Extrapolating Efficacy Across Levels of Disease Severity

Why Severity Extrapolation Requires More Than a Shared Dose

An exposure-response relationship estimated in a less severe population may help predict efficacy in a more severe population, but it does not by itself establish that the prediction transports. The target group may have a different untreated event risk, a different exposure distribution, or a different biological response to a given exposure. The outcome may also be defined differently. A useful analysis therefore asks two separate questions: does severity change the treatment effect at a given exposure, and what does the effect imply for the target population’s absolute outcomes?

The estimand must be clear. In the examples below, it is the probability of achieving a prespecified clinical response by a fixed time, comparing treatment at a specified exposure with no active treatment, within each severity group. This is not automatically the same as an effect on hospitalization, mortality, or a differently defined symptom endpoint.

Definition: Efficacy extrapolation across disease-severity levels is the use of evidence from one severity group to predict a treatment effect in another severity group. It is credible only to the extent that the relevant treatment-effect relationship, endpoint meaning, exposure range, and other effect modifiers are sufficiently comparable or are accounted for in the model.

A Model That Separates Baseline Risk From Treatment Effect

Let \(Y=1\) indicate a favorable clinical response by the analysis time, \(S\) denote disease severity, and \(E\) denote the exposure measure chosen for the analysis. A convenient model separates the response probability without active treatment from the exposure-associated treatment effect:

\operatorname{logit}\{P(Y=1\mid E,S)\} = \operatorname{logit}\{p_0(S)\} + \theta(E) + \delta S\theta(E), \qquad \theta(E)=\theta_{\max}\frac{E}{EC_{50}+E}.

Here \(p_0(S)\) is the response probability without active treatment at severity level \(S\); \(\theta(E)\) is the treatment-associated change in log odds at exposure \(E\) for the reference severity; \(\theta_{\max}\) is its maximum value under this model; and \(EC_{50}\) is the exposure at which \(\theta(E)\) reaches half of \(\theta_{\max}\). The parameter \(\delta\) represents modification of the exposure-response effect by severity. If \(\delta=0\), the model assumes the same log-odds treatment effect at each severity level, while allowing baseline response probabilities to differ.

This is a model, not a guarantee that severity has been fully represented. A more detailed model may include measured prognostic variables, nonlinear severity effects, or a different exposure-response function. The assumptions must fit the scientific question and the available data. In particular, the form above assumes a common endpoint definition and that the chosen exposure metric is relevant to the response.

Key distinction: Equal relative effects do not imply equal absolute benefits. A common odds ratio can produce different response probabilities and risk differences when \(p_0(S)\) differs between severity groups.

Converting a Relative Effect to a Target-Population Benefit

Suppose the treatment effect at a specified exposure corresponds to an odds ratio \(OR\) for response. If the target group’s untreated response probability is \(p_0\), then its treated response probability is obtained by multiplying the untreated odds by \(OR\), then converting the resulting odds back to a probability:

p_T = \frac{OR\,p_0}{1-p_0+OR\,p_0}.

The absolute benefit on the response-probability scale is \(p_T-p_0\). This calculation is valid only if the odds ratio applies to the target group and the endpoint and time horizon match. It is not valid to insert a baseline risk for one endpoint and an effect estimate for a different endpoint without a justified mapping.

Worked Example: Different Baseline Response Probabilities

Hypothetical data. Assume a fitted exposure-response model estimates \(\theta_{\max}=0.80\) on the log-odds scale and \(EC_{50}=1.0\) exposure unit. At the target exposure \(E=2.0\) units, assume no severity interaction for this calculation, so \(\delta=0\). The estimated untreated response probabilities are 0.25 in the less severe group and 0.40 in the more severe group.

At \(E=2.0\), the model’s treatment effect on the log-odds scale is

\theta(2.0) = 0.80\frac{2.0}{1.0+2.0} = 0.5333.

