How to tell the likelihood of getting a false positive?

Question

Say we have a test for a disease we are comparing with the gold standard. You are given the prevalence of the disease, which is 1%, and sensitivity, 80%, and specificity, 90% and a total population of 1000.

Now how would you tell how likely is it to get a false positive result?

And you have a couple of answer options, including A. 10% and B. 93%

When I first saw this question, I remembered there's a thing called "false positive rate" and I had understood it would tell me how likely it is to get a false positive result with the test, and that it was calculated as 1 - Specificity. 1 - 0.9 = 0.1, so the answer should be A. 10%, but turns out I was wrong.

I got explained that to answer this question I first need the Positive Predictive Value and that, as the Positive predictive value is the probability of a positive result with the test being true, then its complement (1 - PPV) would be the probability of a positive test being false, which doing the math would be 93% (PPV is 7% for this test), and it makes sense (of course), but I still don't understand what then is the false positive rate supposed to mean?

So (1-PPV) means (1 - probability of getting a true positive) and then it is the probability of getting a false positive. But what is the false positive rate? What is that 10% you get doing 1-Specificity?

Answer 1

Going through this step by step.

Sensitivity is the rate of correct disease detection.

Specificity is the rate of correct detection of disease absence.

In your example sensitivity is 0.8 which means out of all the sick people investigated on average 80% of them will be correctly detected.

Let's work with your population number - 1000. Disease prevalence is 0.01 so we have 10 sick people and 990 healthy people. Now of 10 sick people we will correctly detect 8 as sick. And we will correctly detect 990 * 0.9 = 891 as healthy.

What about people who will be incorrectly detected as healthy when they are sick and vice-versa? Well it's easy. If our sensitivity is 0.8 - we detect 80% of sick people as sick. This means that 20% of sick people will be incorrectly classified as healthy.

So 1-sensitivity gives us the false negative rate. And likewise 1-specificity is the false positive rate - that is the probability to classify a healthy person as sick.

The false positive rate in the example would be 0.1 (10%).

The second number (93%) is an answer the the question of "If you get a positive result for disease - how likely that the person is actually healthy?"

For that you would look at the ratio of false positives (healthy people classified as sick) and all positives (all people classified as sick).

Working this out in the example: out of 10 sick people 8 are classified as sick. Out of 990 healthy people 99 (10%) are classified as sick. So the required probability is: 99 / (99 + 8) = 0.9252336