Summary statistics
Probability density
Summary statistics
Probability density
Without prior information on the prevalence, a uniform beta prior is used:
Beta(1, 1)
Summary statistics
Probability density
App Options
JAGS Options
If the posterior density distributions have converged, the traces of the MCMC chains should mix well.
Running means should converge if the MCMC chains are mixing well.
If the shrink factor (a.k.a. potential scale reduction factor) stabilizes below a value of about 1.1 or 1.2 then convergence has probably occurred.
This web app provides easy-to-use means for Bayesian Prevalence Estimation under Misclassification (in short, BayesPEM). Misclassifications occur when diagnostic tests are imperfect, with test characteristics —sensitivity and/or specificity— smaller than one.
In a Bayesian framework, probabilities are not interpreted as long-term frequencies but rather as quantities that represent a state of knowledge or belief about the true state of the world. In Bayesian inference, prior knowledge needs to be specified first. Then, a likelihood function for the available data is used to update this knowledge according to Bayes' theorem, yielding the posterior knowledge.
In the BayesPEM web app, prior knowledge or belief about the true prevalence as well as the sensitivity and specificity of the diagnostic test is expressed in terms of probability distributions. Prior knowledge may derive from previous studies or expert opinion. Data from the application of the diagnostic test to the test samples are entered, and upon running the model the web app updates the probability distributions which then describe the posterior knowledge about the true prevalence (and the sensitivity/specificity).
However, the posterior probability distribution can not be calculated analytically. Instead, it is numerically approximated by random sampling algorithms referred to as Monte Carlo Markov Chain (MCMC) methods. To ensure that values are sampled from a stationary distribution, so-called convergence diagnostics are evaluated, both automatically by the algorithm as well as through visual inspection of diagnostic plots by the web app user. If convergence of the chains has been achieved, the posterior probability distribution represents the updated knowledge about the true prevalence as well as the sensitivity/specificity. The mode of the distribution provides a point estimate for the true prevalence. The broader the probability distribution, the less certain the knowledge about the true prevalence is. The 95% highest density interval (HDI) denotes the range of prevalence estimates that together account for 95% of the probability mass of the distribution. Any value outside the 95% HDI has less probability than the values inside of it. Therefore, the 95% HDI constitutes a natural measure of uncertainty for the estimate.
This is the model definition used by JAGS (Just Another Gibbs Sampler) , an implementation of an MCMC method which is used to approximate the posterior probability density distribution. Note that the model string depends on the type of data and on how misclassification parameters are defined.
|
This web application was developed at the Federal Institute for Risk Assessment (BfR) as part of the ZooGloW project. |
|
The application runs JAGS (Just Another Gibbs Sampler) in the background.
Use of the following R packages is acknowledged:
- bde
- Estimation of bounded densities.
- BEST
- Calculation of highest probability density intervals.
- dplyr
- Data frame manipulations.
- ggmcmc
- ggplot2 based plotting of convergence diagnostics and MCMC results (posterior distributions).
- ggplot2
- Plotting of prior density distributions and predictive checks.
- R2jags
- JAGS interface and convergence diagnostics.
- RColorBrewer
- ColorBrewer palettes.
- rriskDistributions
- Fitting of prior density distributions from expert opinion.
- runjags
- JAGS interface, automated control of convergence and sample length diagnostics.
- shiny
- Web application framework.
- shinyBS
- Additional Twitter bootstrap components for the Shiny interface.
- shinyjs
- Common JavaScript operations in Shiny applications.
- shinythemes
- Bootstrap themes for shiny. This app uses the cerulean theme.
- tidyr
- Data frame manipulations.
- xtable
- Nicely formatted summary statistics tables.
Prior predictive check
Prior predictive checks are solely based on repeated sampling from prior distributions and do not involve any JAGS model runs. Data are only used in so far as they are compared to what would be expected from the priors. You may think of these checks as a form of hypothesis test: 'Based on our priors, what is the probability of collecting the observed data or more extreme data?' If the probability falls outside a 95% credible interval then the predictive check issues a warning. I.e., a warning means that your prior beliefs do not fit well with the data, so you may check back on your beliefs. Nevertheless, you can still run the model despite of the warning.
Two plots are generated to inspect prior predictive distributions: A ribbon plot and a density plot . In addition, for individual data samples (i.e. not pooled) the Rogan-Gladen point estimate for the prevalence is calculated using the modes of the prior distributions.
Ribbon plots
For ribbon plots, only sensitivity and specificity priors are used, whereas for the prevalence a fixed true value is assumed. From the resulting sampling distribution of positive data samples or pools, the 95% credible interval is retrieved, and this process is repeated for different assumed true prevalence values such that a ribbon-shaped 95% predictive credible region is formed.
Good
Both the lines for the observed number of positives and the Rogan-Gladen estimate intersect with the ribbon.
Bad
Neither the line for the observed number of positives nor for the Rogan-Gladen estimate intersect with the ribbon. Note that these two checks do not always coincide.
Density plots
For density plots, sensitivity, specificity, and prevalence prior are taken into account. The resulting sampling distribution of positive data samples or pools is plotted as a histogram. The 95% predictive credible interval is shaded in blue.
Good
The line marking the observed number of positives lies within the 95% interval.
Bad
The line marking the observed number of positives lies outside the 95% interval.
MCMC Convergence Diagnostics: Density Distributions
For all monitored variables, the probability density of each Monte-Carlo Markov Chain (MCMC) is plotted. Each of the chains has its own color. They should overlap to a high degree. Large differences between the densities of the chains suggest convergence problems.
Good
If chains have converged to the same stationary distribution then the plots of their densities overlap to a high degree.
Bad
For chains that have failed to converge, plots of their densities overlap to a lesser degree.
MCMC Convergence Diagnostics: Traceplots
Good
Bad
MCMC Convergence Diagnostics: Running Means
Good
Bad
MCMC Convergence Diagnostics: Shrink factors
Good
Bad
