One of the most unique applications of Bayesian statistics is in finding estimates for unknown values that depend upon other unknown values. By taking advantage of the Bayesian ability to integrate prior knowledge into its models, you can develop parameter estimates using priors that are little more than a guess.
This application of Bayesian statistics is commonly seen in diagnostics. When there isn’t a gold standard test that allows simple comparisons, Bayesian models are able to use data on test results to estimate the performance of these tests and the prevalence of the disease. Whether it’s a new test or a new population where the test is unproven, these analyses allow us to glimpse important aspects of diagnostic usage with only scant data.
The pioneering paper that developed these methods is titled “Bayesian estimation of disease prevalence and the parameters of diagnostic tests in the absence of a gold standard” by Lawrence Joseph, Theresa Gyorkos, and Louis Coupal. They collected the results of two tests for the Strongyloides parasite among Cambodian immigrants to Canada in the 70s. Since there was no knowledge of how common the parasite was in this group, they used an uninformative prior for its prevalence, but were able to solicit vague priors about the two tests’ performance from clinical experts. From these priors they built distributions which they then ran through a Gibbs sampler.
A Gibbs sampler is a program that runs repeated sampling to find the parameters – in our case, test performance and prevalence – that would make the most sense in light of the data we have. Because of the way that the sampler moves from parameter estimate to parameter estimate, it devotes most of its samples to high likelihood scenarios. Therefore, the parameter estimates are essentially histograms of the number of samples that the algorithm has run for each parameter value.
JAGS is a commonly used Gibbs sampler, and its name stands for “Just Another Gibbs Sampler.” It’s not the only one, but its got a convenient R interface and a lot of literature to support its use. I recently used JAGS in a tutorial on its R interface that recreates the Joseph, Gyorkos, and Coupal paper. You don’t need any datasets to run it, as you can easily simulate the inputs of the two Strongyloides tests.
The first part deals with gathering estimates from the two different parasite tests independently. This means building models of the test results as Bernoulli samples of a distribution that depends on the tests’ sensitivity and specificity, as well as the disease prevalence.
The second half of the tutorial deals with learning to use the data from the two tests together. This is significantly more complex, as we need to model the joint probability of each possible combination of the two tests together. To do this, we’ll need to read in the results of the tests on each patient. However, since we’re reading in the results directly, we can’t assign a distribution to them. Rather, we’ll learn to create a likelihood that is directly observed from the data and to ensure that our new likelihood affects the model.