These notes are a modified version of K. Friston
(2003) Introduction: experimental design and statistical parametric mapping. In
Frackowiak et al. (Eds.) *Human brain
function*, 2^{nd} Edition. A PDF version of these notes is also available.

A Functional specialization and
segregation

A1 Linear Time Invariant (LTI)
systems and temporal basis functions

B2 Anatomically open hypotheses and levels of inference

This
chapter previews the ideas and procedures used in the analysis of brain imaging
data. It serves to introduce the main
themes covered, in depth, by the following chapters. The material presented in this chapter also
provides a sufficient background to understand the principles of experimental
design and data analysis referred to by the empirical chapters in the first
part of this book. The following
chapters on theory and analysis have been partitioned into four sections. The first three sections conform to the key
stages of analyzing imaging data sequences; computational neuroanatomy,
modeling and inference. These sections
focus on identifying, and making inferences about, regionally specific effects
in the brain. The final section
addresses the integration and interactions among these regions through analyses
of functional and effective connectivity.

Characterizing a regionally specific effect
rests on estimation and inference.
Inferences in neuroimaging may be about differences expressed when
comparing one group of subjects to another or, within subjects, changes over a
sequence of observations. They may
pertain to structural differences (*e.g.*
in voxel-based morphometry - Ashburner and Friston 2000) or neurophysiological
indices of brain functions (*e.g.*
fMRI). The principles of data analysis
are very similar for all of these applications and constitute the subject of
this and subsequent chapters. We will
focus on the analysis of fMRI time-series because this covers most of the
issues that are likely to be encountered in other modalities. Generally, the analysis of structural images
and PET scans is simpler because they do not have to deal with correlated
errors, from one scan to the next.

A
general issue, in data analysis, is the relationship between the
neurobiological hypothesis one posits and the statistical models adopted to
test that hypothesis. This chapter
begins by reviewing the distinction between functional *specialization* and *integration*
and how these principles serve as the motivation for most analyses of
neuroimaging data. We will address the design and analysis of neuroimaging
studies from these distinct perspectives but note that they have to be combined
for a full understanding of brain mapping results.

*Statistical
parametric mapping* is generally used to identify functionally specialized
brain responses and is the most prevalent approach to characterizing functional
anatomy and disease-related changes. The
alternative perspective, namely that provided by functional integration,
requires a different set of [multivariate] approaches that examine the
relationship among changes in activity in one brain area others. Statistical parametric mapping is a
voxel-based approach, employing classical inference, to make some comment about
regionally specific responses to experimental factors. In order to assign an observed response to a
particular brain structure, or cortical area, the data must conform to a known
anatomical space. Before considering
statistical modeling, this chapter deals briefly with how a time-series of
images are realigned and mapped into some standard anatomical space (*e.g*. a stereotactic space). The general ideas behind statistical
parametric mapping are then described and illustrated with attention to the
different sorts of inferences that can be made with different experimental
designs.

fMRI is special, in the sense that the data
lend themselves to a signal processing perspective. This can be exploited to ensure that both the
design and analysis are as efficient as possible. Linear time invariant models provide the
bridge between inferential models employed by statistical mapping and
conventional signal processing approaches.
Temporal autocorrelations in noise processes represent another important
issue, specific to fMRI, and approaches to maximizing efficiency in the context
of serially correlated errors will be discussed. Nonlinear models of evoked hemodynamics are
considered here because they can be used to indicate when the assumptions
behind linear models are violated. fMRI
can capture data very fast (in relation to other imaging techniques), affording
the opportunity to measure event-related responses. The distinction between event and
epoch-related designs will be discussed and considered in relation to
efficiency and the constraints provided by nonlinear characterizations.

Before considering multivariate analyses we
will close the discussion of inferences, about regionally specific effects, by
looking at the distinction between fixed and random-effect analyses and how
this relates to inferences about the subjects studied or the population from
which these subjects came. The final
section will deal with functional integration using models of effective connectivity
and other multivariate approaches.

The
brain appears to adhere to two fundamental principles of functional
organization, *functional integration *and
*functional specialization,* where the
integration within and among specialized areas is mediated by effective
connectivity. The distinction relates to
that between *localisationism* and *[dis]connectionism *that dominated
thinking about cortical function in the nineteenth century. Since the early anatomic theories of Gall,
the identification of a particular brain region with a specific function has
become a central theme in neuroscience.
However functional localization *per
se* was not easy to demonstrate: For example, a meeting that took place on *disconnection syndromes*
and the refutation of localisationism as a complete or sufficient explanation
of cortical organization. Functional
localization implies that a function can be localized in a cortical area,
whereas specialization suggests that a cortical area is specialized for some
aspects of perceptual or motor processing, and that this specialization is
anatomically segregated within the cortex.
The cortical infrastructure supporting a single function may then
involve many specialized areas whose union is mediated by the functional
integration among them. In this view
functional specialization is only meaningful in the context of functional
integration and *vice versa*.

The
functional role played by any component (*e.g.*
cortical area, subarea or neuronal population) of the brain is largely defined
by its connections. Certain patterns of
cortical projections are so common that they could amount to rules of cortical
connectivity. "These rules revolve
around one, apparently, overriding strategy that the cerebral cortex uses -
that of functional segregation" (Zeki 1990). Functional segregation demands that cells
with common functional properties be grouped together. This architectural constraint necessitates
both convergence and divergence of cortical connections. Extrinsic connections among cortical regions
are not continuous but occur in patches or clusters. This patchiness has, in some instances, a
clear relationship to functional segregation.
For example, V2 has a distinctive cytochrome oxidase architecture,
consisting of thick stripes, thin stripes and inter-stripes. When recordings are made in V2, directionally
selective (but not wavelength or color selective) cells are found exclusively
in the thick stripes. Retrograde (*i.e.* backward) labeling of cells in V5
is limited to these thick stripes. All
the available physiological evidence suggests that V5 is a functionally
homogeneous area that is specialized for visual motion. Evidence of this nature supports the notion
that patchy connectivity is the anatomical infrastructure that mediates
functional segregation and specialization.
If it is the case that neurons in a given cortical area share a common
responsiveness (by virtue of their extrinsic connectivity) to some sensorimotor
or cognitive attribute, then this functional segregation is also an anatomical
one. Challenging a subject with the
appropriate sensorimotor attribute or cognitive process should lead to activity
changes in, and only in, the area of interest.
This is the anatomical and physiological model upon which the search for
regionally specific effects is based.

The analysis of functional neuroimaging data
involves many steps that can be broadly divided into; (i) spatial processing,
(ii) estimating the parameters of a statistical model and (iii) making
inferences about those parameter estimates with appropriate statistics (see Figure 1.htm).
We will deal first with spatial transformations: In order to combine
data from different scans from the same subject, or data from different
subjects it is necessary that they conform to the same anatomical frame of
reference.

The
analysis of neuroimaging data generally starts with a series of spatial
transformations. These transformations
aim to reduce unwanted variance components in the voxel time-series that are
induced by movement or shape differences among a series of scans. Voxel-based
analyses assume that the data from a particular voxel all derive from the same
part of the brain. Violations of this
assumption will introduce artifactual changes in the voxel values that may
obscure changes, or differences, of interest.
Even single-subject analyses proceed in a standard anatomical space,
simply to enable reporting of regionally-specific effects in a frame of
reference that can be related to other studies.

The first step is to realign the data to
'undo' the effects of subject movement during the scanning session. After realignment the data are then
transformed using linear or nonlinear warps into a standard anatomical
space. Finally, the data are usually
spatially smoothed before entering the analysis proper.

Changes
in signal intensity over time, from any one voxel, can arise from head motion
and this represents a serious confound, particularly in fMRI studies. Despite restraints on head movement,
co-operative subjects still show displacements of up several millimeters. Realignment involves (i) estimating the 6
parameters of an affine 'rigid-body' transformation that minimizes the [sum of
squared] differences between each successive scan and a reference scan (usually
the first or the average of all scans in the time series) and (ii) applying the
transformation by re-sampling the data using tri-linear, sinc or spline
interpolation. Estimation of the affine
transformation is usually effected with a first order approximation of the *et al* 1995a). For most imaging modalities this procedure is
sufficient to realign scans to, in some instances, a hundred microns or so
(Friston *et al* 1996a). However, in fMRI, even after perfect
realignment, movement-related signals can still persist. This calls for a further step in which the
data are *adjusted* for residual
movement-related effects.

