From volker.franz@tuebingen.mpg.de  Wed Feb 26 23:21:02 2003
From: volker.franz@tuebingen.mpg.de (Volker Franz)
Date: Wed Feb 26 23:21:02 2003
Subject: [Rd] [Package car/data.ellipse]: confidence intervals off
 by factor sqrt(2)??? (PR#2584)
In-Reply-To: <5.0.2.1.0.20030226160733.00afa648@mcmail.cis.mcmaster.ca>
References: <200302262040.h1QKe035015790@pubhealth.ku.dk>
 <5.0.2.1.0.20030226160733.00afa648@mcmail.cis.mcmaster.ca>
Message-ID: <15965.15945.810133.10708@gargle.gargle.HOWL>

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?

Volker

-- 
___________________________________________________________

    Dr. Volker Franz
    Max-Planck-Institute for Biological Cybernetics
    Tuebingen, Germany



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