The OFV forms a surface in ( p + 1)-dimensional space, where p is the number of estimated parameters. More specifically, we focus our investigation on minimization of the approximated − 2log likelihood (objective value function, OFV) using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm ( 12) implementation in NONMEM ( 13), a software package for population pharmacometric modeling that is commonly used for regulatory submission. In this paper, we focus on maximum likelihood-based parameter estimation algorithms where the likelihood is approximated either by the first-order approximation (first order, FO first-order conditional estimate, FOCE) or second-order approximation (Laplace approximation) and then maximized using a gradient-based optimization algorithm. There exist many parameter estimation methods for nonlinear mixed effects models ( 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11). Inaccurately estimated parameter values can introduce bias and inflate uncertainty, which in turn will influence any decisions supported by modeling and simulation results. We demonstrate this using four published pharmacometric models and two models specifically designed to be practically non-identifiable. We have implemented this algorithm in industry standard software for nonlinear mixed effects modeling (NONMEM, version 7.4 and up) and showed that it can be used to avoid termination of parameter estimation at saddle points, as well as unveil practical parameter non-identifiability. In this algorithm, we use the approximated Hessian matrix at the point where BFGS terminates, perturb the point in the direction of the eigenvector associated with the lowest eigenvalue, and restart the BFGS algorithm. We have found that for maximization of the likelihood for nonlinear mixed effects models used in pharmaceutical development, the optimization algorithm Broyden–Fletcher–Goldfarb–Shanno (BFGS) often terminates in saddle points, and we propose an algorithm, saddle-reset, to avoid the termination at saddle points, based on the second partial derivative test. One reason for such failure is that these numerical optimization methods cannot distinguish between the minimum, maximum, and a saddle point hence, the parameters found by these optimization algorithms can possibly be in any of these three stationary points on the likelihood surface. Parameter estimation of a nonlinear model based on maximizing the likelihood using gradient-based numerical optimization methods can often fail due to premature termination of the optimization algorithm.
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