Re: [R] contrast matrix for aov

From: Prof Brian Ripley <>
Date: Thu 10 Mar 2005 - 21:07:22 EST

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,        
Professor of Applied Statistics,
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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Received on Thu Mar 10 21:11:02 2005

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