Estimation of infection prevalence and sensitivity in a stratified two-stage sampling design employing highly specific diagnostic tests when there is no gold standard

In this work, we describe a two-stage sampling design to estimate the infection prevalence in a population. In the first stage, an imperfect diagnostic test was performed on a random sample of the population. In the second stage, a different imperfect test was performed in a stratified random sample of the first sample. To estimate infection prevalence, we assumed conditional independence between the diagnostic tests and develop method of moments estimators based on expectations of the proportions of people with positive and negative results on both tests that are functions of the tests' sensitivity, specificity, and the infection prevalence. A closed-form solution of the estimating equations was obtained assuming a specificity of 100% for both tests. We applied our method to estimate the infection prevalence of visceral leishmaniasis according to two quantitative polymerase chain reaction tests performed on blood samples taken from 4756 patients in northern Ethiopia. The sensitivities of the tests were also estimated, as well as the standard errors of all estimates, using a parametric bootstrap. We also examined the impact of departures from our assumptions of 100% specificity and conditional independence on the estimated prevalence.