Correlation matrices provide a useful way to characterize variable dependencies in many real-world problems. Often, a perturbation in few variables can lead to small differences in multiple correlation coefficients related to these variables. In this paper we propose a low-dimensional representation of these differences as a product of single-variable perturbations that can efficiently characterize such effects; We develop methods for point estimation, confidence intervals and hypothesis tests for this model. Importantly, our methods are tailored for comparing samples of correlation matrices, in that they account for both the inherent variability in correlation matrices and for the variation between matrices in each sample. In simulations, our model shows a substantial increase in power compared to mass univariate approaches.
As a test case, we analyze correlation matrices of resting state functional-MRI (RS-fMRI) in patients with a rare neurological condition - transient global amnesia (TGA) and healthy controls. TGA is characterized by a lesion to a specific brain area and the connectivity matrices supposedly represent changes in only few variables, as in the assumption of our model. In this dataset, our model identifies substantially decreased synchronization in several brain regions within the patient population, which could not be detected using previous methods without prior-knowledge. Our framework shows the advantage of adding informed mean-structure for detecting differences in high-dimensional correlation matrices, and can be adapted for new differential structures. Our methods are available in the open-source package corrpops.
