Mixed effects models that include fixed and factor-specific (also known as random) effects offer a natural framework for studying longitudinal MRI data. This work extends mixed effects models to the setting where the responses lie on curved spaces such as the manifold of symmetric positive definite matrices. By treating the subject-wise diffeomorphic deformations between consecutive time points as a field of Cauchy deformation tensors, our framework can facilitate longitudinal analysis that respects the geometry of such data. While the existing body of work dealing with regression models on manifold-valued data is inherently restricted to cross-sectional studies, the proposed mixed effects formulation significantly expands the operating range of longitudinal analyses.
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