Hi Cyril -
Hope you are well.
>What do you mean by pooled error?
>If I've understand the GLM (a little), for a repeated anova, an
>observation (yij) depends on the mean (u), the effect of the
>factor (Fj)
>and the effect of the subject (Si). The model for all subjects is then
>y=u+F+S+e (mean+factor+subjects+error). The pooled error
>cannot be S+e?
>I think it means that you distinguish different error terms, right?
>(this part of the note is very hard for me) but then how can we
>partition that, i.e. distinguish different error sources?
With only one factor, there is no difference between pooled and
partitioned error. Imagine though a 2x2 ANOVA, with two factors
A,B and three effects (two main, and one interaction). The pooled
error model is:
y = A + B + AxB + S + e
This is (a rotation of) the model you would get in SPM if you
specificed a within-subject ANOVA with 4 conditions.
The partitioned error model however is:
y = A + B + AxB + S + SxA + SxB + SxAxB
The error is split into three separate terms, each of which is
(in fact) the interaction of that effect with the subject factor.
Each effect is tested against its corresponding error term, eg
the F-value for the main effect of A is the df-weighted sum-of-
squares ratio A/AxS. This is what packages like SPSS do. However
SPM does not partition the error for you - it always assumes a
single error term. So to implement such a partitioned error model
in SPM, you need to create a new model for each of the three effects,
by first creating appropriate contrasts of the data y. This is what
the technical note describes.
[In fact, you could emulate such a model in SPM if F-tests were
modified to allow the ratio of two F-contrasts (ie two reduced
but embedded models, specified by appropriate contrasts), but
I don't think anyone can be bothered to pursue this route!]
>>[...]
>Please excuse my ignorance but I really don't understand why this is
>more sensitive to the between-subjects variances. Do you thing you can
>try to explain me (or us, spm users, in the note)?
>
>Thank you very much in advance,
Well, my explanation was confusing, I agree - mainly because I am not
100% sure - but I think it has to do with the fact that the correct
contrasts to specify an effect-specific error term (ie partitioned error)
need to be orthonormal (in addition to the appropriate rank of N-1, for
N levels) to remove the subject means ("S" above).
Also note that the example contrasts I gave for a 3x3 ANOVA in my previous
email were not quite correct (they were orthogonal - so called "Helmert
contrasts"
in fact - but not normed, ie orthonormal). They should have been:
[0.82 -0.41 -0.41; 0 0.71 -0.71]
(which is what you will actually get when typing "orth(diff(eye(3))')'" in
matlab).
The technical note PDF, when available on Monday, will be correct.
Famous last words...
Rik
