Exact divide-and-recombine estimation for negative binomial regression: a simulation study and application to SPARCS hospital length-of-stay data

Hospital length of stay (LOS) is a right-skewed count outcome markedly overdispersed relative to the Poisson benchmark, rendering ordinary least-squares and Poisson regression inappropriate at scale. This paper develops a divide-and-recombine (D&R) framework for negative binomial (NB) regression that enables full-data maximum likelihood inference on datasets exceeding available memory by partitioning observations into sequential chunks and accumulating asymptotically lossless compressed representations (ALCRs) at each pass. We prove the resulting estimator is algebraically identical, not merely asymptotically equivalent, to a fully specified centralised alternating Fisher-scoring/IRLS algorithm at every iteration; conditions for recovering the full-data MLE are stated separately, and we derive the dispersion parameter’s pooled asymptotic variance. Relative to MASS::glm.nb , active memory falls from O ( n p ) to O ( n S p + p 2 ) per iteration. A Monte Carlo study ( n = 5,000,000, 1,000 replications) confirms agreement with MASS::glm.nb , the empirical reference, to within 1.8 × 10 −10 for the coefficients and 8.0 × 10 −10 for the dispersion, leaving relative efficiency indistinguishable from 1.000. Peak memory falls by 96% and computation time by a factor of 3.75, of which partitioning contributes 2.35 (1.15 of 2.58 on the SPARCS data). A forced 50-iteration trace confirms the identity at every iterate, and the plug-in standard error for the dispersion attains 0.947 coverage in a separate study at n = 200, 000. Supplementary experiments show this exactness holds an order of magnitude below the recommended chunk-size floor and that D&R remains faster than a direct fit from n = 10,000 to 5,000,000. In the 2024 New York SPARCS inpatient discharge dataset ( n = 2,196,737), the estimated dispersion \({θ }=1.797\) confirms severe overdispersion (variance-to-mean ratio = 13.4); APR severity dominates, with incidence rate ratios (IRRs) of 4.26, 2.16 and 1.55 for Extreme, Major and Moderate versus Minor admissions. Because discharges cluster within facilities, we report facility-clustered robust standard errors, which inflate model-based errors by a mean factor of 17.8 and move eleven of twenty-nine terms out of nominal significance, while the severity and mortality-risk gradients remain robust. These results establish D&R as a principled, interpretable, and computationally tractable foundation for large-scale count regression in health services research.