I find this whole backlash against p-values pretty confusing. That is probably because I come from particle physics, where we also use a lot of statistics, but in subtly different ways.
Hypothesis testing is not too hard [1]. You pick a cutoff, say p Since we are a cautious bunch, we actually put the threshold for discovery at 5 sigma - pOne other thing that we have to take into account - and many people forget this - is the look-elsewhere effect. If you perform one search, looking for e.g. a Higgs Boson with a mass of 126 GeV, you expect N events in your experiment if it is not there, and N+X if it is there. You know how N is distributed, and the interpretation is straightforward. However if you perform a scan, looking at 120, 121, 122, 123... GeV, then you have to adjust your p-value, since you are basically performing a bunch of different experiments, and by chance alone some of them are bound to turn up "significant".
The same thing applies when hundreds or thousands of Master and PhD students and postdocs do their analyses - even if no one makes a mistake, some of them will "find" a 3 sigma or larger effect that isn't there, just due to the sheer number of independent statistical tests performed. I've "found" new particles myself this way, but when you keep calm, put it into context by looking at other analyses, and try to add more data, you'll often find that your result melts away.
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[1] explaining it is hard, and I will undoubtedly have messed up, especially since I'm tired.