Maximum likelihood estimation in nonlinear mixed effect models: Adaptive Gaussian Quadrature by sparse grid sampling
W.H.O. Clausen (1), B.B.Rønn (1) I.M.Skovgaard (2)
: (1) Biometrics, Genmab a/s, Denmark; (2) Department of Basic Sciences and Environment, University of Copenhagen, Denmark.
Objectives: Compartment models for PK population modeling are often nonlinear mixed effect models (NLME) with normal distributed random effects and residual error structure. The parameters may be found by maximum likelihood estimation, but the log-likelihood function for such models involves an integral, that can not be solved, but has to be approximated. Adaptive Gaussian Quadrature (AGQ) is an approximation of high accuracy [1], but unfortunately the computing costs grow exponentially with the dimension of the random effects. We suggest reducing the growth in cost to be proportional to the dimension of the random effect vector by following the formulation of Smolyak's rule for sparse grid sampling found in [2].
Methods: In NLMEs calculation of the log-likelihood function involves a d-dimensional integral that usually cannot be solved explicitly (d is the dimension of the random parameter). Different approximation methods to the integral have been implemented and AGQ has been shown to be precise, but requires intensive computations. For one-dimensional random effects the integral can be approximated by evaluating the non-linear integrand in m points which are roots in a polynomial of certain degree. However, multiplication of the one-dimensional rule with m points and d-dimensional random effect requires function evaluation in md points. Smolyak developed a tensor product based method of multiplying one-dimensional grids to higher dimensions. A low-order version of this method requires function evaluation in (2d+1) points and gives exact results for normal integrand multiplied by any polynomial of degree 3 or less.
The algorithm is implemented in R, taking censoring problem into consideration, e.g. concentrations observed to be below LLOQ. The method has been tested on literature data on Theophylline [3] and Indomethacin [4]. A small simulation study has been conducted to investigate the accuracy of the suggested method and compare it with ‘full' AGQ.
Results: The sparse grid sampling AGQ resulted in parameter estimates similar to the results obtained with full AGQ. The simulation study revealed that for the considered compartment models both methods results in good approximations for the integral, resulting in similar parameter estimates.
Conclusions: The sparse grid AGQ is a less computational intensive but equally reliable as the full AGQ approximation for maximum likelihood estimation of parameters in NLMEs under normal circumstances.
References:
[1] Pinheiro, J. C. and Bates, D. M. (1995). Approxmiations to the log -likelihood function in the nonlinear mixed-effects model. Journal of Computational and Graphical Statistics 4(1):12-35.
[2] Gertner T. and Griebel, M. (2003) Dimention Adaptive Tensor Product Quadrature. Computing 71. 65-87.
[3] Boekmann, A.J., Sheiner, L.B. and Beal, S.L. (1994). NONMEM Users Guide: Part V, NONMEM Project Group, University of California, San Francisco.
[4] Kwan, K.C., Breault, G.O., Umbenhauer, E.R., McMahon, F.G. and Duggan, D.E. (1976). Kinetics of indomethicin absorption, elimination and enterophetic circulation in man, Journal of Pharmacokinetics and Biopharmaceutics 4:225-280.
