Project Summary With a rapidly aging world population, understanding, diagnosing, and treating Alzheimer's disease (AD) is becoming an international imperative. In recent years, a number of large-scale neuroimaging databases are emerging, which collect multiple imaging modalities from multiple imaging centers, at both the baseline and repeatedly over a number of years of follow-up. Such multicenter multimodal longitudinal neuroimag- ing data are particularly useful to understand neurodegenerative disorders such as AD. However, they pose numerous challenges, including ultrahigh dimensionality, complex spatial and temporal correlations, high proportion of missing values, data heterogeneity, and lack of formal inference or theoretical guaran- tee. These challenges have seriously hindered the application of those large neuroimaging databases to advance our understanding of AD and normal aging. In this proposal, we aim to develop new statistical methods to address those challenges, and to answer some fundamental questions in the field of AD and aging research. Specifically, (1) we develop a new simultaneous covariance inference procedure that provides an explicit quantification of statistical significance, a much improved detection power, a rigorous theoretical support, and a rigid false discovery control in association analysis of multiple imaging modal- ities; (2) we develop an integrative version of linear discriminant analysis for multimodal neuroimaging based classification and disease diagnosis, and aim to show the method is guaranteed to asymptotically improve the classification error rate when using multimodal data than using unimodal data; (3) we develop a dynamic tensor response regression model that can simultaneously handle the longitudinally correlated images and the high proportion of missing scans, through a mixture of sparsity and low-rank structures, fusion regularization and tensor completion; and (4) we propose a heterogeneity correction strategy and embed it with tensor response regression, which models the change of brain images or brain connectiv- ity patterns as the disease status or age changes, meanwhile correcting for potential heterogeneity from multiple imaging sites. Our proposal is motivated by two in vivo studies of AD and normal aging: the Berkeley Aging Cohort Study and the Alzheimer's Disease Neuroimaging Initiative, while it is also appli- cable to studies of other neurological disorders. It addresses a number of overarching challenges facing longitudinal and multimodal neuroimaging analysis, and offers a timely response to the growing demand for analysis of large neuroimaging databases. It also contributes to novel statistical methodology, and advances high-dimensional statistical inference theory. Our proposal is to result in a number of useful tools, in particular, a new computer software, which will be made freely available to both the end users at UC Berkeley and the neuroscience community at large.