The robustness of global optimal designs
Joakim Nyberg and Andrew C. Hooker
Department of Pharmaceutical Biosciences, Uppsala University, Uppsala, Sweden
Objectives: A drawback with local optimal designs (OD), e.g. D-optimal, is that the parameters of the model are assumed known. This is a strong assumption and therefore robust (global) OD has been a suggested approach, i.e. without assuming that the parameters of the model are known but instead distributions of the parameters are known [1-5].
The objective is to compare different design criteria and to suggest an alternative criterion that overcomes some of the issues with other robust design criteria, such as overweighing certain parameter values.
Methods: Six different criteria were investigated; D-optimal, ED-optimal (ED), API-optimal (API), HCD-optimal (HCD), ED-EFF-optimal (EDEFF) and B-API-optimal (BAPI). ED-EFF-optimal is a criterion that weighs each parameter sample by the corresponding D-optimal design. BAPI (Bias-API), tries to spread the design by forcing each sample to be responsible for a portion of the parameter distribution.
Two models were investigated; A one-parameter fixed effect (4 samples between 0-2 & 100 ind), exp decay model (EXP) and a two-parameter mixed effect (3 doses between 0-6 & 100 ind), Emax model (EMAX) were θED50 and ω2ED50 (exp IIV of 30%) were the parameters to estimate. A uniform parameter distribution was assumed for EXP, θk=[2,22] and for EMAX, θED50 = [0.1,6.1]. 200 uniformly spread samples from the parameter distribution were used for the robust criteria and a D-optimal design was found for each sample. Multiple simulations and estimations (SSE)were used to check the performance of the designs.
Results: As expected, the D-optimal designs (which use the optimal design in each SSE) is slightly better (bias and precision) than the robust criteria for both models. All the robust designs except ED perform well for the EXP model, while HCD and BAPI perform best for the EMAX model.
Conclusion: ED is not performing well because the method weighs the information from each parameter sample equally (E|FIM|) and hence is too influenced by more informative samples (large |FIM| values), resulting in a less robust design. API and EDEFF perform better by evening out the importance of the parameter samples. HCD performs very well and is much faster than the other robust approaches; however HCD is likely to have problems if the optimal information over the parameter distribution is non-monotonic. The new BAPI criterion also performs well in both models and might better handle non-monotonic information; however it is slower than the HCD criterion.
Acknowledgement: This work was part of the DDMoRe project.
References:
[1] Atkinson AC and Donev AN (1992) Optimum Experimental Designs Oxford University Press, Oxford.
[2] D'Argenio DZ. Incorporating prior parameter uncertainty in the design of sampling schedules for pharmacokinetic parameter estimation experiments. Math. Biosci. 99:105-118(1990).
[3] Tod M and Rocchisani JM (1997) Comparison of ED, EID, and API Criteria for the Robust Optimization of Sampling Times in Pharmacokinetics. Journal of Pharmacokinetics and Biopharmaceutics, 25(4).
[4] Dodds MG, Hooker AC, and Vicini P (2005) Robust population pharmacokinetic experiment design J Pharmacokinet Pharmacodyn. 32(1): p. 33-64.
[5] Foo LK and Duffull S (2010) Methods of Robust Design of Nonlinear Models with an Application to Pharmacokinetics, Journal of Biopharmaceutical Statistics, 20:4, 886-902
