Evaluating the Performance of Smoothed Hazard Functions Over Study Follow-up and by Numbers at Risk: A Simulation Study

OBJECTIVES: Cost-effectiveness analyses routinely require extrapolation of time-to-event data such as overall survival. Guidelines recommend comparing the within-study empirical hazards to the predicted hazard functions of parametric survival models, to guide the survival extrapolation choice. In this simulation study, we evaluate the reliability of smoothed empirical hazard functions in predicting a simple 'true' hazard rate.

METHODS: Smoothed hazards were estimated using ‘muhaz’ and ‘bshazard’ R packages. Each function was fit to 1,000 simulated cohorts, each with 1,000 patients, using the default function settings. The simulated cohorts were sampled from a ‘true’ underlying Weibull survival distribution, with a constant hazard rate (30 cases/100 person-years) in the base case. Decreasing and increasing hazards were also analyzed. The accuracy of each hazard function was estimated by calculating the mean absolute error (MAE) and mean absolute percentage error (MAPE) between the smoothed hazard function fit to each simulation, and the true underlying hazard. Results were categorized by the number of patients at risk (intervals of 10) and averaged across all simulated cohorts.

RESULTS: Using a constant hazard, both functions performed similarly well until the numbers at risk were low, at which point the smoothed hazards became increasingly inaccurate, particularly the muhaz function. The MAE and MAPE increased from 0.0022 (8.9%) and 0.0018 (7.3%) with between 91 and 100 patients at risk, up to 0.0065 (26%) and 0.0032 (13%) with between 21 and 30 patients at risk, for muhaz and bshazard respectively. Similar trends were observed across increasing and decreasing hazard shapes.

CONCLUSIONS: Both the muhaz and bshazard functions perform well until the numbers at risk were low. Smoothed hazard functions should be interpreted cautiously towards the end of follow-up, particularly with the muhaz function. Further research is required to assess the accuracy of both functions with more complex hazard functions, and differing hazard rates.