In
extreme cases as much as 90%^{ }of the variance, in fMRI time-series,
can be accounted for by the effects of movement *after* realignment (Friston *et
al* 1996a). Causes of these
movement-related components are due to movement effects that cannot be modeled
using a *linear* affine model. These nonlinear effects include; (i) subject
movement between slice acquisition, (ii)
interpolation artifacts (Grootoonk *et al*
2000), (iii) nonlinear distortion due to
magnetic field inhomogeneities (Andersson *et
al* 2001) and (iv) spin-excitation history effects (Friston *et al* 1996a). The latter can be pronounced if the TR
(repetition time) approaches T_{1} making the current signal a function
of movement history. These multiple
effects render the movement-related signal (*y*)
a nonlinear function of displacement (*x*)
in the *n*th and previous scans_{}. By assuming a
sensible form for this function, its parameters can be estimated using the
observed time-series and the estimated movement parameters *x* from the realignment procedure.
The estimated movement-related signal is then simply subtracted from the
original data. This adjustment can be
carried out as a pre-processing step or embodied in model estimation during the
analysis proper. The form for (*x*), proposed in Friston *et al* (1996a), was a nonlinear
auto-regression model that used polynomial expansions to second order. This model was motivated by spin-excitation
history effects and allowed displacement in previous scans to explain the
current movement-related signal.
However, it is also a reasonable model for many other sources of
movement-related confounds. Generally,
for TRs of several seconds, interpolation artifacts supersede (Grootoonk *et al* 2000) and first order terms,
comprising an expansion of the current displacement in terms of periodic basis
functions, are sufficient.

This subsection has considered *spatial*
realignment. In multislice acquisition
different slices are acquired at slightly different times. This raises the possibility of *temporal* realignment to ensure that the
data from any given volume were sampled at the same time. This is usually performed using sinc
interpolation over time and only when (i) the temporal dynamics of evoked
responses are important and (ii) the TR is sufficiently small to permit
interpolation. Generally timing effects
of this sort are not considered problematic because they manifest as
artifactual latency differences in evoked responses from region to region. Given that biophysical latency differences
may be in the order of a few seconds, inferences about these differences are
only made when comparing different trial types at the *same* voxel. Provided the
effects of latency differences are modelled, this renders temporal realignment
unnecessary in most instances.

After
realigning the data, a mean image of the series, or some other co-registered (*e.g. *a T_{1}-weighted) image, is
used to estimate some warping parameters that map it onto a template that
already conforms to some standard anatomical space (*e.g.* Talairach and Tournoux 1988).
This estimation can use a variety of models for the mapping, including:
(i) a 12-parameter affine transformation, where the parameters constitute a
spatial transformation matrix, (ii) low frequency basis spatial functions
(usually a discrete cosine set or polynomials), where the parameters are the
coefficients of the basis functions employed and (ii) a vector field specifying
the mapping for each control point (*e.g.*
voxel). In the latter case, the
parameters are vast in number and constitute a vector field that is bigger than
the image itself. Estimation of the
parameters of all these models can be accommodated in a simple Bayesian
framework, in which one is trying to find the deformation parameters _{}that have the maximum
posterior probability _{}given the data *y,* where
_{}. Put simply, one wants to find the deformation that is most
likely given the data. This deformation
can be found by maximizing the probability of getting the data, assuming the
current estimate of the deformation is true, times the probability of that
estimate being true. In practice the
deformation is updated iteratively using a Gauss-Newton scheme to maximize _{}. This involves
jointly minimizing the likelihood and prior potentials _{}and _{}. The likelihood
potential is generally taken to be the sum of squared differences between the
template and deformed image and reflects the probability of actually getting
that image if the transformation was correct.
The prior potential can be used to incorporate prior information about
the likelihood of a given warp. Priors can be determined empirically or
motivated by constraints on the mappings.
Priors play a more essential role as the number of parameters specifying
the mapping increases and are central to high dimensional warping schemes
(Ashburner *et al* 1997).

In practice most people use an affine or
spatial basis function warps and iterative least squares to minimize the
posterior potential. A nice extension of this approach is that the likelihood
potential can be refined and taken as difference between the index image and
the best [linear] combination of templates (*e.g.*
depicting gray, white, CSF and skull tissue partitions). This models intensity differences that are
unrelated to registration differences and allows different modalities to be
co-registered (see Figure 2.htm).

A special consideration is the spatial
normalization of brains that have gross anatomical pathology. This pathology can be of two sorts (i)
quantitative changes in the amount of a particular tissue compartment (*e.g.* cortical atrophy) or (ii)
qualitative changes in anatomy involving the insertion or deletion of normal
tissue compartments (*e.g.* ischemic
tissue in stroke or cortical dysplasia).
The former case is, generally, not problematic in the sense that changes
in the amount of cortical tissue will not affect its optimum spatial location
in reference to some template (and, even if it does, a disease-specific
template is easily constructed). The
second sort of pathology can introduce substantial 'errors' in the
normalization unless special precautions are taken. These usually involve imposing constraints on
the warping to ensure that the pathology does not bias the deformation of
undamaged tissue. This involves 'hard'
constraints implicit in using a small number of basis functions or 'soft'
constraints implemented by increasing the role of priors in Bayesian
estimation. An alternative strategy is
to use another modality that is less sensitive to the pathology as the basis of
the spatial normalization procedure or to simply remove the damaged region from
the estimation by masking it out.

It
is sometimes useful to co-register functional and anatomical images. However, with echo-planar imaging, geometric
distortions of T_{2}* images, relative to anatomical T_{1}-weighted
data, are a particularly serious problem because of the very low frequency per
point in the phase encoding direction.
Typically for echo-planar fMRI magnetic field inhomogeneity, sufficient
to cause dephasing of 2π through the slice, corresponds to an in-plane
distortion of a voxel. 'Unwarping'
schemes have been proposed to correct for the distortion effects (Jezzard and
Balaban 1995). However, this distortion
is not an issue if one spatially normalizes the functional data.

The
motivations for smoothing the data are fourfold. (i) By the matched filter theorem, the
optimum smoothing kernel corresponds to the size of the effect that one
anticipates. The spatial scale of
hemodynamic responses is, according to high-resolution optical imaging
experiments, about 2 to 5mm. Despite the
potentially high resolution afforded by fMRI an equivalent smoothing is
suggested for most applications. (ii) By
the central limit theorem, smoothing the data will render the errors more
normal in their distribution and ensure the validity of inferences based on
parametric tests. (iii) When making
inferences about regional effects using Gaussian random field theory (see
below) the assumption is that the error terms are a reasonable lattice
representation of an underlying and smooth Gaussian field. This necessitates smoothness to be
substantially greater than voxel size.
If the voxels are large, then they can be reduced by sub-sampling the
data and smoothing (with the original point spread function) with little loss
of intrinsic resolution. (iv) In the
context of inter-subject averaging it is often necessary to smooth more (*e.g.* 8 mm in fMRI or 16mm in PET) to
project the data onto a spatial scale where homologies in functional anatomy
are expressed among subjects.

Spatial
registration and normalization can proceed at a number of spatial scales
depending on how one parameterizes variations in anatomy. We have focussed on the role of normalization
to remove unwanted differences to enable subsequent analysis of the data. However, it is important to note that the
products of spatial normalization are bifold; a spatially normalized image and
a deformation field (see Figure 3.htm). This deformation field contains important
information about anatomy, in relation to the template used in the
normalization procedure. The analysis of
this information forms a key part of computational neuroanatomy. The tensor fields can be analyzed directly
(deformation-based morphometry) or used to create maps of specific anatomical
attributes (e.g. compression, shears *etc*.). These maps can then be analyzed on a voxel by
voxel basis (tensor-based morphometry).
Finally, the normalized structural images can themselves be subject to
satirical analysis after some suitable segmentation procedure. This is known as *voxel-based morphometry*. Voxel-based morphometry is the most commonly
used voxel-based neuroanatomical procedure and can easily be extended to
incorporate tensor-based approaches.

Functional
mapping studies are usually analyzed with some form of statistical parametric
mapping. Statistical parametric mapping
entails the construction of spatially extended statistical processes to test
hypotheses about regionally specific effects (Friston *et al* 1991). Statistical
parametric maps (SPMs) are image processes with voxel values that are, under
the null hypothesis, distributed according to a known probability density
function, usually the Student's T or F distributions. These are known colloquially as T- or
F-maps. The success of statistical
parametric mapping is due largely to the simplicity of the idea. Namely, one analyses each and every voxel
using any standard (univariate) statistical test. The resulting statistical parameters are
assembled into an image - the SPM. SPMs
are interpreted as spatially extended statistical processes by referring to the
probabilistic behavior of Gaussian fields (Adler 1981, Worsley *et al* 1992, Friston *et al* 1994a, Worsley *et al*
1996). Gaussian random fields model both
the univariate probabilistic characteristics of a SPM and any non-stationary
spatial covariance structure. 'Unlikely'
excursions of the SPM are interpreted as regionally specific effects,
attributable to the sensorimotor or cognitive process that has been manipulated
experimentally.

Over the years statistical parametric mapping
has come to refer to the conjoint use of *the
general linear model* (GLM) and *Gaussian
random field* (GRF) theory to analyze and make classical inferences about
spatially extended data through statistical parametric maps (SPMs). The GLM is used to estimate some parameters
that could explain the spatially continuos data in exactly the same way as in
conventional analysis of discrete data.
GRF theory is used to resolve the multiple comparison problem that
ensues when making inferences over a volume of the brain. GRF theory provides a method for correcting *p* values for the search volume of a SPM
and plays the same role for *continuous*
data (*i.e.* images) as the Bonferonni
correction for the number of discontinuous or *discrete* statistical tests.

