[R] contrast matrix for aov

Prof Brian Ripley ripley at stats.ox.ac.uk
Thu Mar 10 11:07:22 CET 2005

On Thu, 10 Mar 2005, Christophe Pallier wrote:

> Prof Brian Ripley wrote:
>>> On Wed, 9 Mar 2005, Darren Weber wrote:
>>> We have a two-factor, repeated measures design, with
>> Where does `repeated measures' come into this?  You appear to have repeated 
>> a 2x2 experiment in each of 8 blocks (subjects).  Such a design is usually 
>> analysed with fixed effects.  (Perhaps you averaged over repeats in the 
>> first few lines of your code?)
>>> roi.aov <- aov(roi ~ (Cue*Hemisphere) + Error(Subject/(Cue*Hemisphere)), 
>>> data=roiDataframe)
>> I think the error model should be Error(Subject).  In what sense are `Cue' 
>> and `Cue:Hemisphere' random effects nested inside `Subject'?
> I do not understand this, and I think I am probably not the only one. That is 
> why I would be grateful if you could give a bit more information.
> My understanding is that the fixed factors Cue and Hemisphere are crossed 
> with the random factor Subject (in other words, Cue and Hemisphere are 
> within-subjects factors, and this is probably why Darren called it a 
> "repeated measure" design).

The issue is whether the variance of the error really depends on the 
treatment combination, which is what the Error(Subject/(Cue*Hemisphere)) 
assumes.  With that model

Error: Subject:Cue
           Df Sum Sq Mean Sq F value Pr(>F)
Cue        1 0.2165  0.2165  0.1967 0.6708
Residuals  7 7.7041  1.1006

Error: Subject:Hemisphere
            Df Sum Sq Mean Sq F value Pr(>F)
Hemisphere  1 0.0197  0.0197  0.0154 0.9047
Residuals   7 8.9561  1.2794

Error: Subject:Cue:Hemisphere
                Df Sum Sq Mean Sq F value Pr(>F)
Cue:Hemisphere  1 0.0579  0.0579  0.0773  0.789
Residuals       7 5.2366  0.7481

you are assuming different variances for three contrasts.

> In this case, it seems to me from the various textbooks I read on Anova, that 
> the appropriate MS to  test the interaction Cue:Hemisphere is 
> Subject:Cue:Hemisphere (with 7 degress of freedom, as there are 8 independent 
> subjects). 
> If you input Error(Subject/(Cue*Hemisphere)) in the aov formula, then the 
> test for the interaction indeed uses the Subject:Cue:Hemisphere source of 
> variation in demoninator. This fits with the ouput of other softwares.
> If you include only 'Subjet', then the test for the interaction has 21 
> degrees of Freedom, and I do not understand what this tests.

It uses a common variance for all treatment combinations.

> I apologize in if my terminology is not accurate.  But I hope you can clarify 
> what is wrong with the Error(Subject/(Cue*Hemisphere)) term,
> or maybe just point us to the relevant textbooks.

Nothing is `wrong' with it, it just seems discordant with the description
of the experiment.

Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272866 (PA)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595

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