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
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