# [R] [OT] factorial design

Rolf Turner r.turner at auckland.ac.nz
Tue Nov 4 23:20:05 CET 2008

```On 4/11/2008, at 6:29 PM, Edna Bell wrote:

> Dear R Gurus:
>
> I vaguely remember reading that if interaction was present in a
> factorial design, then the main effect results were suspect.
>
> However, I was reading a text which now uses the tests for main
> effects even if interaction is present.
>
> Which is correct, please?

Both, neither, whatever.

You have to specify what you mean by ``main effect'' if interaction
is present.  One possible definition is to average the response of
factor A (at each level) over all levels of factor B.  There is an
``A main effect'' (by this definition) if these averages are different.

E.g. (1): mu(1,1) = 1, mu(1,2) = -1, mu(2,1) = -1, mu(2,2) = 1

where mu(i,j) is the population mean when factor A is at level i
and factor B is at level j.  Factor A averages out to 0 at both levels
so there is no main A effect in the sense defined.  (And likewise
there is no main B effect either.)  ``Clearly'' there is interaction!

E.g. (2): mu(1,1) = 1, mu(1,2) = 2, mu(2,1) = -1, mu(2,2) = -2.

Here factor A averages to 1.5 at level 1 and to -1.5 at level 2
so there *is* a main A effect ***in the sense defined***.  Note
that factor B averages to 0 at both levels so there is no main
B effect in the sense defined.

This is the sort of thing that the much maligned ``Type III'' sums
of squares test for.

But is it *really* what you are interested in?  Probably NOT.
Read Bill Venables' ``Exegesis on Linear Models'' for more discussion
of why such things are a bad idea:

http://www.stats.ox.ac.uk/pub/MASS3/Exegeses.pdf

The important thing is to get it clear in your head what you want
to know about, and then conduct your analysis in such a way as to
find about that.  This is sometimes not so easy.  But there's no
avoiding it if you want to do a sensible analysis.  Doing tests
mindlessly without a clear understanding of exactly what it is that
you are testing is a recipe for nonsense if not disaster.

cheers,

Rolf Turner

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