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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

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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

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Complex 2-stage risk-based surveillance - calculation of surveillance sensitivity
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Complex risk-based surveillance - calculation of surveillance sample size
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Estimated true prevalence using one test with a Gibbs sampler
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Pooled Prevalence
Pooled Prevalence Calculator - Demonstration analyses
Pooled Prevalence Calculator - Demonstration analyses - 1
Pooled Prevalence Calculator - Demonstration analyses - 2
Pooled Prevalence Calculator - Demonstration analyses - 3
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Pooled Prevalence Calculator - Demonstration analyses - 12
Pooled Prevalence Calculator - Demonstration analyses - 13
Pooled Prevalence Calculator - Demonstration analyses - 14
Pooled Prevalence Calculator - Demonstration analyses - 15
Pooled Prevalence Calculator - Demonstration analyses - 16
Pooled Prevalence Calculator - Demonstration analyses - 17
Pooled prevalence for fixed pool size and perfect tests
Pooled prevalence for fixed pool size and tests with known sensitivity and specificity
Pooled prevalence for fixed pool size and tests with uncertain sensitivity and specificity
Pooled prevalence for variable pool size and perfect tests
Pooled prevalence using a Gibbs sampler
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Population or cluster level sensitivity using pooled sampling
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Sample size for demonstration of freedom (detection of disease) using pooled testing
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Sample size to detect a significant difference between 2 means with equal sample sizes and variances
Sample size to detect a significant difference between 2 means with unequal sample sizes and variances
Sample size to detect a significant difference between 2 proportions
Sample size to estimate a proportion or apparent prevalence with specified precision
Sample size to estimate a single mean with specified precision
Sample size to estimate a true prevalence with an imperfect test
Sample size to estimate a true prevalence with an imperfect test
Simple 2-stage risk-based surveillance - calculation of sample size
Simple 2-stage risk-based surveillance - calculation of surveillance sensitivity
Simple 2-stage risk-based surveillance - calculation of surveillance sensitivity based on herd testing data
Simple risk-based surveillance - calculation of minimum detectable prevalence
Simple risk-based surveillance - calculation of sample size
Simple risk-based surveillance - calculation of surveillance sensitivity
Simple risk-based surveillance with differential sensitivity - calculation of sample size with two sensitivity groups
Simple risk-based surveillance with differential sensitivity - calculation of surveillance sensitivity
Simulate sampling for fixed pool size and assumed known test sensitivity and specificity
Simulate sampling for fixed pool size and assumed perfect test
Simulate sampling for fixed pool size and uncertain test sensitivity and specificity
Simulate sampling for variable pool sizes
Simulated true prevalence estimates from survey testing with an imperfect test
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User guide - Home
User guide 1 - Introduction
User guide 2 - Overview
User guide 3 - Bayesian vs frequentist methods
User guide 4 - Pooled prevalence for fixed pool size and perfect tests
User guide 5 - Pooled prevalence for fixed pool size and tests with known sensitivity and specificity
User guide 6 - Pooled prevalence for fixed pool size and tests with uncertain sensitivity and specificity
User guide 7 - Pooled prevalence for variable pool size and perfect tests
User guide 8 - Pooled prevalence using a Gibbs sampler
User guide 9 - Estimated true prevalence using one test with a Gibbs sampler
User guide 10 - Estimated true prevalence using two tests with a Gibbs sampler
User guide 11 - Estimation of alpha and beta parameters for prior Beta distributions and summarisation of Beta distributions for specified alpha and beta parameters
User guide 12 - Sample size for fixed pool size and perfect test
User guide 13 - Sample size for fixed pool size and known test sensitivity and specificity
User guide 14 - Sample size for fixed pool size and uncertain test sensitivity and specificity
User guide 15 - Simulate sampling for fixed pool size
User guide 16 - Simulate sampling for variable pool sizes
User guide 17 - Important Assumptions
User guide 18 - Pooled prevalence estimates are biased!