Classical and Bayesian Inference in fMRI - Advance Image Processing in Magnetic Resonance Imaging - page 539

for their relative lack of power is that multivariate approaches (in their most
general form) estimate the model's error covariances using a number of param-
eters (e.g., the covariance between the errors at all pairs of voxels). In general,
the more parameters an estimation procedure has to deal with, the more variable
the estimate of any one parameter becomes. This renders any single estimate less
efficient.
Multivariate approaches consider voxels as different levels of an experimental
or treatment factor and use classical analysis of variance, not at each voxel (c.f.
SPM), but by considering the data sequences from all voxels together, as repli-
cations over voxels. The problem here is that regional changes in error variance,
and spatial correlations in the data, induce profound nonsphericity* in the error
terms. This nonsphericity would require large numbers of parameters to be esti-
mated for each voxel using conventional techniques. In SPM, the nonsphericity
is parameterized in a very parsimonious way with just two parameters for each
voxel. These are the error variance and smoothness estimators. This minimal
parameterization lends SPM a sensitivity that surpasses multivariate approaches.
SPM can do this because GRF theory implicitly imposes constraints on the
nonsphericity implied by the continuous and (spatially) extended nature of the
data. This is something that conventional multivariate and equivalent univariate
approaches do not accommodate, to their cost.
Some analyses use statistical maps based on nonparametric tests that eschew
distributional assumptions about the data e.g., nonparametric approaches (26).
These approaches are generally less powerful (i.e., less sensitive) than parametric
approaches (27). However, they have an important role in evaluating the assump-
tions behind parametric approaches and may supersede in terms of sensitivity
when these assumptions are violated (e.g., when degrees of freedom are very
small and voxel sizes are large in relation to smoothness).

17.4.1 R ANDOM F IELD T HEORY

Classical inferences using SPMs can be of two sorts depending on whether one
knows where to look in advance. With an anatomically constrained hypothesis
about effects in a particular brain region, the uncorrected p value associated with
the height or extent of that region in the SPM can be used to test the hypothesis.
With an anatomically open hypothesis (i.e., a null hypothesis that there is no
effect anywhere in a specified volume of the brain), a correction for multiple
dependent comparisons is necessary. The theory of random fields provides a way
of adjusting the p value that takes into account the fact that neighboring voxels

* Sphericity refers to the assumption of identically and independently distributed error terms (i.i.d.).
Under i.i.d., the probability density function of the errors, from all observations, has spherical
isocontours, hence, sphericity. Deviations from either of the i.i.d. criteria constitute nonsphericity. If
the error terms are not identically distributed then different observations have different error variances.
Correlations among error terms reflect dependencies among the error terms (e.g., serial correlation
in fMRI time series) and constitute the second component of nonsphericity. In fMRI both spatial and
temporal nonsphericity can be quite profound issues.