A Visual Predictive Check for the Evaluation of the Hazard Function in Time-to-Event Analyses
Matthew M Hutmacher
Ann Arbor Pharmacometrics Group (A2PG)
Objectives: To present methods for performing visual predictive checks (VPCs) specifically for evaluating the hazard function while modeling time-to-event (TTE) data. Binned and smoothed hazard estimators will be discussed for continuous single-event TTE data.
Methods: Pharmacometricians are becoming more involved in determining exposure-response relationships for efficacy and safety TTE endpoints. because these can be the most clinically informative for certain indications. Determining the hazard, or instantaneous risk, of an event has great utility. Changes in the absolute risk of an event over time contain information for supporting dosing or titration strategies. Methods for TTE analyses are being discussed (,) and presented more frequently (for example, see ). However, little can be found for simulation-based model evaluation (or VPC) other than using Kaplan-Meier (KM) curves . KM based methods evaluate the model through the survival function, which is an exponential function of the integrated (cumulative) hazard. Thus, hazard evaluation using KM curves does not provide a direct assessment of the hazard's features. It may also lack sufficient sensitivity in some cases . A binned hazard estimate (BHE) approach is presented first. The method essentially considers a piecewise constant hazard for each bin and uses a simple hazard estimator for the bin. Conceptually straightforward extensions can be made using running line smoothers (RLS) with further smoothing using kernel regression. However, going back to Watson and Leadbetter (1964) , kernel smoothers (KS) can be directly applied. This method is more complex conceptually. Literature in this area is quite rich.
Results: Simulations were performed for various hazard functions. The BHE, RLS, and KS methods described above are introduced, implementation and considerations for their use are discussed, and the methods are contrasted to the KM method typically used.
Conclusions: Hazard-based VPCs provide a direct evaluation of the hazard function and provide a valuable simulation-based diagnostic tool for development of TTE models.
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