Bayesian Beta-Binomial Prevalence Estimation Using an Imperfect Test

TL;DR

This paper presents a Bayesian method for estimating disease prevalence from imperfect tests, providing a more accurate posterior distribution than traditional methods, and applies it to COVID-19 data in Santa Clara County.

Contribution

It introduces a simple Bayesian formula and an efficient Monte Carlo algorithm for prevalence estimation with unreliable tests, improving accuracy over previous approaches.

Findings

Bayesian posterior places more mass near zero prevalence.

Estimated COVID-19 infections are significantly lower than previous delta method estimates.

The method is demonstrated with code and additional examples at testprev.com.

Abstract

Following [Diggle 2011, Greenland 1995], we give a simple formula for the Bayesian posterior density of a prevalence parameter based on unreliable testing of a population. This problem is of particular importance when the false positive test rate is close to the prevalence in the population being tested. An efficient Monte Carlo algorithm for approximating the posterior density is presented, and applied to estimating the Covid-19 infection rate in Santa Clara county, CA using the data reported in [Bendavid 2020]. We show that the true Bayesian posterior places considerably more mass near zero, resulting in a prevalence estimate of 5,000--70,000 infections (median: 42,000) (2.17% (95CI 0.27%--3.63%)), compared to the estimate of 48,000--81,000 infections derived in [Bendavid 2020] using the delta method. A demonstration, with code and additional examples, is available at testprev.com.