Testing for interactions in multivariate data

Factorial designs are a mainstay of the scientific paradigm, allowing the effects of multiple experimental factors and their interactions to be efficiently studied within a single experiment. In brain imaging, however, multivariate data analyses commonly proceed using multivariate decoding and we argue that the standard "difference of accuracies" test is not a true test of interactions. To remedy this situation we propose an encoding method based on a Bayesian multivariate linear model which is ideally suited to the factorial analysis of such data. We show how it can be used to test for multivariate main effects and interactions using data from EEG studies of reward learning and declarative memory. This approach additionally allows for null hypotheses to be accepted and allows one to infer whether multivariate effects are driven by collections of univariate effects or voxel dependencies. We also propose that those wishing to pursue tests for interactions using decoding methods use an "accuracy of differences" test.