Why Coarsen Before Matching?
Exact matching on a continuous covariate can be impractical: two people may have nearly identical ages or laboratory measurements but no values that are numerically identical. Coarsened exact matching (CEM) addresses this by mapping each continuous covariate into declared intervals, then requiring treated and control units to agree exactly on the resulting categories. It replaces exact equality of raw measurements with exact equality of coarsened values.
This flexibility has a cost. Units in the same interval can still differ on the original measurement, and the chosen cutpoints determine which units share strata. CEM is therefore a design rule for defining comparable groups, not a guarantee that all confounding has been removed or that a causal effect has been identified.
Defining Intervals and Eligible Strata
For a continuous covariate, the coarsening function is defined by a set of intervals with stated endpoints and a consistent boundary convention. For example, intervals \([35,40)\) and \([40,45)\) include age 35 but not 40 in the first interval; age 40 belongs to the second. A categorical covariate can retain its original categories or use a declared grouping of categories.
When several covariates are coarsened, the stratum is their joint profile, not a separate match on each covariate considered independently. A treated unit and a control unit share a stratum only if every coarsened covariate value agrees. In count notation, define \(n_{1s}\) and \(n_{0s}\) as the numbers of treated and control units in stratum \(s\). Then stratum \(s\) is eligible precisely when:
The retained treated count is the sum of \(n_{1s}\) over eligible strata; the retained control count is the corresponding sum of \(n_{0s}\). These counts do not imply one-to-one pairs. CEM identifies shared strata; how to use units within those strata for an analysis is a separate decision.
Worked Example: Identifying Shared Strata on Age
Suppose a hypothetical study has four treated and four control units. Age, measured in years, is coarsened into \([30,40)\), \([40,50)\), and \([50,60)\).
| Unit | Treatment status | Age | Coarsened age stratum |
|---|---|---|---|
| A | Treated | 34 | [30,40) |
| B | Treated | 47 | [40,50) |
| C | Treated | 53 | [50,60) |
| D | Treated | 38 | [30,40) |
| E | Control | 36 | [30,40) |
| F | Control | 44 | [40,50) |
| G | Control | 55 | [50,60) |
| H | Control | 57 | [50,60) |
The \([30,40)\) stratum contains two treated units (A and D) and one control (E), so it is eligible. The \([40,50)\) stratum contains one treated unit (B) and one control (F), so it is also eligible. The \([50,60)\) stratum contains one treated unit (C) and two controls (G and H), so it is eligible as well.
All four treated units and all four controls are retained. This does not mean that each treated unit has a unique control partner: the first and third strata have unequal group counts. Nor does it mean that units within each interval have identical ages. For example, ages 53, 55, and 57 share a coarsened stratum but are not equal.
What the Retention Rule Guarantees
The central justification for the rule is direct: if a stratum has no treated units or no controls, it cannot provide a within-stratum comparison between treatment groups. Removing its units prevents the design from treating the absence of one group as if a comparison were available. Conversely, if both counts are positive, units from the two groups share the declared coarsened profile and are eligible for a within-stratum comparison.
This is a statement about the design, not a causal result. For causal interpretation, assumptions such as consistency and no interference are still needed. In addition, an observational comparison requires adequate control of confounding. If the analysis relies on comparisons within coarsened strata alone, it needs conditional exchangeability of potential outcomes between treatment groups given the stratum, an assumption that coarsening may not make plausible. Matching on the same bins does not prove that assumption.
A separate overlap condition is visible in the data: both treatment groups must be represented in the strata relevant to the comparison. CEM enforces this empirical requirement for retained strata by construction. It does not establish overlap for units discarded from the design, or guarantee that the retained units represent the original study population.
Worked Example: How Wider Intervals Change Retention
Consider hypothetical age data with two treated units, aged 39 and 41, and two controls, aged 40.5 and 44. First use the intervals \([35,40)\) and \([40,45)\). The treated unit aged 39 is alone in the lower interval, so that stratum is ineligible. The treated unit aged 41 and both controls are in the upper interval, so that stratum is eligible.
The treated retention fraction is \(1/2=0.50\): one of two treated units is retained. The control retention fraction is \(2/2=1.00\): both controls are retained. The overall fraction retained is \(3/4=0.75\), but that is not the treated retention fraction.
Now use the wider interval \([35,45)\). All four units fall into the same stratum, which is eligible. Treated retention rises from \(1/2\) to \(2/2\), while control retention remains \(2/2\). Overall retention rises from \(3/4\) to \(4/4\).
The wider interval retains more treated units here, but it also groups the age-39 treated unit with controls aged 40.5 and 44. The resulting shared category allows larger within-stratum age differences. Thus, retention alone is not sufficient to choose a coarsening rule; the remaining covariate differences must also be examined.
