Module: Obtaining the posterior distribution of true prevalence given the apparent prevalence
This module presents a sample of WinBUGS program code that can be used to obtain the posterior distribution of true prevalence when a "binomial experiment" has been performed to estimate apparent (or true) prevalence. The user specifies prior distributions for test sensitivity and specificity, which may be based either upon expert opinion or previous diagnostic test validation studies. The necessary observed data are a sample of size N, of which, y are test positive.
Diagnostic tests are often imperfect and when a population is sampled and the individuals' disease status determined using a diagnostic test, the quantity estimated--apparent prevalence--may be noticeably different from the true prevalence of disease in the population. The degree to which apparant prevalence and true prevalence differ is a function of the sensitivity and specificity of the diagnostic test.
The conventional approach to calculating true prevalence (TP) based on known sensitivity, specificity, and apparent prevalence is based on the following relationship:
AP = Se * TP + (1 - Sp) (1 - TP)
Rearranging terms, it follows that TP can be estimated by
TP = (AP + Sp - 1) / (Se + Sp - 1)
Some algebra shows that when AP is less than 1 - Sp, TP will be estimated to be negative. Likewise, when AP is greater than Se, then TP will be estimated to be greater than 1.
By applying priors for TP, Sp, and Se, to observed binomial data (say, y individuals in a population test positive out of n tested), WinBUGS can be used obtain posterior distributions for TP, Sp, and Se. Moreover, because these posterior disbributions obey the rules of probability, they remain in the interval from 0 to 1, preventing nonsensical estimates of TP that are negative or greater than 1.
View WinBUGS Code and Worked Example
The 
AP-to-Prev contains the WinBUGS code for obtaining the posterior distribution of prevalence, given prior distributions for sensitivity, specificity, and prevalence, combined with observed binomial data for apparent prevalence.