Dear Donald,

The effective data when (properly) assessing the main effect of group  
is obtained by averaging all within subject measures together (i.e.,  
to obtain a single datum per subject). Thus, the within-subject  
correlation structure is irrelevant to assessing the main effect of  
group (if it is being computed correctly, that is). F-tests for the  
main effect of group and the group x condition interaction have  
different mean squares in the denominator (these are the effect of  
subject within-group and subject x condition interaction,  
respectively; "Experimental Design" by Kirk is a good textbook for  
this.) Thus, one does not want to use the subject x condition  
interaction as the error term when assessing the main effect of group.  
I think Lauren and Stephen have said as much in their recent posts.

Eric

Quoting "MCLAREN, Donald" <[log in to unmask]>:

> While that is true. You are violating the assumptions of the ANOVA,
> specifically --
> Independence <http://en.wikipedia.org/wiki/Statistical_independence> of
> cases - this is a requirement of the design.
>
> I've been told that as soon as you violate the assumptions, the results are
> invalid.
>
> I just did a quick check in SPSS with 2 conditions:
> (1) 2 sample t-test of the the mean of both conditions
> (2) repeated measures ANOVA with 1 within-subject factor (contains
> factor*group term)
> (3) univariate GLM with group, condition, group*condition factors
> (4) 2 sample t-test of the means of both conditions
>
> Results: (1) and (2) are the same; (3) and (4) were different from (1) and
> (2). (3) and (4) violate the independence assumption.
>
> As far as I can tell, SPSS must code subject into the model (otherwise there
> isn't any repeated component).
>
> Thus, it seems like having the subject term is the correct way to go as it
> produces the same results as.
>
> I'm going to check the results of (1) and (2) in SPM this week.
>
>
> On Sat, Apr 4, 2009 at 2:18 PM, Laura Menenti <[log in to unmask]
>> wrote:
>
>> 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
>>
>
>
>
> --
> Best Regards, Donald McLaren
> =====================
> D.G. McLaren
> University of Wisconsin - Madison
> Neuroscience Training Program
> Office: (608) 265-9672
> Lab: (608) 256-1901 ext 12914
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