[Rd] Re: [R] H-F corr.: covariance matrix for interaction effect
Peter Dalgaard
p.dalgaard at biostat.ku.dk
Mon Feb 28 23:46:47 CET 2005
Peter Dalgaard <p.dalgaard at biostat.ku.dk> writes:
> Where I would have expected
>
> > (20*5*0.6917-2)/(5*(19-5*.6917))
> [1] 0.8643953
>
> Does anyone have a clue as to what is going on here? Is mighty SAS
> simply doing the wrong thing? The G-G epsilon depends only on the
> eigenvalues of the observed covariance matrix, so surely the H-F
> correction should depend only on the dimension and the DF for the
> empirical covariance matrix?
Just in case anyone was wondering, I think I now know what SAS is
doing, and yes, it is a bug.
The HF correction is
HFeps = (n * (k-1) * GGeps - 2) / ((k-1) * ((n-1) - (k-1) * GG.eps))
for the simple two-way layout, where the residual SSD matrix has (n-1)
degrees of freedom. For the case with covariates, it looks like (to 4
significant digits) SAS is generalizing the above to
HFeps = (n * (k-1) * GGeps - 2) / ((k-1) * (f - (k-1) * GG.eps))
where f is the degrees of freedom for the SSD. However, the first n
also needs adjustment; the correctly generalized formula should read
HFeps = ((f+1) * (k-1) * GGeps - 2) / ((k-1) * (f - (k-1) * GG.eps))
(The G-G epsilon is essentially the squared mean of the eigenvalues of
a suitably transformed SSD divided by the mean of the squares of the
eigenvalues. This is less than one unless all eigenvalues are
identical. H-F replaces numerator and denominator with bias-corrected
variants. However, since everything is a function of the SSD matrix,
sthe formula can only depend on n via the degrees of freedom.)
--
O__ ---- Peter Dalgaard Blegdamsvej 3
c/ /'_ --- Dept. of Biostatistics 2200 Cph. N
(*) \(*) -- University of Copenhagen Denmark Ph: (+45) 35327918
~~~~~~~~~~ - (p.dalgaard at biostat.ku.dk) FAX: (+45) 35327907
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