Extrapolation

This page explains why extrapolation is needed in health-economic evaluation, how survival extrapolations are constructed and assessed, how long-term outcomes enter economic models, and why uncertainty in the unobserved period must remain visible.

What extrapolation adds to observed evidence

Clinical studies commonly end before all relevant differences in survival, disease progression, resource use, or quality of life have occurred. Economic evaluation may therefore require a model to extend outcomes beyond the observed follow-up so that costs and health effects are estimated over an appropriate time horizon. The extrapolated period is model-based rather than directly observed, even when the fitted curve follows the observed data closely.

Extrapolation is different from interpolation. Interpolation estimates values within the observed range, where data constrain the shape on both sides of the estimate, whereas extrapolation predicts outside that range and relies more heavily on assumptions about future behaviour. As the projection extends farther beyond the data, the choice of model and external evidence usually becomes increasingly influential.

Where extrapolation enters an economic model

Extrapolation may be required for any model input whose relevant consequences continue beyond the available observation period. Survival extrapolation is especially important because projected survival affects life-years, quality-adjusted life-years, treatment duration, subsequent care, and future costs simultaneously. Other applications require the same separation between observed evidence and model-based projection.

The mathematical basis of survival extrapolation

Let $T$ be time to an event, $S(t)=\Pr(T>t)$ the survival function, $h(t)$ the hazard function, and $H(t)$ the cumulative hazard. A fitted survival model defines these quantities within the observed period and then extends them beyond the data cut-off $t_{\mathrm{obs}}$. The extrapolation is the portion of the fitted function evaluated for $t>t_{\mathrm{obs}}$.

The survival and cumulative-hazard functions are related by:

$$ S(t)=\exp[-H(t)] $$

and, for a continuous time-to-event distribution:

$$ H(t)=\int_0^t h(u),du $$

Expected survival up to horizon $\tau$, also called restricted mean survival time, is the area under the survival curve:

$$ \operatorname{RMST}(\tau)=\int_0^{\tau}S(t),dt $$

If the model makes a defensible lifetime projection and the integral is finite, mean survival is:

$$ \operatorname{E}[T]=\int_0^{\infty}S(t),dt $$

These identities do not select the extrapolation model. Different models can fit the observed data similarly while implying different hazards and survival in the unobserved tail, so model choice requires more than an in-sample goodness-of-fit statistic.

Common survival models and their long-term shapes

Candidate models make different assumptions about how the hazard changes over time. Their parameterisations can vary across software, so formulas and parameter labels must be reported with the implementation used. A model's flexibility is useful only when its projected behaviour remains clinically and epidemiologically plausible.

A defensible model-selection process

Selection should combine statistical fit, visual diagnostics, clinical plausibility, external validity, and the consequences for the decision model. Choosing the curve with the lowest Akaike information criterion or Bayesian information criterion is insufficient because these criteria mainly assess relative fit within the observed data. The preferred model and credible alternatives should be justified before their economic consequences are interpreted.

1. Define the estimand and data cut. Specify the outcome, time origin, censoring, treatment groups, follow-up, and point beyond which outcomes are unobserved.
2. Inspect the observed evidence. Review Kaplan–Meier curves, numbers at risk, event counts, censoring, hazards, and potential non-proportional hazards.
3. Fit plausible candidate models. Use distributions or flexible approaches whose shapes could represent the clinical process and treatment mechanism.
4. Assess internal fit. Compare fitted and observed curves, residual or hazard diagnostics, and information criteria without treating any one statistic as decisive.
5. Examine the extrapolated hazards and survival. Check whether long-term patterns, turning points, and tails remain clinically credible.
6. Compare external evidence. Use relevant registries, long-term studies, natural-history data, or population mortality while accounting for differences in population and treatment.
7. Elicit expert judgment carefully. Ask structured, answerable questions about long-term hazards, survival proportions, treatment-effect duration, and clinical limits when empirical evidence is incomplete.
8. Quantify decision impact. Compare costs, effects, and cost-effectiveness results across credible extrapolations and report material model uncertainty.

Treatment effects beyond the trial

Extrapolating baseline risk and extrapolating a relative treatment effect are separate decisions. A proportional-hazards model assumes a constant hazard ratio, whereas other approaches allow the treatment effect to change or fit separate curves to each group. The observed data, mechanism of action, post-treatment evidence, and clinical expectations should determine whether continued, waning, or no effect beyond follow-up is credible.

