Exact Matching and Stratum-Weighted Effects

Why stratum weights matter

Exact matching requires treated and control units to agree on specified covariates. This can make the comparison more transparent: each treated unit is compared only with controls in the same covariate stratum. But an exact-matching rule does not, by itself, say how much each stratum should contribute to the overall effect. That choice determines the target population represented by the aggregate.

For example, suppose effects are larger in one stratum than another. Giving more weight to the stratum where treated units are concentrated can produce a different aggregate from giving each eligible person equal representation in a defined target population. Neither result is automatically the unique “matched effect.” The stratum-specific contrasts and the weights must both be stated.

Definition: Let \(T\in\{0,1\}\) denote treatment, let \(Y(1)\) and \(Y(0)\) be the potential outcomes under treatment and control, and let \(S=f(X)\) be a stratum formed from specified pre-treatment covariates \(X\). Exact matching compares treated and control units only when they have the same value of \(S\). A stratum is shared (or eligible for exact-match comparisons) if it contains units in both treatment groups. Its stratum-specific average treatment effect is \(\tau_s=E[Y(1)-Y(0)\mid S=s]\). For a chosen target distribution with stratum proportions \(q_s\), the stratum-weighted effect is \(\tau_q=\sum_{s\in\mathcal S^*}q_s\tau_s\), where \(\mathcal S^*\) is the set of target strata, \(q_s\geq 0\), and \(\sum_{s\in\mathcal S^*}q_s=1\).

The stratum definition is part of the design. If \(X\) contains age group and baseline severity, for instance, \(S\) might be the joint category formed by those two variables. Exact matching then means equality on both categories, not merely similarity on average. If continuous covariates are grouped into bins, matching is exact on the bins, not on the original measurements; units within one bin can still differ.

Identification within shared strata

A within-stratum treated-control difference has a causal interpretation only under assumptions. First, treatment and strata must be defined before treatment, and consistency requires that the observed outcome equals the potential outcome corresponding to the treatment actually received: \(Y=T Y(1)+(1-T)Y(0)\). Assume also no interference, so one unit’s treatment does not alter another unit’s outcome. For each target stratum, conditional exchangeability requires \((Y(1),Y(0))\perp T\mid S=s\). This means that, within the stratum, treatment assignment is independent of both potential outcomes. Finally, positivity requires \(0<P(T=1\mid S=s)<1\), so both treatment groups are possible in every target stratum.

These assumptions make the observed outcomes in each treatment group representative of the corresponding potential outcomes in that stratum. Exact matching enforces agreement on \(S\), but does not establish exchangeability: unmeasured confounding or remaining variation in covariates within a coarsened stratum may still bias the contrast.

\begin{aligned} E[Y\mid T=1,S=s]-E[Y\mid T=0,S=s] &=E[Y(1)\mid T=1,S=s]-E[Y(0)\mid T=0,S=s]\\ &=E[Y(1)\mid S=s]-E[Y(0)\mid S=s]\\ &=E[Y(1)-Y(0)\mid S=s]=\tau_s. \end{aligned}

The first equality uses consistency, the second uses conditional exchangeability, and the third uses linearity of expectation. Thus, under the stated assumptions, the within-stratum observed contrast identifies \(\tau_s\). Weighting these identified stratum effects by \(q_s\) identifies the average effect for a target population whose stratum distribution is \(q_s\). The weights must describe that target; they are not a consequence of the within-stratum comparison.

Worked Example: A Target Distribution Gives the Aggregate Weight

Suppose a hypothetical study has two shared strata, A and B. In stratum A, the treated and control outcome means are 8 and 6 points; in stratum B, they are 12 and 9 points. Suppose the explicitly chosen target population has 40% of its units in A and 60% in B. Assume consistency, no interference, conditional exchangeability within each stratum, and positivity in both strata.

The estimated stratum contrasts are \(\hat{\tau}_A=8-6=2\) points and \(\hat{\tau}_B=12-9=3\) points. Applying the target proportions gives

\hat{\tau}_q=(0.40)(2)+(0.60)(3)=0.80+1.80=2.60\text{ points}.

Under the assumptions, this estimates the average effect for the stated target distribution: 40% in A and 60% in B. The 2.60-point result is not an unweighted average of the two stratum effects; equal weighting would give \((2+3)/2=2.50\) points and would represent a target with half its units in each stratum.

Weights specify the population represented

Once stratum-specific effects have been estimated, the aggregate is a weighted average. Valid weights are nonnegative and sum to one. Their meaning depends on how they are chosen. For example, weights based on the distribution of treated units target the treated population restricted to the included strata. Weights based on the distribution of controls target the corresponding control population. Weights based on the combined matched sample target that sample’s stratum composition. Alternatively, a prespecified external or study-population distribution can define a different target, provided the required stratum-specific effects are identified there.

The same set of within-stratum contrasts can therefore support several aggregate estimands. Changing \(q_s\) changes the population average being summarized; it does not change \(\tau_s\) within any stratum. If a target assigns positive weight to a stratum without both treatment groups, exact matching alone does not identify that stratum’s effect. The analyst must restrict the target to supported strata or use additional assumptions and methods to extrapolate.

Worked Example: The Treated and Control Distributions Give Different Effects

Consider two shared strata in a hypothetical study. The estimated effects are 1 point in stratum A and 5 points in stratum B. Among treated units, 25% are in A and 75% in B. Among controls, 75% are in A and 25% in B. Assume the causal identification conditions hold in both strata.

Weighting by the treated distribution gives the treated-population target:

\hat{\tau}_{T}=(0.25)(1)+(0.75)(5)=0.25+3.75=4.00\text{ points}.

