On the estimation of variance parameters in non-standard generalised linear mixed models: application to penalised smoothing

Abstract

We present a novel method for the estimation of variance parameters in generalised linear mixed models. The method has its roots in Harville (J Am Stat Assoc 72(358):320-338, 1977)'s work, but it is able to deal with models that have a precision matrix for the random effect vector that is linear in the inverse of the variance parameters (i.e., the precision parameters). We call the method SOP (separation of overlapping precision matrices). SOP is based on applying the method of successive approximations to easy-to-compute estimate updates of the variance parameters. These estimate updates have an appealing form: they are the ratio of a (weighted) sum of squares to a quantity related to effective degrees of freedom. We provide the sufficient and necessary conditions for these estimates to be strictly positive. An important application field of SOP is penalised regression estimation of models where multiple quadratic penalties act on the same regression coefficients. We discuss in detail two of those models: penalised splines for locally adaptive smoothness and for hierarchical curve data. Several data examples in these settings are presented. ; This research was supported by the Basque Government through the BERC 2018-2021 program and by Spanish Ministry of Economy and Competitiveness MINECO through BCAM Severo Ochoa excellence accreditation SEV-2013-0323 and through projects MTM2017-82379-R funded by (AEI/FEDER, UE) and acronym "AFTERAM", MTM2014-52184-P and MTM2014-55966-P. The MRI/DTI data were collected at Johns Hopkins University and the Kennedy-Krieger Institute. We are grateful to Pedro Caro and Iain Currie for useful discussions, to Martin Boer and Cajo ter Braak for the detailed reading of the paper and their many suggestions, and to Bas Engel for sharing with us his knowledge. We are also grateful to the two peer referees for their constructive comments of the paper.