The approach was called SPM for three
reasons; (i) To acknowledge *S*ignificance
*P*robability *M*apping, the use of interpolated pseudo-maps of *p* values used to summarize the analysis
of multi-channel ERP studies. (ii) For consistency with the nomenclature of
parametric maps of physiological or physical parameters (*e.g.* regional cerebral blood flow rCBF or volume rCBV parametric
maps). (iii) In reference to the *parametric*
statistics that comprise the maps.
Despite its simplicity there are some fairly subtle motivations for the
approach that deserve mention. Usually,
given a response or dependent variable comprising many thousands of voxels one
would use *multivariate* analyses as
opposed to the *mass-univariate*
approach that SPM represents. The
problems with multivariate approaches are that; (i) they do not support
inferences about regionally specific effects, (ii) they require more
observations than the dimension of the response variable (*i.e.* number of voxels) and (iii), even in the context of dimension
reduction, they are less sensitive to focal effects than mass-univariate
approaches. A heuristic argument, for
their relative lack of power, is that multivariate approaches estimate the
models error covariances using lots of parameters (*e.g.* the covariance between the errors at all pairs of
voxels). In general, the more parameters
(and hyper-parameters) an estimation procedure has to deal with, the more
variable the estimate of any one parameter becomes. This renders inferences about 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
replications over voxels. The problem
here is that regional changes in error variance, and spatial correlations in
the data, induce profound non-sphericity[1]
in the error terms. This non-sphericity
would again require large numbers of [hyper]parameters to be estimated for each
voxel using conventional techniques. In
SPM the non-sphericity is parameterized in a very parsimonious way with just
two [hyper]parameters for each voxel.
These are the error variance and smoothness estimators (see Figure 2.htm).
This minimal parameterization lends SPM a sensitivity that surpasses
multivariate approaches. SPM can do this
because GRF theory implicitly imposes constraints on the non-sphericity 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
non-parametric tests that eschew distributional assumptions about the
data. These approaches are generally
less powerful (*i.e.* less sensitive)
than parametric approaches (see Aguirre *et
al* 1998). However, they have an
important role in evaluating the assumptions behind parametric approaches and
may supercede 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).

The Bayesian alternative to classical
inference with SPMs rests on conditional inferences about an effect, given the
data, as opposed to classical inferences about the data, given the effect is
zero. Bayesian inferences about
spatially extended effects use Posterior Probability Maps (PPMs). Although less commonly used than SPMs, PPMs
are potentially very useful, not least because they do not have to contend with
the multiple comparisons problem induced by classical inference. In contradistinction to SPM, this means that
inferences about a given regional response do not depend on inferences about
responses elsewhere.

. Next we consider parameter estimation in the
context of the GLM. This is followed by
an introduction to the role of GRF theory when making classical inferences
about continuous data.

Statistical
analysis of imaging data corresponds to (i) modeling the data to partition
observed neurophysiological responses into components of interest, confounds
and error and (ii) making inferences about the interesting effects in relation
to the error variance. This classical
inference can be regarded as a direct comparison of the variance due to an
interesting experimental manipulation with the error variance (*c.f.* the F statistic and other
likelihood ratios). Alternatively, one
can view the statistic as an estimate of the response, or difference of
interest, divided by an estimate of its standard deviation. This is a useful way to think about the T
statistic.

A brief review of the literature may give
the impression that there are numerous ways to analyze PET and fMRI time-series
with a diversity of statistical and conceptual approaches. This is not the case. With very a few exceptions, every analysis is
a variant of the general linear model.
This includes; (i) simple T tests on scans assigned to one condition or
another, (ii) correlation coefficients between observed responses and boxcar
stimulus functions in fMRI, (iii) inferences made using multiple linear regression,
(iv) evoked responses estimated using linear time invariant models and (v)
selective averaging to estimate event-related responses in fMRI. Mathematically, they are all identical can be
implemented with the same equations and algorithms. The only thing that distinguishes among them
is the design matrix encoding the experimental design. The use of the correlation coefficient
deserves special mention because of its popularity in fMRI (Bandettini *et al* 1993). The significance of a correlation is
identical to the significance of the equivalent T statistic testing for a
regression of the data on the stimulus function. The correlation coefficient approach is
useful but the inference is effectively based on a limiting case of multiple
linear regression that obtains when there is only one regressor. In fMRI many regressors usually enter into a
statistical model. Therefore, the T
statistic provides a more versatile and generic way of assessing the
significance of regional effects and is preferred over the correlation
coefficient.

The general linear model is an equation _{}that expresses the
observed response variable *Y* in terms
of a linear combination of explanatory variables *X* plus a well behaved error term (see Figure
4.htm and Friston *et al*
1995b). The general linear model is
variously known as 'analysis of covariance' or 'multiple regression analysis'
and subsumes simpler variants, like the 'T test' for a difference in means, to
more elaborate linear convolution models such as finite impulse response (FIR)
models. The matrix *X* that contains the explanatory variables (*e.g.* designed effects or confounds) is called the *design matrix*. Each column of the design matrix corresponds
to some effect one has built into the experiment or that may confound the
results. These are referred to as
explanatory variables, covariates or regressors. The example in Figure
1.htm relates to a fMRI study of visual stimulation under four
conditions. The effects on the response
variable are modeled in terms of functions of the presence of these conditions
(*i.e.* boxcars smoothed with a
hemodynamic response function) and constitute the first four columns of the
design matrix. There then follows a
series of terms that are designed to remove or model low frequency variations
in signal due to artifacts such as aliased biorhythms and other drift
terms. The final column is whole brain
activity. The relative contribution of
each of these columns is assessed using standard least squares and inferences
about these contributions are made using T or F statistics, depending upon
whether one is looking at a particular linear combination (*e.g.* a subtraction), or all of them together. The operational equations are depicted
schematically in Figure 4.htm. In this scheme the general linear model has
been extended (Worsley and Friston 1995) to incorporate intrinsic
non-sphericity, or correlations among the error terms, and to allow for some
specified temporal filtering of the data with the matrix *S*. This generalization
brings with it the notion of *effective
degrees of freedom*, which are less than the conventional degrees of freedom
under i.i.d. assumptions (see footnote).
They are smaller because the temporal correlations reduce the effective
number of independent observations. The
T and F statistics are constructed using Satterthwaites approximation. This is the same approximation used in
classical non-sphericity corrections such as the Geisser-Greenhouse
correction. However, in the Worsley and
Friston (1995) scheme, Satherthwaites approximation is used to construct the
statistics and appropriate degrees of freedom, not simply to provide a *post hoc* correction to the degrees of
freedom.

A special case of temporal filtering
deserved mention. This is when the
filtering decorrelates (i.e. whitens) the error terms by using _{}. This is the
filtering scheme used in current implementations of software for SPM and
renders the ordinary least squares (OLS) parameter estimates maximum likelihood
(ML) estimators. These are optimal in
the sense that they are the minimum variance estimators of all unbiased
estimators. The estimation of _{}uses expectation
maximization (EM) to provide restricted maximum likelihood (ReML) estimates of _{}in terms of
hyperparameters _{}corresponding to
variance components. In
this case the effective degrees of freedom revert to their maximum that would
be attained in the absence of temporal correlations or non-sphericity.

The equations summarized in Figure 4.htm can be used to implement a vast range of
statistical analyses. The issue is
therefore not so much the mathematics but the formulation of a design matrix *X* appropriate to the study design and
inferences that are sought. The design
matrix can contain both covariates and indicator variables. Each column of *X* has an associated unknown parameter. Some of these parameters will be of interest
(*e.g.* the effect of particular
sensorimotor or cognitive condition or the regression coefficient of
hemodynamic responses on reaction time).
The remaining parameters will be of no interest and pertain to
confounding effects (*e.g.* the effect
of being a particular subject or the regression slope of voxel activity on
global activity). Inferences about the
parameter estimates are made using their estimated variance. This allows one to test the null hypothesis
that all the estimates are zero using the F statistic to give an SPM*e.g*.
a subtraction) of the estimates is zero using an SPM*vector*
would be [-1 1 0 0..... ] to compare the difference in responses evoked by two
conditions, modeled by the first two condition-specific regressors in the
design matrix. Sometimes several
contrasts of parameter estimates are jointly interesting. For example, when using polynomial (Bchel *et al* 1996) or basis function expansions
of some experimental factor. In these
instances, the SPM*matrix*
of contrast weights that can be thought of as a collection of T contrasts
that one wants to test together. An
F-contrast may look like,

_{}

which
would test for the significance of the first *or* second parameter estimates.
The fact that the first weight is 1 as opposed to 1 has no effect on
the test because the F statistic is based on sums of squares.