Coarsening Multiple Covariates
With more than one covariate, the joint profile can create many strata. This permits a transparent rule for requiring agreement on several characteristics, but it also makes empty or one-group-only strata more likely. The following example shows why each covariate must be considered jointly.
Worked Example: Joint Age and Baseline-Score Strata
Suppose age is coarsened into \([40,50)\) and \([50,60)\), and a baseline score is grouped as “low” or “high,” with the grouping rule specified before matching. The hypothetical units have these profiles:
| Unit | Status | Age | Score category | Joint stratum |
|---|---|---|---|---|
| A | Treated | 44 | Low | [40,50), low |
| B | Treated | 46 | High | [40,50), high |
| C | Treated | 53 | Low | [50,60), low |
| D | Control | 48 | Low | [40,50), low |
| E | Control | 49 | Low | [40,50), low |
| F | Control | 45 | High | [40,50), high |
| G | Control | 56 | High | [50,60), high |
The \([40,50)\), low stratum has one treated unit (A) and two controls (D and E), so all three are retained. The \([40,50)\), high stratum has one treated unit (B) and one control (F), so both are retained. The \([50,60)\), low stratum contains treated unit C but no controls, so C is excluded. The \([50,60)\), high stratum contains control G but no treated units, so G is excluded.
The retained counts are three treated units out of three and three controls out of four. If age were considered alone, C and G would share the \([50,60)\) age category and both would be retained. The joint rule excludes them because their score categories differ. This is exactly what exact agreement on the full coarsened profile means.
Choosing Coarsening Rules and Checking the Result
Coarsening decisions should be stated clearly and made without using outcome information to search for a favorable effect estimate. Useful rules may reflect meaningful thresholds, established clinical categories, or a prespecified balance goal. Whatever the rationale, report interval endpoints, boundary conventions, categorical groupings, and the covariates included in the joint profile. If cutpoints are selected after inspecting the outcome, the design decision can become outcome-driven and make the resulting analysis difficult to interpret.
After forming strata, inspect the retained and excluded counts by treatment group and by stratum. Also assess the original, uncoarsened covariates among retained units. Sharing a bin guarantees equality only of the coarsened label: the values within that bin may still differ substantially. Compare distributions or summaries of the original measurements, rather than relying only on a table of matching categories.
- Many single-group strata: this signals that the joint coarsening may be too fine for the available sample or that treatment groups have limited empirical overlap.
- Large within-bin differences: this suggests that the intervals may be too wide to support the desired comparability.
- Unequal retention: report treated and control retention separately. An overall retention percentage can conceal that one group lost a larger share of its units.
- Sensitivity to cutpoints: compare reasonable alternative coarsenings and note whether eligible strata and retained units change materially.
These checks inform design decisions but do not certify causal validity. In particular, balance on measured covariates cannot rule out unmeasured confounding, and discarding units can change the population to which a result applies.
How CEM Differs From Other Matching Rules
| Approach | What defines comparability? | Main trade-off |
|---|---|---|
| Exact matching on key covariates | Equal values on the selected covariates | Strict equality can leave few shared profiles, especially for continuous values. |
| Coarsened exact matching | Equal declared categories or intervals for every selected covariate | More potential overlap than raw exact matching, but differences remain within bins. |
| Nearest-neighbor matching | Closeness under a specified distance or score | Can use nearby rather than identical profiles; the distance rule affects partner selection. |
| Caliper matching | A specified distance restriction that rules out pairs beyond a threshold | Allows variation within the caliper and may leave units without eligible partners. |
CEM is especially useful when investigators want an auditable rule: each unit’s coarsened profile either agrees with another treatment group’s profile or it does not. Its principal limitation is that interval choices control both retention and the resolution of within-stratum similarity. A very fine rule can discard many units; a very broad rule can retain units whose original covariate values are far apart.
Common Mistake
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A treated unit and a control unit have the same coarsened age but different coarsened score categories. Do they share a joint CEM stratum? Explain.
- A stratum contains three treated units and no controls. How many units from it are retained, and why?
- In a design, 8 of 10 treated units and 15 of 20 controls are retained. Calculate each group’s retention fraction and the overall retention fraction.
- State one benefit and one limitation of making continuous-covariate intervals wider.
Solutions. (1) No. A joint stratum requires agreement on every coarsened covariate, including score. (2) None; the stratum is ineligible because \(n_{0s}=0\), so it contains no within-stratum treated-control comparison. (3) Treated retention is \(8/10=0.80\), or 80%; control retention is \(15/20=0.75\), or 75%. Overall retention is \((8+15)/(10+20)=23/30\approx0.767\), or about 76.7%. (4) Wider intervals may increase retention and create more shared strata, but they allow larger differences in the original measurements within each stratum.