[R] calculating area of ellipse

John Fox j|ox @end|ng |rom mcm@@ter@c@
Fri May 7 18:15:03 CEST 2021

Dear David and Jim,

As I explained yesterday, a confidence ellipse is based on a quadratic 
form in the inverse of the covariance matrix of the estimated 
coefficients. When the coefficients are uncorrelated, the axes of the 
ellipse are parallel to the parameter axes, and the radii of the ellipse 
are just a constant times the inverses of the standard deviations of the 
coefficients. The constant is typically the square root of twice a 
corresponding quantile (say, 0.95) of an F distribution with 2 numerator 
df, or a quantile of the chi-square distribution with 2 df.

In the more general case, the confidence ellipse is tilted, and the 
radii correspond to the square roots of the eigenvalues of the 
coefficient covariance matrix, again multiplied by a constant. That 
explains the result I gave yesterday based on the determinant of the 
coefficient covariance matrix, which is the product of its eigenvalues.

These results generalize readily to ellipsoids in higher dimensions, and 
to degenerate cases, such as perfectly correlated coefficients.

For more on the statistics of ellipses, see 


John Fox, Professor Emeritus
McMaster University
Hamilton, Ontario, Canada
web: https://socialsciences.mcmaster.ca/jfox/

On 2021-05-06 10:31 p.m., David Winsemius wrote:
> On 5/6/21 6:29 PM, Jim Lemon wrote:
>> Hi James,
>> If the result contains the major (a) and minor (b) axes of the
>> ellipse, it's easy:
>> area<-pi*a*b
> ITYM semi-major and semi-minor axes.

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