Abstract

Mixed-effects models are defined by the distributions of two vector-valued random variables, an n-dimensional response vector, Y and an unobserved q-dimensional random-effects vector, B. The mean of the conditional distribution, Y|B=b, depends on a linear predictor expression of the form X+Zb where is a p-dimensional fixed-effects parameter vector and the fixed and known model matrices, X and Z, are of the appropriate dimension. For linear mixed-effects models the conditional mean is the linear predictor; for generalized linear mixed-effects models the conditional mean is the value of an inverse link function applied to the linear predictor and for a nonlinear mixed-effects model the conditional mean is the result of applying a nonlinear model function for which the parameter vector is derived from the linear predictor. We describe the formulation of these mixed-effects models and provide computationally effective expressions for the profiled deviance function through which the maximum likelihood parameter estimates can be determined. In the case of the linear mixed-effects model the profiled deviance expression is exact. For generalized linear or nonlinear mixed-effects models the profiled deviance is approximated, either through a Laplace approximation or, at the expense of somewhat greater computational effort, through adaptive Gauss-Hermite quadrature.