This analysis uses a Bayesian approach and Gibbs sampler to estimate the true animal-level prevalence of infection based on testing of individual (not pooled) samples using two (2) tests with imperfect sensitivities and/or specificities. The analysis requires prior estimates of true prevalence and sensitivity and specificity for both tests as Beta probability distributions. Outputs are posterior probability distributions for prevalence, sensitivity and specificity. The analysis assumes that the two tests are independent, conditional on disease status. See Joseph et al. (1995) for more details.

Required inputs for this analysis are:

- the number of samples tested;
- the number of samples in each cell of the 2x2 table of comparative test results;
- alpha and beta parameters for prior Beta distributions for:
- assumed true prevalence;
- estimated sensitivity for both tests; and
- estimated specificity for both tests.
- the number of iterations to be simulated in the Gibbs sampler;
- the number of iterations to be discarded to allow convergence of the model;
- lower and upper probability (confidence) limits for summarising the output distributions; and
- starting values for the number of truly infected individuals in each cell of the 2x2 table of comparative test results.

The number of samples tested must be a positive integer and the number of positive samples must be an integer >=0 and <= the number of samples tested. Alpha and beta parameters for prevalence, sensitivities and specificities must be >0 and upper and lower confidence limits must be >0 and <1. Starting values for the numbers of truly infected individuals in each cell of the 2x2 table must be integers >= zero and <= the number of results in that cell. The number of iterations and the number discarded must both be positive integers (>0) and the number discarded must be less than the number of iterations.

For this analysis, the observed results of testing with two tests concurently can be described in a 2x2 table as follows:

Test 2: |
||

Test 1: |
+ve | -ve |

+ve: | a |
b |

-ve: | c |
d |

where a, b, c & d are the observed number of sample results in each cell. A proportion of these samples in each cell will be from truly infected animals, depending on true prevalence and test sensitivities and specificities. The Gibbs sampler is used to estimate the true number of infected animals represented in each of the cells ( Y1, Y2, Y3 & Y4) and hence to generate posterior probability distributions for true prevalence, and test sensitivities and specificities that best fit the data and the prior distributions provided.

Prior estimates of the true prevalence and test sensitivity and specificity may be based on expert
knowledge or on previous data. These estimates are specified as
Beta probability distributions, with parameters alpha and beta. Beta probability distributions are commonly used to express uncertainty about a proportion
based on a random sample of individuals. In this situation, if x individuals are positive for a
characteristic out of n examined, then the alpha and beta parameters can be calculated as alpha = x + 1
and beta = n - x + 1. Alternatively, alpha and beta can be calculated using the *Beta distribution utility*, provided estimates of
the mode and 5% or 95% confidence limits are available from expert opinion.

Outputs from the Gibbs sampler are posterior probability distributions for:

- animal-level prevalence;
- test sensitivity for both tests;
- test specificity for both tests;
- positive and negative predictive values for both tests; and
- the numbers of truly infected individuals ( Y1, Y2, Y3 & Y4) in each cell of the 2x2 table describing the comparative test results.

These distributions are described by their:

- minimum;
- lower probability limit;
- median;
- upper probability limit;
- maximum;
- mean;
- standard deviation;
- histogram and density chart (download by clicking on the appropriate icon); and
- text files of parameter estimates for all iterations of the Gibbs sampler (download by clicking on the appropriate icon).

Because the Gibbs sampler estimates prevalence iteratively, based on the data and the prior distributions, it may take a number of iterations for the model to converge on the true value. Therefore, a specified number of initial iterations must be discarded (not used for estimation) to allow the model to converge on the true values. This number must be sufficient to allow convergence, and should be at least 2000 - 5000. It is also important to carry out an adequate number of iterations to support inference from the results. Suggested minimum values for the total number of iterations and the number to be discarded are provided, but can be varied if desired.

This analysis may take several minutes to complete, depending on the number of iterations required.

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1-sample z-test for a population proportion

1-Stage Freedom analysis

2-sample t-test for summary data

2-sample z-test to compare sample proportion

2-Stage surveys for demonstration of freedom

Analyse test repeatability

Analyse two-stage prevalence data

Analysis of 2-stage freedom survey data

Analysis of simple 2-stage freedom survey

Bioequivalence analysis - two-period, two-treatment crossover trial

Calculate Cluster-level sensitivity and specificity for range of sample sizes and cut-points for given cluster size and imperfect tests

Calculate confidence limits for a sample proportion

Calculate sample sizes for 2-stage freedom survey where individual cluster details are available

Calculate sample sizes for 2-stage freedom survey where individual cluster details are NOT available

Calculate sample sizes for 2-stage freedom survey with fixed cluster-level sensitivity

