Hi Stephen
You are right, orthogonalization of the gp doesn't make any sense, I
mean to orthogonalize the other covariates.
1. Following the paper's fig. 1 (thx for link), my understanding is
that gp effect on the voxel content (DV) are in areas 5 and 6 - adding
the total grey matter volume as covariate will take away the areas 2
and 5 and therefore gp effects only reflect area 6 (with te error
being areas 7 and 5)- as the authors put it, the problem is often in
the misinterpretation, i.e. gp effect results are maybe bigger than
what we see at the voxel level but because of the total grey matter
difference (cov) the gp effect is underestimated (as I stated earlier)
- unfortunately there is nothing we can do about it - having said that
if we observe a gp effect then we can be be happy because it means
area 6 is big. I think the point in this paper is to understand that
we cannot control the cov in the sense of adjusting for, i.e. the
local effect we will observe on voxel are not adjusted for global
differences (do as if there were no diff.)
2. Regarding orthogonalization, if we have other covariables (some
behavioural measure) as it is often the case in VBM and that there are
also gp differences (which is also often the case) I think that
orthogonalization to the globals could help because you could have
another circle in this fig 1 that account for some variance of the DV
(area 7) but also area 4, 5 and/or 6 - in this case reducing again the
power - orthogonalization to the globals means that only parts of
areas 6 and 7 will be explained by the additional covariates (but
again the effects can be bigger but we cannot know because of the
globals) - my point here is that if you do not orthogonalize, the
shared parts between the additional covariates and the globals (parts
of areas 4 and 5) will go into the error and you have less chances to
observe an effect - does that make sense?
Hope this clarrify things for Harma (not so sure .. )
Season greetings
C
> On Wed, 23 Dec 2009 16:07:38 +0000, Cyril Pernet <[log in to unmask]>
> wrote:
>
>> re Harma
>>
>>> Dear Cyril,
>>> Thank you for your response. I am not a statistician, but I shall
>>> try to explain what I think is the argument of the paper. The
>>> authors state that ANCOVA was developped to ?improve the power of
>>> the test of the independent variable and not ?control? for
>>> anything?. In their paper they argue that an ANCOVA is properly used
>>> when the covariate correlates with the dependent variable (and not
>>> with the independent variable), thereby reducing the variance in the
>>> error term and increasing the power of the test. When an ANCOVA is
>>> conducted in this way the F-test will reflect the ratio of the
>>> residual variance of the dependent variable (the variance that was
>>> attributed to the covariate was taken out) and the sum of the
>>> residual variance of the dependent variable and the variance of the
>>> independent variable. This is possible in true experimental settings
>>> were subjects were randomly assigned to each group.
>>> When doing a non-random assignment to group, which is the case in my
>>> study, you cannot always control the fact that these groups differ
>>> before the test on certain variables. For example on global gray
>>> matter. In this case using global gray matter as a covariate will
>>> also ?take out? some meaningfull variance of the independent
>>> variable (group membership). Or as the authors frame it: ?When group
>>> membership is determined non-randomly, there is typically no
>>> thorough basis for determining whether a given pre-treatment
>>> difference reflects random error or a true group difference?.
>>
>> ok I see there point; so what they say is that the gp regressor that
>> you have (1111-1-1-1-1) and the globals are correlated therefore some
>> variance will be shared and goes into the error, i.e. you are less
>> likely to find differences between groups because part of this is
>> explained by the globals ; similarly if you have other regressors of
>> interest differences will be attenuated because of the globals -
>> having say that, if you do have differences, then be happy because you
>> 'only' underestimate the effect. One option I can think of is to
>> orthogonalize all of your zscored regressors relative to the globals,
>> so there is no shared variance with a maximum of variance attributed
>> to the globals - you end up accounting for more variance overall (ie
>> smaller residuals = stroger effect) than the non orthogonalized matrix.
>
> I read the Miller and Chapman paper a few months ago. (Full citation: G. A.
> Miller, J. P. Chapman, "Misunderstanding Analysis of Covariance," J. Abnormal
> Psychology (2001), v. 110, pp. 40 -- 48.) From my understanding,
> orthogonalization won't fix the problem. The variables (here, group
> and global)
> are confounded, so if you do orthogonalization, you no longer have
> "group" but
> rather something ill-defined. As per the caption to Fig 1 of the paper, "In
> such a case, removing the variance associated with _Cov_ will also alter
> _Grp_ in potentially problematic ways." That would be true regardless of
> whether you do orthogonalization.
>
> There seems to be at least one free copy of the paper (via Google "Scholar");
> one is at
> http://www.usq.edu.au/users/patrick/PAPERS/covariance%201.pdf
>
> Best,
>
> S
>
> <snip>
>
>
>
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