Bounding the Accuracy of Diagnostic Tests, With Application to COVID-19 Antibody Tests

Tests used to diagnose illness commonly have imperfect accuracy, with some false-positive and negative results. For risk assessment and clinical decisions, predictive values are of interest. Positive predictive value (PPV) is the chance that a member of a relevant population who tests positive has been ill. Negative predictive value (NPV) is the chance that someone who tests negative has not been ill. The medical literature regularly reports sensitivity and specificity. Sensitivity is the chance that an ill person receives a positive test result. Specificity is the chance that a nonill person receives a negative result. Knowledge of sensitivity and specificity enables one to predict the test result given a person's illness status. These predictions are not directly relevant to patient care but, given knowledge of sensitivity and specificity, PPV and NPV can be derived if one knows the prevalence of the disease, the population rate of illness. There is considerable uncertainty about the prevalence of some diseases, a notable case being COVID-19. This paper addresses the problem of identification of PPV and NPV given knowledge of sensitivity and specificity and given bounds on prevalence. I explain the problem and show how to bound PPV and NPV as well as the risk ratio and difference, which are functions thereof. I apply the findings to COVID-19 antibody tests. I question the realism of supposing that sensitivity and specificity are known.