E.g. say there is a test T for cancer C, for which n% of positives track to real cancer, and the cancer has a d% 5-year risk of death. However the test preferentially picks up on deadlier forms of the cancer, so given a positive result, your risk of death is higher than n*d/100.
If I say “you have cancer C”, you’ll assume you have a 5 year risk of death of d.
If I say “You have an n% chance of having cancer C”, you’ll assume you have an n*d/100 5 year risk of death.
If I say “You tested positive on test T”, you can discover your actual chance of death over 5 years. So knowing the test result rather than the summary, even the detailed summary, is more informative.
OTOH, your estimates in scenario 3 will be heavily dependent on who gets tested. If the governing body changes the testing recommendations, your chance of death given a positive result on T will change. So knowing “you have n% chance of cancer” is in some ways a more robust result.
E.g. say there is a test T for cancer C, for which n% of positives track to real cancer, and the cancer has a d% 5-year risk of death. However the test preferentially picks up on deadlier forms of the cancer, so given a positive result, your risk of death is higher than n*d/100.
If I say “you have cancer C”, you’ll assume you have a 5 year risk of death of d.
If I say “You have an n% chance of having cancer C”, you’ll assume you have an n*d/100 5 year risk of death.
If I say “You tested positive on test T”, you can discover your actual chance of death over 5 years. So knowing the test result rather than the summary, even the detailed summary, is more informative.
OTOH, your estimates in scenario 3 will be heavily dependent on who gets tested. If the governing body changes the testing recommendations, your chance of death given a positive result on T will change. So knowing “you have n% chance of cancer” is in some ways a more robust result.