At 11:23 PM 2/26/2003 +0100, Volker Franz wrote:
Hi John,
>>>>> "JF" == John Fox <[EMAIL PROTECTED]> 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: [EMAIL PROTECTED] web: http://www.socsci.mcmaster.ca/jfox
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