refitting lm() with same x, different y

From: Brian Ripley

As a first shot, use lm with a matrix response.  That fits them all at once 
with one QR-decomposition.  No analogue for glm or lmer, though, since for 
those the iterative fits run do depend on the response.

Thanks Brian, that's very helpful. Also thanks to Kevin Wright who 
suggested using lsfit(x,Y) as being faster than lm for a Y matrix.

I've since worked out that I can bypass even more lm machinery by basing 
my permutation test significance thresholds on the RSS from qr.resid().
Since,

     y = QRb + e
   Q'y = Rb + Q'e
   RSS = || Q'y - Rb ||

then I can do

   X.qr <- qr(X)

once, and for every instance of y calculate

   e   <- qr.resid(X.qr, y)
   rss <- e %*% e

recording them in

   rss.for.all.fits[i] <- rss

which gives me an empirical distribution of RSS scores. The degrees of 
freedom in my X matrix are constant throughout (I should have said that 
before), so all RSS's are on a level footing and map trivially to the 
p-value. I can therefore take the RSS at, say, the 5th percentile, turn it 
into a p-value and report that as my 5% threshold.

Thanks again,

William

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
Dr William Valdar               ++44 (0)1865 287 717
Wellcome Trust Centre           valdar at well.ox.ac.uk
for Human Genetics, Oxford      www.well.ox.ac.uk/~valdar