Why Standardize Across Strata?
Treatment and control groups can have different distributions across propensity-score strata. If outcome levels also differ across strata, a crude comparison can combine two influences: differences in outcomes within strata and differences in which strata contribute to each treatment group. Direct standardization separates these influences by applying the same chosen stratum proportions to the stratum-specific potential-outcome means under each treatment condition.
The result is a pair of marginal means for a specified target population, followed by their difference. The target matters: study-population proportions answer a question about the study population, while different proportions define a different target and hence a potentially different marginal contrast.
The weights \(q_s\) are proportions in the target population, not automatically the proportions among treated units, the proportions among controls, or equal weights for each stratum. The standardized contrast is a weighted average of stratum-specific causal contrasts, with weights determined by the target's distribution across strata. Its units are the units of the outcome.
Identification and Calculation
Potential outcomes are not both observed for any one unit. To estimate the stratum-specific means from observed data, the comparison of treated and control units within a stratum must identify the corresponding potential-outcome means. A precise sufficient set of conditions for this stratified calculation is:
- Consistency: the observed outcome is \(Y=Y(A)\), where \(A\) is the treatment received.
- Within-stratum exchangeability: \((Y(1),Y(0))\) is independent of \(A\) conditional on \(S\). In particular, treatment status within a stratum does not select units with different potential-outcome means.
- Positivity in target strata: for every stratum with \(q_s>0\), both treatment conditions occur with positive probability, so both stratum-specific potential-outcome means can be estimated.
- Target relevance: the weights \(q_s\) correctly describe the target population. If the target is external to the study, the stratum-specific potential-outcome means estimated in the study must also apply to that target, or an appropriate transportability assumption is needed.
A propensity-score stratum is usually an interval of estimated propensity scores, not a single exact score. Stratifying on intervals does not automatically guarantee exchangeability within each stratum. The assumption must be considered in light of the covariates, the stratum construction, and how treatment groups compare within strata. Standardization changes the distribution of strata used in aggregation; it does not remove confounding that remains within a stratum.
For a study sample, let \(n_{1s}\) and \(n_{0s}\) be the numbers treated and control in stratum \(s\), and let \(\bar Y_{1s}\) and \(\bar Y_{0s}\) be their observed outcome means. Under the conditions above, these means estimate \(\mu_{1s}\) and \(\mu_{0s}\). If the target is the study population, a natural estimate of its stratum proportion is \(q_s=n_s/n\), where \(n_s=n_{1s}+n_{0s}\) and \(n=\sum_s n_s\). For another target, use that target's stratum proportions instead, if the necessary outcome transportability conditions are defensible.
The formula follows by conditioning each target-population potential-outcome mean on \(S\). For treatment condition \(a\), the law of total expectation gives \(E_T[Y(a)]=\sum_s P_T(S=s)E_T[Y(a)\mid S=s]\). Substituting \(q_s=P_T(S=s)\) and the stratum-specific mean \(\mu_{as}\) yields the standardized mean. Subtracting the control mean from the treatment mean gives the weighted sum of within-stratum contrasts. Identification allows the observed treated or control mean in each stratum to estimate its corresponding potential-outcome mean.
Worked Calculations
Worked Example: Standardizing to the Study Population
Suppose a hypothetical study has 60 units in stratum 1 and 40 in stratum 2. The target is the study population, so its stratum proportions are \(q_1=60/100=0.60\) and \(q_2=40/100=0.40\). There are treated and control units in both strata. Their hypothetical observed outcome means, measured in points, are:
| Stratum | Study proportion | Treated mean | Control mean | Within-stratum contrast |
|---|---|---|---|---|
| 1 | 0.60 | 8 | 6 | 2 |
| 2 | 0.40 | 20 | 17 | 3 |
The standardized treated mean is \(0.60(8)+0.40(20)=4.8+8.0=12.8\) points. The standardized control mean is \(0.60(6)+0.40(17)=3.6+6.8=10.4\) points. Thus
Under the identification assumptions, this estimates a study-population average causal contrast of 2.40 points. The weights match the stated target: the study has 60 of 100 units in stratum 1 and 40 of 100 in stratum 2. This calculation does not assert that every unit's effect is 2.40 points; it averages the stratum-specific contrasts according to the study population's stratum composition.
Worked Example: A Different Target Population
Consider hypothetical stratum-specific means from a study: in stratum 1, the treated mean is 10 and the control mean is 7; in stratum 2, they are 18 and 14. Suppose the study has equal numbers in the two strata, but the explicitly specified external target has proportions \(q_1=0.75\) and \(q_2=0.25\). Assume the stratum-specific potential-outcome means apply to this external target and that both treatment conditions are represented in each stratum.
For the external target, the standardized treated mean is \(0.75(10)+0.25(18)=7.50+4.50=12.0\) points. The standardized control mean is \(0.75(7)+0.25(14)=5.25+3.50=8.75\) points. The marginal contrast is \(12.0-8.75=3.25\) points, equivalently \(0.75(10-7)+0.25(18-14)=2.25+1.00=3.25\) points.