External validity and mortality constraints

External data can test whether projected outcomes are plausible, but they are not automatically exchangeable with the trial population. Differences in case mix, calendar period, subsequent treatment, geography, and outcome definition can make a direct overlay misleading. External evidence should therefore be described, adjusted where defensible, and used with an explicit account of its relevance and limitations.

For all-cause survival, projected mortality should be reconciled with appropriate population mortality. A model should not imply that a patient group has lower all-cause mortality than a comparable general population unless a defensible mechanism and evidence support that result. Approaches such as relative survival or excess-hazard modelling can combine disease-related risk with background mortality more coherently than an arbitrary cap.

Worked survival example

Consider an illustrative trial with 36 months of survival follow-up and a lifetime economic-model horizon. Analysts fit several candidates and find that multiple models follow the observed Kaplan–Meier curve adequately, but their projected hazards differ after month 36. A Weibull model is used here only to demonstrate the mechanics, not to establish that it is the preferred model.

With the illustrative parameterisation $S(t)=\exp(-\lambda t^{\gamma})$, let $\lambda=0.08$, $\gamma=1.3$, and time $t$ be measured in years. The projected survival probabilities at 3 and 10 years are:

$$ S(3)=\exp[-0.08(3^{1.3})]\approx0.716 $$

$$ S(10)=\exp[-0.08(10^{1.3})]\approx0.203 $$

The 3-year value is near the boundary of the observed period, whereas the 10-year value depends heavily on the assumed Weibull tail. A different credible distribution could give a similar fit through 3 years but a materially different 10-year survival estimate, changing projected life-years, quality-adjusted life-years, and costs. The example values are illustrative and should not be used for a real population.

Translating the projection into costs and health outcomes

An extrapolated survival curve becomes economically relevant when it determines time alive, time in health states, treatment exposure, resource use, or quality of life. The model must maintain coherent relationships among overall survival, intermediate endpoints, health-state occupancy, and costs. Numerical integration and cycle-based approximations should be sufficiently accurate for the chosen time step and horizon.

If $u(t)$ is expected utility among people alive at time $t$, discounted quality-adjusted life-years through horizon $\tau$ can be represented as:

$$ \operatorname{QALY}(\tau)=\int_0^{\tau}S(t)u(t)\exp(-r_E t),dt $$

If $c(t)$ is the expected cost rate while alive and $r_C$ is the cost discount rate, the corresponding discounted cost is:

$$ C(\tau)=\int_0^{\tau}S(t)c(t)\exp(-r_C t),dt $$

Additional health states, event costs, terminal costs, and treatment stopping rules require their own internally consistent terms. The equations show why even small differences in a long survival tail can accumulate into meaningful differences in lifetime costs and outcomes.

Representing extrapolation uncertainty

Extrapolation contains parameter uncertainty within a chosen model and structural uncertainty across plausible model forms or assumptions. Sampling the parameters of one selected curve in probabilistic sensitivity analysis addresses only the first component. The analysis should also show whether alternative defensible tails, treatment-effect assumptions, or external-data choices change the decision.

• Parameter uncertainty can be propagated by sampling correlated fitted parameters from their joint distribution.
• Structural uncertainty can be explored through scenario analyses, alternative model families, or model averaging when justified.
• External-data uncertainty should reflect both statistical uncertainty and uncertainty about transportability to the target population.
• Treatment-effect duration and waning should be varied when long-term persistence is not established.
• The proportion of total life-years, quality-adjusted life-years, and costs arising after the observed period should be reported when it helps reveal reliance on extrapolation.

Common mistakes and reporting requirements

A smooth projected curve is not evidence that its long-term predictions are correct. Credible reporting makes the boundary between observation and projection visible and explains the evidence supporting the tail. Analysts should provide enough detail to reproduce the fitted models and understand how extrapolation changes the economic result.

• Selecting a model only because it has the best in-sample information criterion ignores the uncertainty most relevant to extrapolation.
• Showing survival curves without hazard plots can conceal implausible long-term hazard behaviour.
• Applying a constant hazard ratio indefinitely can overstate treatment benefit when proportional hazards or persistence is unsupported.
• Fitting treatment arms independently without checking crossings or convergence can produce clinically implausible projections.
• Using population mortality as a simple cap can create discontinuities or obscure how background and excess mortality interact.
• Reporting only the preferred curve hides structural uncertainty when other credible extrapolations change cost-effectiveness.
• Extending a curve to a lifetime horizon does not remove uncertainty; it converts an evidence gap into an explicit modelling assumption.