Dear Jeff,
I don't think that it would be sensible to claim that it *never* makes
sense to multiply quantities measured in different units, but rather
that this would rarely make sense for regression coefficients. James
might have a justification for finding the area, but it is still, I
think, reasonable to point out that doing so may be problematic.
With respect to ratios of areas: I apologize if my examples were
cryptic. Imagine, for example, that the same regression model is fit to
two groups and joint-confidence ellipse for two coefficients computed
for each. The ratio of the two areas would reflect the relative
precision of the estimates in the two groups, which is unaffected by the
units of measurement of the coefficients. This is also the idea behind
generalized variance inflation, where the comparison is to a "utopian"
situation in which the parameters are uncorrelated. For details, see
help("vif", package="car") and in particular Fox, J. and Monette, G.
(1992) Generalized collinearity diagnostics. JASA, 87, 178–183.
Best,
John
On 2021-05-11 10:48 a.m., Jeff Newmiller wrote:
The area is a product, not a ratio. There are certainly examples out there of
meaningful products of different units, such as distance * force (work) or power
" time (work).
If you choose to form a ratio with the area as numerator, you could conceivably
obtain the numerator with force snd distance and then meaningfully form a ratio
with time (power). So this asserted requirement as to homogeneous units seems
inaccurate. But without context I don't know if any of this will aid in
interpretation of variance for the OP.
On May 11, 2021 7:30:22 AM PDT, John Fox <j...@mcmaster.ca> wrote:
Dear Stephen,
On 2021-05-11 10:20 a.m., Stephen Ellison wrote:
In doing meta-analysis of diagnostic accuracy I produce ellipses of
confidence
and prediction intervals in two dimensions. How can I calculate the
area of
the ellipse in ggplot2 or base R?
There are established formulae for ellipse area, but I am curious: in
a 2-d ellipse with different quantities (eg coefficients for salary and
age) represented by the different dimensions, what does 'area' mean?
I answered James's question narrowly, but the point you raise is
correct
-- the area isn't directly interpretable unless the coefficients are
measured in the same units.
It still may be possible to compare areas of ellipsoids for, say,
different regressions with the same predictors, as ratios, however,
since these ratios would be unaffected by rescaling the coefficients.
The generalization of this idea to ellipsoids of any dimension is the
basis for the generalized variance-inflation factors computed by the
vif() function in the car package.
Best,
John
John Fox, Professor Emeritus
McMaster University
Hamilton, Ontario, Canada
web: https://socialsciences.mcmaster.ca/jfox/
S
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