Why Form Sets or Strata?
Matching and subclassification organize observational units into groups that are similar in their observed treatment-related characteristics. The treatment and control outcomes can then be compared within those groups. The final estimate combines these within-group contrasts, giving more influence to groups that represent more of the target population.
Full matching forms sets that each contain at least one treated unit and at least one control unit. A set can contain several treated units and one control, one treated unit and several controls, or multiple units in both groups. Propensity-score subclassification instead divides units into bins, or strata, according to their estimated propensity scores. Each retained stratum must contain both treatment groups for a within-stratum contrast to be calculated.
These are design and estimation strategies, not substitutes for the assumptions that identify causal effects. Let \(D_i\) indicate whether unit \(i\) receives treatment, let \(X_i\) be its observed pretreatment covariates, and let \(Y_i(1)\) and \(Y_i(0)\) be its potential outcomes under treatment and control. The observed outcome is \(Y_i=D_iY_i(1)+(1-D_i)Y_i(0)\). A standard identification basis is consistency, conditional exchangeability \((Y_i(1),Y_i(0))\perp D_i\mid X_i\), and positivity: each covariate profile in the target population has a positive probability of receiving either treatment. The propensity score is \(e(X_i)=P(D_i=1\mid X_i)\). Grouping on estimated propensity scores only approximates comparisons at the same score when groups are finite and scores vary within them.
Full Matching: Building Mixed-Treatment Sets
A full-matching algorithm partitions eligible treated and control units into mixed-treatment sets, often using distances based on estimated propensity scores or covariates. The sets can have different numbers of treated and control units. In practice, the chosen distance measure and any caliper constrain which units can be grouped together; the resulting sets should not be interpreted as exact matches on all covariates.
Once sets are formed, calculate a treated mean and control mean within every set, then subtract. The contrast \(\hat\tau_j\) is the building block. It is not yet a single overall effect: the set contrasts must be weighted to represent the intended target population.
Worked Example: Forming Sets and Estimating an Effect for Treated Units
Suppose a hypothetical study has estimated propensity scores and outcomes as shown below. A full-matching procedure groups units into two sets; each set contains at least one treated and one control unit.
| Set | Treated propensity scores | Control propensity scores | Treated outcomes | Control outcomes |
|---|---|---|---|---|
| 1 | 0.31, 0.34 | 0.33 | 12, 14 | 10 |
| 2 | 0.62 | 0.60, 0.65 | 9 | 7, 9 |
In set 1, the treated outcome mean is \((12+14)/2=13\), and the control mean is \(10\). Thus, \(\hat\tau_1=13-10=3\) outcome units. In set 2, the treated mean is \(9\), and the control mean is \((7+9)/2=8\), so \(\hat\tau_2=9-8=1\) outcome unit.
There are three treated units in total. To estimate the average treatment effect among treated units in these sets, weight each set contrast by its share of treated units:
The estimate is an average difference of about 2.333 outcome units for the treated units represented in the matched sets. This calculation does not by itself establish that the observational assumptions hold.
Weights Depend on the Target Population
The average treatment effect (ATE) targets the average effect for a specified population of treated and control units. The average treatment effect among the treated (ATT) targets the average effect for the treated units in a specified population. For either target, the set or stratum contrast is averaged using that target population’s distribution across groups.
If all \(J\) groups are retained and the target population is the pooled sample, the empirical share in group \(j\) is \(n_j/n\), where \(n=\sum_j n_j\). For the treated-group target, the share is \(n_{1j}/n_1\), where \(n_1=\sum_j n_{1j}\). The resulting estimates are:
These weights need not be the same. A group with many controls but few treated units can have a substantial pooled-sample share, yet a small treated-group share. More generally, if the target population has group shares \(q_j\), use \(\sum_j q_j\hat\tau_j\), with \(q_j\geq 0\) and \(\sum_j q_j=1\). The weights must be chosen from the target, not merely because they are convenient.
Worked Example: Subclassification and the ATE
Consider a hypothetical sample divided into three propensity-score strata. Each stratum contains both treatment groups, and the target is the pooled sample. The table gives the group counts and the mean outcome in each treatment group.
| Stratum | \(n_{1j}\) | \(n_{0j}\) | \(\bar Y_{1j}\) | \(\bar Y_{0j}\) |
|---|---|---|---|---|
| 1 | 20 | 30 | 8 | 6 |
| 2 | 30 | 20 | 11 | 9 |
| 3 | 50 | 50 | 14 | 13 |
The stratum contrasts are \(8-6=2\), \(11-9=2\), and \(14-13=1\) outcome units. The stratum sizes are \(50\), \(50\), and \(100\), for a total of \(n=200\). Therefore, the pooled-sample weights are \(50/200\), \(50/200\), and \(100/200\), and:
Under the stated identification assumptions and the approximation involved in using these strata, the estimated average effect for the pooled sample is 1.5 outcome units. This result gives the largest stratum twice the weight of either smaller stratum because it contains half the sample.
