Dear Stephen,

Thanks for pointing us out again to the tutorial - and to the topic we 
ourselves started when we noted what Radek noted: suspiciously strong 
between-group effects. Since then we got convinced of what we stated in 
our previous email, that you should not model the subject factor when making 
between-subject comparisons, i.e. modeling main effects of group. I remain 
convinced of that after your replies, though IANASE (I am not a statistician 
either) I will try to explain why.

The trouble is not, as you state in a previous email:

"You're absolutely right that specifying the effect of subject will remove the 
effect of group if you're not careful."

The problem is, it removes variance from the error term that should not be 
removed. Your explanation shows this:

>
>The model is
>
>    y_ijk = s_i + g_j + c_k + gc_jk + e_ijk
>
>where
>    s = subject term
>    g = group term (and j = j(i) depends on i)
>    c = condition term
>    gc = interaction of group and condition term
>    e = residual
>
>After estimating and fitting, we have (I'm omitting hats and carats, and not 
>taking into account prewhitening/nonsphericity correction)
>
>    y_ijk = s_i + g_j + c_k + gc_jk
>

So, the subject means are modeled, are therefore not in the residual, which is 
removed after estimating.

Now, I fully agree the above model provides a better model for the y-term. 
However, the whole point of ANOVA's is testing any difference in means 
against the appropriate error term. When testing for within-subjects effects, 
the appropriate term is within-subject variance - it is then appropriate to 
remove the between-subject variance from the residual by modeling it.

However, when modeling between-subject effects, as you state: 

"In terms of breaking down variance, the intersubject variance is split between 
group differences and differences between subjects _within groups_."

The first part is the variance we are trying to explain with a main effect of 
group, the second part is the error variance that gives an indication of how 
reliable the group effect is. By inserting the subject factor, this variance is 
removed from the residual - which leaves only within-subject variance.  Jan 
made the same claim in his reply to us last summer:  

"It is these subject constants that absorb much of the inter-subject
variability present in most imaging data, which in turns leads to more
sensitivity for the experimental effects (including group differences)."

In conclusion, though no doubt including the subject means in the model 
provides a better fit (and therefore stronger group effects), I think this is not 
appropriate _when looking at between-subject effects_. 

Laura