Exact multivariate tests for brain imaging data

In positron emission tomography (PET) and functional magnetic resonance imaging (fMRI) data sets, the number of variables is larger than the number of observations. This fact makes application of multivariate linear model analysis difficult, except if a reduction of the data matrix dimension is performed prior to the analysis. The reduced data set, however, will in general not be normally distributed and therefore, the usual multivariate tests will not be necessarily applicable. This problem has not been adequately discussed in the literature concerning multivariate linear analysis of brain imaging data. No theoretical foundation has been given to support that the null distributions of the tests are as claimed. Our study addresses this issue by introducing a method of constructing test statistics that follow the same distributions as when the data matrix is normally distributed. The method is based on the invariance of certain tests over a large class of distributions of the data matrix. This implies that the method is very general and can be applied for different reductions of the data matrix. As an illustration we apply a test statistic constructed by the method now presented to test a multivariate hypothesis on a PET data set. The test rejects the null hypothesis of no significant differences in measured brain activity between two conditions. The effect responsible for the rejection of the hypothesis is characterized using canonical variate analysis (CVA) and compared with the result obtained by using univariate regression analysis for each voxel and statistical inference based on size of activations. The results obtained from CVA and the univariate method are similar.