Question

A certain disease has an incidence rate of 0.2%. If the false negative rate is 4% and the false positive rate is 2%, compute the probability that a person who tests positive actually has the disease.

Answer 1

“false negative rate” of 6% means that the test has sensitivity of 94%, and that the “false positive rate” of 2% means that the specificity is 98%.

The total number of positive tests, in this background of 0.3% prevalence, will be (as a fraction of all tests):

True positive results plus false positive results:

(0.003 x 0.94) + (0.997 x 0.02) = 0.00282 + 0.01994 = 0.02276

So, if 100,000 people in this population are tested,

2276 will have a positive test result.
Only 282 of those testing positive will actually have the disease.

18 people who have the disease will be missed in this testing.

The predictive value of a positive result is the number of true positives divided by the total number of positives, 282 ÷ 2276 = .123905…….., or 12.4%

87.6% of persons testing positive for the disease will not have the disease in this background of low disease prevalence.

If disease prevalence is different, these numbers will be different.

Let’s suppose we have an epidemic, and the prevalence is now ten percent, thirty-three times the level in the original case. The calculation is now:

True positive results plus false positive results:

(0.10 x 0.94) + (.90 x 0.02) = 0.094 + 0.018 = 0.112

Notice, that with the increase in prevalence of the disease, true positives have increased from .00282 to .094, a factor of 33.3, the same multiple as the increase of the disease in the population, and the false positives have decreased by about 10%, the same decrease as the number of true negatives in the population. All of the false positives came from people who don’t have the disease. Since there are now fewer of them in the population, as they’ve been replaced by people who have the disease, there are now fewer false positives.

The new predictive value of a positive result is: .094 ÷ .112 = .839, or 83.9%. This is 6.77 times the first result.