[R] calculating area of ellipse

[John Fox]

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 <jfox using 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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>