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

From: Kevin Wright <kw.statr_at_gmail.com>
Date: Tue, 18 Dec 2007 11:02:51 -0600

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)



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