Testing for Interactions in Multivariate Data

Abstract 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. Author summary Our understanding of human brain function has been transformed by non-invasive imaging methods such as functional Magnetic Resonance Imaging and Electroencephalography. Statistical modelling has been central to this endeavour with foundational work employing a univariate encoding method in which multiple characteristics of participants behaviour (the stimuli they are exposed to, the decisions they make, the events they remember) are used to predict brain activity at a single point or voxel element (pixel/voxel) in a brain recording, and this process repeats for all voxels in an image. Subsequent work has used multivariate decoding methods which identify categorical behavioural variables from multivariate brain imaging data. Here we propose a new type of multivariate encoding approach in which a Bayesian multivariate linear model is used to predict multivariate image data from multivariate behavioural variables. This approach has three advantages (i) it can correctly test for interactions among experimental factors, (ii) we can quantify the experimental evidence in support of no experimental effects being present and (iii) we can make inferences about the nature of the multivariate effects.