[R] boxplot notches

[Prof Brian Ripley]

On Mon, 1 Mar 2004, Martin Maechler wrote:

> >>>>> "TL" == Thomas Lumley <tlumley at u.washington.edu>
> >>>>>     on Mon, 1 Mar 2004 09:54:48 -0800 (PST) writes:
> 
>     TL> On Mon, 1 Mar 2004, Christoph Scherber wrote:
>     >> Dear list members,
>     >> 
>     >> Can anyone tell me how the notches in boxplot(Y~X,notch=T)  are
>     >> calculated? What do these notches represent exactly? I´d suppose they
>     >> are Conficence Intervals for the median, but I´ve also been told they
>     >> might show Least Significant Difference (LSD) equivalents.
> 
>     TL> The help page says that 
>     TL> " If the notches of two plots do not overlap then
>     TL>   the medians are significantly different at the 5 percent level."
> 
>     TL> The only thing wrong with this is that it isn't true.
>     TL> The code says that the notches are +/- 1.58 IQR/sqrt(n),
>     TL> so I think the claimed confidence level holds only for
>     TL> normal distribuitons with small amounts of contamination.
> 
> I think John Tukey's idea was that this formula (or just the fact of
> using median and quartiles) is still often approximately correct
> for quite a few kinds of moderate contaminations...

It may be approximately correct for the width of a CI (and when I checked 
it was only appproximately correct for a normal), but I would seriously 
doubt if it were approximately correct for a significance level of 5%.
Remember how fast the tails of the asymptotic normal distribution decay: a 
20% error turns 5% into 2%.

BTW, if there is a precise reference for this it would be good to add it
to boxplot.stats.Rd, as the confidence limits are unexplained there.

-- 
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272866 (PA)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595