### Table 9-3Parameterization when the reference test is assumed to be imperfect, and the index and reference test results are assumed independent within the strata of the condition of interest

Reference Test (+)Reference Test (−)
Index Test (+) $p×Seref×Seindex︸α1+(1-p)×(1-Spref)×(1-Spindex)︸α2$ $(1-p)×Spref×(1-Spindex)︸γ1+p×(1-Seref)×Seindex︸γ2$
Index Test (−) $p×Seref×(1-Seindex)︸β1+(1-p)×(1-Spref)×Spindex︸β2$ $(1-p)×Spref×Spindex︸δ1+p×(1-Seref)×(1-Seindex)︸δ2$

We now have five unknowns: the sensitivity and specificity of the index test (Seindex and Spindex, respectively) and of the reference test (Seref and Spref, respectively), and the disease prevalence (p). The under-braces refer to the probabilities of the 8 strata in Figure 9-1. Note that sensitivity and specificity always refer to the (unknown) true status of the condition of interest. Further, the results of the tests are assumed to be independent given the true status of the condition of interest. The cross-tabulation of the results of the index and reference tests is not sufficient to specify the problem, and additional information is necessary. It is easy to see that if the reference test is “perfect” (Seref = 1, Spref = 1), one obtains the parameterization in Table 9-1. If the results of the index and reference tests are not independent among units with or without the condition of interest, the formulas in Table 9-1 change; in fact, several parameterizations are possible.18,2531

Methods Guide for Medical Test Reviews [Internet].
Chang SM, Matchar DB, Smetana GW, et al., editors.

NCBI Bookshelf. A service of the National Library of Medicine, National Institutes of Health.