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Bayesian analysis of tests with unknown specificity and sensitivity

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  • Andrew Gelman
  • Bob Carpenter

Abstract

When testing for a rare disease, prevalence estimates can be highly sensitive to uncertainty in the specificity and sensitivity of the test. Bayesian inference is a natural way to propagate these uncertainties, with hierarchical modelling capturing variation in these parameters across experiments. Another concern is the people in the sample not being representative of the general population. Statistical adjustment cannot without strong assumptions correct for selection bias in an opt‐in sample, but multilevel regression and post‐stratification can at least adjust for known differences between the sample and the population. We demonstrate hierarchical regression and post‐stratification models with code in Stan and discuss their application to a controversial recent study of SARS‐CoV‐2 antibodies in a sample of people from the Stanford University area. Wide posterior intervals make it impossible to evaluate the quantitative claims of that study regarding the number of unreported infections. For future studies, the methods described here should facilitate more accurate estimates of disease prevalence from imperfect tests performed on non‐representative samples.

Suggested Citation

  • Andrew Gelman & Bob Carpenter, 2020. "Bayesian analysis of tests with unknown specificity and sensitivity," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 69(5), pages 1269-1283, November.
    • Handle: RePEc:bla:jorssc:v:69:y:2020:i:5:p:1269-1283
    DOI: 10.1111/rssc.12435
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    References listed on IDEAS

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    1. Carpenter, Bob & Gelman, Andrew & Hoffman, Matthew D. & Lee, Daniel & Goodrich, Ben & Betancourt, Michael & Brubaker, Marcus & Guo, Jiqiang & Li, Peter & Riddell, Allen, 2017. "Stan: A Probabilistic Programming Language," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 76(i01).
    2. Andrew Gelman & Ginger L. Chew & Michael Shnaidman, 2004. "Bayesian Analysis of Serial Dilution Assays," Biometrics, The International Biometric Society, vol. 60(2), pages 407-417, June.
    3. Yair Ghitza & Andrew Gelman, 2013. "Deep Interactions with MRP: Election Turnout and Voting Patterns Among Small Electoral Subgroups," American Journal of Political Science, John Wiley & Sons, vol. 57(3), pages 762-776, July.
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    Cited by:

    1. David McConnell & Conor Hickey & Norma Bargary & Lea Trela-Larsen & Cathal Walsh & Michael Barry & Roisin Adams, 2021. "Understanding the Challenges and Uncertainties of Seroprevalence Studies for SARS-CoV-2," IJERPH, MDPI, vol. 18(9), pages 1-19, April.
    2. Toulis, Panos, 2021. "Estimation of Covid-19 prevalence from serology tests: A partial identification approach," Journal of Econometrics, Elsevier, vol. 220(1), pages 193-213.
    3. Leonid Hanin, 2020. "Estimation of Population Prevalence of COVID-19 Using Imperfect Tests," Mathematics, MDPI, vol. 8(11), pages 1-16, October.
    4. Panos Toulis, 2020. "Estimation of Covid-19 Prevalence from Serology Tests: A Partial Identification Approach," Papers 2006.16214, arXiv.org.
    5. Panos Toulis, 2020. "Estimation of COVID-19 Prevalence from Serology Tests: A Partial Identification Approach," Working Papers 2020-54_Revised, Becker Friedman Institute for Research In Economics.

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