₦airaland Forum

The base rate fallacy.

Say that we have a city with 1 million people, and that 100 of them are terrorists.

Say that we have a terrorist detection machine, that will identify a terrorist with 99% certainty—that is, it will correctly identify a terrorist 99 times out of 100.

The machine will also give “false positives”. It will incorrectly identify a “normal” person as a terrorist 1% of the time; that is, if you use it on 100 people who are not terrorists, it will still flag one of them.

A person is tested by the machine, and it goes “Bing! Terrorist!”

What is the chance that the person actually is a terrorist?

Correct answer: about 1%.

Sadly, the same mathematics apply to medical tests, and most doctors are completely ignorant of this. If you have the same city, and 100 people in it are HIV positive, and your test similarly identifies 99% of those who are, but incorrectly flags 1% of those who aren’t—then this means that if you get your test back and it’s positive, it’s way more likely that you are not HIV positive. But most doctors will say it’s 99% likely. A fairly large proportion will say it’s 50%. Some will even say it’s completely certain.

People have committed suicide over this. When the second test came back and proved they were healthy, it was too late.
And when some of these people go to church and get prayers, they claim that they got cured from HIV. HIV test is not 100% in the first place

This is really nicely said, though lacking in some detail. The probability isn't exactly 1%. It's a little higher, but it's no more than 2-3%. And as you have rightly pointed out, the consequence is a significant shift in people's experiences which often leads them into thinking a miracle is at work. Quite the contrary!

Say the prevalence of HIV in Nigeria is probably no more than 1.5%. This is the probability of having HIV and it significantly affects the probability of testing positive given that you have the disease. Let's call that P(D). Therefore, P(not D)=0.985.

If the sensitivity of the test, like you've pointed out is, 99%, then it means the test is wrong 1% of the time. Let's say the specificity of the test is 97%, which means if someone is not HIV positive, then the HIV test gets that right 97% of the time. You can go from this to obtaining P(D | HIV+) which is the probability that you have HIV given that you test positive. That's (0.99 x 0.015)/(0.99 x 0.015 + 0.97 x 0.985) = 0.0153.

That equals just about 1.53% probability that you have HIV given that you test positive. This is the posterior probability in Bayes theorem. So, the chance is not even great at all and the person may actually not have HIV at all.

So your observation is correct. Many go from here into panick mode. Only by taking the test a second time and using this as your new prior probability does the posterior increase. This is why a second test is recommended after doing a first.

Before extensive use of Bayes theorem, many people didn't understand this. Not even doctors who conducted the test, so it's not totally unreasonable to have many still ignorant about it.

Workch:
The base rate fallacy.

Say that we have a city with 1 million people, and that 100 of them are terrorists.

Say that we have a terrorist detection machine, that will identify a terrorist with 99% certainty—that is, it will correctly identify a terrorist 99 times out of 100.

The machine will also give “false positives”. It will incorrectly identify a “normal” person as a terrorist 1% of the time; that is, if you use it on 100 people who are not terrorists, it will still flag one of them.

A person is tested by the machine, and it goes “Bing! Terrorist!”

What is the chance that the person actually is a terrorist?

Correct answer: about 1%.

Sadly, the same mathematics apply to medical tests, and most doctors are completely ignorant of this. If you have the same city, and 100 people in it are HIV positive, and your test similarly identifies 99% of those who are, but incorrectly flags 1% of those who aren’t—then this means that if you get your test back and it’s positive, it’s way more likely that you are not HIV positive. But most doctors will say it’s 99% likely. A fairly large proportion will say it’s 50%. Some will even say it’s completely certain.

People have committed suicide over this. When the second test came back and proved they were healthy, it was too late.
And when some of these people go to church and get prayers, they claim that they got cured from HIV. HIV test is not 100% in the first place

I don't know how you came up with that probability score, but yes, we rate based on margin of error. HIV test is a test that has come under scrutiny because of its medicolegal and psychological connotations. Now there are many types. The commonest is the HIV screening which basically tests the titer of the human antibody to the HIV antigen. It can give a false-positive or -negative result. And it may give a negative result even though the individual is positive, that is why rescreening is encouraged after about 3 months because the titer may not be high enough to be detected at the initial screen. The other broad (and costlier test) is the DNA detection which is more reliable.
So back to your topic. You did not indicate which of the tests youre alluding to.
By the way, we always go for confirmation tests in a government hospital before one is declared HIV positive(because of the medicolegal ish). So no doctor would declare a patient seropositive on just one test.
Also there may be factors that can give false negatives , like anergies, where a patient is so sick that the antibody levels are too low to be detected by the antibody test. In this case, we go for the DNA test because the patient may not be alive in 3 months for a repeat
I hope I cleared you

Guontey:
This is really nicely said, though lacking in some detail. The probability isn't exactly 1%. It's a little higher, but it's no more than 2-3%. And as you have rightly pointed out, the consequence is a significant shift in people's experiences which often leads them into thinking a miracle is at work. Quite the contrary!

Say the prevalence of HIV in Nigeria is probably no more than 1.5%. This is the probability of having HIV and it significantly affects the probability of testing positive given that you have the disease. Let's call that P(D). Therefore, P(not D)=0.985.

If the sensitivity of the test, like you've pointed out is, 99%, then it means the test is wrong 1% of the time. Let's say the specificity of the test is 97%, which means if someone is not HIV positive, then the HIV test gets that right 97% of the time. You can go from this to obtaining P(D | HIV+) which is the probability that you have HIV given that you test positive. That's (0.99 x 0.015)/(0.99 x 0.015 + 0.97 x 0.985) = 0.0153.

That equals just about 1.53% probability that you have HIV given that you test positive. This is the posterior probability in Bayes theorem. So, the chance is not even great at all and the person may actually not have HIV at all.

So your observation is correct. Many go from here into panick mode. Only by taking the test a second time and using this as your new prior probability does the posterior increase. This is why a second test is recommended after doing a first.

Before extensive use of Bayes theorem, many people didn't understand this. Not even doctors who conducted the test, so it's not totally unreasonable to have many still ignorant about it.

Probability of having HIV in a region is different from prevalence of a the disease. 2 different entities. You need to get your epidemiology right

jaephoenix:
Probability of having HIV in a region is different from prevalence of a the disease. 2 different entities. You need to get your epidemiology right

And how does this appear I'm talking about epidemiology? Take it easy, dude. This is just maths, not epidemiology. It doesn't carry with it all those intricacies you have in your post above. Prevalance in Bayesian statistics is different from prevalence in epidemiology.

Guontey:
And how does this appear I'm talking about epidemiology? Take it easy, dude. This is just maths, not epidemiology. It doesn't carry with it all those intricacies you have in your post above. Prevalance in Bayesian statistics is different from prevalence in epidemiology.

I'm not a mathematician or statistician. Just a physician. And we speak from an epidemiological view