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In the era of open data, Poisson and other count regression models areincreasingly important. Still, conventional Poisson regression has remainingissues in terms of identifiability and computational efficiency. Especially,due to an identification problem, Poisson regression can be unstable for smallsamples with many zeros. Provided this, we develop a closed-form inference foran over-dispersed Poisson regression including Poisson additive mixed models.The approach is derived via mode-based log-Gaussian approximation. Theresulting method is fast, practical, and free from the identification problem.Monte Carlo experiments demonstrate that the estimation error of the proposedmethod is a considerably smaller estimation error than the closed-formalternatives and as small as the usual Poisson regressions. For counts withmany zeros, our approximation has better estimation accuracy than conventionalPoisson regression. We obtained similar results in the case of Poisson additivemixed modeling considering spatial or group effects. The developed method wasapplied for analyzing COVID-19 data in Japan. This result suggests thatinfluences of pedestrian density, age, and other factors on the number of caseschange over periods.

Improved log-Gaussian approximation for over-dispersed Poisson regression: application to spatial analysis of COVID-19