From the blog Three-Toed Sloth by Cosma Shalizi (this also appeared in yesterday's links):
Any P-Value Distinguishable from Zero is Insufficiently Informative: After ten years of teaching statistics, I feel pretty confident in saying that one of the hardest points to get through to undergrads is what "statistically significant" actually means. (The word doesn't help; "statistically detectable" or "statistically discernible" might've been better.) They have a persistent tendency to think that parameters which are significantly different from 0 matter, that ones which are insignificantly different from 0 don't matter, and that the smaller the p-value, the more important the parameter. Similarly, if one parameter is "significantly" larger than another, then they'll say the difference between them matters, but if not, not. If this was just about undergrads, I'd grumble over a beer with my colleagues and otherwise suck it up, but reading and refereeing for non-statistics journals shows me that many scientists in many fields are subject to exactly the same confusions as The Kids, and talking with friends in industry makes it plain that the same thing happens outside academia, even to "data scientists". ... To be fair, one meets some statisticians who succumb to these confusions.
One reason for this, I think, is that we fail to teach well how, with enough data, any non-zero parameter or difference becomes statistically significant at arbitrarily small levels. The proverbial expression of this, due I believe to Andy Gelman, is that "the p-value is a measure of sample size". More exactly, a p-value generally runs together the size of the parameter, how well we can estimate the parameter, and the sample size. The p-value reflects how much information the data has about the parameter, and we can think of "information" as the product of sample size and precision (in the sense of inverse variance) of estimation, say n/σ2. In some cases, this heuristic is actually exactly right, and what I just called "information" really is the Fisher information.
Rather than working on grant proposals Egged on by a friend As a public service, I've written up some notes on this... [The mathematics comes next.]