Solutions to problems of nonexistence of parameter estimates and sparse data bias in Poisson regression

Poisson regression can be challenging with sparse data, in particular with certain data constellations where maximum likelihood estimates of regression coefficients do not exist. This paper provides a comprehensive evaluation of methods that give finite regression coefficients when maximum likelihood estimates do not exist, including Firth's general approach to bias reduction, exact conditional Poisson regression, and a Bayesian estimator using weakly informative priors that can be obtained via data augmentation. Furthermore, we include in our evaluation a new proposal for a modification of Firth's approach, improving its performance for predictions without compromising its attractive bias-correcting properties for regression coefficients. We illustrate the issue of the nonexistence of maximum likelihood estimates with a dataset arising from the recent outbreak of COVID-19 and an example from implant dentistry. All methods are evaluated in a comprehensive simulation study under a variety of realistic scenarios, evaluating their performance for prediction and estimation. To conclude, while exact conditional Poisson regression may be confined to small data sets only, both the modification of Firth's approach and the Bayesian estimator are universally applicable solutions with attractive properties for prediction and estimation. While the Bayesian method needs specification of prior variances for the regression coefficients, the modified Firth approach does not require any user input.