The general linear model (GLM) is one of the most commonly utilized statistical platform that is currently used in analyzing task-based fMRI data. In this talk we will introduce the general over view and basic concepts of GLM and how it is used in this very specific application of clinical neuroimaging. We will briefly review the history of introduction of GLM into the fMRI community and later use some examples to demonstrate the utility in analyzing fMRI data. In the end we will discuss some of its limitations.
Target Audience: This lecture is intended for the clinical and basic science researchers, students in psychology, bioengineering and other disciplines interesting in analyzing the fMRI data and clinicians involved in task based fMRI data analysis. There is no prerequisites required other than basic knowledge of BOLD imaging.
Outcome and Objectives: At the end of this lecture the attendees should be able to understand and explain the rationale for using the GLM for task based fMRI analysis, linear regression and ANOVA, understanding of design matrix, and limitation of this model in fMRI analysis.
Purpose: In task-based fMRI, the BOLD response that is measured in subjects following the introduction or exposure of a task is considered an indirect measure of neuronal activity. The time course that we measure from a single voxel in the images obtained from these experiments (with and without task) forms the fundamental unit of this measurement. Accurate extraction of reliable information from these experiments poses some challenges. These challenges vary from subject motion, signal drift in the scanners, pulsatile motion of the vessels and thermal noise introduced in the scanner. In order to address these challenges and analyze data in a meaningful manner, GLM was introduced in the 1990’s and continued to be widely used by the fMRI community in the field. In GLM one estimates the BOLD effect size from our observed data and the hypothesized data and later assess the statistical significance of this observed effect using various statistical methods.
Methods: GLM is a model. In this model the term “General” means that we model all the effects that are involved in our data together for the all the voxels and time series present in the BOLD data. Specifically we model the effects of interest (our hypothesized BOLD activation) and confounds (e.g., age of the subject involved in the data collection) together and each effect becomes a regressor in the model. Hence the term general implies a multiple linear regression with more than one dependent variable. The GLM encompasses linear regression and ANOVA. The term “Linear” simply refers to the belief that our data (in this case the hypothesized model) is a superposition (additions or subtractions) of weighted known data (in this case our observed BOLD data). The BOLD signal scales proportionally with the modeled effect. The word model can have several definitions in different contexts but here in this context it means that we know something about how our data (task based BOLD activation) are related to other data or knowledge (such as the age of a subject). The model is the expression of the form of this belief with mathematical symbols. Mathematically, we can write the assumptions of the GLM as a matrix equation Y = Xβ+ε Where Y represents the voxel time series, a column vector (of N time points); X represents the design matrix, where regressors modeling effects and confounds (both of length N) are stacked into M different columns; β, refers to the parameter estimates, a column vector (M elements), one element (effect size) for each modeled regressor; and ε refers to the residual time series, i.e. any unmodeled fluctuation, a column vector (of N time points). The parameter estimation process finds the least-square solution to this matrix equation, i.e. the betas that minimize the squared error or residual (ε). The best fit or parameter estimate, i.e. closest point to the data one can reach within the plane, is the point where the residual is an orthogonal vector pointing from the plane towards our measured data point. This model along with statistical testing is what is commonly used in fMRI analysis.
Results: A design matrix created based on the model will be used to generate BOLD activations that accurately represent fMRI signal changes in the specific regions of interest in the brain.
Discussion and Conclusion: GLM is a powerful and widely model used model based tool for analyzing task-based BOLD fMRI data. However like any models that are used in the scientific world it has certain limitations. Multiple comparison problem, noise estimation assumptions are some of the limitations that needs attention when using the GLM models for fMRI analysis. It is very important for the users of these models in clinical and research applications of fMRI to understand these limitations and judiciously utilize these models using appropriate settings.
Below is collection of important references that proposed the idea of utilization of GLM in fMRI analysis and some references showing its limitations and utility.
Friston, K. J., Frith, C. D., Turner, R., & Frackowiak, R. S. Characterizing evoked hemodynamics with fMRI. NeuroImage, 2(2), 157–165, 1995.
Worsley, K. J., Poline, J. B., Vandal, A. C., & Friston, K. J. Tests for distributed, nonfocal brain activations. NeuroImage, 2(3), 183–194, 1995.
Friston, K.J., Holmes, A.P., Worsley, K.J., Poline, J.-P., Frith, C.D., Frackowiak, R.S.J. Statistical parametric maps in functional imaging: A general linear approach. Hum. Brain Mapp. 2, 189–210, 1994.
Friston, K. J., Holmes, A. P., Poline, J. B., Grasby, P. J., Williams, S. C., Frackowiak, R. S., & Turner, R. Analysis of fMRI time-series revisited. NeuroImage, 2(1), 45–53, 1995.
Friston, K.J., Penny, W., Phillips, C., Kiebel, S., Hinton, G., Ashburner, J. Classical and Bayesian Inference in Neuroimaging: Theory. NeuroImage 16, 465–483, 2002.
Friston, K. J., Frith, C. D., Frackowiak, R. S., & Turner, R. Characterizing dynamic brain responses with fMRI: a multivariate approach. NeuroImage, 2(2), 166–172, 1995.
Poline, J.-B., Brett, M. The general linear model and fMRI: Does love last forever? NeuroImage 62, 871–880, 2012.
Stephan, K.E., Iglesias, S., Heinzle, J., Diaconescu, A.O. Translational Perspectives for Computational Neuroimaging. Neuron 87, 716–732, 2015.
The General Linear Model, in: Karl Friston, John Ashburner, Stefan Kiebel, Thomas Nichols and William Penny - Karl Friston, J.A., William Penny (Eds.), Statistical Parametric Mapping. Academic Press, London, pp. 101–125, 2007.