Scalable logistic regression with crossed random effects

Abstract

The cost of both generalized least squares (GLS) and Gibbs sampling in a crossed random effects model can easily grow faster than N 3/2 for N observations. Ghosh et al. ( 2022) develop a backfitting algorithm that reduces the cost to O(N ). Here we extend that method to a generalized linear mixed model for logistic regression. We use backfitting within an iteratively reweighted penalized least squares algorithm. The specific approach is a version of penalized quasi-likelihood due to Schall (1991). A straightforward version of Schall's algorithm would also cost more than N 3/2 because it requires the trace of the inverse of a large matrix. We approximate that quantity at cost O(N ) and prove that this substitution makes an asymptotically negligible difference. Our backfitting algorithm also collapses the fixed effect with one random effect at a time in a way that is analogous to the collapsed Gibbs sampler of Papaspiliopoulos et al. (2020). We use a symmetric operator that facilitates efficient covariance computation. We illustrate our method on a real dataset from Stitch Fix. By properly accounting for crossed random effects we show that a naive logistic regression could underestimate sampling variances by several hundred fold.