Dear Kochiyama,

Thank you for your mail.

> (1) The robustness of SPM approach to higher order autocorrelation.

This is difficult to assess because robustness is only specified in
relation to a violation of the assumptions in a particular data set.
In your case substantial high-order correlations may be sufficient to
overcome the pre-filtering device in SPM99 used to ensure robustness in
fMRI.

You can only assess robustenss by simulating data with known (or no)
signal and emulating real correlations (e.g. using phase shuffling
techniques).  You would then assess the false positive rate in
simulations.  It seems that you have done something like this and the
filtering adopted was not sufficent.

> (2) The robustness of SPM approach to spatially varying autocorrelation.

The same argument applies here.  In principle, with sufficent
pre-filtering, spatial differences should be attenuated but this
depends on the filter and the intrinsic autocorrelations.

> (3) When a whitening procedure is used, even if I adopt a different order of
> AR model for every channels of NIRS, df is apparently equal by all
> channels.  In this case, are there any problem about performing
> multiple comparisons between channels?

I am afraid there is - assuming you use the same design matrix for each
channel.  Pre-whitening the data gives biased estimates of the parameters.
Take the general linear model with non-spherical errors e ~ N{0,V}.

                y   = X*B + e

        =>      <B> = <pinv(X)*y> = <pinv(X)*(X*B + e)> = B


Apply a whitening filter W = V^(-1/2) to the data to give a new model
with i.i.d errors e' ~ N{0,W*V*W} = N{0,1}

                W*y  = X*B' + e' = W*X*B + W*e

        =>      <B'> = <pinv(X)*W*y> = <pinv(X)*(W*X*B + W*e)>
                     =  pinv(X)*W*X*B

i.e. the OLS estimator B', based on whitened data, is not an unbiased
estimator of B.  This means you are making inferences about different
contrasts at each channel.  The way round this is to use a different
design matrix at each channel, which is a whitened X -> W*X

        =>      <B'> = <pinv(W*X)*W*y> = <pinv(W*X)*(W*X*B + W*e)>
                     =  pinv(W*X)*W*X*B = B


But, of course, now the design matix has changed so the d.f. will also
change for each channel.

Your question suggests that you want to compare responses in different
channels.  If you simply want to compare t values in different
channels, anecdotally then I would use the pre-whitening of both data
and design so that the t values are testing for the same thing.
However, if you want to do a formal comparison then you require channel
by effect interactions and you have to model the non-sphericity in a
single model.  This is quite complicated and usually needs iterative
schemes that use ReML estimates of the covariance components.  There a
two papers currently under submission that discuss this in depth.  You
can download them from


/home/ftp/out/methods/Bayes/B1:
B1.doc      B1figs.doc

/home/ftp/out/methods/Bayes/B2:
B2.doc      B2_11.ps    B2_14.ps    B2_4.ps     B2_7.ps
B2_1.ps     B2_12.ps    B2_2.ps     B2_5.ps     B2_8.ps
B2_10.ps    B2_13.ps    B2_3.ps     B2_6.ps

logging in as 'anonymous' with ftp www.fil.ion.ucl.ac.uk


I hope this helps - Karl







----- Begin Included Message -----

From [log in to unmask] Thu Nov 29 03:54:09 2001
From: Takanori Kochiyama <[log in to unmask]>
Date: Thu, 29 Nov 2001 12:54:35 +0900
To: [log in to unmask]
Subject: About estimating autocorrelation 2

Dear Karl

I study the multi-channel NIRS time series (as one of the themes of my
doctor thesis$B!D(B) At present, I think that NIRS signal is
characterized by the higher order and spatially varying temporal
autocorrelation and try to take the following two approaches .  One is
SPM based (band pass filtere & AR model), another is whitening based on
PW-transformation.

The problems arise when I apply SPM99 approach to NIRS data.
Then I want to know following:

(1) The robustness of SPM approach to higher order autocorrelation.

(2) The robustness of SPM approach to spatially varying autocorrelation.

- The number of channels of NIRS is dozens mostly
- and their autocorrelation matrix (i.e. V matrix) highly varies among channels.
- In my analysis, taking a spatial average of autocorrelation lead to
increase false positive.
- However, without spatial averaging, df naturally differs from position
to position.
- Hence we cannot easily compare statistical significance between
channels.

Next is a more general question. I wish your advice $B!D (B

(3) When whitening procedure is used, even if I adopt a different order of
AR model for every channels of NIRS, df is apparently equal by all
channels.  In this case, are there any problem about performing
multiple comparisons between channels?

Kochiyama

------------------------------------------------------
Takanori Kochiyama
Graduate School of Human and Environmental Studies
Kyoto University
E-mail [log in to unmask]
Tel +81 75 753 7862


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