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In books on probability there are statements such as

The population, for a disease D, has a true rate of T%

Some Test ST, has false positive rate of FP% and a false negative rate of FN%.

T, FP, and FN are elements of the set of real numbers.

Was the number T determined by some 100% accurate and possibly expensive test? My reaction is that the source is not adequately discussed. Maybe a few simple sentences would do to explain this. One explanation of this possibly perfect test that could be stated once per text document.

Is ST some low cost test?

VIRTUAL-VOID
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The problem for which $T$, $FP$ and $FN$ are the input data usually asks for a calculation of the probability that a patient actually has disease $D$ given that they test positive.

The source of those numbers is not part of the problem. $T$ might come from data kept by departments of health, hospitals and insurance companies. You could get it by giving that hypothetical perfect test to a large random sample of the population, but that's not a likely source for the number. $FP$ and $FN$ result are found by actually trying the test on people who for some other reason you know do or don't have the disease.

Many people are surprised by the answer when $T$ is small, which is (much) less than $FN$ since then most of the positives will be false positives.

Relevant: Applied Probability- Bayes theorem

Ethan Bolker
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  • I am asking about the source of T specifically. – VIRTUAL-VOID Sep 13 '21 at 00:09
  • As I wrote, $T$ usually comes from public health data. In the real world the actual source will vary from application to application. When you see the problem in a textbook you must just assume $T$ is known "somehow". – Ethan Bolker Sep 13 '21 at 00:19
  • That seems inadequate to me. Perhaps compounded by the current pandemic. If T comes from public health data .. does the public health department use a different test? – VIRTUAL-VOID Sep 13 '21 at 00:22
  • The inadequacy in not in the mathematics, so this site can't provide an answer. Search how is covid rate calculated and you will find lots of (sometimes confusing) information. Good luck. – Ethan Bolker Sep 13 '21 at 00:27