Well, "equal statistical power" is probably not a good term in this
context. What I was trying to say is that in the 2 x 2 x ... case,
interactions can at least sometimes be interpreted as main effects, not
necessarily as a difference of differences. If you think of A(2) x B(2) as
4 conditions the 3 degrees of freedom each contrast half of the
observations against the other half.

A1B1  A1B2 A2B1 A2B2
  1          1     -1      -1        # main effect of A
  1         -1      1      -1        # main effect of B
  1         -1     -1       1        # Interaction of A and B  or
 (implied) main effect of "C"
So if A1B1 and A2B2 correspond to level 1 of C and A1B2 and A2B1 correspond
to level 2 of C, conceptually you don't have a difference of differences,
you have a "third" main effect, just like A and B. If you interpret C as a
main effect, A may represent the interaction of B and C or B may represent
the interaction of A and C. It was in this sense that I talked about equal
statistical power.



On Sat, Nov 9, 2013 at 10:37 AM, peter dalgaard <pdalgd@gmail.com> wrote:

>
> On 08 Nov 2013, at 19:49 , Reinhold Kliegl <reinhold.kliegl@gmail.com>
> wrote:
>
> > If I understand your question correctly, I think there is one exception.
> > Whenever you are dealing with balanced 2 x 2 x 2 x ... design all main
> > effects and interactions have equal statistical power.
>
> Say what? A main effect in that case is a difference between two averages.
> A (1st order) interaction effect is a difference between two differences of
> means, each based on half as many observations. So the standard error in
> the latter case is twice that of the former. Of course, you can scale the
> interaction so that the s.e. is the same as that of the main effects, but
> it is then not clear that similarly sized effects are equal in importance.
> At the very least, you need to be very specific about what you mean by
> "equal power".
>
>
>
> --
> Peter Dalgaard, Professor,
> Center for Statistics, Copenhagen Business School
> Solbjerg Plads 3, 2000 Frederiksberg, Denmark
> Phone: (+45)38153501
> Email: pd.mes@cbs.dk  Priv: PDalgd@gmail.com
>
>
>
>
>
>
>
>
>

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