Averaging Counterfactual Predictions Over the Target Population

Why Average Predictions Over a Population?

A fitted outcome model can produce a predicted outcome under a specified treatment regime for a person with given baseline characteristics. But a causal question usually concerns a population, not one person: for example, the mean outcome if everyone in an eligible clinic population followed a particular treatment policy. Empirical standardization answers that question by predicting for units in a defined target population and averaging their predictions.

The choice of target population is part of the estimand. Averaging predictions over the analysis sample describes that sample’s baseline-characteristic distribution; averaging over a separate target sample describes that target sample’s distribution. The same conditional outcome model can therefore produce different population means when applied to different target populations.

Identification Before Prediction

A prediction is not automatically a causal prediction. To interpret \(m_d(w)\) as a mean under an intervention, the causal assumptions used for the longitudinal g-formula must apply. In outline, consistency says that a person’s observed outcome equals the counterfactual outcome for the treatment history actually received. Sequential exchangeability says that, at each treatment time, treatment is conditionally independent of future regime-specific outcomes given the measured past history. Positivity requires that the treatment choices required by \(d\) have positive probability at histories represented in the target population.

These assumptions identify regime-specific means from observed data, but an empirical standardization also relies on estimating the required conditional outcome distributions or means adequately. If the target population differs from the model-fitting population, the conditional causal response model must be applicable to that target, and the target covariate profiles must be supported by the data used to fit it. A model cannot reliably provide evidence for covariate profiles absent from its support.

In a longitudinal setting, \(m_d(w)\) must account for intermediate covariates that evolve after treatment begins. The g-formula does this by averaging over their distributions as they would arise under the specified regime, while applying the relevant conditional outcome models. It is generally not enough to set treatment values to those in \(d\) while leaving each person’s observed post-baseline covariate history fixed: that history may itself have been affected by observed treatment.

The Empirical Standardization Calculation

Suppose the target population is represented by \(M\) units with baseline covariates \(W_1,\ldots,W_M\). A fitted sequential outcome-model procedure supplies a prediction \(\hat m_d(W_i)\) for each unit: the estimated conditional mean outcome if that unit’s baseline profile were followed under regime \(d\), with the longitudinal g-formula’s averaging over intermediate histories carried out as needed. The empirical standardized mean is

If target units represent the target population unequally, use normalized weights \(q_i\), with \(q_i\geq 0\) and \(\sum_{i=1}^{M}q_i=1\):

For a contrast between regimes \(d_1\) and \(d_0\), predict under each regime for the same target units and subtract the two target means. This yields an estimate of the target-population mean contrast, not a contrast between two separately selected populations.

Worked Examples

Why the Standardization Formula Works

The justification is the law of total expectation applied to the regime-specific outcome. For a target population with baseline distribution \(Q\), partition or condition on baseline covariates:

Thus, once the conditional regime-specific mean \(m_d(w)\) is identified and estimated, the target mean is obtained by averaging it over the target distribution of \(W\). Empirical standardization replaces the expectation over that distribution with an empirical average (or a weighted empirical average when units represent different amounts of the target population). This calculation concerns the population mean; it does not require the fitted prediction for a given person to equal that person’s eventual outcome.

For a finite set of target profiles, the distinction can be made explicit. If outcomes are treated as stochastic conditional on baseline covariates, the average realized counterfactual outcome decomposes as

Predictions of the conditional means address the first term. The second term is the average deviation of realized counterfactual outcomes from their conditional expectations; averaging conditional predictions does not generally reveal it. This is why a statement about a mean conditional expected outcome must not be rewritten as a claim that the fitted values recover the target units’ realized counterfactual outcomes.

Common Mistakes and Practical Choices

• Wrong: “Each fitted value is that person’s counterfactual outcome.” A conditional prediction is an estimated expected outcome under the regime, given the modeled information. Correction: describe the average as a standardized mean of conditional predictions, not as the realized average of unobserved outcomes.
• Wrong: “The analysis sample always defines the target.” The target may instead be an eligible population, a clinic, or another population represented by target covariates. Correction: state the target population and use its covariate distribution in the averaging step.
• Wrong: “For each person, use the observed intermediate covariates and change only treatment.” Intermediate covariates may be affected by earlier treatment. Correction: obtain regime-specific predictions by averaging over intermediate histories as required by the longitudinal g-formula.
• Wrong: “A conditional model can be averaged into a valid target mean even when the target has unsupported covariate profiles.” Such predictions may be extrapolations. Correction: check target support and whether the conditional response model is applicable to the target.

A useful report states the regime, outcome and units, target population, method used to represent its baseline-covariate distribution, and resulting standardized mean or contrast. These details make clear which population the estimate describes. The standardized calculation itself does not remove uncertainty from estimating the models or from sampling; those are separate inferential questions.