.
In most analysis the design matrix contains indicator variables or parametric
variables encoding the experimental manipulations. These are formally identical to classical
analysis of [co]variance (*i.e.*
AnCova) models. An important instance of
the GLM, from the perspective of fMRI, is the linear time invariant (LTI)
model. Mathematically this is no
different from any other GLM. However,
it explicitly treats the data-sequence as an ordered time-series and enables a
signal processing perspective that can be very useful.

In
Friston *et al* (1994b) the form of the
hemodynamic impulse response function (HRF) was estimated using a least squares
de-convolution and a time invariant model, where evoked neuronal responses are
convolved with the HRF to give the measured hemodynamic response (see Boynton *et al* 1996). This simple linear framework is the
cornerstone for making statistical inferences about activations in fMRI with
the GLM. An impulse response function is
the response to a single impulse, measured at a series of times after the
input. It characterizes the input-output
behavior of the system (*i.e.* voxel)
and places important constraints on the sorts of inputs that will excite a
response. The HRFs, estimated in Friston
*et al* (1994b) resembled a Poisson or
Gamma function, peaking at about 5 seconds.
Our understanding of the biophysical and physiological mechanisms that
underpin the HRF has grown considerably in the past few years (*e.g.* Buxton and Frank 1997). Figure 5.htm
shows some simulations based on the hemodynamic model described in Friston *et al* (2000a). Here, neuronal activity induces some
auto-regulated signal that causes transient increases in rCBF. The resulting flow increases dilate the
venous balloon increasing its volume (*v*)
and diluting venous blood to decrease deoxyhemoglobin content (*q*).
The BOLD signal is roughly proportional to the concentration of
deoxyhemoglobin (*q*/*v*) and follows the rCBF response with
about a seconds delay.

Knowing the forms that the HRF can take is
important for several reasons, not least because it allows for better
statistical models of the data. The HRF
may vary from voxel to voxel and this has to be accommodated in the GLM. To allow for different HRFs in different
brain regions the notion of temporal basis functions, to model evoked responses
in fMRI, was introduced (Friston *et al*
1995c) and applied to event-related responses in Josephs *et al* (1997) (see also Lange and Zeger 1997). The basic idea behind temporal basis
functions is that the hemodynamic response induced by any given trial type can
be expressed as the linear combination of several [basis] functions of
peristimulus time. The convolution model
for fMRI responses takes a stimulus function encoding the supposed neuronal
responses and convolves it with an HRF to give a regressor that enters into the
design matrix. When using basis
functions, the stimulus function is convolved with all the basis functions to
give a series of regressors. The
associated parameter estimates are the coefficients or weights that determine
the mixture of basis functions that best models the HRF for the trial type and
voxel in question. We find the most
useful basis set to be a canonical HRF and its derivatives with respect to the
key parameters that determine its form (*e.g.*
latency and dispersion). The nice thing
about this approach is that it can partition differences among evoked responses
into differences in magnitude, latency or dispersion, that can be tested for
using specific contrasts and the SPM*et al* 1998b).

Temporal basis functions are important
because they enable a graceful transition between conventional multi-linear
regression models with one stimulus function per condition and FIR models with
a parameter for each time point following the onset of a condition or trial
type. Figure
6.htm illustrates this graphically (see Figure legend). In summary, temporal basis functions offer
useful constraints on the form of the estimated response that retain (i) the
flexibility of FIR models and (ii) the efficiency of single regressor models.
The advantage of using several temporal basis functions (as opposed to an
assumed form for the HRF) is that one can model voxel-specific forms for
hemodynamic responses and formal differences (*e.g.* onset latencies) among responses to different sorts of events. The advantages of using basis functions over
FIR models are that (i) the parameters are estimated more efficiently and (ii)
stimuli can be presented at any point in the inter-stimulus interval. The latter is important because time-locking
stimulus presentation and data acquisition gives a biased sampling over
peristimulus time and can lead to differential sensitivities, in multi-slice
acquisition, over the brain.

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 are not independent by virtue of
continuity in the original data.
Provided the data are sufficiently smooth the GRF correction is less
severe (*i.e.* is more sensitive) than
a Bonferroni correction for the number of voxels. As noted above GRF theory
deals with the multiple comparisons problem in the context of continuous,
spatially extended statistical fields, in a way that is analogous to the
Bonferroni procedure for families of discrete statistical tests. There are many ways to appreciate the
difference between GRF and Bonferroni corrections. Perhaps the most intuitive is to consider the
fundamental difference between a SPM and a collection of discrete T
values. When declaring a connected
volume or region of the SPM to be significant, we refer collectively to all the
voxels that comprise that volume. The
false positive rate is expressed in terms of connected [excursion] sets of
voxels above some threshold, under the null hypothesis of no activation. This is not the expected number of false
positive voxels. One false positive
region may contain hundreds of voxels, if the SPM is very smooth. A Bonferroni correction would control the
expected number of false positive *voxels*,
whereas GRF theory controls the expected number of false positive *regions*.
Because a false positive region can contain many voxels the corrected
threshold under a GRF correction is much lower, rendering it much more
sensitive. In fact the number of voxels
in a region is somewhat irrelevant because it is a function of smoothness. The GRF correction discounts voxel size by
expressing the search volume in terms of smoothness or resolution elements (*Resels*). See Figure
7.htm. This intuitive perspective is
expressed formally in terms of differential topology using the *Euler characteristic* (Worsley *et al* 1992). At high thresholds the
Euler characteristic corresponds to the number of regions exceeding the
threshold.

There are only two assumptions underlying
the use of the GRF correction: (i) The error fields (but not necessarily the
data) are a reasonable lattice approximation to an underlying random field with
a multivariate Gaussian distribution.
(ii) These fields are continuous, with a differentiable and invertible
autocorrelation function. A common
misconception is that the autocorrelation function has to be Gaussian. It does not.
The only way in which these assumptions can be violated is if; (i) the
data are not smoothed (with or without sub-sampling to preserve resolution),
violating the reasonable lattice assumption or (ii) the statistical model is
mis-specified so that the errors are not normally distributed. Early
formulations of the GRF correction were based on the assumption that the
spatial correlation structure was wide-sense stationary. This assumption can now be relaxed due to a
revision of the way in which the smoothness estimator enters the correction
procedure (Kiebel *et al* 1999). In other words, the corrections retain their
validity, even if the smoothness varies from voxel to voxel.

When
making inferences about regional effects (*e.g.*
activations) in SPMs, one often has some idea about where the activation should
be. In this instance a correction for
the entire search volume is inappropriate.
However, a problem remains in the sense that one would like to consider
activations that are 'near' the predicted location, even if they are not
exactly coincident. There are two
approaches one can adopt; (i) pre-specify a small search volume and make the
appropriate GRF correction (Worsley *et al*
1996) or (ii) used the uncorrected *p*
value based on spatial extent of the nearest cluster (Friston 1997). This probability is based on getting the
observed number of voxels, or more, in a given cluster (conditional on that
cluster existing). Both these procedures
are based on distributional approximations from GRF theory.

To
make inferences about regionally specific effects the SPM is thresholded, using
some height and spatial extent thresholds that are specified by the user. Corrected *p*-values
can then be derived that pertain to; (i) the number of activated regions (*i.e.* number of clusters above the height
and volume threshold) -* set level
inferences*, (ii) the number of activated voxels (*i.e.* volume) comprising a particular region - *cluster level inferences* and (iii) the *p*-value for each voxel within that cluster - *voxel level inferences*.
These *p*-values are corrected
for the multiple dependent comparisons and are based on the probability of
obtaining *c*, or more, clusters with *k*, or more, voxels, above a threshold *u* in an SPM of known or estimated
smoothness. This probability has a
reasonably simple form (see Figure 7.htm for
details).

Set-level refers to the inference that the
number of clusters comprising an observed activation profile is highly unlikely
to have occurred by chance and is a statement about the activation profile, as
characterized by its constituent regions.
Cluster-level inferences are a special case of set-level inferences,
that obtain when the number of clusters *c*
= 1. Similarly voxel-level inferences
are special cases of cluster-level inferences that result when the cluster can
be small (*i.e. k* = 0). Using a theoretical power analysis (Friston *et al* 1996b) of distributed activations,
one observes that set-level inferences are generally more powerful than
cluster-level inferences and that cluster-level inferences are generally more
powerful than voxel-level inferences.
The price paid for this increased sensitivity is reduced localizing
power. Voxel-level tests permit individual voxels to be identified as
significant, whereas cluster and set-level inferences only allow clusters or
sets of clusters to be declared significant.
It should be remembered that these conclusions, about the relative power
of different inference levels, are based on distributed activations. Focal activation may well be detected with
greater sensitivity using voxel-level tests based on peak height. Typically, people use voxel-level inferences
and a spatial extent threshold of zero.
This reflects the fact that characterizations of functional anatomy are
generally more useful when specified with a high degree of anatomical
precision.

This
section considers the different sorts of designs that can be employed in
neuroimaging studies. Experimental
designs can be classified as *single
factor* or *multifactorial *designs,
within this classification the levels of each factor can be *categorical* or *parametric*. We will start by
discussing categorical and parametric designs and then deal with multifactorial
designs.

The
tenet of cognitive subtraction is that the difference between two tasks can be
formulated as a separable cognitive or sensorimotor component and that
regionally specific differences in hemodynamic responses, evoked by the two
tasks, identify the corresponding functionally specialized area. Early applications of subtraction range from
the functional anatomy of word processing (Petersen *et al* 1989) to functional specialization in extrastriate cortex
(Lueck *et al* 1989). The latter studies involved presenting visual
stimuli with and without some sensory attribute (*e.g. *color, motion *etc*). The areas highlighted by subtraction were
identified with homologous areas in monkeys that showed selective
electrophysiological responses to equivalent visual stimuli.

Cognitive conjunctions (Price and Friston
1997) can be thought of as an extension of the subtraction technique, in the
sense that they combine a series of subtractions. In subtraction ones tests a *single* hypothesis pertaining to the
activation in one task relative to another.
In conjunction analyses *several*
hypotheses are tested, asking whether all the activations, in a series of task
pairs, are jointly significant. Consider
the problem of identifying regionally specific activations due to a particular
cognitive component (*e.g.* object
recognition). If one can identify a
series of task pairs whose differences have only that component in common, then
the region which activates, in all the corresponding subtractions, can be associated
with the common component. Conjunction
analyses allow one to demonstrate the context-invariant nature of regional
responses. One important application of
conjunction analyses is in multi-subject fMRI studies, where generic effects
are identified as those that are conjointly significant in all the subjects
studied (see below).

The
premise behind parametric designs is that regional physiology will vary
systematically with the degree of cognitive or sensorimotor processing or
deficits thereof. Examples of this
approach include the PET experiments of Grafton *et al* (1992) that demonstrated significant correlations between
hemodynamic responses and the performance of a visually guided motor tracking
task. On the sensory side Price *et al* (1992) demonstrated a remarkable
linear relationship between perfusion in peri-auditory regions and frequency of
aural word presentation. This
correlation was not observed in Wernicke's area, where perfusion appeared to
correlate, not with the discriminative attributes of the stimulus, but with the
presence or absence of semantic content.
These relationships or *neurometric
functions* may be linear or nonlinear.
Using polynomial regression, in the context of the GLM, one can identify
nonlinear relationships between stimulus parameters (*e.g.* stimulus duration or presentation rate) and evoked
responses. To do this one usually uses a
SPM*et al* 1996).

The example provided in Figure 8.htm illustrates both categorical and
parametric aspects of design and analysis.
These data were obtained from a fMRI study of visual motion processing
using radially moving dots. The stimuli
were presented over a range of speeds using *isoluminant*
and *isochromatic* stimuli. To identify areas involved in visual motion a
stationary dots condition was subtracted from the moving dots conditions (see
the contrast weights on the upper right).
To ensure significant motion-sensitive responses, using both color and
luminance cues, a conjunction of the equivalent subtractions was assessed under
both viewing contexts. Areas V5 and V3a
are seen in the ensuing SPM_{min}}*u* in each subtraction separately.
This *conjunction *SPM has an
equivalent interpretation; it represents the intersection of the excursion
sets, defined by the threshold *u*, of
each *component* SPM. This intersection is the essence of a
conjunction. The expressions in Figure 7.htm pertain to the general case of the
minimum of *n* T values. The special case where *n* = 1 corresponds to a conventional SPM

The responses in left V5 are shown in the
lower panel of Figure 8.htm and speak to a
compelling inverted 'U' relationship between speed and evoked response that
peaks at around 8 degrees per second. It
is this sort of relationship that parametric designs try to characterize. Interestingly, the form of these
speed-dependent responses was similar using both stimulus types, although
luminance cues are seen to elicit a greater response. From the point of view of a factorial design
there is a *main effect* of cue
(isoluminant vs. isochromatic), a main [nonlinear] effect of speed, but no
speed by cue *interaction*.

Clinical neuroscience studies can use
parametric designs by looking for the neuronal correlates of clinical (*e.g.* symptom) ratings over
subjects. In many cases multiple
clinical scores are available for each subject and the statistical design can
usually be seen as a multilinear regression.
In situations where the clinical scores are correlated principal
component analysis or factor analysis is sometimes applied to generate a new,
and smaller, set of explanatory variables that are orthogonal to each
other. This has proved particularly
useful in psychiatric studies where syndromes can be expressed over a number of
different dimensions (*e.g.* the degree
of psychomotor poverty, disorganization and reality distortion in
schizophrenia. See Liddle *et al* 1992). In this way, regionally specific correlates
of various symptoms may point to their distinct pathogenesis in a way that
transcends the syndrome itself. For
example psychomotor poverty may be associated with left dorsolateral prefrontal
dysfunction irrespective of whether the patient is suffering from schizophrenia
or depression.

Factorial designs are becoming more
prevalent than single factor designs because they enable inferences about
interactions. At its simplest an
interaction represents a change in a change.
Interactions are associated with factorial designs where two or more
factors are combined in the same experiment.
The effect of one factor, on the effect of the other, is assessed by the
interaction term. Factorial designs have
a wide range of applications. An early
application, in neuroimaging, examined physiological adaptation and plasticity
during motor performance, by assessing time by condition interactions (Friston *et al* 1992a). Psychopharmacological activation studies are
further examples of factorial designs (Friston *et al* 1992b). In these
studies cognitively evoked responses are assessed before and after being given
a drug. The interaction term reflects
the pharmacological modulation of task-dependent activations. Factorial designs have an important role in the
context of cognitive subtraction and additive factors logic by virtue of being
able to test for interactions, or context-sensitive activations (*i.e.* to demonstrate the fallacy of 'pure
insertion'. See Friston *et al* 1996c). These interaction effects can sometimes be
interpreted as (i) the integration of the two or more [cognitive] processes or
(ii) the modulation of one [perceptual] process by another. See Figure 9.htm
for an example. From the point of view
of clinical studies interactions are central.
The effect of a disease process on sensorimotor or cognitive activation
is simply an interaction and involves replicating a subtraction experiment in
subjects with and without the pathophysiology studied. Factorial designs can also embody parametric
factors. If one of the factors has a
number of parametric levels, the interaction can be expressed as a difference
in regression slope of regional activity on the parameter, under both levels of
the other [categorical] factor. An
important example of factorial designs, that mix categorical and parameter
factors, are those looking for *psychophysiological
interactions*. Here the parametric
factor is brain activity measured in a particular brain region. These designs have proven useful in looking
at the interaction between bottom-up and top-down influences within processing
hierarchies in the brain (Friston *et al *1997).

In
this section we consider fMRI time-series from a signal processing perspective
with particular focus on optimal experimental design and efficiency. fMRI time-series can be viewed as a linear
admixture of signal and noise. Signal
corresponds to neuronally mediated hemodynamic changes that can be modeled as a
[non]linear convolution of some underlying neuronal process, responding to
changes in experimental factors, by a hemodynamic response function (HRF). Noise has many contributions that render it
rather complicated in relation to other neurophysiological measurements. These include neuronal and non-neuronal
sources. Neuronal noise refers to
neurogenic signal not modeled by the explanatory variables and has the same
frequency structure as the signal itself.
Non-neuronal components have both white (*e.g.* R.F. Johnson noise) and colored components (*e.g.* pulsatile motion of the brain
caused by cardiac cycles and local modulation of the static magnetic field B_{0}
by respiratory movement). These effects
are typically low frequency or wide-band (*e.g.*
aliased cardiac-locked pulsatile motion).
The superposition of all these components induces temporal correlations
among the error terms (denoted by _{}in Figure 4.htm) that can effect sensitivity to
experimental effects. Sensitivity
depends upon (i) the relative amounts of signal and noise and (ii) the
efficiency of the experimental design.
Efficiency is simply a measure of how reliable the parameter estimates
are and can be defined as the inverse of the variance of a contrast of parameter
estimates (see Figure 4.htm). There are two important considerations that
arise from this perspective on fMRI time-series: The first pertains to optimal
experimental design and the second to optimum [de]convolution of the
time-series to obtain the most efficient parameter estimates.

As
noted above, an LTI model of neuronally mediated signals in fMRI suggests that
only those experimentally induced signals that survive convolution with the
hemodynamic response function (HRF) can be estimated with any efficiency. By convolution theorem the frequency
structure of experimental variance should therefore be designed to match the
transfer function of the HRF. The
corresponding frequency profile of this transfer function is shown in Figure 10.htm - solid line). It is clear that frequencies around 0.03 Hz
are optimal, corresponding to periodic designs with 32-second periods (*i.e.* 16-second epochs). Generally, the first objective of
experimental design is to comply with the natural constraints imposed by the
HRF and ensure that experimental variance occupies these intermediate
frequencies.

This
is quite a complicated but important area.
Conventional signal processing approaches dictate that whitening the
data engenders the most efficient parameter estimation. This corresponds to filtering with a
convolution matrix *S* (see Figure 3.htm) that is the inverse of the intrinsic
convolution matrix *K* (_{}). This *whitening* strategy renders the least
square estimator in Figure 4.htm equivalent to the
ML or Gauss-Markov estimator. However, one generally does not know the form of
the intrinsic correlations, which means they have to be estimated. This estimation usually proceeds using a
restricted maximum likelihood (ReML) estimate of the serial correlations, among
the residuals, that properly accommodates the effects of the residual-forming
matrix and associated loss of degrees of freedom. However, using this estimate of the intrinsic
non-sphericity to form a Gauss-Markov estimator at each voxel is not easy. First the estimate of non-sphericity can
itself be imprecise leading to bias in the standard error (Friston *et al* 2000b). Second, ReML estimation requires a
computationally prohibitive iterative procedure at every voxel. There are a number of approaches to these
problems that aim to increase the efficiency of the estimation and reduce the
computational burden. The approach adopted in current versions of our software
is use ReML estimates based on all voxels that respond to experimental
manipulation. This affords very
efficient hyperparameter estimates[2]
and, furthermore, allows one to use the same matrices at each voxel when
computing the parameter estimates.

Although we usually make _{}, using a first-pass ReML estimate of the serial
correlations, we will deal with the simpler and more general case where *S* can take any form. In this case the parameter estimates are *generalized* least square (GLS)
estimators. The GLS estimator is
unbiased and, luckily, is identical to the Gauss-Markov estimator if the
regressors in the design matrix are periodic[3]. After GLS estimation, the ReML estimate of _{}enters into the
expressions for the standard error and degrees of freedom provided in Figure 4.htm.

fMRI noise has been variously characterized
as a 1/f process (Zarahn *et al* 1997)
or an autoregressive process (Bullmore *et
al* 1996) with white noise (Purdon and Weisskoff 1998). Irrespective of the exact form these serial
correlations take, treating low frequency drifts as fixed effects can finesse
the hyper-parameterization of serial correlations. Removing low frequencies from the time series
allows the model to fit serial correlations over a more restricted frequency
range or shorter time spans. Drift removal can be implemented by including
drift terms in the design matrix or by including the implicit residual forming
matrix in S to make it a high-pass filter.
An example of a high-pass filter with a high-pass cut-off of 1/64 Hz is
shown in inset of Figure 8.htm. This filters transfer function (the broken
line in the main panel) illustrates the frequency structure of neurogenic
signals after high-pass filtering.

Implicit
in the use of high-pass filtering is the removal of low frequency components that
can be regarded as confounds. Other
important confounds are signal components that are artifactual or have no
regional specificity. These are referred
to as *global confounds* and have a
number of causes. These can be divided
into physiological (*e.g. *global
perfusion changes in PET, mediated by changes in pCO_{2}) and
non-physiological (*e.g.* transmitter
power calibration, B_{1} coil profile and receiver gain in fMRI). The latter generally scale the signal before
the MRI sampling process. Other
non-physiological effects may have a non-scaling effect (*e.g.* Nyquist ghosting, movement-related effects *etc*).
In PET it is generally accepted that regional changes in rCBF, evoked
neuronally, mix additively with global changes to give the measured
signal. This calls for a global
normalization procedure where the global estimator enters into the statistical
model as a confound. In fMRI,
instrumentation effects that scale the data motivate a global normalization by
proportional scaling, using the whole brain mean, before the data enter into
the statistical model.

It is important to differentiate between
global confounds and their estimators.
By definition the global mean over intra-cranial voxels will subsume all
regionally specific effects. This means
that the global estimator may be partially collinear with effects of interest,
especially if the evoked responses are substantial and widespread. In these situations global normalization may
induce apparent deactivations in regions *not*
expressing a physiological response.
These are not artifacts in the sense that they are real, relative to
global changes, but they have little face validity in terms of the underlying
neurophysiology. In instances where
regionally specific effects bias the global estimator, some investigators
prefer to omit global normalization.
Provided drift terms are removed from the time-series, this is generally
acceptable because most global effects have slow time constants. However, the issue of normalization-induced
deactivations is better circumnavigated with experimental designs that use
well-controlled conditions, which elicit differential responses in restricted
brain systems.

So
far we have only considered LTI models and first order HRFs. Another signal processing perspective is
provided by nonlinear system identification (Vazquez and Noll 1998). This section considers nonlinear models as a
prelude to the next subsection on event-related fMRI, where nonlinear
interactions among evoked responses provide constraints for experimental design
and analysis. We have described an
approach to characterizing evoked hemodynamic responses in fMRI based on
nonlinear system identification, in particular the use of *Volterra series* (Friston *et
al* 1998a). This approach enables one
to estimate Volterra kernels that describe the relationship between stimulus
presentation and the hemodynamic responses that ensue. Volterra series are essentially high order
extensions of linear convolution models.
These kernels therefore represent a nonlinear characterization of the
HRF that can model the responses to stimuli in different contexts and
interactions among stimuli. In fMRI, the
kernel coefficients can be estimated by (i) using a second order approximation
to the Volterra series to formulate the problem in terms of a general linear
model and (ii) expanding the kernels in terms of temporal basis functions. This allows the use of the standard
techniques described above to estimate the kernels and to make inferences about
their significance on a voxel-specific basis using SPMs.

One important manifestation of the nonlinear
effects, captured by the second order kernels, is a modulation of
stimulus-specific responses by preceding stimuli that are proximate in
time. This means that responses at high
stimulus presentation rates saturate and, in some instances, show an inverted U
behavior. This behavior appears to be
specific to BOLD effects (as distinct from evoked changes in cerebral blood
flow) and may represent a *hemodynamic
refractoriness*. This effect has
important implications for event-related fMRI, where one may want to present
trials in quick succession.

The results of a typical nonlinear analysis
are given in Figure 11.htm. The results in the right panel represent the
average response, integrated over a 32-second train of stimuli as a function of
stimulus onset asynchrony (SOA) within that train. These responses are based on the kernel
estimates (left hand panels) using data from a voxel in the left posterior
temporal region of a subject obtained during the presentation of single words
at different rates. The solid line
represents the estimated response and shows a clear maximum at just less than
one second. The dots are responses based
on empirical data from the same experiment.
The broken line shows the expected response in the absence of nonlinear
effects (*i.e.* that predicted by
setting the second order kernel to zero).
It is clear that nonlinearities become important at around two seconds
leading to an actual diminution of the integrated response at sub-second
SOAs. The implication of this sort of
result is that (i) SOAs should not really fall much below one second and (ii)
at short SOAs the assumptions of linearity are violated. It should be noted that these data pertain to
single word processing in auditory association cortex. More linear behaviors may be expressed in
primary sensory cortex where the feasibility of using minimum SOAs as low as
500ms has been demonstrated (Burock *et al*
1998). This lower bound on SOA is
important because some effects are detected more efficiently with high
presentation rates. We now consider this
from the point of view of event-related designs.

A
crucial distinction, in experimental design for fMRI, is that between *epoch *and* event*-related designs. In
SPECT and PET only epoch-related responses can be assessed because of the
relatively long half-life of the radiotracers used. However, in fMRI there is
an opportunity to measure event-related responses that may be important in some
cognitive and clinical contexts. An important
issue, in event-related fMRI, is the choice of inter-stimulus interval or more
precisely SOA. The SOA, or the
distribution of SOAs, is a critical factor in and is chosen, subject to
psychological or psychophysical constraints, to maximize the efficiency of
response estimation. The constraints on
the SOA clearly depend upon the nature of the experiment but are generally
satisfied when the SOA is small and derives from a random distribution. Rapid presentation rates allow for the
maintenance of a particular cognitive or attentional set, decrease the latitude
that the subject has for engaging alternative strategies, or incidental
processing, and allows the integration of event-related paradigms using fMRI
and electrophysiology. Random SOAs
ensure that preparatory or anticipatory factors do not confound event-related
responses and ensure a uniform context in which events are presented. These constraints speak to the
well-documented advantages of event-related fMRI over conventional blocked
designs (Buckner *et al* 1996, Clark *et al* 1998).

In
order to compare the efficiency of different designs it is useful to have some
common framework that encompassed all of them.
The efficiency can then be examined in relation to the parameters of the
designs. Designs can be *stochastic* or
*deterministic* depending on whether
there is a random element to their specification. In stochastic designs (Heid *et al* 1997) one needs to specify the
probabilities of an event occurring at all times those events could occur. In deterministic designs the occurrence probability
is unity and the design is completely specified by the times of stimulus
presentation or trials. The distinction
between stochastic and deterministic designs pertains to how a particular
realization or stimulus sequence is created.
The efficiency afforded by a particular event sequence is a function of
the event sequence itself, and not of the process generating the sequence (*i.e.* deterministic or stochastic).
However, within stochastic designs, the design matrix *X*, and associated efficiency, are random variables and the *expected* or average efficiency, over
realizations of *X* is easily computed.

