Worked Case of Transporting Efficacy to a New Region

From a Source Trial to a Target-Region Estimate

A treatment effect observed in one region is not automatically the effect expected in another. Even when the endpoint and treatment are the same, the regions may differ in baseline characteristics that predict outcome or modify treatment effect. A regional efficacy bridge makes this issue explicit: estimate treatment and control outcomes conditional on relevant covariates in the source trial, then average those conditional outcomes over the target region’s covariate distribution.

The worked case below uses a hypothetical randomized trial and a binary unfavorable outcome, assessed at a fixed follow-up time. A lower risk on treatment indicates benefit. The earlier material on endpoint harmonization is essential here: the source and target outcome must represent the same clinical endpoint over a comparable time horizon. The case also assumes that important trial-conduct differences have been considered, rather than treating region as the sole possible source of discrepancy.

Definition: Let $Y(a)$ denote the binary outcome that would be observed under treatment $a$, where $a=1$ is active treatment and $a=0$ is control. Let $X$ be a set of measured baseline covariates and $R$ denote region. The target-region risk difference is $$ \Delta_T = E\{Y(1)-Y(0)\mid R=T\}. $$ A negative $\Delta_T$ means the treatment reduces the target-region risk of the unfavorable outcome. Covariate standardization estimates this quantity by averaging source-trial conditional outcome risks over the target population’s distribution of $X$.

For a discrete covariate pattern $x$, define $m_a(x)=P(Y=1\mid A=a,X=x,R=S)$, the source-region risk under treatment $a$. If the source trial is randomized and the conditional outcome relationship transports to the target region, the target risks are estimated as

\widehat{\mu}_{a,T}=\sum_x q_T(x)\widehat{m}_a(x), \qquad \widehat{\Delta}_T=\widehat{\mu}_{1,T}-\widehat{\mu}_{0,T},

where $q_T(x)$ is the proportion of the target population with covariate pattern $x$. With continuous covariates, the sum is replaced by an integral, or approximated by averaging predictions over target-population records.

Assumptions Behind the Bridge

Standardization is a calculation; its interpretation as a target-region effect depends on assumptions. First, treatment allocation in the source trial must identify the treatment-specific risks conditional on $X$. Randomization supports this, provided outcome ascertainment and missingness do not introduce relevant bias. Second, the covariates used for standardization must capture the measured differences that matter for the conditional outcome risks. In particular, the bridge assumes conditional transportability: after conditioning on $X$, the source-trial outcome risks under each treatment apply in the target region. This is not guaranteed by randomization, which applies to treatment assignment within the source trial, not to membership in the target region.

Third, there must be covariate overlap: each target covariate pattern used in the calculation must be represented sufficiently in the source trial to estimate its treatment-specific risks. Finally, the treatment, comparator, endpoint definition, follow-up, and relevant care conditions must be sufficiently comparable. If these assumptions are doubtful, the estimate should be presented as conditional on them and accompanied by appropriate sensitivity analyses—not as a direct observation of efficacy in the target region.

Worked Example: Standardizing Three Risk Strata

Hypothetical data. A randomized source-region trial reports the following unfavorable-outcome counts by baseline risk stratum. The target region’s stratum proportions come from a separate representative data source.

Baseline stratum	Source treatment: events / n	Source control: events / n	Target proportion
Low risk	30 / 200	25 / 100	0.20
Moderate risk	60 / 200	45 / 100	0.50
High risk	45 / 100	60 / 100	0.30

1. Estimate the conditional risks. The treatment risks are $30/200=0.150$, $60/200=0.300$, and $45/100=0.450$. The control risks are $25/100=0.250$, $45/100=0.450$, and $60/100=0.600$. These observed stratum-specific risks estimate $m_a(x)$ in a saturated three-stratum model.

2. Average over the target distribution.

\widehat{\mu}_{1,T} =(0.20)(0.150)+(0.50)(0.300)+(0.30)(0.450) =0.030+0.150+0.135=0.315,

\widehat{\mu}_{0,T} =(0.20)(0.250)+(0.50)(0.450)+(0.30)(0.600) =0.050+0.225+0.180=0.455.

