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Register a new userScalable logistic regression with crossed random effects
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Added by
Swarnadip Ghosh
Publication date
2021
fields
Mathematical Statistics
and research's language is
English
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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. (2020) 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 square algorithm. The specific approach is a version of penalized quasi-likelihood due to Schall (1991). A straightforward version of Schalls 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.
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Regression models with crossed random effect errors can be very expensive to compute. The cost of both generalized least squares and Gibbs sampling can easily grow as $N^{3/2}$ (or worse) for $N$ observations. Papaspiliopoulos et al. (2020) present a collapsed Gibbs sampler that costs $O(N)$, but under an extremely stringent sampling model. We propose a backfitting algorithm to compute a generalized least squares estimate and prove that it costs $O(N)$. A critical part of the proof is in ensuring that the number of iterations required is $O(1)$ which follows from keeping a certain matrix norm below $1-delta$ for some $delta>0$. Our conditions are greatly relaxed compared to those for the collapsed Gibbs sampler, though still strict. Empirically, the backfitting algorithm has a norm below $1-delta$ under conditions that are less strict than those in our assumptions. We illustrate the new algorithm on a ratings data set from Stitch Fix.
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