One of the more common errors made by would-be statisticians is assuming two events are independent, when they are not. For example, while two flips of a coin are independent (one “head” doesn’t influence the likelihood of another head coming up next toss), stock market moves are not independent across geography. The latter exhibits correlation, while the former does not.
I was reminded of this issue by a letter in the current issue of the journal Nature. The author wrote in to say that statistical analysis in a recent U.K. cot-death trial was wildly flawed:
Another aspect of his testimony that should be discussed more widely is the use of statistical arguments to conclude that
the probability of two children in the same family dying a cot death was 1 in 73 million.
The writer argued that this testimony was premised on two cot-deaths being independent in the same family. That, he and others correctly argue, is almost certainly untrue.
By definition, no-one knows what causes cot death – and probably many factors are involved – but it would be very surprising if shared factors such as genetics and household environment played no part at all.
Analysis by Ray Hill, a mathematician at Salford University, UK, suggests the correct probabilities of multiple cot-deaths given one death are much higher, perhaps 10 to 22 times. A gentle application of Bayes Theorem showed that, far from being a one-in-73-million improbability, there was roughly a two-thirds chance that the woman convicted in a recent U.K. double cot-death trial was actually not guilty.