Standardization for Transporting Treatment Effects

Why Standardize Treatment Effects?

A treatment effect observed in one population need not describe the effect in another. The populations may differ in covariates that predict outcomes or modify treatment effects. For example, a source trial may include relatively few patients with severe disease, while the intended treatment decision concerns a target population with more severe disease. A single source-population average can then misrepresent the target effect, even if the conditional outcome relationships apply in both populations.

Standardization, also called g-computation in this setting, addresses this problem by predicting outcomes under each treatment for covariate profiles represented in the target population, then averaging those predictions over the target covariate distribution. It is a model-based way to calculate the target-population effect; it does not make the source and target populations interchangeable without assumptions.

Definition: Let \(S=1\) denote membership in the source population, where treatment and outcomes are observed, and \(S=0\) denote membership in the target population. Let \(A\in\{0,1\}\) be treatment, \(X\) the covariates used for adjustment, and \(Y^a\) the outcome that would be observed under treatment \(a\). Define \(m_a^S(x)=E(Y\mid A=a,X=x,S=1)\). The standardized target mean under treatment \(a\) is \(\mu_a^T=E_T\{m_a^S(X)\}\), where \(E_T\) averages over the target population’s covariate distribution. The standardized target average treatment effect is \(\Delta^T=\mu_1^T-\mu_0^T=E_T\{m_1^S(X)-m_0^S(X)\}\).

For a binary outcome, \(\mu_a^T\) is a target-population risk and \(\Delta^T\) is a risk difference. For a continuous outcome, \(\mu_a^T\) is a target-population mean and \(\Delta^T\) is a difference in means, in the outcome’s units. The direction of subtraction is treatment 1 minus treatment 0 throughout.

When Does the Standardized Quantity Identify a Target Effect?

The formula is an identification result only when its assumptions connect the observed source data to the target counterfactual outcomes. The covariates \(X\) must be measured comparably in the two populations and must capture the relevant differences for transporting the treatment effect. In particular, if a covariate changes the conditional treatment effect and its distribution differs between populations, it must be represented in the standardization or otherwise addressed.

Consistency: a person receiving treatment \(a\) has observed outcome \(Y=Y^a\), with treatment versions sufficiently well defined for this equality to be meaningful.
Treatment exchangeability in the source: \(Y^a\) is independent of \(A\) conditional on \(X\) among source participants, for each \(a\). Random assignment can support this condition; in an observational source it requires adequate control of treatment-outcome confounding.
Transport exchangeability: \(E(Y^a\mid X=x,S=0)=E(Y^a\mid X=x,S=1)\) for covariate profiles relevant to the target. This mean condition is sufficient for the mean effect defined here; full equality of outcome distributions is not required.
Treatment positivity in the source: within relevant source covariate profiles, both treatment strategies have a positive probability of being observed. Otherwise the conditional outcome mean for one treatment is not supported by data there.
Population overlap (transport positivity): target covariate profiles must be represented in the source data well enough to estimate the required conditional means. This is the overlap concern in bridging populations, not a property that a more elaborate outcome model can guarantee.
Correct estimation: the fitted conditional outcome models, or another method used to estimate their means, must be adequate over the target covariate profiles. Correct model specification is not itself a causal assumption, but model misspecification can make the estimated standardization inaccurate.

The source and target must also refer to compatible treatment strategies, outcome definitions, and follow-up times. Standardizing a source model for a different endpoint or treatment regimen does not resolve that mismatch. If the target covariate distribution is estimated from a sample, that sample must represent the population named by the target estimand.

Identification: Why Averaging Works

The result follows by conditioning on the target covariates and then applying transport exchangeability, source treatment exchangeability, and consistency. The sequence matters: first express the target mean as an average of conditional counterfactual means; then use assumptions to replace those unobserved means with source quantities that can be estimated.

\begin{aligned} E(Y^a\mid S=0) &= \int E(Y^a\mid X=x,S=0)f_T(x)\,dx\\ &= \int E(Y^a\mid X=x,S=1)f_T(x)\,dx\\ &= \int E(Y\mid A=a,X=x,S=1)f_T(x)\,dx\\ &= E_T\{m_a^S(X)\}. \end{aligned}

The first equality is the law of total expectation, with \(f_T(x)\) denoting the target covariate distribution. The second uses transport exchangeability. For the third, source treatment exchangeability equates the conditional mean under treatment assignment with the conditional counterfactual mean, and consistency links that observed outcome to \(Y^a\). Applying the result to \(a=1\) and \(a=0\), then subtracting, identifies \(\Delta^T\). Positivity is needed so the conditional source means in this calculation are estimable for the profiles being averaged.

