Maximum likelihood estimation of nonlinear mixed-effects models with crossed random effects by combining first-order conditional linearization and sequential quadratic programming

Nonlinear mixed-effects (NLME) models have become popular in various disciplines over the past several decades. However, the existing methods for parameter estimation implemented in standard statistical packages such as SAS and R/S-Plus are generally limited to single- or multi-level NLME models that only allow nested random effects and are unable to cope with crossed random effects within the framework of NLME modeling. In this study, we propose a general formulation of NLME models that can accommodate both nested and crossed random effects, and then develop a computational algorithm for parameter estimation based on normal assumptions. The maximum likelihood estimation is carried out using the first-order conditional expansion (FOCE) for NLME model linearization and sequential quadratic programming (SQP) for computational optimization while ensuring positive-definiteness of the estimated variance-covariance matrices of both random effects and error terms. The FOCE-SQP algorithm is evaluated using the height and diameter data measured on trees from Korean larch (L. olgensis var. Changpaiensis) experimental plots as well as simulation studies. We show that the FOCE-SQP method converges fast with high accuracy. Applications of the general formulation of NLME models are illustrated with an analysis of the Korean larch data.