[R] Lack of independence in anova()

Duncan Murdoch murdoch at stats.uwo.ca
Thu Jul 7 03:10:29 CEST 2005

Spencer Graves wrote:
> Hi, Göran:  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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