A Worked Example of Identification by Stratification

From Stratum-Specific Outcomes to a Causal Contrast

When treatment groups contain different mixes of people, their crude outcome averages compare both treatment status and covariate composition. Stratification makes that composition visible: calculate an outcome mean for each treatment group within each covariate stratum, then average those means using one common set of stratum weights. The result is a pair of treatment-specific standardized means and, by subtraction, a standardized contrast.

This tutorial works through that calculation. The weights determine the population to which the contrast refers, so they are not a cosmetic detail. The introductory material on covariate standardization gives the general identification argument; here the emphasis is on carrying it out from a table and checking what the resulting number means.

For a sample representing the target population, a common choice is \(w_x=\hat{P}(X=x)=N_x/N\), where \(N_x\) is the number of sample members in stratum \(x\) and \(N\) is the total sample size. These weights make both treatment-specific means refer to the covariate distribution of the whole sample. Other target populations can call for different weights, but the intended target must be stated.

When the Standardized Contrast Is Causal

The arithmetic produces a standardized association whether or not the assumptions for causal interpretation hold. For a causal interpretation, suppose the treatment is well-defined, \(Y(a)\) denotes the potential outcome under treatment state \(a\), and the following conditions apply to the target population:

• Consistency: for a person who receives treatment \(A=a\), the observed outcome equals that person’s potential outcome under that treatment: \(Y=Y(a)\).
• Conditional exchangeability: within each stratum \(x\), treatment assignment is independent of the potential outcomes: \(Y(a)\perp A\mid X=x\), for \(a=0,1\).
• Positivity: each stratum with positive weight in the target population contains individuals who could receive either treatment; in the observed data, both treatment groups must be represented to calculate both stratum means.

Under these conditions, the population mean potential outcome under treatment state \(a\) can be expressed using observed outcomes within strata:

The first equality averages over the target distribution of \(X\). Conditional exchangeability gives the second equality, and consistency gives the third. In a table, the within-stratum observed means estimate the conditional expectations, while the chosen weights represent the target distribution. Subtracting the standardized mean for \(a=0\) from the standardized mean for \(a=1\) estimates the average treatment effect in that target population. This is an identification argument, not a guarantee that the assumptions are true.

For the calculation to match this argument, the strata must be defined using pretreatment covariates and the same weights must be used for both treatment-specific means. The outcome scale also matters: if the outcome is measured in days, both means and their difference are in days. The calculation itself does not provide a standard error or confidence interval.

Reading the Table Before Calculating

A reliable calculation starts by identifying three pieces of information: which variables define the strata, the target-population share of each stratum, and the outcome mean for each treatment state within each stratum. A table may report counts, means, or both. Counts are useful for checking the weights and for seeing whether a cell is empty; the stratum mean and the target weight are the quantities used directly in standardization.

The target weights need not equal the proportion of treated participants in each stratum or the proportion of controls in each stratum. Those are treatment-specific compositions. If the goal is to compare treatment-specific means for the full sample, use the full-sample composition for both. Also verify that the weights sum to one. If they do not, check whether a stratum was omitted or whether the counts use a different target population than intended.

Weights Define the Target

The same stratum-specific means can produce different standardized means if the target population has a different distribution of covariates. That is not a contradiction: the average causal effect can vary across populations when treatment effects differ across strata. State the target distribution whenever reporting a standardized contrast.

Checks, Alternatives, and Common Errors

Before interpreting a result, check the calculation and its causal basis. Confirm that the stratum weights correspond to the named target population and sum to one; confirm that each cell mean is paired with the correct treatment label; and verify that every positively weighted stratum has observations in both treatment groups. A missing treatment-by-stratum cell means its corresponding observed mean cannot be computed from the table. That is a warning that the standardized calculation is not directly supported for that target, not a reason to silently assign the missing mean a value.

Also consider whether the covariates defining the strata were measured before treatment. Conditioning on a variable affected by treatment can change the causal question and does not serve as ordinary pretreatment adjustment. Conditional exchangeability is a substantive assumption: a balanced-looking table of measured strata cannot establish that all confounding has been removed.

The crude difference in means is simpler but leaves treatment groups with their observed, potentially different covariate compositions. Regression adjustment is another way to adjust for covariates, but its result can depend on model specification. Matching constructs a comparison using treated and control units with similar covariates and can target a different population. These approaches are not interchangeable merely because each is called “adjustment”; identify the target estimand and assumptions before comparing their results.

• Pitfall: averaging stratum effects without weights. The unweighted average gives every stratum equal importance, even when the target population has very different proportions in those strata. Use \(\sum_x w_x(\bar{Y}_{1,x}-\bar{Y}_{0,x})\) with weights chosen for the target.
• Pitfall: using different weights for treatment and control. This compares means for different covariate compositions. Use the same \(w_x\) in both sums if the intended contrast standardizes both treatment states to one target distribution.
• Pitfall: calling every adjusted contrast causal. The weighted arithmetic alone cannot establish exchangeability, consistency, or positivity. Describe the computed result as a standardized contrast unless its causal assumptions are defended.

A standardized point estimate also does not communicate sampling uncertainty by itself. Inference requires an appropriate variance calculation that reflects how the data and weights were obtained; do not read precision or statistical significance from the weighted mean alone.