Positive Predictive Value (PPV)

This is the number the person holding the result actually wants, and it is the one most often confused with sensitivity. Sensitivity reads down a column of the table; predictive value reads along a row. Of everyone who tested positive, how many really have it? That change of direction is the whole subject, because a row of the table contains people from both columns, and how many it draws from each depends entirely on how common the condition is.

Work the arithmetic once and it is impossible to unsee. Take an excellent test — 99 % sensitive and 99 % specific — and run it on a million people in a population where 1 in 1,000 has the condition. A thousand people have it, and the test finds 990 of them. Nine hundred and ninety-nine thousand people do not have it, and the test wrongly flags 1 % of them: 9,990 false positives. So 10,980 people get a positive result, and only 990 of them are real. The positive predictive value is $990/10{,}980 = 9.0\,\%$. More than nine positives in every ten are wrong, from a test that is wrong only once in a hundred.

Nothing about the test changed. Take the same test into a clinic where 20 % of the people walking in have the condition and its PPV climbs past 95 %. Predictive value is not a property of a test at all; it is a property of a test and the population it is pointed at, which is why a manufacturer can advertise sensitivity and specificity honestly and a user can still be badly misled. It is also why moving a test from a diagnostic setting to a screening one — the same assay, a different population — can turn a good test into a machine for generating false alarms.

Two consequences follow that are easy to state and hard to accept. First, this is exactly Bayes' theorem, with prevalence as the prior; the odds form on the post-test odds page does the same job in one multiplication. Second, the failure of intuition here is robust and well documented — clinicians, statisticians and medical students asked to estimate this figure typically answer with something close to the specificity, off by an order of magnitude. Knowing the trap does not reliably protect against it. Doing the arithmetic does.