The importance of sensitivity is obvious to most people, but why should it matter how good a test is at detecting people who don't have the disease? The answer is this: Without being able to accurately detect someone's negative state, the test results would be overwhelmed with false positives.
A false positive result is when a test says that a person has a disease when they do not. Conversely, a false negative result is when a test incorrectly says a person does not have a disease. Depending on the severity of the disease, and doctors' ability to treat it, one or the other type of false result may have more severe consequences. For example, in a non-contagious disease where treatment delay doesn't have any long-term consequences but the treatment itself is grueling, false positives are far worse than false negatives. On the other hand, if early treatment is important for good outcomes, false negatives will cause more significant problems.
How often a test gives a false positive or false negative result depends not only on the sensitivity and specificity of the test but how common the disease is. I could show you the math to prove it (and do in this piece over here), but since I figure that the vast majority of my readers would skip right over it I'll ask you to take me at my word. It is important to understand this, however, since it's why testing companies and physicians can't just give you a simple answer as to how likely an incorrect result is. It depends not only on the test but on the population it is used in.
So what do you do if you get two different results from two different diagnostic tests? It depends on the disease. If it's easy enough to treat, and the treatment doesn't have any serious side effects, just go with the flow and take the drugs prescribed for you. If not, then take yet another test. Depending on the type of tests involved, it becomes less and less likely that you would continue to have false results with each subsequent test that you take.
This is actually the principal behind most HIV testing protocols. False negatives aren't that common on HIV tests (although they do occur), but false positives can be more of a problem. That's why most labs do a second test for anyone who initially turns out to be HIV positive. If both tests are positive, the individual in question is almost certainly infected. Rapid tests are an exception to this rule, which is why they are primarily available in high prevalence settings. In areas where HIV is relatively common, for instance, a rapid test does a relatively good job of correctly diagnosing positive individuals and not vastly over diagnosing negative individuals.
Source:
Walensky RP, Paltiel AD."Rapid HIV testing at home: does it solve a problem or create one?" Ann Intern Med. 2006 Sep 19;145(6):459-62.
