A statistical framework for optimal design matrix generation with application to fMRI

The general linear model (GLM) is a well established tool for analyzing functional magnetic resonance imaging (fMRI) data. Most fMRI analyses via GLM proceed in a massively univariate fashion where the same design matrix is used for analyzing data from each voxel. A major limitation of this approach is the locally varying nature of signals of interest as well as associated confounds. This local variability results in a potentially large bias and uncontrolled increase in variance for the contrast of interest. The main contributions of this paper are two fold: 1) we develop a statistical framework that enables estimation of an optimal design matrix while explicitly controlling the bias variance decomposition over a set of potential design matrices and 2) we develop and validate a numerical algorithm for computing optimal design matrices for general fMRI data sets. The implications of this framework include the ability to match optimally the magnitude of underlying signals to their true magnitudes while also matching the "null" signals to zero size thereby optimizing both the sensitivity and specificity of signal detection. By enabling the capture of multiple profiles of interest using a single contrast (as opposed to an F-test) in a way that optimizes for both bias and variance enables the passing of first level parameter estimates and their variances to the higher level for group analysis which is not possible using F-tests. We demonstrate the application of this approach to in vivo pharmacological fMRI data capturing the acute response to a drug infusion, to task-evoked, block design fMRI and to the estimation of a haemodynamic response function (HRF) in event-related fMRI. Although developed with motivation from fMRI, our framework is quite general and has potentially wide applicability to a variety of disciplines.