Replies to post #128847 on Biotech Values

[DewDiligence]

Re: Drug A or drug B?

Without loss of generality, say that 100 people in this patient pool take the test; within this group of 100, there will be 1 person who actually has the disease in question.

For the 1 person with the disease, the test will generate 0.1 false negatives and 0.9 true positives (a 10% false-negative rate).

For the 99 people without the disease, the test will generate 29.7 false positives and 69.3 true negatives (a 30% false-positive rate).

Thus, the test’s positive predictive value (in this patient pool) = true positives / total positives = 0.9/(0.9+29.7) = 0.9/30.6 = 2.94%. In other words, if a person in this patient pool tests positive, the probability that he actually has the disease is 2.94%.

(Although not necessary for solving your problem, the test’s negative predictive value in this patient pool = true negatives / total negatives = 69.3/(69.3+0.1) = 69.3/69.4 = 99.86%.)

If a person who tests positive in this patient pool takes drug A, he will be killed by side effects 20% of the time and will live the other 80% of the time for a survival rate of 80%.

If he takes drug B, he will live if he either: i) does not have disease in the first place (probability 97.06%); or ii) has disease and is cured (probability 2.94% x 70% = 2.06%). The sum of these cases is 99.12%, which is the survival rate for taking drug B.

Hence, drug B is clearly the better choice.

[TastyTheElf]

Thank you, iwfal, for the restatement, which for everyone's benefit I'm copying in:

1) disease is 100% fatal in one week if not treated

2) the test for the disease has a 30% false pos rate

3) the test for the disease has a 10% false neg rate

4) the true incidence of the disease among those tested is 1%

5) if you test positive for the disease do you:

A) take a drug that always cures disease but kills 20% of patients from it's side effects

OR

B) take a drug that saves only 70 % from disease mort - but with relatively benign side effects (I.e. No one dies from side effects)

Let's try to keep this simple. First, the false negative rate doesn't matter. I tested positive.

The true incidence of the disease doesn't matter either. I tested positive, and know my chances of having the disease....

There is a 70% chance I have the disease, and a 30% chance that I don't.

If I take drug A, regardless of all the parameters, there is a 20% chance I die. None of the other stuff matters.

If I take drug B:

If I have the disease (70%), I'll be cured 70% of the time, but still die 30% of the time -- 21% of the time overall.

If I don't have the disease, nothing happens.

It's still 20% vs 21%. Nothing has changed.

One might try to impose the following perspective, however:

There's only a 1% chance I have this disease, generally. So if I get a positive result, I should presume that the chances of a false positive massively outweigh the actual probability of having the disease (1%). (Of course we have to consider why I'm taking this test at all if it's only 1%.... I must have had some symptoms that actually pushed that percentage way up.... but never mind that.)

If the test were actually randomly administered, where the expected incidence in my case were genuinely 1%, I should always assume that a false positive is more likely. The percentage of positives that are concurrent with actual disease is very small -- 1% x 90% = 0.9% vs 99% x 30% = 29.7%.... so if the whole population got the test, only 3.0303% of the positives would be associated with actual cases of the disease.

Given that, I would take drug B, because I still assume there's only a 3.0303% chance I have the disease.... so I get a 70% insurance policy against the test being accurate without the 20% killer side effects.

But of course the presumption that I'd be getting this test randomly is suspect.