Classical and Bayesian Inference in fMRI - Advance Image Processing in Magnetic Resonance Imaging - page 535

design matrix corresponds to some effect built into the experiment or that may
confound the results. These are referred to as explanatory variables, covariates,
or regressors.
The design matrix can contain both covariates and indicator variables. Each
column of

has an associated unknown 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 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).

X

The example in Figure 17.1 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 or Bayesian estimation. Classical 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. Baye-
sian inferences are based on the posterior or conditional probability that the
contribution exceeded some threshold, usually zero.
Due primarily to the presence of aliased biorhythms and unmodeled neuronal
activity, the errors in the GLM will be temporally autocorrelated. To accommodate
this, the GLM has been extended (12) to incorporate intrinsic nonsphericity, or
correlations among the error terms. 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.
More recently, a restricted maximum likelihood (ReML) algorithm for estimation
of the autocorrelation, variance components, and regression parameters has been
proposed (13).

17.3.2 C

ONTRASTS

To assess effects of interest that are spanned by one or more columns in the
design matrix one uses a contrast (i.e., a linear combination of parameter esti-
mates). An example of a contrast weight vector would be [ ] 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 polyno-

− 1 1 0 0 …

mial (14) or basis function expansions (see Subsection 17.3.1 ) of some experi-

mental factor. In these instances, a matrix of contrast weights is used that can be