Development and performance of npde for the evaluation of time-to-event model
M. Cerou [1,2,3], M. Lavielle[4] and E. Comets[1,2]
[1] Inserm, IAME, UMR 1137, F-75018, Paris, France [2] Inserm, CIC 1414, Clinical Investigation Center, Rennes, France [3] Division of Clinical Pharmacokinetics and Pharmacometrics, Institut de Recherches Internationales Servier, Suresnes, France [4] Iria Saclay & CMAP, Ecole Polytechnique, University Paris-Saclay, Palaiseau, France
Objectives: Normalised prediction distribution errors (npde) are used to evaluate graphically and statistically continuous responses in non-linear mixed effect models [1]. Here, our aim was to extend npde for time-to-event (TTE) models and to evaluate their performances.
Methods: Let V denote a dataset with TTE observations. The null hypothesis H0 is that observations in V can be described by a model M. Prediction discrepancies (pd) are defined as the quantile of the observation within its predictive distribution, which is approximated through Monte-Carlo simulations. pd for unobserved (censored) event times were imputed using a similar method as the one developed to handle data under the Lower Quantification Limits (LOQ) [2]. npde are then obtained using the inverse function of the normal cumulative density function. Under H0, they follow a normal N(0,1) distribution, which was tested through a combined test [1].
We evaluated the performance of npde for TTE data through a simulation study inspired by the work of Desmée et al. [3]. They characterised the relationship between the biomarker PSA (prostate specific antigen) and survival in 500 metastatic castration-resistant prostate cancer patients via joint modelling. We simulated event times from the joint model, based on the predicted PSA trajectories, for different sample sizes. We evaluated the type I error and power of npde to detect different types of model misspecifications for the TTE component in several scenarios.
Results: Type I error was found to be close to the expected 5% both with true and with censored event times, for all tested sample sizes. npde were able to detect misspecifications in the baseline hazard as well as in the link between the longitudinal variable and survival. The power to detect model misspecifications was more important as the difference of survival was large. As expected, the power also increased as sample size increased. The percentage of rejection was closer to 5% when censored events were considered.
Conclusion: npde can be readily extended to TTE data, and we found that they performed well with an adequate type I error. The next step will be to evaluate the npde for the joint model, also considering the longitudinal measurements and its impact on TTE. We will also extend npde to evaluate repeated TTE models including inter-individual variability on the parameters of the hazard function.
References:
[1] Brendel, K., Comets, E., Laffont, C., Laveille, C., & Mentré, F. (2006). Metrics for external model evaluation with an application to the population pharmacokinetics of gliclazide. Pharmaceutical research, 23(9), 2036-2049.
[2] Nguyen, T. H. T., Comets, E., & Mentré, F. (2012). Extension of NPDE for evaluation of nonlinear mixed effect models in presence of data below the quantification limit with applications to HIV dynamic model. Journal of pharmacokinetics and pharmacodynamics, 39(5), 499-518.
[3]Desmée, S., Mentré, F., Veyrat-Follet, C., & Guedj, J. (2015). Nonlinear mixed-effect models for prostate-specific antigen kinetics and link with survival in the context of metastatic prostate cancer: a comparison by simulation of two-stage and joint approaches. The AAPS journal, 17(3), 691-699.
