[S] [R] confidence ellipsoid for model parameters

[John Fox]

Dear Duncan,

Yes, you're correct -- I was careless with the multipler: Using 2*qf(.95, 
2, df) to scale the ellipse produces the usual 95% confidence ellipse for 
the two parameters. Using 3*qf(.95, 3, df) produces a larger Scheffe 
ellipse, which is just the 2D projection of the 3D ellipsoid -- apparently 
what was wanted. That is, the ellipsoid whose shadows are the 2D confidence 
ellipses is inside the 3D confidence ellipsoid.

Sorry (and thanks for the correction),
  John

At 03:54 PM 1/4/2002 -0500, Duncan Murdoch wrote:
>On Fri, 04 Jan 2002 15:29:23 -0500, you wrote in message
><5.1.0.14.2.20020104152709.02e2f418 at mcmail.cis.mcmaster.ca>:
>
> >At 01:49 PM 1/4/2002 -0500, Duncan Murdoch wrote:
> >>John Fox wrote:
> >>
> >> >The confidence ellipse for a pair of coefficients is just (with an
> >> >adjustment for size) the perpendicular shadow of the joint ellipsoid, in
> >> >the same sense that the confidence intervals for individual coefficients
> >> >are shadows of the joint ellipsoid or ellipse. The usual confidence 
> ellipse
> >> >for 2 of q coefficients uses qf(.95, 2, df.error) to scale the 
> ellipse; if
> >> >you want to scale the ellipse as a literal shadow of the joint confidence
> >> >region for all q coefficients, use qf(.95, q, df.error) instead (which
> >> >produces a smaller ellipse). You can do this by simply editing
> >> >confidence.ellipse.lm in car.
> >>
> >>Doesn't it produce a larger ellipse, not a smaller one?
> >
> >No, using q > 2 produces a smaller ellipse, e.g.:
> >
> > > qf(.95, 2, 100)
> >[1] 3.0873
> > > qf(.95, 3, 100)
> >[1] 2.6955
>
>But isn't the relevant formula
>
>  RSS/s^2 <= df*qf(0.95, df, 100)
>
>where df is 2 or 3?  That gives
>
> > 2*qf(.95, 2, 100)
>[1] 6.174592
> > 3*qf(.95, 3, 100)
>[1] 8.086603
>
> >>If you are just doing a confidence region for 2 parameters, then the
> >>other parameters can take on any values; if you are projecting a p-dim
> >>region onto 2, then there are still restrictions on the other p-2
> >>parameters which your graphic will not show.  So the projection
> >>corresponds to a smaller region of parameter space than an equal sized
> >>2-dim confidence region, hence it needs to be bigger to give the same
> >>confidence level.
> >
> >If I understand properly what you're saying, I think that we agree.
>
>I'm pretty sure we don't, but re-reading my paragraph above, I can
>understand why you think we might!
>
>I think the argument is a lot easier to visualize if we are comparing
>a 1 dimensional confidence interval to a projection of a 2 dimensional
>confidence region.
>
>Say the parameters are T1 and T2.  Then a confidence interval for T1
>can also be thought of as a rectangle with no limits on T2 at all.  A
>joint confidence region for both parameters will be an ellipse.  The
>ellipse will be a bit wider than the rectangle, i.e. it will project
>to a *longer* interval.

Yes -- the longer projections of the 2D confidence ellipse are the Scheffe 
intervals. (I'm ashamed to say that I used the picture you suggest is in a 
book that I wrote!)

>This makes sense, because we want the probability of the rectangle
>covering the true parameter value to match the probability that the
>ellipse does the same; the ellipse can't be a proper subset of the
>rectangle.
>
>Duncan

-----------------------------------------------------
John Fox
Department of Sociology
McMaster University
Hamilton, Ontario, Canada L8S 4M4
email: jfox at mcmaster.ca
phone: 905-525-9140x23604
web: www.socsci.mcmaster.ca/jfox
-----------------------------------------------------

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