Thus, $\widehat{\Delta}_T=0.315-0.455=-0.140$, or a 14.0-percentage-point lower estimated risk under treatment in the target population. The target standardized risk ratio is $0.315/0.455=0.692$, corresponding to an estimated relative risk reduction of about 30.8%. These are different effect scales; the risk difference is the primary estimand in this example.

Uncertainty and a Short Derivation

With fixed target proportions and independent treatment and control groups, a plug-in variance estimate for the standardized risk difference is

\widehat{\operatorname{Var}}(\widehat{\Delta}_T) = \sum_x q_T(x)^2 \left\{ \frac{\widehat{m}_1(x)[1-\widehat{m}_1(x)]}{n_{1x}} + \frac{\widehat{m}_0(x)[1-\widehat{m}_0(x)]}{n_{0x}} \right\},

where $n_{ax}$ is the source-trial sample size in treatment arm $a$ and stratum $x$. To see why, write the estimate as the sum of weighted stratum-specific treatment risks minus weighted control risks. Under independent binomial sampling, each stratum risk estimate has variance $p(1-p)/n$. Multiplying an estimate by $q_T(x)$ multiplies its variance by $q_T(x)^2$; independent strata and arms contribute additively. Replacing unknown risks with observed risks gives the formula above.

For the hypothetical data, the treatment contribution to the variance is

(0.20)^2\frac{0.150(0.850)}{200} +(0.50)^2\frac{0.300(0.700)}{200} +(0.30)^2\frac{0.450(0.550)}{100} =0.0005108.

The control contribution is

(0.20)^2\frac{0.250(0.750)}{100} +(0.50)^2\frac{0.450(0.550)}{100} +(0.30)^2\frac{0.600(0.400)}{100} =0.0009098.

The estimated variance is $0.0014205$, so $SE=\sqrt{0.0014205}=0.03769$. A normal-approximation 95% confidence interval is

-0.140\ \mathbin{\pm}\ 1.96(0.03769) = [-0.2139,\,-0.0661].

Under the stated sampling model, this interval procedure has approximately 95% coverage over repeated samples. In this hypothetical analysis, the interval corresponds to a target-region risk difference between 21.4 and 6.61 percentage points lower under treatment. It reflects sampling uncertainty in the trial risks, but not uncertainty in the target proportions, model choice, or transportability assumption. If the target distribution is estimated rather than fixed, its uncertainty should also be accounted for, for example by resampling the trial and target data together.

Overlap: Where Standardization Is Supported

The target distribution determines which source-trial predictions receive weight. A target stratum absent from the trial cannot be rescued by assigning it a large weight: there is no observed source information from which to estimate its conditional risks. Sparse strata are also concerning because their estimates can be noisy and can dominate the target average. Practical diagnostics include a source-versus-target comparison of covariate distributions, the target weight assigned to each stratum, arm-specific sample sizes, and inspection of the range of covariate values used for predictions. For continuous covariates, plots and summaries of the distributions should reveal extrapolation beyond the source data.

Worked Example: Detecting a Missing Target Stratum

Hypothetical overlap check. Suppose a second candidate source trial includes low- and moderate-risk participants, but no high-risk participants. Its covariate proportions are 0.60 low risk, 0.40 moderate risk, and 0 high risk; the target proportions remain 0.20, 0.50, and 0.30.

For represented strata, the target-to-source proportion ratios are $0.20/0.60=0.333$ for low risk and $0.50/0.40=1.25$ for moderate risk. For high risk, the ratio $0.30/0$ is undefined: there are no source participants in that stratum. The target distribution assigns 30% of its population to a group with no source-trial support.

One tempting calculation is to renormalize the represented target proportions: $0.20/(0.20+0.50)=0.286$ and $0.50/(0.20+0.50)=0.714$. That would estimate an effect for the covered low- and moderate-risk portion of the target population, not for the full target region. It cannot be reported as the full target estimate unless the high-risk group is addressed by additional data or an explicit extrapolation model and its assumptions are justified. The correct diagnostic conclusion is that the requested full-population bridge is unsupported by this source trial alone.

