Dismounting Saddles on the Likelihood Surface
Henrik Bjugård Nyberg[1,2], Andrew C. Hooker, Yasunori Aoki
 Department of Pharmaceutical Biosciences, Uppsala University, Uppsala, Sweden  Mango Solutions Ltd, Chippenham, United Kingdom
Objectives: One issue with parameter estimation in nonlinear mixed-effects models is saddle points on the likelihood surface. A saddle point is characterized by at least one eigenvector along which the objective function is at a maximum rather than a minimum. Methods that work by minimizing the gradient will not be able to distinguish saddle points from minima, and may therefore produce final parameter estimates that actually lie in saddle points. The point may be a minimum in all other directions, making it potentially hard to find a way out of the point.
In this work we propose a method that characterizes a point using the R-matrix, and based on the computed R-matrix selects new parameter values to dismount from a saddle.
The Proposed Algorithm
The best acquired approximation of the R matrix in a point is eigen-decomposed. The lowest eigenvalue is determined, positive or negative, and the corresponding eigenvector is identified. A new set of population parameter values is then selected such that they lie along the identified eigenvector. The new parameter values are determined by finding dOFV=1 using a second order Taylor series approximation of the likelihood surface. Parameter estimation is then re-initiated.
Two published models and datasets were selected for their stability, relative simplicity and short estimation times:
1. Jönsson et al 
2. Bergmann et al 
Initial estimates were randomly perturbed within an order of magnitude around the published parameter values. Parameter estimation was then performed using the FOCEI method in NONMEM 7.3 . Our algorithm was then applied once as described above.
Results: With the first model 201 out of 1,000 estimations produced a higher OFV than the lowest known. For the second model the number was 293 out of 1,000. After applying the proposed algorithm, only 25 estimations remained with higher than minimum OFVs for model 1, and 187 remained for model 2. Some estimations were not moved by our procedure at all, which may be explained by local minima.
Our procedure caused no rise in OFV for any of the estimations.
Conclusion: We have proposed a method that efficiently reduces the issue with saddle points in parameter estimation.
This work was supported by the DDMoRe (www.ddmore.eu) project.
 Beal, S., Sheiner, L.B., Boeckmann, A., & Bauer, R.J., NONMEM User's Guides. (1989-2014), Icon Development Solutions, Ellicott City, MD, USA, 2014.
 Jönsson et al, Population pharmacokinetic modelling and estimation of dosing strategy for NXY-059, a nitrone being developed for stroke, Clinical Pharmakocinetics (2005), 44(8):863-78
 Bergmann et al, Impact of CYP2C8*3 on paclitaxel clearance: a population pharmacokinetic and pharmacogenomic study in 93 patients with ovarian cancer, The Pharmacogenomics Journal (2011) 11, 113–120