# Pooled Prevalence Calculator – Demonstration analyses

## Pooled prevalence using a Gibbs sampler

This method estimates prevalence for a fixed pool size and tests with uncertain sensitivity and specificity, using a Bayesian approach and a Gibbs sampler. It assumes that the true values of both sensitivity and specificity are not known exactly but can be estimated as Beta probability distributions. This method should be used if you are uncertain about the true values of sensitivity and specificity but can estimate their values from existing data or expert opinion. It is also useful if you already have some information on probable prevalence, which can also be included in the analysis as a prior probability distribution. This method also produces revised estimates of sensitivity and specificity, consistent with the observed data.

For this analysis, input values similar to those for the frequentist method for fixed pool size and uncertain sensitivity and specificity were used, so that results from the two approaches can be compared. It was assumed that samples from 300 individual fruit bats were aggregated into 60 pools of 5 samples each, that 29 pools produced a positive test result and that the test sensitivity was 90% and specificity was 100%.  An assumed sensitivity of less than 100% was used to demonstrate the possible effect of dilution on sensitivity of the pooled test. To allow for uncertainty about the true values of test sensitivity and specificity, alpha and beta values for the prior distributions were calculated assuming that sample sizes of 50 and 10,000, respectively, were used to estimate these values. A uniform prior distribution (all values between 0 and 1 occur with equal probability) was assumed for prevalence because there was no prior information on which to base an estimate. Input values and results for this analysis are summarised in the tables below.

 Input Value 5 Number of pools tested 60 Number of pools positive 29 1 1 46 6 10001 1 25000 5000

The prior Beta distributions defined above are equivalent to:

 Distribution Alpha value Beta value 2.5% percentile Median 97.5% percentile Mean Mode Standard deviation Prevalence 1 1 0.025 0.5 0.975 0.5 0.2887 Sensitivity 46 6 0.7859 0.8895 0.9556 0.8846 0.9 0.0439 Specificity 10001 1 0.9996 0.9999 1 0.9999 1 0.0004

The simulation was run for 25,000 iterations, with 5,000 iterations discarded to allow for convergence. Posterior probability distributions for prevalence, sensitivity and specificity are summarised below. Median and upper and lower 95% probability limits from this analysis were all slightly higher than the corresponding values from the frequentist approach.

 Summary results 0.0651 0.6492 0.999 0.1015 0.7775 0.9996 0.1509 0.8847 0.9999 0.226 0.9536 1 0.3267 0.9951 1 0.1533 0.8808 0.9999 0.032 0.0454 0.0004 20000 20000 20000