Estimating the Standardized Effect

In practice, fit or otherwise estimate \(m_0^S(x)\) and \(m_1^S(x)\) using source data. Then predict both conditional means for each target covariate profile and average each set of predictions across the target. If the target covariate data are \(X_1^T,\ldots,X_{n_T}^T\), the empirical standardization is

\widehat{\mu}_a^T=\frac{1}{n_T}\sum_{i=1}^{n_T}\widehat{m}_a^S(X_i^T), \qquad \widehat{\Delta}^T=\widehat{\mu}_1^T-\widehat{\mu}_0^T.

The same target profiles are used for both treatment predictions. This creates a comparison over a common target covariate distribution rather than comparing treatment and control groups with different covariate mixes. When the target distribution is known in strata, the sum can instead use the known stratum proportions. For a continuous target distribution, integration may be done analytically when possible or approximated by averaging over a representative target sample.

Worked Example: Standardizing a Binary Outcome From a Logistic Model

Hypothetical data: Suppose a source trial supports the fitted conditional risk model \(\operatorname{logit}\{m_a^S(x)\}=-1.4+1.2x-0.85a+0.45ax\), where \(X=1\) indicates a baseline risk factor and \(X=0\) indicates its absence. The target population has 25% with \(X=0\) and 75% with \(X=1\). Assume treatment was randomized in the source, the risk factor captures the relevant effect modification and population differences, both target profiles occur in both source treatment groups, and the logistic model adequately estimates source risks for those profiles.

The inverse-logit function is \(\operatorname{expit}(z)=1/(1+\exp(-z))\). Evaluating the model for each treatment and covariate profile gives

\begin{aligned} m_0^S(0)&=\operatorname{expit}(-1.4)=0.1978161, & m_0^S(1)&=\operatorname{expit}(-0.2)=0.4501660,\\ m_1^S(0)&=\operatorname{expit}(-2.25)=0.0953495, & m_1^S(1)&=\operatorname{expit}(-0.6)=0.3543437. \end{aligned}

Average the unrounded predicted risks using the target proportions:

\begin{aligned} \widehat{\mu}_0^T &=(0.25)(0.1978161)+(0.75)(0.4501660) =0.0494540+0.3376245 =0.3870785\approx0.3871,\\ \widehat{\mu}_1^T &=(0.25)(0.0953495)+(0.75)(0.3543437) =0.0238374+0.2657578 =0.2895951\approx0.2896,\\ \widehat{\Delta}^T &=0.2895951-0.3870785 =-0.0974834\approx-0.0975. \end{aligned}

Under the fitted model, the estimated target risk is about 38.71% under control and 28.96% under treatment. The standardized risk difference is approximately \(-0.0975\), or 9.75 percentage points lower risk under treatment for this target population. This is a model-based estimate conditional on the assumptions above, not evidence by itself that the effect will transport.

Worked Example: Standardizing a Binary Risk Difference by Stratum

Hypothetical data: Suppose source estimates are available for two covariate strata, and the target population is 40% in stratum 1 and 60% in stratum 2. Assume treatment exchangeability in the source, transport exchangeability within these strata, positivity, and adequate estimation of each listed risk. The outcome is an adverse event.

Stratum	Target proportion	Control risk	Treatment risk
1	0.40	0.3000	0.1800
2	0.60	0.5000	0.4433

Use the target proportions to average each treatment-specific risk. Retaining the unrounded stratum 2 treatment risk of \(0.4433333\) in the calculation:

\begin{aligned} \widehat{\mu}_0^T &=(0.40)(0.3000)+(0.60)(0.5000) =0.1200+0.3000=0.4200,\\ \widehat{\mu}_1^T &=(0.40)(0.1800)+(0.60)(0.4433333) =0.0720+0.2660=0.3380,\\ \widehat{\Delta}^T &=0.3380-0.4200=-0.0820. \end{aligned}

The standardized target risk difference is \(-0.0820\), or 8.2 percentage points lower risk under treatment. The two stratum-specific risk differences are \(-0.1200\) and approximately \(-0.0567\); their target-weighted average is \((0.40)(-0.1200)+(0.60)(-0.0567)\approx-0.0820\). The target effect is not an unweighted average of the stratum effects.

