Hi Roman,
beta's and "variance" are not synonymous things, and treating them as
such will get confusing.  Statistically, when you have correlated
regressors the standard error of the beta is increased (goggle "variance
inflation factor"), which is how the statistics account for the fact
that the betas become less stable in the presence of correlated
regressors.

cheers,
-MH

On Sun, 2011-08-21 at 16:59 +0000, Roman M wrote:
> Dear Mark, Jeanette, 
> 
> Thank you for the response. So the one thing that isn't clear to me,
> then, is how to reconcile Mark's comment with those simulations. I
> agree with Mark that typically in multiple regression the beta is
> interpreted as the "unique" variance of each variable, yet those
> simulations (and Jeanette's very clear example) say that in the
> presence of correlation that overlap can be assigned in very unstable
> ways (though the fact that residuals don't change tells me that all
> the variance is equally explained). Are these two different things?
> 
> More pragmatically speaking, yes I will look at subsets with equal
> ages and see what happens
> 
>  thank you
> 
>  Roman
> 
> 
> ______________________________________________________________________
> Date: Fri, 19 Aug 2011 21:22:55 -0500
> From: [log in to unmask]
> Subject: Re: [FSL] correlations in Vertex Analysis
> To: [log in to unmask]
> 
> Hi,
> 
> I wonder if you're referring to one of my lectures, since it sounds
> familiar and I ran a simulation to show how the variability of betas
> occurred with collinearity.  I may not have explained it well, the
> parameter estimates only model the portion unique to that specific
> regressor, but I didn't want to imply that the shared variability was
> ignored, because it isn't.
> 
> The reason for variability in parameter estimates is due to
> collinearity.  Basically you are approaching a condition where there
> isn't a unique solution to the GLM because your two regressors are
> trying to do exactly the same thing.  It is sort of like me saying, "I
> have written down 2 numbers that sum to 10, what are the numbers?"
> There are an infinite number of solutions.  
> 
> My guess is that your collinearity isn't bad enough to cause this and
> you can check it by looking at the collinearity stats in FSL (click
> the efficiency button at the design setup/GLM window and look there).
> The issue with your model is that age is stealing the show and there
> is no remaining group effect.  
> 
> Sorry, no good answer!  I do know one person who ran into this and
> managed to collect some more subjects and it balanced out the ages.  
> 
> Jeanette
> 
> On Fri, Aug 19, 2011 at 4:52 PM, Roman M <[log in to unmask]> wrote:
>         Mark, thank you this is helpful. One question, though: I agree
>         that in multivariate regression the beta shows the 'unique'
>         variance to a regressor, but I've done (and seen) simulations
>         showing that in the presence of correlations between variables
>         it's not that the betas get "reduced" to the non-overlapping
>         amount of variance, but rather the shared variance appears to
>         be "randomly" allocated between the two variables, creating
>         often very unstable estimates [while obviously the residuals
>         remain unchanged].
>         
>         Perhaps the best thing is to take a subset of my sample that
>         equalizes the mean age of the two groups and see if the effect
>         is still there..
>         
>         Thank you
>         
>          Roman
>         
>         > Date: Fri, 19 Aug 2011 08:25:36 +0100
>         > From: [log in to unmask]
>         > Subject: Re: [FSL] correlations in Vertex Analysis
>         > To: [log in to unmask]
>         
>         > 
>         > Dear Roman,
>         > 
>         > It is hard to know how much the effect is due to correlation
>         or just whether
>         > there is a slight change in statistic to be above/below
>         threshold in the two
>         > cases. It is true that FIRST still cannot take general
>         contrasts and so you
>         > need to have your effect of interest in a single EV, and
>         that it also demeans
>         > all EVs for you.
>         > 
>         > If the mean age between your different groups is actually
>         different then 
>         > you could easily be in the situation where an age EV *on its
>         own* could
>         > show an effect driven by group difference. However, if you
>         include both
>         > age *and* a group difference EV then this should not happen
>         as you 
>         > should only see the *unique* parts of the effect and not
>         whatever is shared.
>         > 
>         > In the case that there is a mean age difference between your
>         groups
>         > then there is no way to actually separate the effect beyond
>         just putting
>         > both EVs in the analysis (group difference and age) and
>         seeing what
>         > unique effects are found by the GLM. Anything else that you
>         do is 
>         > artificially forcing the difference to be treated only in
>         one way. For 
>         > example, you can demean age within each group and then put
>         this
>         > into one EV, but it is artificially saying that any
>         difference seen between
>         > the groups could not, under any circumstances, be due to age
>         and must
>         > be only due to group difference. It is hard to know how this
>         could be
>         > justified, as can you really be sure that the differences
>         are not due to age?
>         > The safest and truest thing is just to put both in and see
>         what the GLM 
>         > gives you.
>         > 
>         > One thing with FIRST is that you cannot get the equivalent
>         of a joint
>         > F-test in the standard GLM, where it shows you the effects
>         that could
>         > be due to one or other or *both* of the EVs in the model. If
>         you do
>         > an analysis without one of the EVs then you get some of this
>         information
>         > and combining across both gives a reasonable estimation of
>         what this
>         > combined effect is like (although it is not exact). This
>         might help you to
>         > see what is jointly driven by age *and* group difference,
>         and you should
>         > only look at these in terms of what they are like when taken
>         together
>         > and not consider them separately.
>         > 
>         > Sorry for the rather long answer.
>         > Correlated regressors is never easy, and the current setup
>         in FIRST
>         > makes it a little more difficult.
>         > 
>         > By the way, the best way of dealing with this issue is to
>         only recruit
>         > subjects so that the mean age in each group is matched!
>         Alternatively,
>         > you could exclude certain individuals to reduce the age
>         difference
>         > between the groups if the age difference is not large and
>         you have 
>         > enough subjects. But nothing beats designing the experiment
>         so that
>         > there is never a correlation between effects of interest and
>         effects of
>         > no interest.
>         > 
>         > I hope this helps though.
>         > All the best,
>         > Mark
>         > 
>         > 
>         > 
>         > 
>         > 
>         > 
>         > On 19 Aug 2011, at 03:55, Roman M wrote:
>         > 
>         > > Dear FSLers,
>         > > 
>         > > I am having a little problem with a vertex analysis. I
>         have 2 groups (e.g., control/patients) and several covariates.
>         Here is the issue: when I put BOTH control and patients in one
>         analysis, and use age as a covariate, I see an effect of age.
>         Yet, if I look at controls alone and patients alone, the
>         effect of age isn't there at all, which tells me it's not a
>         real thing. Indeed, I see age and group are correlated -- so
>         is age "stealing" some of the variance that should go in the
>         Group effect?
>         > > 
>         > > 1. What is the right way of orthogonalizing (e.g.,
>         ortogonalize age with group, or viceversa?) I'm not sure I
>         appreciate which is the correct way of doing this. 
>         > > 
>         > > 2. Is still not possible to specify 2 different groups (in
>         the Glm) for a vertex analysis, and is it still the case that
>         I don't need to de-mean the covariates (e.g., age), since
>         first_utils does it automatically?
>         > > 
>         > > Any other/better way of dealing with the correlation
>         issue?
>         > > 
>         > > Thank you
>         > > 
>         > > Roman
>         
> 
>