  In an increasing number of neuroimaging studies, brain images, which are in
the form of multidimensional arrays (tensors), have been collected on multiple
subjects at multiple time points. Of scientific interest is to analyze such
massive and complex longitudinal images to diagnose neurodegenerative disorders
and to identify disease relevant brain regions. In this article, we treat those
problems in a unifying regression framework with image predictors, and propose
tensor generalized estimating equations (GEE) for longitudinal imaging
analysis. The GEE approach takes into account intra-subject correlation of
responses, whereas a low rank tensor decomposition of the coefficient array
enables effective estimation and prediction with limited sample size. We
propose an efficient estimation algorithm, study the asymptotics in both fixed
$p$ and diverging $p$ regimes, and also investigate tensor GEE with
regularization that is particularly useful for region selection. The efficacy
of the proposed tensor GEE is demonstrated on both simulated data and a real
data set from the Alzheimer's Disease Neuroimaging Initiative (ADNI).
