FOCE generalized log-likelihood using nlmixr2
Matthew Fidler (1, 7), William S. Denney (2, 7), John Harrold (7), Richard Hooijmakers (4, 7), Max Taubert (1, 7), Mirjam N. Trame (5, 7), Theodoros Papathanasiou (6, 7), Rik Schoemaker (3, 7), Justin Wilkins (3, 7)
1) Novartis Pharmaceuticals, USA, 2) Human Predictions, USA, 3) Occams, The Netherlands, 4) LAP&P Consultants, The Netherlands, 5) Certara Strategic Consulting, USA, 6) GSK, Switzerland, 7) The nlmixr2 team
Objectives:
Create a generalized log-likelihood that allows specification of common probability distributions or user-defined likelihood distributions in nlmixr2’s first-order conditional estimation method with interaction (FOCEI)[1].
Methods:
For FOCEI, the likelihoods of the individual parameters were adjusted to use the numerically approximated individual Hessian instead of using the standard normal approximation as applied in NONMEM[2]/nlmixr2. Since this uses numerical approximation, the step size is optimized to give the most accurate Hessian. First the gradients of the likelihood are calculated using a Stan-based automatic differentiation[3] and forward-sensitivities of the ODE system[4]. The Hessian is then calculated by numerical differentiation. The numerical differentiation step size for the Hessian is tuned with the harmonic mean of every observations’ gradient[5]. This step size then gives a more accurate Hessian approximation.
To make sure that the methodology is correct we validated using the generalized likelihood method to make sure the results were similar in the normal case. While we expected longer run times because of a higher computation burden, if the methodology is reasonable with the normal likelihood, we expect it to be reasonable with other likelihoods. Hence, we validated the generalized likelihood method using the same models to compare nlmixr2 to NONMEM and Monolix and comparing the results. Note that this is a new methodology for optimizing numerical gradient step size released in 2011 and therefore it is unlikely that other modeling tools use the same methods for generalized likelihood.
Results:
The nlmixr2 generalized likelihood methods had similar parameter estimates as the normal likelihood methods. The generalized likelihood generally took longer and resulted in similar objective functions when comparing the two methods. Additionally, the covariance calculation for the generalized likelihood method was more likely to have an unsuccessful covariance step and more likely to give small values for standard errors. Overall, the values were still comparable for the standard errors. With these results we can have confidence that the method will give reasonable results for non-normal likelihoods. Future work on this is to test other non-normal distributions using to see how this new method performs.
Conclusions:
The generalized likelihood, as implemented in nlmixr2, uses new methods that provide reasonable estimates for the tested models, and is ready to be used with other likelihood-based problems.
References:
[1] https://cran.r-project.org/package=nlmixr2
[2] Beal SL et al. 1989-2011. NONMEM Users Guides. Icon Development Solutions, Ellicott City, Maryland, USA.
[3] https://arxiv.org/abs/1509.07164
[4] https://doi.org/10.1007/s10928-015-9409-1
[5] https://arxiv.org/pdf/2110.06380.pdf
