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

TL;DR

This paper introduces a new semi-analytical linear regression method for Poisson count data, providing explicit solutions for parameters and covariance, demonstrated on COVID-19 data, and compared with existing methods.

Contribution

It offers a novel analytical approach for linear regression with Poisson data, improving simplicity and accuracy over traditional methods.

Findings

The new method yields accurate parameter estimates and covariance matrices.

Compared to OLS and chi-squared regressions, it performs better on COVID-19 data.

The approach is broadly applicable across fields dealing with count data.

Abstract

This paper 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 Bonamente and Spence (2022) maximum-likelihood solution for the best-fit parameters, and this paper introduces a simple analytical solution for the covariance matrix that completes the problem of linear regression with Poisson data. The analytical nature for both parameter estimates and their covariance matrix is made possible by a convenient factorization of the linear model proposed by J. Scargle (2013). 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, constitute 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 new linear regression method and two alternative methods often used for the regression of count data -- the ordinary least-square regression and the $\chi^2$ regression -- is provided with the application of these methods to the analysis of recent COVID-19 count data. The paper also discusses the relative advantages and disadvantages among these methods for the linear regression of Poisson count data.