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Appendix 1: Bayesian approach for estimating “true” prevalence from “apparent” prevalence
Appendix 1: Bayesian approach for estimating “true” prevalence from “apparent” prevalence
To introduce the Bayesian approach for estimating “true” prevalence (pi) from “apparent” prevalence (p), we build on the illustrations from the main text. In these examples, we had assumed a test sensitivity (SE) of 80 % and a test specificity (SP) of 90 %. We then estimated “true” prevalence from an “apparent” prevalence of 24 and 5 %, using the Rogan–Gladen estimator (Eq. (2)). This resulted in estimates of, respectively, 20 and −48 %.
To parameterize the Bayesian model, we need information on the sample size (n), the number of positive test results (x), and an a priori probability distribution for “true” prevalence (pi). As a first illustration, we will continue to assume that SE and SP are known, fixed values (Model 1). This assumption will be relaxed in a further illustration (Model 2).
For simplicity, we will assume that our sample size was 500. An “apparent” prevalence of 24 % would thus have resulted from observing 120 positive test results. Likewise, an “apparent” prevalence of 5 % would have resulted from observing 25 positive results. As prior probability distribution for “true” prevalence, we will apply a uniform distribution ranging from 0 to 100 %. This is a common choice, as it expresses our belief before having observed any data, that the “true” prevalence can take any possible value and that each possible value is equally likely.
This information can now be used to establish the following Bayesian model:
Model 1 Bayesian estimation of true prevalence from apparent prevalence, based on fixed sensitivity and specificity; comments, denoted by the “#” symbol, translate the code in words
We fitted the model in WinBUGS using two chains, each containing 6,000 samples, of which the first 1,000 were discarded as “burn-in”. Typical output is presented in Table 3. As expected, the Bayesian model succeeded in yielding only positive values, thus providing a useful “true” prevalence estimate for our second case.
The Bayesian approach makes it possible to flexibly account for uncertainty in the values for SE and SP. Instead of assuming fixed values, we assume in Bayesian model 2 that SE can take any possible value between 70 and 90 %, and that SP can take any possible value between 85 and 95 %. For the sake of comparability, these values were chosen so that their means would correspond to the fixed values applied before.
Model 2 Bayesian estimation of true prevalence from apparent prevalence, based on stochastic sensitivity and specificity; comments, denoted by the “#” symbol, translate the code in words
Again, the model was fitted in WinBUGS using two chains, each containing 1,000 “burn-in” samples and 5,000 retained samples. Typical (numerical and graphical) output is presented in Table 4. The results are similar to those obtained by Model 1, but show wider credibility intervals, owing to the additional uncertainty introduced by defining SE and SP as stochastic nodes.
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Speybroeck, N., Devleesschauwer, B., Joseph, L. et al. Misclassification errors in prevalence estimation: Bayesian handling with care. Int J Public Health 58, 791–795 (2013). https://doi.org/10.1007/s00038-012-0439-9
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DOI: https://doi.org/10.1007/s00038-012-0439-9