# [R] Lack of independence in anova()

Kjetil Brinchmann Halvorsen kjetil at acelerate.com
Thu Jul 7 04:43:46 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?
>
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
>>
>>______________________________________________
>>R-help at stat.math.ethz.ch mailing list
>>https://stat.ethz.ch/mailman/listinfo/r-help
>>PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html
>>
>>
>
>
>

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Kjetil Halvorsen.

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--  Mahdi Elmandjra

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