2.2 Are negative and positive test results always equally informative?

Positive and negative and positive test results are not generally equally informative. Examples include a Gram stain of cerebrospinal fluid to diagnose bacterial meningitis and a sputum smear for acid-fast bacilli to diagnose tuberculosis. In each case, a positive test rules in the disease, but a negative test does not rule it out.

As a nonmedical example, suppose Tom cannot find his bicycle where the thinks he parked it and wonders if it was stolen. A very specific but insensitive test to determine if his bicycle was stolen is to see whether a lock that has been cut in half and which his key opens remains at the parking meter where he left his bicycle.

A characteristic of tests that are generally more helpful when positive than negative is that they have high specificity and low sensitivity. This makes their positive likelihood ratios much farther from one (on a multiplicative scale) than their negative likelihood ratios. This means that on the log scale of the likelihood ratio slide rule, their arrows pointing to the right (for positive test results) are a lot longer than their arrows pointing to the left (for negative results). (See the likelihood ratio slide rule for a visual demonstration.)

Note that we need to say generally more informative because, depending on your definition of “informative,” there may be some situations in which a test that is much more specific than it is sensitive is still more informative when negative than positive. For example, you could argue that a test with a negative LR of 0.5 and a positive LR of 100 is more informative when negative if the prior probability is 99%!