Sensitivity to Regional Effect Differences

Even strong measured-covariate overlap does not establish conditional transportability. A sensitivity analysis can show how much an unmeasured regional modifier or residual regional effect difference would need to change the estimate. This is not a correction learned from the source trial; it is a transparent calculation under specified departures from the primary assumption. The direction of the departure must be defined on the same effect scale as the estimand.

Worked Example: Additive Attenuation on the Risk-Difference Scale

Return to the standardized estimate $\widehat{\Delta}_T=-0.140$. Suppose, as a sensitivity scenario, that the target-region treatment effect is $\delta$ risk-difference units less favorable than predicted from the source trial. Then the adjusted scenario estimate is $\widehat{\Delta}_{T,\delta}=\widehat{\Delta}_T+\delta$. For $\delta=0.040$, representing a 4-percentage-point attenuation, the estimate is $-0.140+0.040=-0.100$, or 10.0 percentage points lower risk. For $\delta=0.100$, it is $-0.140+0.100=-0.040$, or 4.0 percentage points lower risk.

The point-estimate tipping value for no difference is $\delta=0.140$, because $-0.140+0.140=0$. This does not establish that a 14-point regional shift is plausible; that judgment requires clinical and external evidence. If $\delta=0.100$ is treated as fixed and the primary standard error is retained, the shifted interval is $[-0.2139+0.100,\,-0.0661+0.100]=[-0.1139,\,0.0339]$. It includes zero. This interval shifts sampling uncertainty but does not include uncertainty about the chosen sensitivity value.

Model Checks, Alternatives, and Limits

For more than a few strata, conditional risks are often estimated with a regression model and then standardized over target covariates. Diagnostics should address both prediction and support: compare observed and predicted risks within the source trial, check calibration across important covariates and treatment arms, inspect influential target records or strata, and verify that target covariates lie within the source data’s range. A model that fits source data well can still extrapolate poorly when target covariates are unsupported. Sensitivity to reasonable alternative model forms is important; the earlier material on model-form sensitivity and residual heterogeneity motivates treating such differences as uncertainty rather than hiding them in one fitted estimate.

The main alternatives answer related but not identical practical questions. A crude source-trial treatment effect applies the source trial’s covariate mix, so it is not generally a target-region estimate when the covariate distributions differ. Standardization directly averages conditional predictions over the target distribution and is often straightforward to interpret. Reweighting instead gives source participants weights intended to reproduce the target covariate mix; it can be unstable when weights are extreme and still requires overlap and transportability assumptions. A model-based regression approach can handle continuous covariates, but its result depends on model specification, particularly outside well-supported regions. Local target-region trial data can directly inform target efficacy and test transportability, but may be limited or imprecise. None of these approaches removes the need to define the endpoint, population, and assumptions clearly.

Common pitfall: “The source trial was randomized, so its treatment effect applies to the target region.” Randomization supports a causal comparison between treatment arms in the source trial; it does not ensure that conditional risks or treatment effects are the same across regions. The correction is to define a target estimand, standardize over the target covariate distribution, check overlap, and state the additional transportability assumptions and sensitivity analyses.

The resulting estimate is therefore an evidence-based prediction, not a substitute for target-region observations. Its credibility depends on endpoint comparability, adequate source support, measured covariates that capture relevant differences, and robustness to reasonable modeling and transportability departures. Reporting the standardized risks as well as the risk difference helps readers see what the estimated contrast is built from.

Check Your Understanding

Answer each question, then compare with the solutions below.

If the standardized target risks are 0.22 on treatment and 0.34 on control, what is the risk difference and how should its sign be interpreted?
A target stratum has proportion 0.25 but is absent from the source trial. Can renormalizing the other target strata produce a full-target estimate?
Why does randomization in the source trial not by itself justify applying the source treatment effect to another region?
A primary risk difference is $-0.08$. Under an additive attenuation scenario of $0.03$, what is the scenario estimate?

Solutions. (1) $0.22-0.34=-0.12$, a 12-percentage-point lower estimated risk on treatment. (2) No. Renormalization estimates an effect for the represented portion, not the full target population; the absent stratum needs data or an explicit, justified extrapolation. (3) Randomization identifies treatment comparisons within the source trial, not equality of outcome risks or treatment effects across regions. (4) $-0.08+0.03=-0.05$, a 5-percentage-point lower risk under that scenario.