5 ms·

I'm pretty sure it is quite right. It's standard Bayesian statistics. If the math is wrong, show us.

by SamBam 5y ago

giantg2 5y ago
If the right, show the proof (or at least the real math). Specifically, why would the 10,001st test carry a 50% chance of being a false positive?
- tetromino_ 5y ago
  You misunderstood the scenario: 10,000 people would receive a test. Out of those, ~100 would be expected to test positive. Only those ~100 people would receive a second test. Thus, the population for tests 1..10,000 is different than the population for tests 10,001..10,100.
  - giantg2 5y ago
    Ok... and why would that second population carry a 50% risk of the second test being a false positive? Edit: I misunderstood the scenario.
    - rndgermandude 5y ago
      In this example? Because we know that one is a true positive. We tested 100 people with a false positive, and 1 with a true positive. If we test again, and the test is "fair", then of these 101 people, we will retest 99% (or 99) true negative, 1% (or 1) false positive, and 1 true positive. So two positives, one false positive and one true positive. 1 out of 2 is 50%. (Tho, there is a chance also that the true positive might come back as a false negative, then 1 out of 1 or 100% of positive results would be false positive)
    - jgeralnik 5y ago
      The second population contains 100 people without the disease. Of them we expect one person to have a false positive. In addition the person who actually has the disease will test positive. So on average 2 people will test positive, one of whom has the disease and one of whom doesn't. If you are one of those two people you have a 50% chance of your (positive) diagnosis being false.
    - hervature 5y ago
      There is a 1/10000 (the OP said 0.01%) chance of having the disease. That is, in 10000 people, one person will have the disease. But also, if you test everyone with a 99% accurate test, 100 people will test positive. If you test them again, 1 person will get another positive test. So, at the end of the day, your test will identify the one person with the disease with a 99% certainty (or 98.01 if you require two tests to be positive). But also identify another individual as being doubly positive. So, after two rounds of tests, 50% of the marked individuals truly have the disease. The silly thing with this demonstration is that accuracy is poorly defined. Tests are normally conditional on the actual presence of the disease or not.
    - dontreact 5y ago
      Yes, it's not a 50% chance of false positive. It's 50% chance of false, given positive test.
rndgermandude 5y ago
It forgets to consider false negatives, which will not really change much for this example as the accuracy at 99% is pretty high and the incidence is pretty low, and false negatives thus would not really change much. However, when it comes to mammograms, where the false negative rate is up to 12.5% and the false positive rate is up to 50%, and the incidence of breast cancer in women (12.5% over a lifetime) is rather high, false negatives would play more of a role.
- SamBam 5y ago
  It doesn't need to consider false-negatives. The premise is that you received a positive test. The math is, given the test shows a positive, what is the probability that you actually have the disease?
  - csee 5y ago
    Well, if we assume there is a 0% chance of false negatives (which seems implicit in the post, even though the definition of "test accuracy" doesn't make that assumption), then we don't need to consider it. If we are not making that assumption, then the 50% chance claim at the end is incorrect, since there's a chance that both of the two positive results are false positives and there is a false negative somewhere else in the remaining 99 (out of 101) people. The 50% chance of it being a false positive (conditional on two positive results being observed) is actually a lower bound, and will increase as a function of the false negative rate.
    - Dylan16807 5y ago
      I'm reasonably sure the false negative rate is 1%, because the least misleading definition of "99% accurate" means it fails 1% of the time at random. A 1% false negative rate has a negligible impact on the math. 50% isn't perfect down to the decimal point, but it's very close. After the first round you have 99.99 false positives and .99 true positives. After the second round, if independent, you have .9999 false positives and .9801 true positives. And in fact the lower bound, if you had no false negatives, is slightly below 50%. .9999 vs 1.

I'm pretty sure it is quite right. It's standard Bayesian statistics. If the math is wrong, show us.

by SamBam 5y ago

I'm pretty sure it is quite right. It's standard Bayesian statistics. If the math is wrong, show us.

If the right, show the proof (or at least the real math). Specifically, why would the 10,001st test carry a 50% chance of being a false positive?

tetromino_ 5y ago
You misunderstood the scenario: 10,000 people would receive a test. Out of those, ~100 would be expected to test positive. Only those ~100 people would receive a second test. Thus, the population for tests 1..10,000 is different than the population for tests 10,001..10,100.
- giantg2 5y ago
  Ok... and why would that second population carry a 50% risk of the second test being a false positive? Edit: I misunderstood the scenario.
  - rndgermandude 5y ago
    In this example? Because we know that one is a true positive. We tested 100 people with a false positive, and 1 with a true positive. If we test again, and the test is "fair", then of these 101 people, we will retest 99% (or 99) true negative, 1% (or 1) false positive, and 1 true positive. So two positives, one false positive and one true positive. 1 out of 2 is 50%. (Tho, there is a chance also that the true positive might come back as a false negative, then 1 out of 1 or 100% of positive results would be false positive)
  - jgeralnik 5y ago
    The second population contains 100 people without the disease. Of them we expect one person to have a false positive. In addition the person who actually has the disease will test positive. So on average 2 people will test positive, one of whom has the disease and one of whom doesn't. If you are one of those two people you have a 50% chance of your (positive) diagnosis being false.
  - hervature 5y ago
    There is a 1/10000 (the OP said 0.01%) chance of having the disease. That is, in 10000 people, one person will have the disease. But also, if you test everyone with a 99% accurate test, 100 people will test positive. If you test them again, 1 person will get another positive test. So, at the end of the day, your test will identify the one person with the disease with a 99% certainty (or 98.01 if you require two tests to be positive). But also identify another individual as being doubly positive. So, after two rounds of tests, 50% of the marked individuals truly have the disease. The silly thing with this demonstration is that accuracy is poorly defined. Tests are normally conditional on the actual presence of the disease or not.
  - dontreact 5y ago
    Yes, it's not a 50% chance of false positive. It's 50% chance of false, given positive test.

rndgermandude 5y ago

It forgets to consider false negatives, which will not really change much for this example as the accuracy at 99% is pretty high and the incidence is pretty low, and false negatives thus would not really change much. However, when it comes to mammograms, where the false negative rate is up to 12.5% and the false positive rate is up to 50%, and the incidence of breast cancer in women (12.5% over a lifetime) is rather high, false negatives would play more of a role.

SamBam 5y ago
It doesn't need to consider false-negatives. The premise is that you received a positive test. The math is, given the test shows a positive, what is the probability that you actually have the disease?
- csee 5y ago
  Well, if we assume there is a 0% chance of false negatives (which seems implicit in the post, even though the definition of "test accuracy" doesn't make that assumption), then we don't need to consider it. If we are not making that assumption, then the 50% chance claim at the end is incorrect, since there's a chance that both of the two positive results are false positives and there is a false negative somewhere else in the remaining 99 (out of 101) people. The 50% chance of it being a false positive (conditional on two positive results being observed) is actually a lower bound, and will increase as a function of the false negative rate.
  - Dylan16807 5y ago
    I'm reasonably sure the false negative rate is 1%, because the least misleading definition of "99% accurate" means it fails 1% of the time at random. A 1% false negative rate has a negligible impact on the math. 50% isn't perfect down to the decimal point, but it's very close. After the first round you have 99.99 false positives and .99 true positives. After the second round, if independent, you have .9999 false positives and .9801 true positives. And in fact the lower bound, if you had no false negatives, is slightly below 50%. .9999 vs 1.