2*log(likelihood ratio) is approximately chi-square.
For the (multivariate) normal distribution with known (co)variance
(matrix), this is exact and becomes the following:
t(x.bar-mu) %*% Sigma.x.bar.inverse %*% (x.bar-mu)
Set this equal to the (1-alpha) percentage point of the chi-square
distribution with degrees of freedom = dim(mu). If Sigma.x.bar is
estimated, then you may need an F distribution instead of a chi-square.
See Draper and Smith, Applied Regression Analysis or "profiling" or
"profile likelihood" in Bates and Watts, Nonlinear Regression or
Pinhiero and Bates, Mixed Effects Models in S and S-Plus, or look
elsewhere for "profile likelihood.
For a prediction interval, you need to add the variability of an
individual observation. This also should be discussed in the above
references -- at least in Draper & Smith.
Hope this helps.
Spencer Graves
Kyle Armstrong wrote:
Hi,
I am at my wits end trying to solve this. I would like to know if S-plus or
any other program can provide the equation of a confidence ellipse, or at
least information that I can easily build this equation from. The equation
is important for another application.
I have three measured variables that I can use to run a discriminant
function analysis to define groups. My data gives me two functions that I
can use to construct a scatterplot showing the separation and variation of
species groups. I have used Systat to draw both 'centroid' and 'sample'
confidence ellipses around the variation in each group. However, Systat
does not provide a formula for the ellipses.
While I am asking, there seems to be many types of confidence ellipses- what
is the difference between a data ellipse and a confidence ellipse, and where
does a predictive ellipse fit in?
In the event you are unable to help me, if there are any websites or other
people etc that you know could help, I would be very grateful for the info.
I don¹t own S-plus, but I may be able to access it at a local university.
Best regards,
Kyle
------------------------------------------------
Kyle Armstrong kyle@biota.net.au
Zoologist
Biota Environmental Sciences Pty Ltd
First Floor Suite 2, 186 Scarborough Beach Road
Mount Hawthorn Western Australia 6016
Tel. (08) 9201 9955, +61 8 9201 9955
Fax (08) 9201 9599, +61 8 9201 9599
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