[R] regression towards the mean, AS paper November 2007

[Kevin Wright]

On Dec 17, 2007 3:10 PM, hadley wickham <h.wickham at gmail.com> wrote:
> >         This has nothing to do really with the question that Troels asked,
> >         but the exposition quoted from the AA paper is unnecessarily confusing.
> >         The phrase ``Because X0 and X1 have identical marginal
> > distributions ...''
> >         throws the reader off the track.  The identical marginal distributions
> >         are irrelevant.  All one needs is that the ***means*** of X0 and X1
> >         be the same, and then the null hypothesis tested by a paired t-test
> >         is true and so the p-values are (asymptotically) Uniform[0,1].  With
> >         a sample size of 100, the ``asymptotically'' bit can be safely ignored
> >         for any ``decent'' joint distribution of X0 and X1.  If one further
> >         assumes that X0 - X1 is Gaussian (which has nothing to do with X0 and
> >         X1 having identical marginal distributions) then ``asymptotically''
> >         turns into ``exactly''.
>
> Another related issue is that uniform distributions don't look very uniform:
>
> hist(runif(100))
> hist(runif(1000))
> hist(runif(10000))
>
> Be sure to calibrate your eyes (and your bin width) before rejecting
> the hypothesis that the distribution is uniform.
>
> Hadley

Thanks for the example, Hadley.  To me, this suggests we should stop
teaching histograms in Stat 101 and instead use quantile plots, which
give excellent results for n=100 and even surprisingly good results
for n=10:

par(mfrow=c(2,2))
for(i in c(10, 100, 1000, 10000)) {
  qqplot(runif(i), qunif(seq(1/i, 1, length=i)), main=i,
         xlim=c(0,1), ylim=c(0,1),
         xlab="runif", ylab="Uniform distribution quantiles")
  abline(0,1,col="lightgray")
}

Kevin (drifting even further off topic)