Open Access
2022 Scalable logistic regression with crossed random effects
Swarnadip Ghosh, Trevor Hastie, Art B. Owen
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Electron. J. Statist. 16(2): 4604-4635 (2022). DOI: 10.1214/22-EJS2047


The cost of both generalized least squares (GLS) and Gibbs sampling in a crossed random effects model can easily grow faster than N32 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 N32 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.


We thank Stitch Fix for sharing some data with us and are especially grateful to Bradley Klingenberg and Sven Schmit. This work was supported in part by a National Science Foundation BIGDATA grant IIS-1837931.


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Swarnadip Ghosh. Trevor Hastie. Art B. Owen. "Scalable logistic regression with crossed random effects." Electron. J. Statist. 16 (2) 4604 - 4635, 2022.


Received: 1 December 2021; Published: 2022
First available in Project Euclid: 21 September 2022

MathSciNet: MR4489236
zbMATH: 07603094
Digital Object Identifier: 10.1214/22-EJS2047

Keywords: backfitting , generalized linear mixed models , quasi-likelihood , Schall’s algorithm

Vol.16 • No. 2 • 2022
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