Parameters:
(1) disease prevalence (p)
(2) actual sensitivity of the screening test (S1)
(3) sensitivity of the reference method (S2)
(4) apparent sensitivity of the screening test (S3)
(5) actual specificity of the screening test (E1)
(6) specificity of the reference method (E2)
(7) apparent specificity of the screening test (E3)
apparent sensitivity of the screening test = S3 =
= ((p * S2 * S1) + ((1 - p) * (1 - E2) * (1 - E1))) / ((p * S2) + ((1 - p) * (1 - E2)))
apparent specificity of the screening test = E3 =
= ((p * (1 - S2) * (1 - S1)) + ((1 - p) * E2 * E1)) / ((p * (1 - S2)) + ((1 - p) * E2))
These can be rearranged to the following:
actual sensitivity of the screening test = S1 =
= ((S3 * ((p * S2) - p - E2 + (p * E2) + 1) + (E2 * (1 - E1 - p + (p * E1))) + (E1 * (1 - p)) + p - 1) / (p * S2)
actual specificity of the screening test = E1 =
= ((E3 * (p - (p * S2) + (E2 * (1 - p)) + (S1 * p * (1 -S2)) - (p * (1 - S2))) / (E2 * (1 - p))
Unfortunately the actual sensitivity cannot be determined until the actual specificity is known, and vice versa.
