[R] Lack of independence in anova()

[Göran Broström]

On Thu, Jul 07, 2005 at 11:18:09AM +0100, Ted Harding wrote:
> My first reaction to Duncan's example was "Touché -- with apologies
> to Göran for suspecting on over-trivial example"! 

No need to apologize; that was of course my first reaction to Thomas'
statement. 

> I had not thought
> long enough about possible cases. Duncan is right; and maybe it is
> the same example as Göran was thinking of.

On second thought it was not difficult to find: (X, Y) bivariate standard
normal, P(Z = 1) = P(Z = -1) = 1/2.

[...]
 
> However, interesting though it maybe, this is a side-issue
> to the original question concerning independence of the F-ratios
> in an ANOVA. Here, numerators and denominator are all positive,
> so examples like the above are not relevant.
> 
> The original argument (that increasing Z diminishes both X/Z
> and Y/Z simultaneously) applies; but it is also possible to
> demonstrate analytically that P(X/Z <= v and Y/Z <= w) is
> greater than P(X/Z <= v)*P(Y/Z <= w).

Maybe it is simplest to calculate Cov(X/Z, Y/Z), which turns out to be
equal to E(X)E(Y)V(1/Z) (given total independence). So, a necessary
condition for independence is that at least one of these three terms is
zero. Which is impossible in the F-ratios case.

Göran