On Mon, 28 Dec 2009 10:40:19 +0000, Cyril Pernet <[log in to unmask]>
wrote:
>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 don't think it's so simple. Chapman and Miller claim that this kind of analysis
threatens "concept validity." While that's abstract and not concrete, they
eventually cite the classic example of Lord's paradox (p. 44).
Getting back to abstract claims, on p. 43, some of the quotes disapproving of
this use of a covariate agree with your point that one possible consequence is
minimizing the impact of the group factor. But there's also claims of the
potential for the introduction of spurious effects (Wildt and Ahtola, "...or may
produce spurious treatment effects..."; Elashoff, "...or may produce a spurious
treatment effect").
There's a lot of literature related to Lord's paradox; it's pretty interesting.
See for example "Simpson's Paradox, Lord's Paradox, and Suppression Effects
are the same phenomenon – the reversal paradox," _Emerging Themes in
Epidemiology_ 2008, 5:2.
Best, and happy New Year,
S
>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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>
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>The University of Edinburgh is a charitable body, registered in
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