Thus \(OR=\exp(0.5333)=1.704\). For the less severe group,

p_T = \frac{1.704(0.25)}{1-0.25+1.704(0.25)} = \frac{0.4260}{1.1760} = 0.3622.

The predicted response-probability increase is \(0.3622-0.25=0.1122\), or 11.22 percentage points. For the more severe group,

p_T = \frac{1.704(0.40)}{1-0.40+1.704(0.40)} = \frac{0.6816}{1.2816} = 0.5318.

The predicted increase is \(0.5318-0.40=0.1318\), or 13.18 percentage points. These are model-based predictions, not observed trial results. The larger absolute increase in the more severe group follows from its different baseline response probability under the assumed common odds ratio; it does not demonstrate that severity makes the drug biologically more effective.

Testing Whether Severity Modifies the Exposure Response

The interaction term addresses whether the exposure-associated effect differs with severity on the model’s chosen scale. Its uncertainty matters. A small estimated interaction is not proof that the treatment effect is identical across severity groups, especially when the sample provides little information about the interaction. Nor does a nonsignificant test establish that extrapolation is acceptable.

Consider a hypothetical estimate \(\hat{\delta}=0.05\) with standard error \(SE=0.18\). Under an approximate normal sampling distribution, a 95% confidence interval is

0.05\pm 1.96(0.18) = 0.05\pm 0.3528 = (-0.3028,\ 0.4028).

This interval includes zero, but also includes interaction values that could imply meaningful differences across severity levels. The estimate therefore does not rule out effect modification. Interpretation should consider the clinically meaningful range of interactions, not just whether the interval includes zero. The interaction is also scale-dependent: no interaction on the log-odds scale does not mean no interaction on the risk-difference scale.

Worked Example: Evaluating an Interaction Estimate

Hypothetical data. Suppose the log-odds interaction estimate is \(\hat{\delta}=0.05\), with \(SE=0.18\), and the analysis uses a two-sided 5% test of \(H_0:\delta=0\) against \(H_a:\delta\ne 0\). The approximate test statistic is

z=\frac{0.05}{0.18}=0.278.

The two-sided p-value is approximately 0.781. The data do not provide evidence against \(H_0\) at \(\alpha=0.05\). However, the confidence interval from the calculation above ranges from \(-0.303\) to \(0.403\), so interactions of potentially important size remain compatible with the data. The correct conclusion is that this analysis has not detected a severity interaction, not that the interaction is absent or that efficacy has been proven equivalent across groups.

Before interpreting this estimate, check whether both severity groups have adequate exposure data over the range being used for prediction. If the more severe group has little or no exposure overlap with the source group, the model’s target prediction may be extrapolation in both severity and exposure. Also check whether the estimated interaction is stable to reasonable model choices and whether baseline severity or other prognostic factors have been represented appropriately.

Endpoint Differences Can Break the Bridge

The preceding calculations presume that \(Y\) has the same clinical meaning, definition, and assessment time in both groups. If the less severe study measures symptom response while the target evidence concerns hospitalization, the response odds ratio cannot simply be carried across. The endpoints may differ in clinical importance, measurement properties, timing, and relationship to underlying disease activity.

A bridge between different endpoints requires evidence that relates them. One possibility is a validated endpoint-specific model using data in which both outcomes were measured, with uncertainty in the relationship carried into the target prediction. Another is to obtain direct evidence on the target endpoint. A biological rationale or a correlation between endpoints alone does not establish that the treatment effect transfers from one endpoint to the other.

Worked Example: A Different Target Endpoint

Hypothetical data. A less severe population has a symptom-response endpoint with an estimated treatment odds ratio of 1.70. The proposed severe-disease decision concerns hospitalization, and the estimated untreated hospitalization risk over the same period is 0.40. The symptom-response odds ratio does not determine the hospitalization effect: it is an effect on a different outcome.

