Medical imaging data with thousands of spatially correlated data points are common in many fields. Methods that account for spatial correlation often require cumbersome matrix evaluations which are prohibitive for data of this size, and thus current work has either used low‐rank approximations or analyzed data in blocks. We propose a method that accounts for nonstationarity, functional connectivity of distant regions of interest, and local signals, and can be applied to large multi‐subject datasets using spectral methods combined with Markov Chain Monte Carlo sampling. We illustrate using simulated data that properly accounting for spatial dependence improves precision of estimates and yields valid statistical inference. We apply the new approach to study associations between cortical thickness and Alzheimer's disease, and find several regions of the cortex where patients with Alzheimer's disease are thinner on average than healthy controls.