Organizers:

Douglas Bates, bates@stat.wisc.edu Mary Lindstrom, lindstro@biostat.wisc.edu

Speakers

3:45 p.m.
Nonlinear Mixed Models: A Future Direction
Russell D. Wolfinger, SAS Institute Inc.

Statistical models in which both fixed and random effects parameters enter nonlinearly are becoming increasingly popular. These models have a wide variety of applications, two of the most common being nonlinear growth curves and overdispersed binomial data. A new SAS procedure, NLMIXED, fits these models using likelihood-based methods. This talk discusses some of the primary features of PROC NLMIXED and illustrates its use with two prototypical examples.

4:15 p.m.
Maximum Likelihood for Generalized Linear Mixed Models via High Order Multivariate LaPlace Approximation
Steve Raudenbush, University of Michigan
Meng Li Yang, Nan Tai Institute of Technology, Taiwan

Hierarchical models are often used to represent similar processes occurring in each of many clusters. Suppose that, given cluster-specific effects, $b$, the data, $y$, follow a likelihood, $L(y | b,\theta)$, while $b$ follows a density, $p(b | \theta)$. Likelihood inference requires maximization of $\int L(y | b,\theta)p(b | \theta) db$ with respect to $\theta$. Evaluation of this integral often proves difficult, making likelihood inference difficult to obtain. We propose a multivariate Taylor series approximation of the log of the integrand that can be made as accurate as desired if the integrand and all its partial derivatives with respect to $b$ are continuous in the neighborhood of the posterior mode of $b | \theta, y$. We then apply a high-order Laplace approximation to the integral and maximize the approximate integrated likelihood via Fisher scoring. We apply this approach to generalized linear models with random coefficients and illustrate it in the special case of binary outcomes. A simulation study reveals that the method is competitive with adaptive or non-adaptive gaussian quadrature.

4:45 p.m.
Computing REML or ML Estimates for Linear or Nonlinear Mixed-Effects Models
Douglas Bates, University of Wisconsin - Madison
Jose Pinheiro, Bell Labs, Lucent Technologies

We describe several methods of streamlining the optimization of the REML and/or the maximum-likelihood parameter estimation criteria for linear mixed-effects models. First, we condition on the ratio of the variance components or, in general, the relative precision factors. Conditional on these values, the criteria can be easily calculated from a penalized least squares problem which is a non-iterative calculation. When optimizing this profiled log-likelihood or restricted log-likelihood with respect to the parameters in the relative precision factor, the dimension of the optimization problem is considerably reduced. By taking advantage of the special structure of the penalized least squares problem the amount of information that must be stored and the amount of computation to be performed can be substantially reduced. Second, we use the matrix-logarithm parameterization for the relative precision factors to provide a more stable optimization problem. Third, we derive both the EM and the Newton-Raphson updates so the optimization method can begin with the EM iterations then switch to the Newton-Raphson updates when near the optimum.

All of these techniques can also be used in the optimization of the estimation criteria in nonlinear mixed-effects models. These methods are implemented in the 3.0 version of the NLME library for S.