For illustration only, suppose a separate, hypothetical validation analysis measured both endpoints and supports a hospitalization odds ratio of 0.75 for treatment versus control. The untreated hospitalization odds are \(0.40/(1-0.40)=0.6667\). Applying the endpoint-specific odds ratio gives treated odds \(0.6667(0.75)=0.5000\), and thus

p_T = \frac{0.75(0.40)}{1-0.40+0.75(0.40)} = \frac{0.3000}{0.9000} = 0.3333.

The predicted hospitalization risk reduction is \(0.40-0.3333=0.0667\), or 6.67 percentage points over the stated period. This prediction relies on the separate hospitalization-effect evidence, not on the symptom-response odds ratio of 1.70. In a real analysis, uncertainty in the baseline risk, endpoint mapping, and treatment-effect estimate must also be represented. Without credible evidence for the target endpoint, the hospitalization prediction is unsupported.

Decision Checks, Diagnostics, and Alternatives

A severity extrapolation should be treated as a conditional prediction. Its credibility depends on the data and assumptions that connect the source population to the target. A practical assessment includes the following:

Exposure support: compare exposure distributions across severity groups and avoid relying on predictions far outside observed exposure ranges.
Effect modification: estimate and interpret severity-by-exposure effects, including uncertainty and clinically meaningful interaction ranges.
Baseline risk: estimate or justify the target group’s untreated risk for the same endpoint and time horizon.
Endpoint comparability: verify consistent definitions, assessment times, and analysis rules, or use a defensible endpoint-specific bridge.
Model adequacy: inspect calibration, residual or predictive checks appropriate to the outcome, influential observations, and sensitivity to alternative plausible model forms.
Exposure interpretation: recognize that exposure is generally not randomized. Disease severity, adherence, dose adjustment, or other factors may influence both exposure and outcome; an exposure-response association alone does not establish a causal exposure effect.

When the exposure-response model is not credible, alternatives include using a randomized treatment contrast within severity groups, fitting a model with explicit severity-specific effects, or collecting direct evidence in the target group. These approaches answer different questions and have different data requirements. A randomized contrast may be less dependent on modeling exposure-response, while a detailed model may use exposure information more efficiently but rely on stronger structural and confounding assumptions. No approach repairs an endpoint mismatch without evidence connecting the outcomes.

A common mistake is to reason, “The severity interaction was not statistically significant, so the efficacy estimate is the same in both groups.” A nonsignificant result can reflect limited precision; it does not demonstrate equivalence. The correction is to examine the interval for the interaction against a prespecified range of clinically unimportant differences, assess support for the target prediction, and report how baseline risk changes absolute benefit. Another mistake is to apply a source-group response effect directly to a different target endpoint. The correction is to require a validated endpoint bridge or direct evidence for that endpoint.

Key takeaway: Severity extrapolation is strongest when the endpoint is comparable, the target exposures are supported by data, the severity interaction is sufficiently constrained, and the target baseline risk is appropriate. Report both the relative effect and the resulting absolute prediction, with uncertainty and assumptions made explicit.

Check Your Understanding

Answer each question, then compare with the solutions below.

A common odds ratio is predicted across severity groups. Does that imply the same risk difference in each group? Explain.
An interaction estimate is near zero, but its confidence interval is wide. What can and cannot be concluded?
Why is an odds ratio for symptom response insufficient by itself to predict a hospitalization effect?
At a target exposure, the response odds ratio is 1.50 and the target untreated response probability is 0.20. Calculate the predicted treated response probability and risk difference.

Solutions. (1) No. The same odds ratio can yield different treated probabilities and risk differences when untreated probabilities differ. (2) The analysis has not identified a clear interaction, but it has not ruled out important effect modification; the interval’s range and the data support must be assessed. (3) The endpoints have different definitions and clinical meanings, so a treatment effect on one does not specify the effect on the other without a validated bridge or direct target-endpoint evidence. (4) \(p_T=1.50(0.20)/[0.80+1.50(0.20)]=0.30/1.10=0.2727\). The predicted risk difference is \(0.2727-0.20=0.0727\), or 7.27 percentage points, conditional on the odds ratio applying to this population and endpoint.