# [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
<http://euclid.psych.yorku.ca/datavis/papers/ellipses-STS402.pdf>.

Best,
John

John Fox, Professor Emeritus
McMaster University
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.
>
>

```