If we start with the following expressions for Baye's theorem:
post-test probability disease present given a positive test result =
= ((pretest probability) * S) / (((pretest probability) * S) + ((1 – E) * (1 – (pretest probability))))
post-test probability disease present given a negative test result =
= ((pretest probability) * (1 – S)) / ((E * (1 – (pretest probability))) + ((pretest probability) * ( 1 – S)))
To achieve a certain post-test probability the following sensitivity and specificity are required:
S for a positive test result =
= ((posttest probability) * (1 – E) * (1 – (pretest probability))) / ((pretest probability) * (1 – (posttest probability)))
E for a positive test result =
= 1 – (((pretest probability) * S * (1 – (posttest probability))) / ((posttest probability) * (1 – (pretest probability))))
If S and E are approximately equal:
S or E for a positive test =
= (posttest probability) * (1 - (pretest probability)) / ((pretest probability) + (posttest probability) - (2 * (pretest probability) * (posttest probability)))
S for a negative test result =
= (((posttest probability) * E * (1 – (pretest probability))) + ((pretest probability) * ((posttest probability) – 1))) / ((pretest probability) * ((posttest probability) – 1))
E for a negative test result =
= (((1 – S) * (pretest probability)) * (1 – (posttest probability))) / ((posttest probability) * (1 – (pretest probability)))
