Is there a test to compare a variable across time When using a cutoff value to select for observations with a particularly high or low value.

I can think of many simple examples for this:

  • Did the richest persons (99th percentile) get richer between 2000 and 2010?
  • Do persons with a blood pressure above 140mmHg show an increase in blood pressure when watching their favourite football team?
  • Did the fastest runners get faster in the Olympics 2004 in comparison to 2000?

My own naive guess is something like a regression to the mean (I hope I am using the term correctly): If the times for running in the Olympics are just randomly correlated across years, then I would expect that those who were fast in 2000 are likely to be slower in 2004.
The other case when the rich become richer from year to year holds rather true for money.

  • $\begingroup$ thickest looks like a typo for richest. (In some versions of English, thickest is informal for most stupid and so pejorative, if not offensive, unless used with modest self-reference.) $\endgroup$ – Nick Cox May 14 '18 at 12:47
  • $\begingroup$ These are just special cases of quantile regression which is the key term you seek. As the predictor is time, worry about dependence of errors might be appropriate. $\endgroup$ – Nick Cox May 14 '18 at 12:48
  • $\begingroup$ Thanks a lot for your comments so far. Although I was refering to the waist size, I chose a more suitable example. $\endgroup$ – NicolasBourbaki May 14 '18 at 13:03
  • $\begingroup$ That would be fattest except that circumlocution about circumference using a term like waist size would often be preferred. $\endgroup$ – Nick Cox May 14 '18 at 13:17
  • $\begingroup$ "a blood pressure above 140 mmHg" would usually mean different percentiles in different years. $\endgroup$ – Nick Cox May 14 '18 at 13:44

I think a t-test would fit here. You can, if we take the rich folks example, compute an independent sample t-test to compare between their value in 2000 and their value in 2010. Perhaps even better, to see they got richer than other folks, you can compute an independent sample t-test between (rich 2010 - rich 2000) and (ordinary 2010 - ordinary 2000). This way you test if the richest gained more.

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    $\begingroup$ No; comparing means is not the question. It's comparing upper (more generally particular) percentiles. $\endgroup$ – Nick Cox May 14 '18 at 13:43
  • $\begingroup$ I see, so you don't want to test if the top is different than the rest. The examples imply so. Do you want to get 100 means, one for each percentile, and see if they fit a linear or some curve? $\endgroup$ – Yuval Harpaz May 14 '18 at 13:56
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    $\begingroup$ Not my question. I'm just the interpreter. Flag @NicolasBourbaki to get their attention. (However, it's rare to want curves for each percentile in 1(1)99 unless the data set is enormous.) $\endgroup$ – Nick Cox May 14 '18 at 14:00
  • $\begingroup$ Thanks for your comments. Comparing the means would be easy, but this is exactly what I don't wan't. Actually, I would like to do a paired comparison, but as I said, I'm afraid to get a false effect because of the regression to the mean. $\endgroup$ – NicolasBourbaki May 14 '18 at 17:43
  • $\begingroup$ using a percentile divides the observations to groups. I can't see how this helps you unless you compare the groups somehow, like analyzing their means or the variance (ANOVA). $\endgroup$ – Yuval Harpaz May 15 '18 at 6:22

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