Worked Example: Standardizing a Continuous Outcome

Hypothetical data: Consider a symptom score measured at a fixed follow-up, with lower scores indicating fewer symptoms. Suppose fitted source conditional means are \(m_0^S(x)=20+1.5x\) and \(m_1^S(x)=17+1.2x\), where \(X\) is baseline severity measured in score units. Assume treatment exchangeability, transport exchangeability conditional on \(X\), positivity, and adequate linear conditional-mean models over the target range. The target population has mean baseline severity \(E_T(X)=8\).

Because both fitted means are linear in \(X\), their target averages depend only on the target mean of \(X\):

\begin{aligned} \widehat{\mu}_0^T &=E_T(20+1.5X)=20+1.5(8)=32.0\ \text{score units},\\ \widehat{\mu}_1^T &=E_T(17+1.2X)=17+1.2(8)=26.6\ \text{score units},\\ \widehat{\Delta}^T &=26.6-32.0=-5.4\ \text{score units}. \end{aligned}

The estimated target mean symptom score is 5.4 units lower under treatment. Equivalently, the conditional effect is \(-3-0.3X\) score units, whose target average is \(-3-0.3(8)=-5.4\). This shortcut works because the conditional means are affine functions of \(X\). With nonlinear outcome models, the target mean of \(X\) alone is generally insufficient; the predictions must be averaged over the target distribution itself.

Diagnostics, Alternatives, and Limitations

Standardization can produce a precise-looking answer even when its essential conditions are doubtful. Diagnostics should therefore examine both the source outcome model and how well the target covariate profiles are supported by source data. Useful checks include:

Compare the target covariate ranges and combinations with those observed in the source. Sparse or absent source support means some target predictions depend heavily on extrapolation.
Assess source-model fit and calibration, including whether important nonlinear terms or treatment-by-covariate interactions are needed. In the logistic example, omitting the interaction would force a common conditional odds ratio and could misstate the stratum-specific predictions.
Check whether the covariates used for transport include plausible effect modifiers and important predictors of outcomes whose distributions differ between populations. An omitted modifier can make the conditional treatment effect fail to carry across populations.
Verify harmonized covariate and outcome definitions, follow-up, and treatment versions. Apparent statistical agreement cannot repair incompatible measurements or interventions.

A crude source-population treatment contrast averages over the source covariate distribution, not the target distribution, so it answers a different population question when those distributions differ. Standardization directly predicts outcomes at target covariate profiles. A weighting-based approach is a main alternative: it reweights source observations to represent the target rather than relying on the same outcome-prediction calculation. Each approach has its own sensitivity to model specification, support, and data quality; neither eliminates the identification assumptions.

Standardization is limited by the information in the source data. If the target includes profiles absent from the source, predictions there are extrapolations, and averaging them does not create evidence. A correct conditional outcome model can reduce dependence on the target’s full distribution in special cases, such as the linear example, but it cannot substitute for untestable transport assumptions. Uncertainty should also reflect estimation of the outcome model and, when applicable, sampling of the target covariate distribution; a single set of predicted averages does not communicate that uncertainty.

Common pitfall: “The source trial was randomized, so its treatment effect automatically applies to the target.” Randomization supports treatment exchangeability within the source; it does not establish transport exchangeability or target-population overlap. The correction is to state the target estimand, standardize over its covariate distribution, and justify the conditions that let source conditional outcomes represent target conditional outcomes.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

A target population has 30% of a binary covariate in stratum 1 and 70% in stratum 2. The treatment-minus-control risk differences are \(-0.10\) and \(-0.04\). What is the standardized target risk difference?
Why does randomization in the source population not, by itself, establish that a source treatment effect applies to the target?
For conditional mean models \(m_0(x)=5+2x\) and \(m_1(x)=4+x\), the target mean of \(X\) is 3. Calculate the standardized target mean under each treatment and the treatment-minus-control effect.
What is a key warning sign if many target covariate profiles lie outside the source data’s observed range?

Solutions. 1. Weight the stratum effects by target proportions: \((0.30)(-0.10)+(0.70)(-0.04)=-0.030-0.028=-0.058\), or 5.8 percentage points lower risk under treatment. 2. Randomization supports treatment exchangeability in the source, but transport also requires conditional outcome means to be comparable across source and target and requires adequate source support for target profiles. 3. By linearity, \(\mu_0^T=5+2(3)=11\), \(\mu_1^T=4+3=7\), and \(\Delta^T=7-11=-4\) outcome units. 4. The standardized predictions may rely on extrapolation rather than source evidence; assess support and reconsider whether the target effect is identifiable with the available data and assumptions.