Generalized Linear Mixed Models via Fully Exponential Laplace in EM

Fit generalized linear mixed models (GLMMs) with normal random effects using first-order Laplace, fully exponential Laplace (FEL) with mean-only corrections, and FEL with mean and variance-diagonal corrections in the E-step of an expectation-maximization (EM) algorithm. The current development version provides a matrix-based interface (y, X, Z) and supports binary logit and probit, and Poisson log-link models. An EM framework is used to update fixed effects, random effects, and a single variance component tau^2 for G = tau^2 I, with staged approximations (Laplace -> FEL mean-only -> FEL full) for efficiency and stability. A pseudo-likelihood engine glmmFEL_pl() implements the working-response / working-weights linearization approach of Wolfinger and O'Connell (1993) , and is adapted from the implementation used in the 'RealVAMS' package (Broatch, Green, and Karl (2018)) . The FEL implementation follows Karl, Yang, and Lohr (2014) and related work (e.g., Tierney, Kass, and Kadane (1989) ; Rizopoulos, Verbeke, and Lesaffre (2009) ; Steele (1996) ). Package code was drafted with assistance from generative AI tools.