N-dimensional Likelihood Profiling: An Efficient Alternative to Bootstrap
William S. Denney (1)
PTx Clinical Pharmacology, Pfizer, Inc. Cambridge, MA, USA
Objectives: Bootstrap is the reference method for obtaining robust estimates of the confidence intervals (CI) of model parameters and simulation estimates. However, bootstrapping requires significant computational resources with typically ≥1000 iterations (each taking approximately the same CPU time of a single run) and ≥ 30 independent subjects to provide 95-99% CI. [1] The objective of this work is to develop a time effective alternative to bootstrapping: N-dimensional Likelihood Profiling (NLP) is proposed here as the extension of log-likelihood profiling [2-3] to multiple dimensions with applications analogous to bootstrapping.
Methods: The basis of NLP is integration of the log-likelihood surface with regions defining the deviation from normality. The method iteratively estimates the n-dimensional log-likelihood by adaptive sampling of the surface and refining regions in areas of large uncertainty until the changes are below a user-provided tolerance. At completion, a parametric probability surface is provided to the user with confidence intervals on each estimated parameter. This parametric surface can be sampled with algebraic integration for further model simulation (similar to a bootstrapped VPC). The algorithm logical steps are:
0) Initialize with multivariate normal surface assumption
1) While Δ∫χ2(logLik(θ)) dθ > tol
2) Choose new θ for refinement
3) Estimate logLik(θ)
Where θ is the vector of model parameters; logLik is the log-likelihood as a function of model parameters; χ2 is the p-value from the chi-squared distribution for the given log-likelihood relative to the model minimum; and tol is the user-selected tolerance.
Results: Simulated PK and PD datasets allowed estimation of the likelihood surface using NLP with ≥10-fold reduction in the computational time compared to bootstrap (using 1% as tolerance); when a high number of dimensions are required for uncertainty estimates, the efficiency approaches parity with bootstrapping. Results were similar to bootstrapped estimates when examined visually and bootstrapped points outside an estimated confidence region.
Conclusions: In a variety of PK and PD model examples, NLP provides excellent agreement with bootstrapping in model simulation confidence intervals using NONMEM 7.1.2 FOCE-I. NLP markedly decreases computation time relative to bootstrapping.
References:
[1] Parke J, Holford NH, Charles BG. A procedure for generating bootstrap samples for the validation of nonlinear mixed-effects population models. Comput Methods Programs Biomed. 1999; 59:19-29.
[2] Lindbom L, Pilgren P, Jonsson N. PsN-Toolkit-A collection of computer intensive statistical methods for non-linear mixed effect modeling using NONMEM. Computer Methods and Programs in Biomedicine. 79(3):241-57.
[3] Sheiner LB. Analysis of pharmacokinetic data using parametric models. III. Hypothesis tests and confidence intervals. J Pharmacokinet Biopharm. 14(5):539-55.
