Design evaluation in nonlinear mixed effect models: influence of covariance between random effects
Cyrielle Dumont (1), Marylore Chenel (2), France Mentré (1)
(1) UMR 738, INSERM, University Paris Diderot, Paris, France; (2) Department of Clinical Pharmacokinetics, Institut de Recherches Internationales Servier, Paris, France
Objectives: Nonlinear mixed effect models (NLMEM) are increasingly used during drug development and for analysis of longitudinal data obtained in clinical trials or cohorts. For design evaluation, an alternative to cumbersome simulations is to use the Fisher information matrix (MF). Its expression for NLMEM was derived using a first-order approach [1,2] and is implemented in the R function PFIM [3,4] as well as in several software. Our aims were i) to study the impact of the size of covariance on the standard errors (SE) and on the amount of information; ii) to show how we can analytically predict the SE in the framework of rich individual data without using the model; iii) to study the influence of the covariance on the optimal design and on the corresponding SE. We illustrate the results applying this extension to the design of a pharmacokinetic (PK) model of a molecule in development in children.
Methods: The development of MF for NLMEM including covariance between random effects was implemented in a working version of PFIM. For the PK example, we predicted the SE on fixed effects and on variance components assuming different values of correlations between the two random effects. We also evaluated the total information through the determinant of MF (det (MF)). Assuming rich individual data, one can assume that individual parameters could be observed, and we then derived analytically the predicted SE for fixed effects and variance components. Lastly, we compared optimal designs with and without covariance and their respective SE.
Results: We found that changes in covariance between the two random effects of the PK model did not affect the values of the SE of the fixed parameters nor of the variance parameters for design evaluation. However, the amount of information (i.e. det(MF)) increases when covariance increases. We also found, on the rich individual design, that the SE obtained directly are similar to those given by PFIM. These values are lower bound of SE that could be obtained by population approach. In the framework of optimization, the results showed that optimal designs and the SE are different if the covariance is taken into account or not.
Conclusions: This extension of MF taking into account covariance between random effects will be included in the next version of PFIM. For design evaluation, including covariance has no influence on the predicted SE but has one on the amount of information and therefore on the optimal designs [5].
References:
[1] Mentré F, Mallet A, Baccar D. Optimal design in random effect regression models. Biometrika, 1997; 84(2): 429-442.
[2] Bazzoli C, Retout S, Mentré F. Fisher information matrix for nonlinear mixed effects multiple response models: Evaluation of the appropriateness of the first order linearization using a pharmacokinetic/pharmacodynamic model. Statistics in Medicine, 2009; 28(14): 1940-1956.
[3] Bazzoli C, Retout S, Mentré F. Design evalution and optimisation in multiple response nonlinear mixed effect models: PFIM 3.0. Computer Methods and Programs in Biomedicine, 2010; 98(1): 55-65.
[4] www.pfim.biostat.fr.
[5] Ogungbenro K, Graham G, Gueorguieva I, Aarons L. Incorporating correlation in interindividual variability for the optimal design of multiresponse pharmacokinetic experiments. Journal of Biopharmaceutical Statistics, 2008; 18(2): 342-358.