These are not study-population weights: the study has proportions 0.50 and 0.50. If the target were instead that study population, the contrast would be \(0.50(3)+0.50(4)=3.50\) points. The two answers differ because they describe different target populations, not because one is an arithmetic correction to the other. The external-target interpretation additionally depends on the stated transportability assumption.
Worked Example: Three Strata and Unequal Effects
Suppose a hypothetical target population has stratum proportions \(q_1=0.20\), \(q_2=0.50\), and \(q_3=0.30\). Estimated treated and control means, in hours, are respectively 5 and 4 in stratum 1, 8 and 6 in stratum 2, and 12 and 9 in stratum 3. Assume valid within-stratum identification and adequate treatment representation in all three target strata.
The stratum contrasts are 1, 2, and 3 hours. The standardized treated mean is \(0.20(5)+0.50(8)+0.30(12)=1.0+4.0+3.6=8.60\) hours. The standardized control mean is \(0.20(4)+0.50(6)+0.30(9)=0.8+3.0+2.7=6.50\) hours. Therefore the standardized contrast is \(8.60-6.50=2.10\) hours, also \(0.20(1)+0.50(2)+0.30(3)=0.20+1.00+0.90=2.10\) hours. The largest stratum effect receives only 30% weight because that is its share of the specified target.
Diagnostics, Alternatives, and Limitations
Before interpreting a standardized contrast, check the data and the target definition. The stratum proportions must be nonnegative and sum to one. Confirm whether they represent the study population or another explicitly named population. Within each stratum with positive target weight, verify that both treated and control observations are present and that their counts are not so small that the means are unstable. Review covariate balance within strata: large residual differences in prognostic covariates suggest that within-stratum exchangeability may be doubtful. Also inspect whether stratum-specific contrasts vary substantially; that variation makes the choice of target weights especially consequential.
Direct standardization is related to, but distinct from, several alternatives. A crude treated-versus-control difference uses the observed treatment groups' own stratum mixtures, which can differ and therefore does not generally target the same marginal contrast. Inverse probability weighting instead reweights individual observations using treatment probabilities; direct standardization aggregates stratum-specific outcome means using target stratum proportions. Both approaches can target a marginal causal contrast, but their finite-sample behavior and vulnerabilities differ. A regression-based standardization approach predicts potential-outcome means for individuals and averages those predictions over a target population; it can standardize over richer covariate patterns, but depends on its outcome model. Subclassification first forms strata and estimates within-stratum comparisons; direct standardization specifies how those stratum-level quantities are combined for a chosen target.
Standardization is not a remedy for empty treatment groups, severe lack of overlap, or unmeasured confounding. If a target stratum has \(q_s>0\) but no treated or no control units, its two potential-outcome means cannot both be estimated from the study without additional modeling or assumptions. Omitting that stratum changes the target. Likewise, using a more finely divided set of strata may improve within-stratum comparability but leave sparse cells; using fewer strata may conceal meaningful covariate imbalance. The causal interpretation rests on identification assumptions, not on the weighted arithmetic alone.
Common Mistakes
- Using the wrong target weights. Equal weights or treated-group proportions are not automatically appropriate. Use the stated target population's proportions.
- Mixing target definitions and sample counts. If the study contains 50% of its units in each stratum, study-population weights are 0.50 and 0.50. Weights of 0.75 and 0.25 are valid only for a target with those proportions, such as a specified external population.
- Ignoring treatment-effect heterogeneity. When stratum contrasts differ, changing \(q_s\) can change the marginal contrast. Report the target alongside the estimate.
- Claiming standardization removes all confounding. It reweights stratum-specific means across strata; residual confounding within strata can remain.
- Averaging stratum contrasts without weights. An unweighted average targets no usual population unless equal stratum proportions define the target.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A target has proportions 0.25 and 0.75 across two strata. The stratum contrasts are 2 and 6 points. What is the standardized contrast?
- A study has 80 units in stratum 1 and 20 in stratum 2. What weights standardize to the study population? Would weights 0.50 and 0.50 describe that target?
- In one target stratum, no control units were observed, although its target proportion is positive. Which assumption or requirement is problematic for the direct estimate, and what would dropping the stratum do?
- Why can a crude contrast differ from the standardized contrast even when both use the same study data?
Solutions. (1) \(0.25(2)+0.75(6)=0.50+4.50=5.00\) points. (2) The study weights are \(80/100=0.80\) and \(20/100=0.20\); weights 0.50 and 0.50 describe an equal-stratum target, not this study population. (3) Positivity fails empirically for estimating both means in that target stratum. Dropping it and renormalizing the remaining weights changes the target population rather than recovering the original target contrast. (4) The crude contrast combines treated and control observations using their own potentially different stratum mixtures; standardization applies the same target mixture to both arms, so the difference may change when outcome levels vary across strata.