Weighting by the control distribution gives a different target:

\hat{\tau}_{C}=(0.75)(1)+(0.25)(5)=0.75+1.25=2.00\text{ points}.

Both calculations use the same stratum effects, but they average them over different distributions. The difference of 2 points is an estimand shift due to composition, not evidence that the effect within either stratum changed. Without specifying a target population, reporting one of these aggregates as “the” matched effect is ambiguous.

Support, diagnostics, and failure modes

Exact matching can make unsupported comparisons visible: a stratum with only treated units or only controls cannot supply a within-stratum contrast. The target must be restricted to strata with both groups if the effect is to be estimated by these comparisons. This restriction can change the target from the original population to the population in shared strata. It does not justify extending the resulting estimate to excluded strata.

Before interpreting a stratum-weighted result, check the design and the calculation:

Confirm stratum membership. Use only the specified pre-treatment covariates and verify that units classified together really meet the matching rule.
Tabulate treatment counts by stratum. Identify empty treatment cells and very small cells. A stratum with both groups satisfies empirical overlap, but a tiny group can still produce an imprecise contrast.
Inspect balance within strata. Exact agreement on recorded categories does not guarantee balance on continuous covariates that were binned, unrecorded confounders, or other prognostic variables.
Report the weights and their source. Check that they are nonnegative, sum to one, and correspond to the stated target population.
Compare target composition with the original sample. Report how much of the target is excluded when unsupported strata are removed; large exclusions limit generalizability.

A common mistake is to calculate each stratum contrast correctly, then average the contrasts equally simply because there are several strata. Equal weighting is valid only when the intended target gives every included stratum the same proportion. Another mistake is to interpret exact agreement on a coarse category as eliminating confounding. The correction is to state the within-stratum exchangeability assumption and remember that matching controls only the covariates and resolution used to form the strata.

Worked Example: A Stratum Without Controls Limits the Target

A hypothetical study has three covariate strata. Strata A and B contain both treated and control units, with estimated contrasts of 2 and 4 outcome units. Stratum C contains treated units only. Suppose the intended target is the original population, whose proportions are 40% in A, 30% in B, and 30% in C.

The exact-match contrasts identify effects in A and B, subject to the assumptions above. There is no within-stratum control mean in C, so positivity fails there and the exact-match comparison does not identify \(\tau_C\). The original-population effect would be

\tau_{\text{original}}=(0.40)\tau_A+(0.30)\tau_B+(0.30)\tau_C,

which cannot be evaluated from the available exact-match contrasts because \(\tau_C\) is unknown. If the target is instead restricted to the shared strata, their total original-population mass is \(0.40+0.30=0.70\). The normalized weights are \(q_A=0.40/0.70=4/7\) and \(q_B=0.30/0.70=3/7\). The restricted estimate is

\hat{\tau}_{\text{shared}}=\left(\frac{4}{7}\right)(2)+\left(\frac{3}{7}\right)(4) =\frac{20}{7}\approx 2.857\text{ outcome units}.

This is an effect for the population in A and B, not for the original population including C. Assigning C an effect by silently dropping it or by reusing the shared-strata estimate would require an unsupported assumption.

How exact matching compares with alternatives

Exact matching is attractive when the covariates defining the strata have a manageable number of categories and direct within-stratum comparisons are substantively useful. It avoids comparing units that differ on those exact categories. Its main cost is sparsity: as the number of covariates or categories grows, many joint strata may contain only one treatment group or few observations. Coarsening continuous covariates can preserve more comparisons, but it weakens the guarantee of covariate agreement.

Caliper matching instead permits comparisons within a specified distance on a matching score, so it can retain units when exact joint strata are sparse; the comparison is then approximate rather than exact on those covariates. Model-based adjustment can use information across strata, but its result depends on modeling assumptions and may extrapolate where a treatment group is absent. Neither approach automatically solves confounding. The choice should reflect the covariates, support, target population, and assumptions that are defensible.

Exact matching is therefore not only a way to select comparable units. It also creates a set of stratum-specific contrasts whose aggregation must be designed. A transparent analysis reports the stratum definition, which strata were shared, the within-stratum contrasts, the weights and their target interpretation, and the population excluded by the support restriction.

Key takeaway: Exact matching determines where treated-control comparisons are made; the stratum weights determine which population the aggregate represents. Causal interpretation additionally requires consistency, no interference, within-stratum exchangeability, and positivity in every target stratum.

Check Your Understanding

Answer each question, then compare with the solutions below.

Two shared strata have effects of 2 and 6 points. The target has proportions 0.70 and 0.30 in those strata. What is the stratum-weighted effect?
In a study, one stratum has treated units but no controls. Can exact matching identify the effect in that stratum? What happens to an aggregate targeting the full population?
Why does an equal-weight average of stratum effects not generally represent the matched-sample effect?
What causal assumption is not guaranteed merely by matching exactly on recorded covariates?
Suppose treated-distribution weights are 0.20 and 0.80 across two strata, with effects of 1 and 4 units. Calculate the treated-target effect and state its interpretation.

Solutions. (1) \((0.70)(2)+(0.30)(6)=1.4+1.8=3.2\) points. (2) No: without controls, the within-stratum contrast is unavailable and positivity fails. An aggregate targeting the full population includes an unidentified stratum effect unless additional assumptions or methods provide it; restricting to shared strata changes the target. (3) Equal weights represent a target with equal stratum proportions, which need not match the matched-sample composition. (4) Conditional exchangeability may fail because of unmeasured confounding or residual differences within coarsened strata. (5) \((0.20)(1)+(0.80)(4)=0.20+3.20=3.40\) units; under the identification assumptions, this is the average effect for the treated-population stratum distribution represented by those weights.