Bayesian interval estimation for the difference of two independent Poisson rates using data subject to under‐reporting

Abstract

Comparing occurrence rates of events of interest in science, business, and medicine is an important topic. Because count data are often under-reported, we desire to account for this error in the response when constructing interval estimators. In this article, we derive a Bayesian interval for the difference of two Poisson rates when counts are potentially under-reported. The under-reporting causes a lack of identifiability. Here, we use informative priors to construct a credible interval for the difference of two Poisson rate parameters with underreported data. We demonstrate the efficacy of our new interval estimates using a real data example. We also investigate the performance of our newly derived Bayesian approach via simulation and examine the impact of various informative priors on the new interval.

A Hierarchical Framework for Correcting Under-Reporting in Count Data

“…One way of achieving this is a hierarchical framework consisting of a Binomial model for the recorded counts z i,t,s and a latent Poisson model for the true counts y i,t,s . This approach, often called the Poisson-Logistic (Winkelmann and Zimmermann, 1993) or Pogit model, has been used across a variety of fields including economics (Winkelmann (2008), Winkelmann (1996)), criminology (Moreno and Girn, 1998), natural hazards (Stoner, 2018) and epidemiology (Greer et al (2011), Dvorzak and Wagner (2016), Shaweno et al (2017)). The observed count z i,t,s is assumed a Binomial realisation out of an unobserved total (true) count y i,t,s .…”