View source for Statistical parametric mapping (SPM)

== Statistical parametric mapping ==

Brain <nowiki>mapping</nowiki> studies are usually analyzed with some form of statistical parametric <nowiki>mapping</nowiki>. This entails the construction of continuous statistical processes to test hypotheses about regionally specific effects (Friston et al. 1991). Statistical Parametric Maps (SPM) are images or fields with 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 (''i.e.'', image volume element) using any standard (univariate) statistical test, usually based on a General Linear Model (GLM) of the data. The resulting statistics are assembled into an image - the SPM. SPMs are interpreted as continuous statistical processes by referring to the probabilistic behaviour of random fields (Adler 1981, Worsley et al. 1992, Friston et al. 1994, Worsley et al. 1996). Random fields model both the univariate probabilistic characteristics of an SPM and any non-stationary spatial covariance structure. 'Unlikely' topological features of the SPM, like peaks or clusters, are interpreted as regionally specific effects, attributable to the experimental manipulation. In short, the GLM is used to explain continuous (image) data in exactly the same way as in conventional analyses of discrete data. Random Field Theory (RFT) is used to resolve the multiple-comparison problem when making inferences over the volume analysed. RFT provides a method for adjusting ''p''-values for the search volume and plays the same role for SPMs as the Bonferroni correction for discrete statistical tests.

=== The general linear model ===

[[Image:analysis.jpg|thumb|600px|right|F1|This schematic depicts the transformations that start with an imaging data sequence and end with a statistical parametric map (SPM). An SPM can be regarded as an 'X-ray' of the significance of regional effects. Voxel-based analyses require the data to be in the same anatomical space: this is effected by realigning the data. After realignment, the images are subject to non-linear warping so that they match a spatial model or template that already conforms to a standard anatomical space. After smoothing, the general linear model is employed to estimate the parameters of a temporal model (encoded by a design matrix) and derive the appropriate univariate test statistic at every voxel. The test statistics (usually ''t'' or ''F''-statistics) constitute the SPM.  The final stage is to make statistical inferences on the basis of the SPM and Random Field Theory and characterize the responses observed using the fitted responses or parameter estimates.]]

Statistical analysis of imaging data corresponds to inverting generative models of data. Inferences are then pursued using statistics that assess the significance of interesting effects. A brief review of the literature may give the impression that there are numerous ways to analyze neuroimaging time-series (''e.g.'', from Positron emission tomography (PET), functional magnetic resonance imaging (fMRI) and electroencephalography (EEG)). This is not the case; with very few exceptions, every analysis is a variant of the general linear model. This includes: simple ''t''-tests on scans assigned to one condition or another, correlation coefficients between observed responses and boxcar stimulus functions in fMRI, inferences made using multiple linear regression and evoked responses estimated using linear time-invariant models. Mathematically, they are all formally identical and can be implemented with the same equations and algorithms. The only thing that distinguishes among them is the design matrix encoding the temporal model or experimental design. 

The general linear model is an equation <math>Y=X\beta+\epsilon</math> that expresses the observed response variable in terms of a linear combination of explanatory variables <math>X</math> plus a well behaved error term (see <figref>Analysis.jpg</figref> and Friston et al. 1995). 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 that contains the explanatory variables (e.g. designed effects or confounds) is called the ''design matrix''. Each column of the design matrix corresponds to an 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 <figref>Analysis.jpg</figref> relates to an fMRI study of visual stimulation, under four conditions. The effects on the response variable are modelled 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 artefacts 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 maximum likelihood 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.  

=== Testing for contrasts ===
The GLM can be used to implement a vast range of statistical analyses. The issue is therefore not the mathematics but the formulation of a design matrix appropriate to the study design and inferences sought. The design matrix can contain both covariates and indicator variables. Each column has an associated unknown or free parameter. Some of these parameters will be of interest (''e.g.'' the effect of a particular sensorimotor or cognitive condition or the regression coefficient of hemodynamic responses on reaction time). The remaining parameters will 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 or that some contrast or linear mixture (''e.g.'' a subtraction) of the estimates is zero using an SPM{''t''}. An example of a contrast weight vector would be <math>\begin{bmatrix}-1 & 1 & 0 & \ldots\end{bmatrix}</math> to compare the difference in responses evoked by two conditions, modelled by the first two regressors in the design matrix. Sometimes several contrasts of parameter estimates are jointly interesting. For example, when using polynomial (Büchel et al. 1996) or basis function expansions of some experimental factor. In these instances, the SPM{''F''} is used and is specified with a matrix of contrast weights that can be thought of as a collection of ‘''t''-contrasts’. An F-contrast may look like:

:<math>
\begin{bmatrix}
-1 & 0 & 0 & 0 & \cdots\\
0  & 1 & 0 & 0 & \cdots
\end{bmatrix}
</math>

This would test for the significance of the first or second parameter estimates.  The fact that the first weight is negative has no effect on the test because the ''F''-statistic is based on sums of squares.

In most analyses the design matrix contains indicator variables or parametric variables encoding the experimental manipulations.  These are formally identical to classical analysis of covariance (i.e. ANCOVA) models. An important instance of the GLM, from the perspective of time-series data (''e.g.'', functional magnetic resonance imaging or fMRI), is the linear time-invariant model or convolution model.