Fundamentals of Data Analysis Methods in Functional MRI - Advance Image Processing in Magnetic Resonance Imaging - page 477

activation maps are produced by using univariate parametric or nonparametric
statistical methods. The commonly used parametrical z - or t -test maps are formed
by computing on a voxel-by-voxel basis the value of statistical significance for
the difference of the means between two conditions. Voxels with a significance
value below a given threshold (e.g., P

< 0.001) are considered activated by the

task (see e.g., Reference 22 ). In so-called correlation maps, each voxel is associated

with the value of the linear cross-correlation coefficient ( r ) between the time course
of the voxel intensity and a reference function [19]. Separation of activated and
nonactivated voxels is achieved by imposing a threshold value for r . Correlation
maps measure the similarity between the shape of the gray-level time course of a
voxel and the expected hemodynamic response; thus, the sensitivity and specificity
of this method strongly depends on the resemblance between the reference function
and the “real” shape of the BOLD response. A simple on-off response [19], and
the convolution of this ideal function with the impulse response of a linear model
of the hemodynamics [18,23], are commonly used.
It can be shown that the t -test, correlation, and most other parametric tests
can be regarded as special cases of the general linear model (GLM). The GLM
is a standard statistical tool, which was introduced to imaging data analysis by
Friston and coworkers [18,24]. The method allows the analysis of factorial designs
that are expressed in a “design matrix” containing the description of all factors
of interest as well as confounders (e.g., a linear trend) of an experiment
[18,24-26]. The GLM is the most commonly used method for the voxel-by-voxel
statistical analysis of the functional time series and is thus reviewed in more detail
in the following subsection.

15.3.1 T HE GLM

The GLM “explains” or “predicts” the variation of the observed time courses in
terms of a linear combination of several regressor variables (or predictors) plus
an error term

y t = X t1 β 1 ++ X tl β l ++ X tL β L + e t. (15.8)

number of measurements) is the observed
signal time course at a given voxel, X tl (l

In Equation 15.8, y t (t = 1,…, T ; T =

T ) are a set of L
“explanatory” variables or “predictors” (functions of measurements),

= 1,…, L ; L <

β l 's are the
unknown weights (or regressor values)—one for each predictor, and the e t 's
denote error terms that are assumed to be independent and identically normally
distributed with zero mean and variance

2 .
Writing Equation 15.8 for each observation t gives the equation system:

σ

e 1
……………………………………………..
y t

y 1 = X 11 β 1 ++ X 1l β l ++ X 1L β L +

e t
……………………………………………..
y T

= X t1 β 1 ++ X tl β l ++ X tL β L +

= X T1 β 1 ++ X Tl β l ++ X TL β L + e T