Hi David,

> I am in the process of comparing results from different analysis packages,
> and am new to SPM.  Can one of the SPM gurus explain the difference between
> "Worsley's pooled variance" and the voxel variance approach used by SPM? One
> of our packages-- Minoshima & Koeppe-- uses the former, and I understand
> that this can lead to more stable variance estimates for small subject
> numbers.

This relates to the problem of finding a suitable estimator of the
voxel-variance (described e.g. by Worsley et al. 1996). In short, to obtain a
good estimator it is necessary to pool information about the variance from
different sources under some appropriate assumption about the ''true'' ensemble
variance. The "Worsley's pooled variance" (Worsley et al. 1992) estimates the
ensemble variance under the assumption that the voxel-variance is the same in
all voxels of the brain (and subjects) for a given condition, while the SPM
approach estimates the ensemble variance under the assumption that the
voxel-variance is the same in all conditions (and subjects). The advantage of
the latter approach is that it allows for unequal voxel standard deviations,
i.e. it handles the situation when the ensemble variance is regionally
dependent. However, if for example the standard deviation is (sufficiently)
different between conditions then this approach is not valid.

> Under what other conditions would you recommend it?

I suppose the short answer is to use the "Worsley's pooled variance" when the
variance field is translationally invariant, in other words, when the
voxel-variance is the same in all voxels of the brain. This issue is discussed
in Petersson, K. M., Nichols, T. E., Poline, J.-B., and Holmes, A. P. 1999.
Statistical limitations in functional neuroimaging. II. Signal detection and
statistical inference. Phil. Trans. R. Soc. Lond. B 354: 1261-1282.

Briefly, the assumption of unequal voxel-variance across the brain volume has
not always been rejected in PET data (e.g. Worsley et al., 1992). This allows
for a straightforward use of (smooth) Gaussian random field (GRF) theory.
However, there is evidence that this assumption is not generally tenable (e.g.
Holmes et al., 1996; Worsley et al., 1996), particularly for FMRI data (Worsley
et al., 1997). The pooled variance approach seems to tolerate variations in the
voxel-variance of about 8%, while the local approach seems to tolerate
variations in the variance between experimental conditions of about 6% (Worsley
et al., 1996) reasonably well. If the assumption of an equal voxel-variance
across the brain is not reasonable then the use of t field theory is an
appropriate alternative to the GRF theory (e.g. as implemented in SPM99; or
perhaps GRF theory after Gaussianization, when the number of effective df is
sufficiently large [earlier versions of SPM]).

Now, a t statistic image is constructed by dividing the estimated signal image
with the estimated standard deviation, so noise in the variance image is
propagated to the t statistic image. In particular, this is a problem for t
statistic images with low df (even though the signal image may be smooth). In
the case of low df it would be attractive to use the pooled variance estimate,
when this is valid, thereby obtaining a more reliable variance estimate and
there are some indications that using a pooled variance estimate may give
result that are more reproducible compared to using voxel variance estimates
for PET data (Hunton et al., 1996; Strother et al., 1997). In addition, the
properties of such noisy statistic images may not be well approximated by those
of a smooth RF (because the smooth RF have features at a sub-voxel resolution).
The net result is that the smooth RF approximation approach becomes
increasingly conservative at smaller df. An alternative to the pooled variance
estimate (and perhaps a preferable strategy when this is not valid) is to pool
the variance estimates locally, effectively smoothing the variance image.
However, this requires a non-parametric approach to statistical inference
(Holmes et al., 1996), as implemented in for example SnPM (Nichols and Holmes
Submitted); available at http://www.fil.ion.ucl.ac.uk/spm/snpm/), or
alternatively for FMRI, as in Ledberg et al. (2000/Submitted).

See further:
Grabowski, T. J., Frank, R. J., Brown, C. K., Damasio, H., Boles Ponto, L. L.,
Watkins, G. L., and Hichwa, R. D. 1996. Reliability of PET activation across
statistical methods, subject groups and sample sizes. Hum. Brain Map. 4: 23-46.

Holmes, A., Blair, R. C., Watson, J. D. G., and Ford, I. 1996. Nonparametric
analysis of statistic images from functional mapping experiments. J. Cereb.
Blood Flow Metab. 16: 7-22.

Hunton, D. L., Miezin, F. M., Buckner, R. L., van Mier, H. I., Raichle, M. E.,
and Petersen, S. E. 1996. An assessment of functional-anatomical variability in
neuroimaging studies. Hum. Brain Map. 4: 122-139.

Ledberg, A., Larsson, J., and Petersson, K. M. 2000. 4D Analysis of functional
brain images. NeuroImage 11: S925

Ledberg, A., Fransson, P., Larsson, J., and Petersson, K. M. (Submitted). A 4D
approach to the analysis of functional brain images: Applications to fMRI data.

Nichols, T. E., and Holmes, A. P. (Submitted). Nonparametric permutation tests
for functional neuroimaging experiments: A primer with examples. Available at
http://www.fil.ion.ucl.ac.uk/spm/snpm/.

Strother, S. C., Lange, N., Anderson, J. R., Schaper, K. A., Rehm, K., Hansen,
L. K., and Rottenberg, D. A. 1997. Activation pattern reproducibility:
Measuring the effect of group size and data analysis models. Hum. Brain Map. 5:
312-316.

Worsley, K. J., Evans, A. C., Marrett, S., and Neelin, P. 1992. A
three-dimensional statistical analysis for CBF activation studies in human
brain. J. Cereb. Blood Flow Metab. 12: 900-18.

Worsley, K. J., Marrett, S., Neelin, P., Vandal, A. C., Friston, K. J., and
Evans, A. C. 1996. A unified statistical approach for determining significant
signals in images of cerebral activation. Hum. Brain Map. 4: 58-73.

Worsley, K. J., Wolforth, M., and Evans, A. C. 1997. Scale space searches for a
periodic signal in fMRI data with spatially varying hemodynamic response.
Proceedings of BrainMap'95 Conference

All the best,

karl magnus

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karl magnus petersson
Cognitive Neurophysiology Research Group R2-01
Department of Clinical Neuroscience, Karolinska Institute
Karolinska Hospital, S-171 76 Stockholm, Sweden

SANS Research Group, Department of Numerical Analysis and Computing
Royal Institute of Technology, S-100 44 Stockholm, Sweden

Tel: +46-8-517 720 39; Fax: +46-8-34 41 46;
Email: [log in to unmask]
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