Hello Antonia,
The statistical validity of correlating individual differences in BOLD
response amplitude and behavior is best dealt with from first prinicples, as
opposed to looking for some sort of empirical demonstration (obviously,
showing that there is a published result of a correlation between fMRI
signal and behavior has nothing to do with proving its validity). I do not
know what the reviewer's specific concern was, but I'll focus on bias. In
particular, we will see if the expected correlation between fMRI signal
amplitude and behavior can be non-zero can be even when there is a true zero
correlation between neural response amplitude and behavior.
Assume the following model (which seems to be an appropriate mathematization
of your problem, sans the measurement error, which would only make things
more complicated):
fMRI response amplitude = (neural response amplitude)*(fMRI "excitability")
,
behavior = constant*fMRI response amplitude.
An important detail of the above is that neural response and excitability
are multiplicative, not additive. Now assume that neural response amplitude
and fMRI excitability have zero covariance (and are otherwise independent).
This is reasonable, because any such non-zero correlations would tend to
simply increase or decrease the intersubject variability of the fMRI
response amplitude, without affecting the bias of the covariance between
neural response amplitude and behavior (well this isn't exactly true, but it
is ignorable for this problem). Additionally allow fMRI exictability and
behavior to have a non-zero covariance. Finally, assume that neural response
amplitude and behavior have a zero covariance (i.e., the null hypothesis of
interest is true). Then,
COV(fMRI response amplitude, behavior) = E<behavior*neural response
amplitude*fMRI excitability > - E<behavior>*E<neural response amplitude*fMRI
excitability >
= E<behavior*fMRI excitability>*E<neural response amplitude> -
E<behavior>*E<neural response amplitude>*E<fMRI excitability >
= E<neural response amplitude>*(E<behavior*fMRI excitability> -
E<behavior>*E<fMRI excitability >)
= E<neural response amplitude>*COV(fMRI excitability, behavior) .
Thus, it seems that your reviewer's concern (if this indeed was his concern)
is legitimate. Under the null hypothesis, if the expected neural response
amplitude is non-zero, and fMRI excitability is correlated with behavior,
then the covariance of fMRI response amplitude and behavior will be
non-zero.
I hope this helps,
Eric
----- Original Message -----
From: "antonia hamilton" <[log in to unmask]>
To: <[log in to unmask]>
Sent: Friday, July 01, 2005 12:51 PM
Subject: [SPM] brain-behaviour correlation
> Hi,
>
> I have a study which uses a correlation between individual differences
> in performance and signal change in the analysis, and the reviewer is
> worrying that this could be invalid because there are also individual
> differences in genearal BOLD 'excitability' between subjects. I've
> found lots of other papers which use the same method, but none give
> references saying if the method is valid. Can anyone supply the
> reference that will satisfy my reviewer please?
>
> Thank you!
>
> Antonia.
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