Two nice statistical analogies
I'm always trying to find good analogies to help explain statistical topics. Here's a couple of good ones I've come across lately, that I'll add to the working list I keep in my brain.
Here's Paul Bruno, explaining why r-squared is not necessarily a good indicator of whether or not something is actually important in real life:
"Consider 'access to breathable oxygen'. If you crunched the numbers, you would likely find that access to breathable oxygen accounts for very little – if any – of the variation in students' tests scores. This is because all students have roughly similar access to breathable oxygen. If all students have the same access to breathable oxygen, then access to breathable oxygen cannot 'explain' or 'account for' the differences in their test scores.
"Does this mean that access to breathable oxygen is unimportant for test scores? Obviously not. On the contrary: access to breathable oxygen is very important for kids’ test scores, and this is true even though access to breathable oxygen explains ≈ 0% of their variation."
Great way to explain it, and an easy way to understand why, if you want to see if a factor is important in a "breathable oxygen" kind of way, you need to look at the regression coefficient, not the r-squared.
This sentence comes from Jordan Ellenberg's mathematics book, "How Not To Be Wrong," which I talked a bit about last post:
"The significance test is the detective, not the judge."
I like that analogy so much I wanted to start by putting it by itself ... but I should add the previous sentence for context:
"A statistically significant finding [only] gives you a clue, suggesting a promising place to focus your research energy. The significance test is the detective, not the judge." [page 161, emphasis in original]
(By the way, Ellenberg doesn't put a hyphen in the phrase "statistically-significant finding," but I normally do. Is the non-hyphen a standard one-off convention, like "Major League Baseball?")
(UPDATE: this question now answered in the comments.)
The point is: one in twenty experiments would produce a statistically-significant result just by random chance. So, statistical significance doesn't mean you can just leap to the conclusion that your hypothesis is true. You might be one of that "lucky" five percent. To be really confident, you need to wait for replications, or find other ways to explore further.
I had previously written an analogy that's kind of similar:
"Peer review is like the police deciding there's enough evidence to lay charges. Post-publication debate is like two lawyers arguing the case before a jury."
Well, it's really the district attorney who has the final say on whether to lay charges, right? In that light, I like Ellenberg's description of the police better than mine. Adding that in, here's the new version:
"Statistical significance is the detective confirming a connection between the suspect and the crime. Peer review is the district attorney deciding there's enough evidence to lay charges. Post-publication debate is the two lawyers arguing the case before a jury."
Much better, I think, with Ellenberg's formulation in there too.