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Covariates and the G matrix

The centerpiece of a general linear model analysis is the design matrix, G. It is a two-dimensional array, with each column corresponding to an independent variable in the analysis, and each row corresponding to one of the observations that make up the dependent data.

Independent variables (also called parameters or covariates) are vectors designed to explain some component of the variance present in the dependent data. The operation of the GLM assigns a parameter estimate (β) to each independent variable. The β value is, in effect, a measure of the unique variance within the dependent data that the independent variable is able to explain. It contributes to the numerator of the t-test. By scaling each independent variable by its parameter estimate, and summing the vectors, one obtains the fit to the data. The difference between the fit and the actual data is the residual error. The sum of the squares of these residuals contributes to the denominator of the t-test.

We may distinguish, generally, between two types of independent variables: those of interest and those of no interest. Independent variables of interest model effects in the data about which we wish to test hypotheses. An independent variable of interest might model the difference in the mean level of fMRI signal between two task conditions. We would use this independent variable to test the hypothesis that, for example, periods of flashing light evoke greater fMRI signal within visual cortex than do periods of darkness.

By contrast, an independent variable of no interest is one that models variance components about which we do not wish to test hypotheses. If independent variables of no interest are able to parsimoniously explain variance within the data, then their inclusion will improve the sensitivity and validity of the analysis by denying the nuisance variance to the error term. Despite this conceptual distinction, it is important to note that the GLM is blind to the designation of a covariate as being of interest or not—parameter estimates are obtained in the same manner for both. The distinction is one that exists only in interpretation, and one scientist's nuisance variance may be another's Nature paper. Finally, it should be noted that independent variables of no interest can be classified themselves as nuisance variables and as confounds. In short, a confound is an independent variable of no interest that is correlated with an independent variable of interest. Inclusion of confounds (of which a global signal covariate is an example; Aguirre et al., 1998c) can alter the interpretation of data.

Another axis along which independent variables may be divided is that of categorical vs. ordinal. Many neuroimaging experiments test for differences in fMRI signal between different categorical conditions. In condition-based designs, periods of time in an experiment are assigned to one discrete, non-ordinal state or another. This can be contrasted to parametric designs in which the experimental state of interest adopts an ordinal value along a continuous range. Often, fMRI experiments are composed of multiple, categorical conditions, with hypotheses asked regarding comparisons between these conditions (e.g., different stimulus categories such as faces, objects, houses, etc.). The Condition Function file is used to code the experimental state of each observation in an analysis, and routines are present within the G Design Workshop to make it a simple matter to create covariates that are contrasts between these different conditions.

Some covariates of no interest

Intercept - An independent variable where all values are unity. This independent variable is needed in all analysis designs that are reference coded.

Trial effects - A set of independent variables that model the mean level of the signal during each “trial” of a sparse event-related design. The inclusion of these independent variables of no interest ensure that comparisons between different trial types are not confounded by poorly modeled low-frequency fluctuations in signal. Note that the number of covariates produced is one fewer than the total number of trials. This is to prevent the set of covariates from being perfectly colinear with the intercept term.

Scan effects - A set of independent variables that model the mean level of signal during a given “scan”. These only need to be added when data from more than one scan has been concatenated for analysis. Note that scan effect covariates should not be included in a model that also includes trial effect covariates, as the two sets will be perfectly colinear.

Global signals - A set of independent variables derived from the .GS file calculated for each TES. The appropriate use of global signal covariates in the analysis of fMRI data is the subject of some debate. In particular, when there is some degree of correlation between the global signal covariate and the experimental task, this independent variable acts as a confound and may be an undesirable component of the analysis (Aguirre et al., 1998c).

Movement params - Adds a set of independent variables derived from the .MoveParams file calculated for each TES. There will be a total of six covariates added for each TES file, corresponding to X,Y, and Z translation, and pitch, roll and yaw rotation. The inclusion of these covariates is the subject of some debate, as these movement parameters will reflect the time course of task related activation, even in the absence of task correlated movement. As a result, these parameters can be colinear with task parameters of interest.

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