Odd odds

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was produced by the Winton programme for the public understanding of risk based in the Statistical Laboratory in the University of Cambridge. The aim was to help improve the way that uncertainty and risk are discussed in society, and show how probability and statistics can be both useful and entertaining.

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Ben McGarry pointed out this blog entry about an article published in Sexually Transmitted Infections Online that says some rather odd things.

The article says that men sent text messages to remind them to have HIV tests had double the testing rate compared to men not sent messages, and after statistical adjustment for difference between the groups "re-testing was 4.4 times more likely". But since the testing rate was 30% in the group not sent text messages, how could testing be 4 times more likely in the group sent the messages?

The answer lies in the tricky issue of odds, as explained by our own Kevin McConway in a recent blog. The testing rate was 64% in the group sent a text message, so the testing rate roughly doubled from 30% to 64%. This means that the odds on testing went up from 0.3/0.7 = 0.43 to 0.64/0.36 = 1.77. So the odds ratio associated with being sent a text message is 1.77/0.43 = 4.1. When the authors did their statistical adjustment, they obtained an odds ratio of 4.4 - almost exactly the same as the unadjusted odds ratio. But they interpreted this as meaning that testing was 4 times more likely, which is not very sensible.

The moral of this story is that odds ratios are convenient measures for statistical models but almost uninterpretable unless the rates are low. The other moral is that this was a non-randomised study in which we do not know why some men were not sent text messages, and so perhaps the conclusions should be interpreted cautiously.

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