Maximum Likelihood for Generalized Linear Models With Nested Random Effects Via High-Order, Multivariate Laplace Approximation

摘要

Nested random effects models are often used to represent similar processes occurring in each of many clusters. Suppose that, given cluster-specific random 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 integral 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 respec to b are continuous in the neighborhood of the posterior mode of b|#theta#, y. We then apply a Laplace approximation to the integral and maximize the approximate integrated likelihood via Fisher scoring. We develop computational formulas that implement this approach for two-level generalized linear models with canonical link and multivariate normal random effects. A comparison with approximations based on penalized quasi-likelihood, Gauss-Hermite quadrature, and adaptive Gauss-hermite quadrature reveals that, for the hierarchical logistic regression model under the simulated conditions, the sixth-order Laplace approach is remarkably accurate and computationally fast.