Chapter Two Statistic

In our discussion of linear models relating rCBF to design factors and gCBF, we have not considered the computational problems of simultaneously fitting the model for thousands of voxels.32 Viewing the problem from a multivariate perspective provides efficient computation, and offers some insight into the problem. In this section we demonstrate the relationship between the simultaneous general linear models for PET data (which we term image regression), and multivariate regression. Some basic multivariate regression theory is reviewed. Under an assumption of multivariate normality for the rCBF images the PET scenario is a multivariate regression, but the high dimensionality of the data precludes any multivariate analysis. It is for this reason that we concentrate on simultaneous univariate tests. Whilst there is nothing new in this section for the statistical reader, the advantages of the multivariate perspective are only just dawning on the PET community, and this section is included for the benefit of the reader in the latter category. 2.5.1. Sums of squares approaches We have a single model to be fitted at each voxel. In a typical data set there are 77000 voxels, making individual fitting on a voxel-by-voxel basis using statistical packages prohibitive. Working with the images as row vectors, and using matrix manipulation routines, the appropriate sums of squares can be computed for all voxels simultaneously. This is the approach taken in older versions of the SPM software (Friston et al., 1991a), using the exposition of ANCOVA provided by Wildt & Atholla (1978). This approach is rather inelegant, being rather slow and requiring a purpose written program for each possible design. Clearly direct fitting of general linear models for images is possible by viewing the problem from a multivariate perspective and using matrix methods. This point was noted by the author, passed on to Friston et al. (1994b), and is implemented in SPM94. 32A substantial (and unseen) part of the work undertaken during this Ph.D. has been the development of software for interactive analysis of PET image data. A number of the author’s routines are part of the SPM software. A Multivariate Perspective 77 2.5.2. Multivariate regression formulation Consider the general linear model for the data at voxel k: Yjk = xj1 β1k + ... + xjQ βQk + εjk where εjk ~ iid N(0,σk) (27) Where Yjk denotes the rCBF (rA) measurement at voxel k =1,...,K; of scan j =1,...,N; and let xjq q = 1,...,Q be a set of Q explanatory variables for scan j, either covariates (such as gCBF), dummy variables indicating levels of a factor, or a combination (for interactions or for covariates with effect dependent on the level of a factor). In matrix form the model is: