Image by Matt Buck, under Attribution-ShareAlike 2.0 Generic.
Note: Snout (Reckless Endangerment) has made some good arguments in the comment to this post. The gist is that HIV/AIDS denialists overestimate the false positive rate by assuming that the initial test is all there is, when in fact, it is just the beginning of the diagnostic process. Snout also points out that it is probably wrong to say that most people who get tested have been involved in some high-risk behavior, as a lot of screening goes on among e. g. blood donors etc. I have made some changes (indicated by del or ins tags) in this post because I find myself convinced by the arguments Snout made.
There have already been several intuitive introductions to Bayes’ theorem posted online, so there is little point in writing another one. Instead, let us apply elementary medical statistics and Bayes’ theorem to HIV tests and explode some of the flawed myths that HIV/AIDS denialists spread in this area.
The article will be separated into three parts: (1) introductory medical statistics (e. g. specificity, sensitivity, Bayes’ theorem etc.), (2) applying Bayes’ theorem to HIV tests to find the posterior probability of HIV infection given a positive test result in certain scenarios and (3) debunking HIV/AIDS denialist myths about HIV tests by exposing their faulty assumptions about medical statistics. For those that already grasp the basics of medical statistics, jump to the second section.
(1) Introductory medical statistics
A medical test usually return a positive or a negative result (or sometimes inconclusive). Among the positive results, there are true positives and false positives. Among the negative results, there are true negatives and false negatives.
True positive: positive test result and have the disease.
False positive: positive test result and do not have the disease
True negative: negative test result and do not have the disease.
False negative: negative test result and have the disease.
For the purpose of this discussion, will indicate a positive test, will indicate a negative test, will indicate having HIV and will indicate not having HIV.
is the probability of an event A, say, the probability that a fair dice will land on three. Conditional probabilities, such as , represents the probability of event A, given that event B has occurred. If A and B are statistically independent events, then , if (because the definition of has in the denominator).
Let us define some conditional probabilities that are relevant for HIV tests and Bayes theorem: Read more of this post