Sensitivity is defined as probability of positive result in population with interested disease.

Specificity is said to be probalility of negative result in population without interested disease.

Sensitivity and specificity had been thought to be unrelated to disease prevalence in population.

However, let we think such a scenario that 3 health states exists, as A(disease A), B(disease B), C(healthy).

When test U is performed to examine, positive rate is d in population A, e in population B, f in population C.

Prevalence of A, B, C is g,h,i in general population and g+h+i=1. 

Then, positive rate will be (dg+eh+fi) in general population.

If the interested disease is A, and we performs test U for that, 

No doubt that sensitivity should be d.

But when consider specificity, prevalence of "no disease A" is (h+i), 

and prevalence of such population with positive result is (eh+fi),so (1-specificity) becomes (eh+fi)/(h+i).

If e is unequal to f,  specificity will vary with h/i ratio, namely, prevalence ratio.

Maybe we should redefine specificity to "negative result in healthy population", which is (1-f).

Positive likelihood ratio,LRa+,LRb+,LRc+ is d/f, e/f, 1.

Negative likelihood ratio,LRa-,LRb-,LRc- is (1-d)/(1-f),(1-e)/(1-f), 1.

Relative positive LR to clarify disease A & B is LRa+/LRb+.

Practically, we could just multiply pretest probalility ratio to LR ratios of health states.

Pretest ratio of general population: g:h:i

Posttest ratio of positive result: [gd/f : he/f : i] OR [gd : he : if]

Posttest ratio of negative result: [g(1-d)/(1-f) : h(1-e)/(1-f) : i] OR [g(1-d) : h(1-e) : i(1-f)]

If one state outstands all other states after serial tests, then, we could diagnose the patient.

Finally, if the social cost due to failure to diagnosis in date are j,k,l,

Damage ratio could be obtained by multifying social cost to posttest ratio:[ gdj : hek : ifl ]

If the expected value of damage could be lowered to tolerable level, maybe further expensive tests could be stop temporarily.

This could be expanded to more than 3 health states.

However, there still may be different positive rate for healthy people of different age,sex,and so on.