Higgs: is it one-sided or two-sided?
Announcements about the Higgs Boson are invariably framed in terms of the number of sigmas, with 5-sigmas needed for a ‘discovery’. Media outlets helpfully explain what this means by translating 5-sigmas to a probability, which is almost invariably misreported as a probability of the hypothesis that it is all just statistical error e.g. “meaning that it has just a 0.00006% probability that the result is due to chance” [Nature] (see bottom of this blog for comments about the misinterpretation).
But the Daily Telegraph says that 5 sigma is equivalent "to meaning it is 99.99997 per cent likely to be genuine rather than a fluke" - this is a P-value of 0.00003%. So is 5-sigmas equivalent to 0.00003% ( 1 in 3,500,000) or 0.00006% (1 in 1,750,000)?
This reflects whether one is quoting a probability of a Normal observation being more than 5-sigmas away from the expected value in the direction of interest (one-sided), or either direction (two-sided). The two-sided P-value is twice the one-sided, and therefore looks less interesting. The Telegraph is using a one-sided, Nature uses two-sided, who is right?
It’s best to go back to a paper from CERN, eg the ATLAS team announcing their previous results. There they say that "The significance of an excess is quantified by the probability (p0) that a background-only experiment is more signal-like than that observed."
which is excellent and clear. The global P-value is calculated through a sophisticated method that allows for the multiple tests that have been done (the 'look-elsewhere' effect), and the sigma interpretation given afterwards using the graphs in Fig 3 of the paper. The translation is clearly equivalent to a one-sided test - for example they quote 1.4% as being equivalent to 2.2 sigma. And so Nature is wrong: 5-sigmas should be interpreted as a 1 in 3,500,000 chance that such results would happen, if it were all just a statistical fluke.
This is all rather bizarre: the correct (2-sided) P-value is calculated by the scientists, which they translate into sigmas (using a 1-sided interpretation), but then the sigma is then translated back by journalists to a P-value, often wrongly.
What is a P-value anyway?
As discussed previously, the P-values are almost invariably interpreted incorrectly. The probability, or P-value, refers to the probability of getting such an extreme result, were there really nothing special going on. The probability should be applied to the data, not the hypothesis. This may seem pedantic, but people have been convicted of murder (Sally Clark) because of this mistake being made in court. This quantumy blog gets it right and has got more explanation. The BBC website now has a reasonably good, if slightly ambiguous, definition
“The number of sigmas measures how unlikely it is to get a certain experimental result as a matter of chance rather than due to a real effect”
but would be much much better if there were a comma after the word ‘result’.