Alternative Method of Estimating False Rate of Diagnostic Screening Test for a Condition in a Population

Abstract

Diagnostic screening tests are essential tools in clinical medicine and epidemiology for detecting the presence or absence of a disease condition. Their quality is conventionally assessed using Sensitivity (Se), Specificity (Sp), False Positive Rate (FPR), False Negative Rate (FNR), True Positive Rate (TPR), and True Negative Rate (TNR). A critical and often overlooked distinction is that Se and Sp are conditional probabilities given the true disease state, whereas FPR, FNR, TPR, and TNR as used in practice are conditional probabilities given the observed test result. Standard estimation of the latter group requires prior knowledge of the population prevalence rate, data that are frequently unavailable, particularly in developing nations.
This paper proposes, develops, and illustrates a novel statistical method for estimating all of the above indices using only directly observable cell frequencies from a 2×2 contingency table of screening results, without requiring the population prevalence rate. The method introduces a concordance index ω that measures the net relative difference between concordant and discordant test outcomes and derives closed-form estimators with established theoretical properties, including exact expressions for their asymptotic standard errors and 95% confidence intervals via the delta method. A simulation study across varying sample sizes (n = 50, 100, 200, 500) and prevalence levels confirms that the estimators are nearly unbiased, converge rapidly, and maintain nominal confidence interval coverage. Applied to a real prostate cancer screening dataset (n = 135), the method yields Se = 33.33%, Sp = 97.44%, FPR = 33.33%, TPR = 66.67%, FNR = 9.52%, and TNR = 90.48%. Comparison with the traditional Bayesian prevalence dependent method confirms the practical superiority of the proposed approach in low prevalence and data scarce settings, while also clarifying the scenarios in which the Bayesian approach remains indispensable.