# 5 - 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
Pool size 5
Number of pools tested 60
Number of pools positive 29
Prior prevalence alpha 1
Prior prevalence beta 1
Prior Se alpha 46
Prior Se beta 6
Prior Sp alpha 10001
Prior Sp beta 1
Iterations 25000
Discard 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 Prevalence Sensitivity Specificity
Minimum 0.0651 0.6492 0.999
0.025 0.1015 0.7775 0.9996
Median 0.1509 0.8847 0.9999
0.975 0.226 0.9536 1
Maximum 0.3267 0.9951 1
Mean 0.1533 0.8808 0.9999
SD 0.032 0.0454 0.0004
Iterations 20000 20000 20000

Contents
1 Fixed pool size and perfect tests
2 Fixed pool size and tests with known sensitivity and specificity
3 Fixed pool size and tests with uncertain sensitivity and specificity
4 Variable pool size and perfect test
5 Pooled prevalence using a Gibbs sampler
6 Estimated true prevalence using one test (unpooled) with a Gibbs sampler
7 Estimated true prevalence using two tests (unpooled) with a Gibbs sampler
8 Sample size calculation for fixed pool size and perfect tests
9 Sample size calculation for fixed pool size and tests with known sensitivity and specificity
10 Sample size calculation for fixed pool size and tests with uncertain sensitivity and specificity
11 Simulate sampling for fixed pool size and assumed perfect test
12 Simulate sampling for fixed pool size and test with known sensitivity and specificity
13 Simulate sampling for fixed pool size and test with uncertain sensitivity and specificity
14 Simulate sampling for variable pool size and assumed perfect test
15 Demonstration of freedom using pooled testing with tests of known sensitivity and fixed pool size
16 Estimation of alpha and beta Parameters for Prior Beta distributions
17 Estimation of Beta probability distributions for specified alpha and beta parameters

Search