Calculate test Sensitivity and Specificity and ROC curves

Capture-Recapture analysis

Chi-squared test for contingency table from original data

Chi-squared test for homogeneity of a sample

Chi-squared test for r x c contingency table

Chi-squared test for trend

Cluster-level sensitivity and specificity with variable cut-points

Compare prevalence values

Compare two tests

Complex 2-stage risk-based surveillance - calculation of surveillance sample size

Complex 2-stage risk-based surveillance - calculation of surveillance sensitivity

Complex 2-stage risk-based surveillance - calculation of surveillance sensitivity based on herd testing data

Complex risk-based surveillance - calculation of surveillance sample size

Complex risk-based surveillance - calculation of surveillance sensitivity

Confidence of population freedom (NPV) for a surveillance system

Confidence of population freedom for multiple time periods

Contact

Design prevalence required to achieve target population (cluster or system) sensitivity

Diagnostic test evaluation and comparison

Estimate 95% confidence limits for a median

Estimate alpha and beta Parameters for Beta distributions from count data

Estimate confidence limits for a mean

Estimate parameters for multiple Beta probability distributions or summarise distributions for specified parameters

Estimated true prevalence and predictive values from survey testing

Estimated true prevalence using one test with a Gibbs sampler

Estimated true prevalence using two tests with a Gibbs sampler

Estimating prevalence

Estimation of alpha and beta parameters for prior Beta distributions

"EUFMD - Demonstration of FMD freedom": 2-stage risk-based surveillance with 1 herd-level risk factor, 1 animal-level risk factor and multiple surveillance components

FreeCalc: Analyse results of freedom testing

FreeCalc: Calculate sample size for freedom testing with imperfect tests

Get P and critical values for the Chi-squared distribution

Get P and critical values for the F distribution

Get P and critical values for the normal distribution

Get P and critical values for the t distribution

Glossary

HerdPlus utilities

HerdPlus: Calculate SeH and SpH for a single herd

HerdPlus: SeH and SpH comparison for varying herd sizes

HerdPlus: SeH and SpH for listed herd sizes and optimised sample sizes

HerdPlus: SeH and SpH for optimised sample sizes for range of herd sizes

HerdPlus: SeH and SpH for range of sample sizes and cut-points for given herd size

HerdPlus: SeH and SpH for varying sample sizes

HerdPlus: SeH for fixed sample size and cut-point

HerdPlus: SeH for optimised sampling strategy

HerdPlus: SeH for varying design prevalence

Home

Likelihood ratios and probability of infection in a tested individual

Mantel-Haenszel chi-square test for stratified 2 by 2 tables

McNemar's chi-squared test for association of paired counts

Numbers of false positives to a test

One-sample test to compare sample mean or median to population estimate

Paired t-test or Wilcoxon signed rank test on numeric data

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

Pooled Prevalence Calculator - Demonstration analyses - 4

Pooled Prevalence Calculator - Demonstration analyses - 5

Pooled Prevalence Calculator - Demonstration analyses - 6

Pooled Prevalence Calculator - Demonstration analyses - 7

Pooled Prevalence Calculator - Demonstration analyses - 8

Pooled Prevalence Calculator - Demonstration analyses - 9

Pooled Prevalence Calculator - Demonstration analyses - 10

Pooled Prevalence Calculator - Demonstration analyses - 11

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

Population (or cluster) sensitivity for varying unit sensitivity

Population level (or herd, flock, cluster, or other grouping) sensitivity

Population or cluster level sensitivity using pooled sampling

Positive and Negative Predictive Values for a test

Probability of infection in a test-negative sample

Random Geographic Coordinates Sampling

Random Number Sampling

Random sampling from a sampling frame

Random sampling from populations

Random sampling of animals

References

Risk-based surveillance

Sample size calculation for fixed pool size and perfect tests

Sample size calculation for fixed pool size and uncertain sensitivity and specificity

Sample size calculations

Sample size for a case-control study

Sample size for a cohort study

Sample size for demonstration of freedom (detection of disease) using pooled testing

Sample Size for survival analysis to compare median times since last outbreak

Sample size required to achieve target confidence of freedom

Sample size to achieve specified population level (or herd, flock, cluster, etc) sensitivity

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

Statistical analysis of numeric data

Stochastic analysis of 2-stage freedom survey data

Summarise Beta probability distributions for specified alpha and beta parameters

Summarise Binomial probability distributions for specified sample size and probability

Summarise categorical or continuous data

Summarise continuous data (ungrouped)

Summarise continuous data by single grouping variable

Summarise measures of association from a 2x2 table

Summarise Pert probability distributions for specified minimum, mode and maximum values

Summarise probability distributions

Survey Toolbox for livestock diseases

Survival analysis of herd incidence data

Test evaluation against a gold standard

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!