[R] boxplot notches

[Prof Brian Ripley]

On Tue, 2 Mar 2004, Michael Friendly wrote:

> >
> >
> >>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.
> >
> >  
> >
> 
> The factor 1.58 for H-spr/\sqrt{n} comes from the product of three 
> approximations going from a 95%
> confidence interval for a difference in means, to one for a difference 
> in medians, using the H-spr=IQR
> instead of the standard deviation:
> 
>     H-spr/1.349  \approx \sigma in a N(0,1) dist/n
>     \sqrt{ \pi / 2} \approx std error of a median
>    1.7 / sqrt{n}  is the average of 1.96 and 1.39=1.96/\sqrt{2}, factors 
> for the standard error of the difference
>          between two means, in the cases where one variance is tiny, and 
> where both are equal.
> 
> I believe this is explained in
> 
> @Article{McGill-etal:78,
>   author =       "R. McGill and J. W. Tukey and W. Larsen",
>   year =         "1978",
>   title =        "Variations of Box Plots",
>   journal =      TAS,
>   volume =       "32",
>   pages =        "12--16",
> }

Yes it is (see earlier messages in the thread), but note that 1.7 is 
pretty unprincipled and leads to quite large errors in the nominal 5% 
significance level.

The appropriate help pages have been updated.

-- 
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