $\tau_0(H)=6$. What is the ATC?
Matching retains 50% $L$ controls and 50% $H$ controls in Question 2. What is the retained-control effect?
Why does covariate balance in the matched sample not by itself establish that the matched contrast estimates the ATC?
What positivity condition is relevant when targeting the retained-control effect?

Solutions. (1) The ATC averages treatment effects over all controls; the retained-control effect averages them only over controls selected into the matched analysis. (2) $\operatorname{ATC}=0.75(2)+0.25(6)=1.5+1.5=3$ outcome units. (3) $\tau_{\mathrm{ret}}=0.50(2)+0.50(6)=1+3=4$ outcome units. The targets differ because retention changes the profile distribution. (4) Balance concerns the matched groups, not excluded controls; it neither changes the target population nor establishes that retained controls represent all controls. (5) Treated units must be available at the covariate values represented among retained controls, so $P(D=1\mid X=x)>0$ for those relevant $x$.