Pseudo likelihood or Quadrature? What We Thought We Knew, What We Think We Know, and What We Are Still Trying to Figure Out

Two predominant computing methods for generalized linear mixed models (GLMMs) are linearization – e.g. pseudo likelihood (PL) and penalized quasi likelihood (PQL) – and integral approximation – e.g. Gauss-Hermite quadrature and Laplace. The primary GLMM package in R, LME4, uses a one-point quadrature algorithm. R also has a PQL package. In SAS®, PROC GLIMMIX was originally developed using the PL algorithm. Laplace and quadrature options were added in the 2008 release. The choice of methods presents a dilemma for GLMM users: which approximation for GLMM estimation and inference should one use, and why? Linearization methods are more versatile and able to handle both conditional and marginal GLMMs. On the other hand, GLMM software documentation and the literature on which it is based tend to focus on the limitations of linearization. Stroup (2013) reiterated this theme in his GLMM textbook, featuring examples of bias in estimates from PL. As a result, a “conventional wisdom” has arisen that integral approximation – quadrature when possible – is always best. However, despite the 2013 copyright, Stroup’s textbook was written in 2011. Meanwhile, experience with GLMMs and research about its small sample behavior has been on-going. Much “conventional wisdom” circa 2011 turns out not to be true. Above all, it is clear that there is no one-size-fits-all best method. The purpose of this presentation is to provide an updated look at what we now know about quadrature and PL, and to offer a “30,000 foot view” of some general operating principles for making an informed choice between the two. This presentation will included updates based on feedback and discussion at the 2018 Conference on Applied Statistics in Agriculture.