Introduction to General Linear Models

In this chapter, we introduce general linear models (GLM) that have been widely used in brain imaging applications. The GLM is a very flexible and general statistical framework encompassing a wide variety of fixed effect models such as the multiple regressions, the analysis of variance (ANOVA), the multivariate analysis of variance (MANOVA), the analysis of covariance (ANCOVA) and the multivariate analysis of covariance (MANCOVA) [5]. Note that the term linear is misleading in a sense that the model can also include mathematically nonlinear model terms such as the higher degree polynomials. This chapter is mainly based on [2]. The GLM provides a framework for testing various associations and hypotheses while accounting for nuisance covariates in the model in a straightforward fashion. The effect of age, sex, brain size and possibly IQ can have severe confounding effects on the final outcome of many anatomical and functional imaging studies. Older population’s reduced functional activation could be the consequence of age-related atrophy of neural systems [3]. Brain volumes are significantly larger for children with autism 12 years old and younger compared with normally developing children [1]. Therefore, it is desirable to account for various confounding factors such as age and sex in the model. This can be done using GLM. The parameters of the GLM are mainly estimated by the least squares estimation and has been implemented in many statistical packages such as R (www.r-project.org) or Splus [4] and brain imaging packages such as SPM (www.fil.ion.ucl.ac.uk/spm) and fMRISTAT (www.math.mcgill.ca/keith/fmristat).