Re: [R] Lack of independence in anova()

From: Duncan Murdoch <murdoch_at_stats.uwo.ca>
Date: Thu 07 Jul 2005 - 11:10:29 EST

Spencer Graves wrote:
> Hi, Göran: I'll bite:
>
> (a) I'd like to see your counterexample.
>
> (b) I'd like to know what is wrong with my the following, apparently
> defective, proof that they can't be independent: First consider
> indicator functions of independent events A, B, and C.
>
> P{(AC)&(BC)} = P{ABC} = PA*PB*PC.
>
> But P(AC)*P(BC) = PA*PB*(PC)^2. Thus, AC and BC can be independent
> only if PC = 0 or 1, i.e., the indicator of C is constant almost surely.
>
> Is there a flaw in this?

I don't see one.

 > If not, is there some reason this case
> cannot be extended the product of arbitrary random variables X, Y, and
> W=1/Z?

Because you can't? The situations are different?

If C is a non-trivial event independent of A, then AC is strictly a subset of A. However, as the example I just posted shows (with constant 1), you can have a non-trivial random variable W where XW has exactly the same distribution as X.

Duncan Murdoch



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