How Uncertain is the Survival Extrapolation? A Study of the Impact of Different Parametric Survival Models on Extrapolated Uncertainty About Hazard Functions, Lifetime Mean Survival and Cost Effectiveness

Abstract

Background and objective: The extrapolation of estimated hazard functions can be an important part of cost-effectiveness analyses. Given limited follow-up time in the sample data, it may be expected that the uncertainty in estimates of hazards increases the further into the future they are extrapolated. The objective of this study was to illustrate how the choice of parametric survival model impacts on estimates of uncertainty about extrapolated hazard functions and lifetime mean survival.

Methods: We examined seven commonly used parametric survival models and described analytical expressions and approximation methods (delta and multivariate normal) for estimating uncertainty. We illustrate the multivariate normal method using case studies based on four representative hypothetical datasets reflecting hazard functions commonly encountered in clinical practice (constant, increasing, decreasing, or unimodal), along with a hypothetical cost-effectiveness analysis.

Results: Depending on the survival model chosen, the uncertainty in extrapolated hazard functions could be constant, increasing or decreasing over time for the case studies. Estimates of uncertainty in mean survival showed a large variation (up to sevenfold) for each case study. The magnitude of uncertainty in estimates of cost effectiveness, as measured using the incremental cost per quality-adjusted life-year gained, varied threefold across plausible models. Differences in estimates of uncertainty were observed even when models provided near-identical point estimates.

Conclusions: Survival model choice can have a significant impact on estimates of uncertainty of extrapolated hazard functions, mean survival and cost effectiveness, even when point estimates were similar. We provide good practice recommendations for analysts and decision makers, emphasizing the importance of considering the plausibility of estimates of uncertainty in the extrapolated period as a complementary part of the model selection process.

Fig. 1

Visualisation of the Kaplan–Meier survival function estimate (with 95% confidence interval) and empirical hazard estimates in the observed 12-month period

Fig. 2

Visualisation of the estimated hazard (and 95% confidence interval) in the observed and extrapolated periods for seven commonly used statistical time-to-event models studied in four hypothetical datasets. The dotted line shows the observed (smoothed) hazard and the vertical dashed line denotes the end of the observed time period

Fig. 3

Visualisation of the estimated survival (and 95% confidence interval) in the observed and extrapolated periods for seven commonly used statistical time-to-event models studied in four hypothetical datasets. The dotted line indicates the observed survival and the vertical dashed line denotes the end of the observed time period

Fig. 4

Estimates of lifetime mean survival and uncertainty (95% confidence interval) for seven commonly used statistical time-to-event models studied in four hypothetical datasets