Linear regression for Poisson count data: a new semi-analytical method with applications to COVID-19 events

Abstract

This study presents the application of a new semi-analytical method of linear regression for Poisson count data to COVID-19 events. The regression is based on the maximum-likelihood solution for the best-fit parameters presented in an earlier publication, and this study introduces a simple analytical solution for the covariance matrix that completes the problem of linear regression with Poisson data for one independent variable. The analytical nature of both parameter estimates and their covariance matrix is made possible by a convenient factorization of the linear model proposed by J. Scargle. The method makes use of the asymptotic properties of the Fisher information matrix, whose inverse provides the covariance matrix. The combination of simple analytical methods to obtain both the maximum-likelihood estimates of the parameters and their covariance matrix constitutes a new and convenient method for the linear regression of Poisson-distributed count data, which are of common occurrence across a variety of fields. A comparison between this maximum-likelihood linear regression method for Poisson data and two alternative methods often used for the regression of count data—the ordinary least–square regression and the χ2 regression—is provided with the application of these methods to the analysis of recent COVID-19 count data. The study also discusses the relative advantages and disadvantages among these methods for the linear regression of Poisson count data.