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You open a fresh pack of cards, shuffle the pack and watch him carefully. The Amazing Significo deals you five cards and you find that you do indeed have three of a kind.
According to Wikipedia, the chance of this happening by chance when dealing from an unbiased deck of cards is around 2 per cent - so you are likely to be impressed. You may go public to endorse The Amazing Significo's claim to have supernatural abilities.
But then I tell you that The Amazing Significo has actually dealt five cards to 49 other people that morning, and you are the first one to get three of a kind. Your excitement immediately evaporates: in the context of all the hands he dealt, your result is unsurprising.
Let's take it a step further and suppose that The Amazing Significo was less precise: he just promised to give you a good poker hand without specifying the kind of cards you would get. You regard your hand as evidence of his powers, but you would have been equally happy with two pairs, a flush, or a full house. The probability of getting any one of those good hands goes up to 7 per cent, so in his sample of 50 people, we'd expect three or four to be very happy with his performance.
So context is everything. If The Amazing Significo had dealt a hand to just one person and got a three-of-a-kind hand, that would indeed be amazing. If he had dealt hands to 50 people, and predicted in advance which of them would get a good hand, that would also be amazing. But if he dealt hands to 50 people and just claimed that one or two of them would get a good hand without prespecifying which ones it would be - well, he'd be rightly booed off the stage.
When researchers work with probabilities, they tend to see p-values as measures of the size and importance of a finding. However, as The Amazing Significo demonstrates, p-values can only be interpreted in the context of a whole experiment: unless you know about all the comparisons that have been made (corresponding to all the people who were dealt a hand) they are highly misleading.
In recent years, there has been growing interest in the phenomenon of p-hacking - selecting experimental data after doing the statistics to ensure a p-value below the conventional cutoff of .05. It is recognised as one reason for poor reproducibility of scientific findings, and it can take many forms.
I've become interested in one kind of p-hacking, use of what we term 'ghost variables' - variables that are included in a study but not reported unless they give a significant result. In a recent paper (preprint available here), Paul Thompson and I simulated the situation when a researcher has a set of dependent variables, but reports only those with p-values below .05. This would be like The Amazing Significo making a film of his performances in which he cut out all the cases where he dealt a poor hand**. It is easy to get impressive results if you are selective about what you tell people. If you have two groups of people who are equivalent to one another, and you compare them on just one variable, then the chance that you will get a spurious 'significant' difference (p < .05) is 1 in 20. But with eight variables, the chance of a false positive 'significant' difference on any one variable is 1-.95^8, i.e. 1 in 3. (If variables are correlated these figures change: see our paper for more details).
Quite simply p-values are only interpretable if you have the full context: if you pull out the 'significant' variables and pretend you did not test the others, you will be fooling yourself - and other people - by mistaking chance fluctuations for genuine effects. As we showed with our simulations, it can be extremely difficult to detect this kind of p-hacking, even using statistical methods such as p-curve analysis, which were designed for this purpose. This is why it is so important to either specify statistical tests in advance (akin to predicting which people will get three of a kind), or else adjust p-values for the number of comparisons in exploratory studies*.
Unfortunately, there are many trained scientists who just don't understand this. They see a 'significant' p-value in a set of data and think it has to be meaningful. Anyone who suggests that they need to correct p-values to take into account the number of statistical tests - be they correlations in a correlation matrix, coefficients in a regression equation, or factors and interactions in Analysis of Variance, is seen as a pedantic killjoy (see also Cramer et al, 2015). The p-value is seen as a property of the variable it is attached to, and the idea that it might change completely if the experiment were repeated is hard for them to grasp.
This mass delusion can even extend to journal editors, as was illustrated recently by the COMPare project, the brainchild of Ben Goldacre and colleagues. This involves checking whether the variables reported in medical studies correspond to the ones that the researchers had specified before the study was done and informing journal editors when this was not the case. There's a great account of the project by Tom Chivers in this Buzzfeed article, which I'll let you read for yourself. The bottom line is that the editors of the Annals of Internal Medicine appear to be people who would be unduly impressed by The Amazing Significo because they don't understand what Geoff Cumming has called 'the dance of the p-values'.
*I am ignoring Bayesian approaches here, which no doubt will annoy the Bayesians
**PS.27th Jan 2016. Marcus Munafo has drawn my attention to a film by Derren Brown called 'the System' which pretty much did exactly this! http://www.secrets-explained.com/derren-brown/the-system