In the framework considered here (Friston *et al* 1999a) the occurrence probability *p* of any event* *occurring is specified at each time that it could occur (*i.e.* every SOA). Here *p*
is a vector with an element for every SOA.
This formulation engenders the distinction between *stationary* stochastic designs, where the occurrence probabilities
are constant and *non-stationary *stochastic
designs, where they change over time.
For deterministic designs the elements of *p* are 0 or 1, the presence of a 1 denoting the occurrence of an
event. An example of *p* might be the boxcars used in
conventional block designs. Stochastic
designs correspond to a vector of identical values and are therefore stationary
in nature. Stochastic designs with
temporal modulation of occurrence probability have time-dependent probabilities
varying between 0 and 1. With these
probabilities the expected design matrices and expected efficiencies can be
computed. A useful thing about this
formulation is that by setting the mean of the probabilities *p* to a constant, one can compare
different deterministic and stochastic designs given the same number of
events. Some common examples are given
in Figure 12.htm (right panel) for an SOA of 1
second and 32 expected events or trials over a 64 second period (except for the
first deterministic example with 4 events and an SOA of 16 seconds). It can be seen that the least efficient is
the sparse deterministic design (despite the fact that the SOA is roughly
optimal for this class), whereas the most efficient is a block design. A slow modulation of occurrence probabilities
gives high efficiency whilst retaining the advantages of stochastic designs and
may represent a useful compromise between the high efficiency of block designs
and the psychological benefits and latitude afforded by stochastic
designs. However, it is important not to
generalize these conclusions too far. An
efficient design for one effect may not be the optimum for another, even within
the same experiment. This can be
illustrated by comparing the efficiency with which evoked responses are
detected and the efficiency of detecting the difference in evoked responses
elicited by two sorts of trials:

Consider a stationary stochastic design with
two trial types. Because the design is
stationary the vector of occurrence probabilities, for each trial type, is
specified by a single probability. Let
us assume that the two trial types occur with the same probability **p**. .By varying **p**^{ }and SOA one can find the most efficient design
depending upon whether one is looking for evoked responses *per se* or differences among evoked responses. These two situations are depicted in the left
panels of Figure 12.htm. It is immediately apparent that, for both
sorts of effects, very small SOAs are optimal.
However, the optimal occurrence probabilities are not the same. More infrequent events (corresponding to a
smaller** p** = 1/3) are required to
estimate the responses themselves efficiently.
This is equivalent to treating the baseline or control condition as any
other condition (*i.e.* by including
null events, with equal probability, as further event types). Conversely, if we are only interested in
making inferences about the differences, one of the events plays the role of a
null event and the most efficient design ensues when one or the other event
occurs (*i.e.* **p** = 1/2). In short, the most
efficient designs obtain when the events subtending the differences of interest
occur with equal probability.

Another example, of how the efficiency is
sensitive to the effect of interest, is apparent when we consider different
parameterizations of the HRF. This issue
is sometimes addressed through distinguishing between the efficiency of
response *detection* and response *estimation*. However, the principles are identical and the
distinction reduces to how many parameters one uses to model the HRF for each
trail type (one basis function is used for detection and a number are required
to estimate the shape of the HRF). Here
the contrasts may be the same but the shape of the regressors will change
depending on the temporal basis set employed.
The conclusions above were based on a single canonical HRF. Had we used a more refined parameterization
of the HRF, say using three-basis functions, the most efficient design to
estimate one basis function coefficient would not be the most efficient for
another. This is most easily seen from
the signal processing perspective where basis functions with high frequency
structure (*e.g.* temporal derivatives)
require the experimental variance to contain high frequency components. For these basis functions a randomized
stochastic design may be more efficient than a deterministic block design,
simply because the former embodies higher frequencies. In the limiting case of FIR estimation the
regressors become a series of stick functions (see Figure
6.htm) all of which have high frequencies. This parameterization of the HRF
calls for high frequencies in the experimental variance. However, the use of FIR models is
contraindicated by model selection procedures that suggest only two or three
HRF parameters can be estimated with any efficiency. Results that are reported in terms of FIRs
should be treated with caution because the inferences about evoked responses
are seldom based on the FIR parameter estimates. This is precisely because they are estimated
inefficiently and contain little useful information.

In this section we consider some issues
that are generic to brain mapping studies that have repeated measures or
replications over subjects. The critical
issue is whether we want to make an inference about the effect in relation to
the *within-subject variability* or
with respect to the *between subject
variability*. For a given group of
subjects, there is a fundamental distinction between saying that the response
is significant relative to the precision[4]
with which that response in measured and saying that it is significant in
relation to the inter-subject variability.
This distinction relates directly to the difference between *fixed* and *random*-effect analyses. The
following example tries to make this clear.
Consider what would happen if we scanned six subjects during the
performance of a task and baseline. We
then constructed a statistical model, where task-specific effects were modelled
separately for each subject. Unknown to
us, only one of the subjects activated a particular brain region. When we examine the contrast of parameter
estimates, assessing the mean activation over all the subjects, we see that it
is greater than zero by virtue of this subject's activation. Furthermore, because that model fits the data
extremely well (modelling no activation in five subjects and a substantial
activation in the sixth) the error variance, on a scan to scan basis, is small
and the T statistic is very significant.
Can we then say that the group shows an activation? On the one hand, we can say, quite properly,
that the mean group response embodies an activation but clearly this does not
constitute an inference that the group's response is significant (*i.e.* that this sample of subjects shows
a consistent activation). The problem
here is that we are using the *scan to
scan* error variance and this is not necessarily appropriate for an
inference about group responses. In
order to make the inference that the group showed a significant activation one
would have to assess the variability in activation effects from *subject to subject* (using the contrast
of parameter estimates for each subject).
This variability now constitutes the proper error variance. In this example the variance of these six
measurements would be large relative to their mean and the corresponding T
statistic would not be significant.

The distinction, between the two approaches
above, relates to how one computes the appropriate error variance. The first represents a fixed-effect analysis
and the second a random-effect analysis (or more exactly a mixed-effects
analysis). In the former the error
variance is estimated on a scan to scan basis, assuming that each scan
represents an independent observation (ignoring serial correlations). Here the degrees of freedom are essentially
the number of scans (minus the rank of the design matrix). Conversely, in random-effect analyses, the
appropriate error variance is based on the activation from subject to subject
where the effect *per se* constitutes
an independent observation and the degrees of freedom fall dramatically to the
number of subjects. The term
random-effect indicates that we have accommodated the randomness of
differential responses by comparing the mean activation to the variability in
activations from subject to subject.
Both analyses are perfectly valid but only in relation to the inferences
that are being made: Inferences based on fixed-effects analyses are about the
particular subject[s] studied.
Random-effects analyses are usually more conservative but allow the
inference to be generalized to the population from which the subjects were
selected.

The
implementation of random-effect analyses in SPM is fairly straightforward and
involves taking the contrasts of parameters estimated from a *first-level* (fixed-effect) analysis and
entering them into a *second-level*
(random-effect) analysis. This ensures
that there is only one observation (*i.e. *contrast)
per subject in the second-level analysis and that the error variance is
computed using the subject to subject variability of estimates from the first
level. The nature of the inference made
is determined by the contrasts entered into the second level (see Figure 13.htm).
The second-level design matrix simply tests the null hypothesis that the
contrasts are zero (and is usually a column of ones, implementing a single
sample T test).

The reason this multistage procedure
emulates a full mixed-effects analyses, using a hierarchical observation model,
rests upon the fact that the design matrices for each subject are the same (or
sufficiently similar). In this special
case the estimator of the variance at the second level contains the right
mixture of variance induced by observation error at the first level and
between-subject error at the second. It
is important to appreciate this because the efficiency of the design at the
first level percolates to higher levels.
It is therefore important to use efficient strategies at all levels in a
hierarchical design.

In
some instances a fixed effects analysis is more appropriate, particularly to
facilitate the reporting of a series of single-case studies. Among these single cases it is natural to ask
what are common features of functional anatomy (*e.g.* the location of V5) and what aspects are subjectspecific (*e.g.* the location of ocular dominance
columns). One way to address
commonalties is to use a conjunction analysis over subjects. It is important to understand the nature of
the inference provided by conjunction analyses of this sort. Imagine that in 16 subjects the activation in
V5, elicited by a motion stimulus, was great than zero. The probability of this occurring by chance, in
the same area, is extremely small and is the *p*-value returned by a conjunction analysis using a threshold of *p* = 0.5 (T = 0) for each subject. This
result constitutes evidence that V5 is involved in motion processing. However, note that this is not an assertion
that each subject activated significantly (we only require the T value to be
greater than zero for each subject). In
other words, a significant conjunction of activations is not synonymous with a
conjunction of significant activations.

