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

**Deepayan Sarkar
**
deepayan@stat.wisc.edu

*Wed Feb 26 23:41:02 2003*

On Wednesday 26 February 2003 04:23 pm, 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].
*
Why so ? Let Y be an independent copy of X (i.e., Y ~ N(0,1) too, independent
of X). Then P(Y is in [-Inf , Inf]) = 1. Now, think of the 2-D confidence
region [-1, 1] x [-Inf, Inf]. This will have (by independence of X and Y)
probability 0.68.
Now, how can you expect an ellipse that will have the same X-range, that is a
strict subset of this region, to still have joint probability 0.68 ?
Hope that helps,
Deepayan