Jan 10, 2011 9:18 PM GMT
So modern HIV Testing claims to be over 98% accurate. What this actually means is that if you have HIV, there is a 98% chance that the test will actually show up positive. The opposite of that however is not the case. In other words, it is not true that if your test shows up positive, you have a 98 percent chance of having HIV. This is often assumed to be the case though.
If we use Bayes' Theorem [url]http://en.wikipedia.org/wiki/Bayes%27_theorem[/url] we can determine the probability of having HIV if your test shows up positive.
Baye's Theorem looks like this:
P(F|E) = ( P(E|F) x P(F) ) / ( P(E|F) x P(F) + P(E|F^c) x P(F^c) )
Here's a key for what the terms mean (in regard to this scenario):
E & F are events that respectively refer to testing positive and actually being positive. P stands for "probability).
P(F) = You Actually Have HIV (0.5% of the US population has HIV)
P(F|E) = Probability of actually having HIV if you test positive for HIV (this is what we're solving for)
P(E|F) = Probability of testing positive if you have HIV (98% as mentioned above)
P(E|F^c) = Probability of getting a false positive (roughly 1%).
P(F^c) = Probability of not having HIV (99.5%).
So if you plug in all those numbers into the original equation:
P(F|E) = ( 0.98 x 0.005) / ( .98 x .005 + 0.01 x 0.995 ) = (roughly) 0.330
This means that if you test positive for HIV, there is only roughly a 33% chance that you actually have HIV.
What accounts for this very counter-intuitive phenomenon?
Basically what happens is that even though the rate of false positives is extremely low (1%), SO MANY people get tested that all of those false positives start to add up and eventually outnumber the true positives, since the percentage of people who actually have HIV is very very small.
On the flip-side however, if you test negative, the probability of getting a false-negative are so low they are almost non-existent. Without writing out all the math here, the chance of getting a false-negative is 0.01%.
This is why its still good to get tested, because if you test negative, its pretty much a sure bet that you are in fact negative (assuming that you waited the appropriate amount of time after getting infected that doctors recommend).
Just thought I'd share this.
If we use Bayes' Theorem [url]http://en.wikipedia.org/wiki/Bayes%27_theorem[/url] we can determine the probability of having HIV if your test shows up positive.
Baye's Theorem looks like this:
P(F|E) = ( P(E|F) x P(F) ) / ( P(E|F) x P(F) + P(E|F^c) x P(F^c) )
Here's a key for what the terms mean (in regard to this scenario):
E & F are events that respectively refer to testing positive and actually being positive. P stands for "probability).
P(F) = You Actually Have HIV (0.5% of the US population has HIV)
P(F|E) = Probability of actually having HIV if you test positive for HIV (this is what we're solving for)
P(E|F) = Probability of testing positive if you have HIV (98% as mentioned above)
P(E|F^c) = Probability of getting a false positive (roughly 1%).
P(F^c) = Probability of not having HIV (99.5%).
So if you plug in all those numbers into the original equation:
P(F|E) = ( 0.98 x 0.005) / ( .98 x .005 + 0.01 x 0.995 ) = (roughly) 0.330
This means that if you test positive for HIV, there is only roughly a 33% chance that you actually have HIV.
What accounts for this very counter-intuitive phenomenon?
Basically what happens is that even though the rate of false positives is extremely low (1%), SO MANY people get tested that all of those false positives start to add up and eventually outnumber the true positives, since the percentage of people who actually have HIV is very very small.
On the flip-side however, if you test negative, the probability of getting a false-negative are so low they are almost non-existent. Without writing out all the math here, the chance of getting a false-negative is 0.01%.
This is why its still good to get tested, because if you test negative, its pretty much a sure bet that you are in fact negative (assuming that you waited the appropriate amount of time after getting infected that doctors recommend).
Just thought I'd share this.
