Two-Sample Testing for Multivariate Cross-Correlation Functions with Applications to Gut-Brain Reward Learning

Cross-correlation functions (CCFs) are classical tools for studying lead-lag relationships between paired time series, but they are most often used descriptively rather than inferentially. Motivated by mouse experiments on gut-brain interactions in reward learning, we carry out a two-sample hypothesis test for formal statistical inference on collections of subject-specific CCF curves. In our application, each experimental session yields two related CCFs describing the temporal association of dopamine activity with locomotor velocity and acceleration, which leads naturally to a multivariate functional data formulation. We treat each empirical CCF as a functional observation indexed by lag and test equality of mean multivariate CCF functions across groups using integrated and maximum-type global statistics, $F_{\mathrm{int}}$ and $F_{\max}$, constructed from pointwise Hotelling $T^2$ statistics. The integrated test targets broad differences across the lag domain, whereas the maximum test is sensitive to local differences. Applied to free-feeding and intragastric infusion datasets, the proposed methods detect substantial differences in dopamine-locomotion coupling across brain region and biological sex in the free-feeding experiment, with more selective effects in the infusion setting. The proposed framework provides a flexible and rigorous FDA-based approach for comparing dynamic dependence structures across experimental conditions.