Why These Weighted Contrasts Work
The weighting formula follows from the definition of an average over groups. Suppose the target population has group \(j\) share \(q_j\), and let \(\tau_j\) denote its average treatment effect in that group. The population average effect is the average of the group effects, weighted by the fraction of the target population in each group:
If each group’s treated and control outcome means identify the corresponding potential-outcome means, their difference estimates \(\tau_j\). Replacing \(\tau_j\) by \(\hat\tau_j\) and the target shares \(q_j\) by the relevant sample shares gives the estimators above. For the pooled-sample ATE, those sample shares are \(n_j/n\). For the ATT, the target consists of treated units, so the shares are \(n_{1j}/n_1\).
This justification also clarifies the limitation. Conditional exchangeability given \(X_i\) does not automatically imply that treated and control units are exchangeable after placing a range of propensity scores into one broad bin. Subclassification approximates adjustment by the score: if scores vary within a stratum, treated and control units may still differ in their covariate distributions. Narrower strata can make the approximation closer, but may leave too few observations from one treatment group to calculate a stable contrast.
Worked Example: The Same Contrasts, a Different Target
Return to the three strata in the preceding example. There are \(100\) treated units: \(20\), \(30\), and \(50\) across the strata. To target the average effect among treated units, use treated-group shares rather than pooled-sample shares:
Here the ATT estimate also equals 1.5 outcome units, but that equality is specific to these stratum contrasts and weights. It does not mean the ATE and ATT are generally identical. For example, if the third stratum’s contrast were 4 rather than 1, the pooled-sample estimate would be \(0.25(2)+0.25(2)+0.50(4)=3\), whereas the treated-group estimate would be \(0.20(2)+0.30(2)+0.50(4)=3\). In this particular sample they still coincide; with different treated and pooled stratum shares they need not. The estimand is determined by the target and its weights, not by the numerical value alone.
Choosing and Reporting the Target
Full matching and subclassification can be used to estimate different effects, but the target must be stated explicitly. For a matched sample that retains only units represented in mixed-treatment sets, the pooled-sample ATE weights describe that retained sample, not necessarily the original population. Similarly, ATT weights describe the treated units retained in the sets. If units or strata are excluded, the target changes unless the analysis supplies a justified way to represent the excluded portion.
A clear report identifies how the sets or strata were formed, gives the within-group contrasts and their weights, names the target population, and states the assumptions under which the contrasts are interpreted causally. The calculation is an estimate; it does not demonstrate that exchangeability or adequate overlap holds. Statistical uncertainty is a separate part of inference and is not represented by these point-estimate formulas.
Common Mistakes
- Giving every set equal weight without justification. A simple average of set contrasts targets an equal-weight average of sets. It does not generally estimate the pooled-sample ATE or the ATT, whose weights are based on the appropriate population shares.
- Using treated shares for an ATE, or pooled shares for an ATT. The right weights follow from the target: pooled-sample shares for the sample ATE, treated-unit shares for the ATT.
- Assuming that creating strata removes confounding. A finite stratum can contain a range of propensity scores and covariate profiles. Grouping is an approximation; causal interpretation still requires the identification assumptions and credible within-stratum comparisons.
- Calling an estimate for retained units an estimate for everyone. If the procedure excludes units, the retained sample may define a narrower target population. Report that target rather than implying the estimate applies to the original population automatically.
Check Your Understanding
Work through these questions using the definitions and formulas above.
- A full-matching set contains four treated units and two controls. What are its treated and control shares for the ATT weighting scheme across all sets?
- Why must a propensity-score stratum contain both treated and control units to have a within-stratum treatment contrast?
- Three strata have contrasts of 1, 3, and 2 outcome units and pooled-sample shares of 0.2, 0.3, and 0.5. Calculate the pooled-sample ATE estimate.
- Explain why the weights for an ATT estimate can differ from those for an ATE estimate.
- What target population is represented if a matching procedure excludes units that cannot be placed in mixed-treatment sets?