The motivations for conjunction analyses, in
the context of multi-subject studies are twofold. (i) They provide an
inference, in a fixed-effect analysis testing the null hypotheses of no
activation in any of the subjects, that can be much more sensitive than testing
for the average activation. (ii) They
can be extended to make inferences about the population as described next.

If, for any given contrast, one can establish
a conjunction of effects over *n*
subjects using a test with a specificity of *a*
and sensitivity *b*, the
probability of this occurring by chance can be expressed as a function of g. g is the
proportion of the population that would have activated (see the equation in Figure 13.htm - lower right panel). This probability has an upper bound *a** _{c}*
corresponding to a critical proportion

In practice, a conjunction analysis of a
multi-subject study comprises the following steps: (i) A design matrix is constructed where the
explanatory variables pertaining to each experimental condition are replicated
for each subject. This subject-separable
design matrix implicitly models subject by condition interactions (*i.e. *different condition-specific
responses among sessions). (ii)
Contrasts are then specified that test for the effect of interest in each
subjects to give a series of SPM*u *(corresponding
to the specificity *a*
in Figure 12.htm) to give a SPM* _{min}*}

Imaging
neuroscience has firmly established functional specialization as a principle of
brain organization in man. The
integration of specialized areas has proven more difficult to assess. Functional integration is usually inferred on
the basis of correlations among measurements of neuronal activity. Functional connectivity has been defined as
statistical dependencies or correlations*
among remote neurophysiological events*.
However correlations can arise in a variety of ways: For example in
multi-unit electrode recordings they can result from stimulus-locked transients
evoked by a common input or reflect stimulus-induced oscillations mediated by
synaptic connections (Gerstein and Perkel 1969). Integration within a distributed system is
usually better understood in terms of effective connectivity: Effective
connectivity refers explicitly to *the
influence that one neural system exerts over another*, either at a synaptic
(*i.e.* synaptic efficacy) or
population level. It has been proposed
that "the [electrophysiological] notion of effective connectivity should
be understood as the experiment- and time-dependent, simplest possible circuit
diagram that would replicate the observed timing relationships between the
recorded neurons" (Aertsen and Preil 1991). This speaks to two important points: (i) Effective connectivity is dynamic, *i.e.* activity- and time-dependent and
(ii) it depends upon a model of the interactions. The estimation procedures employed in
functional neuroimaging can be classified as (i) those based on linear
regression models (McIntosh *et al*
1994, Friston* et al *1995d) or (ii)
those based on nonlinear dynamic models.

There is a necessary relationship between
approaches to characterizing functional integration and multivariate analyses
because the latter are necessary to model interactions among brain
regions. Multivariate approaches can be
divided into those that are inferential in nature and those that are data led
or exploratory. We will first consider multivariate approaches that are universally
based on functional connectivity or covariance patterns (and are generally
exploratory) and then turn to models of effective connectivity (that usually
allow for some form of inference).

Most
analyses of covariances among brain regions are based on the singular value
decomposition (SVD) of between-voxel covariances in a neuroimaging
time-series. In Friston *et al* (1993) we introduced voxel-based
principal component analysis (PCA) of neuroimaging time-series to characterize
distributed brain systems implicated in sensorimotor, perceptual or cognitive
processes. These distributed systems are
identified with principal components or *eigenimages*
that correspond to spatial modes of coherent brain activity. This approach represents one of the simplest
multivariate characterizations of functional neuroimaging time-series and falls
into the class of exploratory analyses.
Principal component or eigenimage analysis generally uses SVD to
identify a set of orthogonal spatial modes that capture the greatest amount of
variance expressed over time. As such
the ensuing modes embody the most prominent aspects of the variance-covariance
structure of a given time-series. Noting
that covariance among brain regions is equivalent to functional connectivity
renders eigenimage analysis particularly interesting because it was among the
first ways of addressing functional integration (*i.e.* connectivity) with neuroimaging data. Subsequently, eigenimage analysis has been
elaborated in a number of ways. Notable
among these is canonical variate analysis (CVA) and multidimensional scaling^{
}(Friston et al 1996d,e). Canonical
variate analysis was introduced in the context of ManCova (multiple analysis of
covariance) and uses the generalized eigenvector solution to maximize the
variance that can be explained by some explanatory variables relative to
error. CVA can be thought of as an
extension of eigenimage analysis that refers explicitly to some explanatory
variables and allows for statistical inference.

In fMRI, eigenimage analysis (*e.g.* Sychra *et al* 1994) is generally used as an exploratory device to
characterize coherent brain activity.
These variance components may, or may not be, related to experimental
design and endogenous coherent dynamics have been observed in the motor system
(Biswal *et al* 1995). Despite its exploratory power eigenimage
analysis is fundamentally limited for two reasons. Firstly, it offers only a linear decomposition
of any set of neurophysiological measurements and secondly the particular set
of eigenimages or spatial modes obtained is uniquely determined by constraints
that are biologically implausible. These
aspects of PCA confer inherent limitations on the interpretability and
usefulness of eigenimage analysis of biological time-series and have motivated
the exploration of nonlinear PCA and neural network approaches (*e.g.* Mrch et al 1995).

Two other important approaches deserve
mention here. The first is independent
component analysis (*independent*. This is a stronger requirement than *orthogonality* in PCA and involves
removing high order correlations among the modes (or dynamics). It was initially introduced as *spatial* *et al* 1998) in which the independence
constraint was applied to the modes (with no constraints on their temporal
expression). More recent approaches use,
by analogy with magneto- and electrophysiological time-series analysis, *temporal*

Linear
models of effective connectivity assume that the multiple inputs to a brain
region are linearly separable. This
assumption precludes activity-dependent connections that are expressed in one
context and not in another. The resolution
of this problem lies in adopting models that include interactions among inputs.
These interactions or bilinear effects can be construed as a context- or
activity-dependent modulation of the influence that one region exerts over
another, where that context is instantiated by activity in further brain
regions exerting modulatory effects.
These nonlinearities can be introduced into structural equation modeling
using so-called 'moderator' variables that represent the interaction between
two regions in causing activity in a third^{ }(Bchel *et al* 1997). From the point of view of regression models
modulatory effects can be modeled with nonlinear input-output models and in
particular the Volterra formulation described above. In this instance, the inputs are not stimuli
but activities from other regions.
Because the kernels are high-order they embody interactions over time
and among inputs and can be thought of as explicit measures of effective
connectivity (see Figure 14.htm). An important thing about the Volterra
formulation is that it has a high face validity and biological
plausibility. The only thing it assumes
is that the response of a region is some analytic nonlinear function of the
inputs over the recent past. This function
exists even for complicated dynamical systems with many [unobservable] state
variables. Within these models, the
influence of one region on another has two components; (i) the direct or *driving* influence of input from the first (*e.g.* hierarchically lower) region, irrespective of the activities
elsewhere and (ii) an activity-dependent, *modulatory*
component that represents an interaction with inputs from the remaining (*e.g.* hierarchically higher)
regions. These are mediated by the first
and second order kernels respectively. The
example provided in Figure 15.htm addresses the
modulation of visual cortical responses by attentional mechanisms (*e.g.* Treue and Maunsell 1996) and the
mediating role of activity-dependent changes in effective connectivity. The
right panel in Figure 15.htm shows a
characterization of this modulatory effect in terms of the increase in V5
responses, to a simulated V2 input, when posterior parietal activity is zero
(broken line) and when it is high (solid lines). The estimation of the Volterra
kernels and statistical inference procedure is described in Friston and
Bchel (2000c).

The key thing about this example is that the
most interesting thing is the change in effective connectivity from V2 to
V5. Context-sensitive changes in effective
connectivity transpire to be the most important aspect of functional
integration and have two fundamental implications for experimental design and
analysis. First, experimental designs
for analyses of effective connectivity are generally multifactorial. This is because one factor is needed to evoke
responses and render the coupling among brain areas measurable and a second
factor is required to induce changes in that coupling. The second implication is that models of
effective connectivity should embrace changes in coupling. These changes are usually modeled with *bilinear* terms or interactions. Bilinear terms appear in the simplest models
of effective connectivity (*e.g.*
psychophysiological interactions) through to nonlinear dynamic causal models.

In
this chapter we have reviewed the main components of image analysis and have
touched briefly on ways of assessing functional integration in the brain. The key principles of functional specialization
and integration were used to motivate the various approaches considered. In the remaining chapters of this book we
will revisit these procedures and disclose the details that underpin each
component.

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[1] 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 iso-contours, hence *sphericity*. Deviations from
either of the i.i.d. criteria constitute non-sphericity. 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
non-sphericity. In Neuroimaging both
spatial and temporal non-sphericity can be quite profound issues.

[2]The efficiency scales with the number of voxels

[3]
More exactly, the GLS and ML estimators are the same if *X* lies
within the space spanned by the eigenvectors of Toeplitz autocorrelation
matrix

[4] Precision is the inverse of the variance.