# Thinking Probabilistically – HIV screening in the U.S. and South Africa

How can the same HIV test and screening process have the dramatic differences that show up between the U.S. and South Africa?

Thinking carefully about probability is so often useful in our every-day lives. I was reminded of this once again after talking with a young South African mountain guide high in the hills of the Western Cape. Conversation had turned to medical issues, and the topic of HIV-awareness and HIV-screening came up. There was a degree of mutual incomprehension regarding testing policies in the U.S. and South Africa. For a young South African, annual screening is a matter of course. For an American, widespread testing of the general population seems odd – it is not the norm.

It seems to me that thinking about probabilities and what the tests can tell us goes far in explaining differences in national screening policies. Consider testing someone from the general population in the U.S. versus South Africa. A positive test result for the U.S. person provides little information about whether the individual actually has HIV, while a positive test for the South African is a very good indication that the individual has HIV, and thus is a candidate for treatment. In the U.S. screening has less individual and public health benefit than one might think initially, while in South Africa it has potentially large benefits.

How does this happen, that the same test can have such differences in the two countries? It has to do with the underlying infection rates in the two countries, and to understand we turn to Bayes’ theorem. And using Gigerenzer’s “natural frequencies” makes it easy to see what is happening. (See Gigerenzer’s Calculated Risks. Also chapter 2, p 48 ff of my book Quantitative Risk Management or chapter 2 of A Practical Guide to Risk Management.)

The HIV test (the inexpensive enzyme immunoassay test commonly used in initial screening) is roughly 98.5% accurate, in the sense that 15 out of 1000 tests give a false positive (the test shows positive for someone who is not infected with HIV). Consider screening 1000 individuals from the general U.S. population. There will be roughly 15 false positives. How many true positives? The underlying infection rate in the U.S. is about 0.6% so we should expect roughly 6 true positives. In sum, 21 positive results but only 6 out of 21 true positives – 29%. In other words, a positive test for someone from the general population in the U.S. provides little useful information on whether the person is truly infected with HIV – a positive test means less than 30% chance you are actually infected with HIV.

Now consider testing 1000 individuals from the general South African population. There will still be roughly 15 false positives. But the underlying infection rate is more like 15%, so there will be roughly 150 true positives. In sum, 165 positive results with 150 out of 165 true positives – 91%. This provides good evidence that the individual is infected and would benefit from treatment of one sort or another. (For reference, properly applying Bayes’ rule gives probabilities of 28.6% for the U.S. and 92.1% for South Africa, assuming the HIV test is 99.7% accurate in reporting true positives.)

Further testing to confirm the test will, of course, reduce the false positive rate. But the fact remains that the information provided by initial screening is different for the U.S. and South Africa and may help explain differences in approaches to screening. Simply thinking probabilistically helps make differences that initially seemed peculiar more understandable.