A Bayesian Method for Evaluating Medical Test operating Characteristics When Some Patients' Conditions Fail to Be Diagnosed by the Reference Standard

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Abstract
The evaluation of a diagnostic test when the reference standard fails to establish a diagnosis in some patients is a common and difficult analytical problem. Conventional operating char acteristics, derived from a 2 x 2 matrix, require that tests have only positive or negative results, and that disease status be designated definitively as present or absent. Results can be displayed in a 2 x 3 matrix, with an additional column for undiagnosed patients, when it is not possible always to ascertain the disease status definitively. The authors approach this problem using a Bayesian method for evaluating the 2 x 3 matrix in which test operating characteristics are described by a joint probability density function. They show that one can derive this joint probability density function of sensitivity and specificity empirically by applying a sampling algorithm. The three-dimensional histogram resulting from this sampling pro cedure approximates the true joint probability density function for sensitivity and specificity. Using a clinical example, the authors illustrate the method and demonstrate that the joint probability density function for sensitivity and specificity can be influenced by assumptions used to interpret test results in undiagnosed patients. This Bayesian method represents a flexible and practical solution to the problem of evaluating test sensitivity and specificity when the study group includes patients whose disease could not be diagnosed by the reference standard. Key words: Bayesian analysis; test operating characteristics; probability density functions. (Med Decis Making 1990;10:102-111)