Varying Coefficient Models For Functional Responses In Image Analysis

Recently, massive functional data have been widely collected over space across a set of grid points in various imaging studies. It is interesting to correlate functional data with various clinical variables, such as age and gender, in order to address scientific questions of interest. Motivated by recent work studying massive imaging data in the neuroimaging literature, in this paper we propose three varying coefficient models to varying association between functional responses (e.g., image) and a set of covariates.In the second Chapter, we propose nonlinear varying mixed effects models (NVMEM) for modeling the relation between longitudinal functional responses and covariates of in-terest, which is more flexible than existing linear methods. We develop efficient estimation methods and several statistical inference procedures for NVMEM and systematically s-tudy their theoretical properties. We establish the weak convergence of estimated fixed effects functions, as well as their asymptotic bias and variance. We establish the uni-form convergence rate of the estimated spatial-temporal covariance operators and their associated eigenvalues and eigenfunetions, and then we derive asymptotic bias and mean integrated squared error of estiamted random effects functions and their uniform conver-gence rate. We propose a global test for linear hypotheses of fixed effects functions, and derive its asymptotic distribution under the null hypothesis. We also propose a simulta-neous confidence band for each fixed effects function. We conduct Monte Carlo simulation to examine the finite-sample performance of the proposed procedures. We apply NVMEM to investigate the spatial-temporal dynamics of white-matter fiber skeletons in a national database for autism research.In the third Chapter, we propose a transformation varying coefficient model (TVCM) for modeling the relation between functional response and a set of covariates. The lineari-ty between the response and the covariates is commonly used in the traditional regression. However, such linearity is not always satisfied. We consider the estimation and inference for transformation varying coefficient model to accommodate such non-linearity, we con-struct asymptotic simultaneous confidence bands for each coefficient function and discuss the theoretical properties of various estimators. Simulation studies are conducted to as-sess the finite sample performance of the proposed methods. We also apply TVCM to an imaging data for illustration.In the forth Chapter, we develop a single-index varying coefficient model (SIVCM) for establishing a varying association between functional responses (e.g., image) and a set of covariates. It enjoys several unique features of both varying-coefficient and single-index models. An estimation procedure is developed to estimate varying coefficient functions, the index function, and the covariance function of individual functions. The optimal integration of information across different grid points arc systematically delineated and the asymptotic properties (e.g., consistency and convergence rate) of all estimators are examined. We also propose a simultaneous confidence band for each coefficient functions. Simulation studies are conducted to assess the finite-sample performance of the proposed estimation procedure. Furthermore, our real data analysis of a white matter tract dataset obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) study confirms the advantage and accuracy of SIVCM over the popular varying coefficient model.