Influence of Covariance Step Success on Final Parameter Estimates
Henrik Bjugård Nyberg (1,2), Andrew C. Hooker (1), Yasunori Aoki (1)
(1) Department of Pharmaceutical Biosciences, Uppsala University, Uppsala, Sweden (2) Mango Solutions Ltd, Chippenham, United Kingdom
Objectives: To determine if computational stability influences final parameter estimates, and if so how we can assess the quality of the parameter estimates from NONMEM output. Specifically we want to determine whether a completed covariance step in NONMEM has any bearing on the quality of the final parameter estimates.
Background: For years there has been debate in the pharmacometric community around whether the different success criteria in NONMEM have any bearing on the quality of the final parameter estimates. One study by Holford et al. using bootstrap methodology concluded termination status was not an indicator of the quality of parameter estimates . Preconditioning  lets us generate linearly reparameterized models resulting in different computational stability. As the likelihood of the nonlinear mixed effect model is invariant under linear re-parameterization, the minimum OFVs of these models are the same.
Method: As an initial example, data for a single dose experiment with 8 samples for each of 32 individuals were simulated with a two-compartment, linear absorption PK model. We generated 10,000 reparameterised models with various levels of computational stability using the precond tool in PsN [3,4]. The condition number of the theoretical variance covariance matrix was varied between 10^6 to 10^15. In addition, we perturbed the initial estimates with a random uniform distribution within +/-20% of the theoretical maximum likelihood parameters. Two groups of runs were analyzed: 1. Minimization and Covariance step successful. 2. Minimization successful and Covariance step failed. A Wilcoxon rank sum test was performed to determine if groups 1 and 2 differ in OFV and theoretical condition numbers.
Results: The OFV of runs in group 2 differed significantly from those in group 1 with a p-value < 2.2e-16. Group 2 also represents a significantly higher theoretical condition number with p=1.1e-5. Mean OFVs for the groups were -79.6 for group 1 (n=7997) and -36.2 for group 2 (n=116). Mean condition numbers were 2.8e+18 (group 1) and 2.1e+20 (group 2).
Conclusion: The quality of parameter estimates is clearly influenced by computational stability, as measured by theoretical condition number. Our investigation shows that a failed covariance step in NONMEM estimation is a good predictor of lower quality parameter estimates.
Acknowledgement: This work was supported by the DDMoRe (www.ddmore.eu) project.
 Holford N et al, NONMEM Termination Status is Not an Important Indicator of the Quality of Bootstrap Parameter Estimates, PAGE 15 (2006) Abstr 992 [www.page-meeting.org/?abstract=992]
 Aoki et al, Preconditioning of Nonlinear Mixed Effect model for Stabilisation of Covariance Matrix Computation, PAGE 24 (2015) submitted abstract
 Keizer RJ, Karlsson MO, Hooker A. (2013). Modeling and Simulation Workbench for NONMEM: Tutorial on Pirana, PsN, and Xpose. CPT Pharmacometrics Syst Pharmacol, 2: e50.
 [http://psn.sourceforge.net/] (accessed on March 13th 2015)