[Rd] [Package car/data.ellipse]: confidence intervals off by factor sqrt(2)??? (PR#2584)

[John Fox]

Dear Volker,

At 11:23 PM 2/26/2003 +0100, Volker Franz wrote:
>Hi John,
>
> >>>>> "JF" == John Fox <jfox@mcmaster.ca> writes:
>     JF> Dear Volker, If the data ellipse (or, in this case, circle) is
>     JF> scaled so that its shadows (projections) on the axes each
>     JF> includes 68% of the data (that is of the marginal distribution
>     JF> of each variable), then the ellipse will include less than 68%
>     JF> of the data (i.e., of the joint distribution of the two
>     JF> variables). Conversely, to include 68% of the data in the
>     JF> ellipse, the shadows of the ellipse have to be larger.
>     JF> Did I understand your point correctly?
>
>I am not sure. I will try to rephrase my initial request:
>
>Let X by a one--dimensional random variable (standard normal
>distribution; mean=0; std=1). The 68% confidence intervall of X will
>approximately be: [-1,1]. Now, if I combine X with a stochastically
>independent second random variable Y, the marginal distribution of X
>should not change. Therefore, the projections of the error ellipse on
>the X--axis should still be: [-1,1].
>
>If I do this with the function data.ellipse:
>
>    data.ellipse(rnorm(10000),rnorm(10000),levels=0.68,plot.points=F)
>
>I get a projection on the X-axis which is larger than [-1,1]. In fact,
>it is a little bit larger than [-sqrt(2),+sqrt(2)].
>
>My interpretation is that this is due to the construction of the
>radius in data.ellipse:
>
>    dfn<-2
>    radius <- sqrt ( dfn * qf(level, dfn, dfd ))
>
>I would expect a dfn<-1 here (such that the radius would correspond to
>the t-distribution).
>
>Does this make sense?

This is a data ellipse, not a confidence ellipse, but the same point arises 
in both cases: For the ellipse to enclose 68 percent of the joint 
distribution of the two variables, its projections on the axes must include 
more than 68% of each marginal distribution. Just think about projecting 
the individual points onto the axes -- there are points outside of the 
ellipse that are inside its shadow on an individual axis.

I hope that this helps,
  John

____________________________
John Fox
Department of Sociology
McMaster University
email: jfox@mcmaster.ca
web: http://www.socsci.mcmaster.ca/jfox