[R] Lack of independence in anova()

[Kjetil Brinchmann Halvorsen]

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? 
>
As far as I can see, this is correct.

> If not, is there some reason this case 
>cannot be extended the product of arbitrary random variables X, Y, and 
>W=1/Z?
>  
>
Yes. Random variables are independent if all events which can be defined 
in terms of them are independent.
If Z is non-constant, it must be some event defined by Z with 
probability strictly beteween 0 and 1
and the above argument cannot be used.

Kjetil

>	  Thanks,
>	  spencer graves
>
>Göran Broström wrote:
>
>  
>
>>On Wed, Jul 06, 2005 at 10:06:45AM -0700, Thomas Lumley wrote:
>>(...)
>>
>>    
>>
>>>If X, Y, and Z are 
>>>independent and Z takes on more than one value then X/Z and Y/Z can't be 
>>>independent.
>>>      
>>>
>>Not really true. I  can produce a counterexample on request (admittedly
>>quite trivial though).
>>
>>Göran Broström
>>
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>>    
>>
>
>  
>


-- 

Kjetil Halvorsen.

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               --  Mahdi Elmandjra





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