Statistical methods for data with different dimensions

This thesis addresses the joint analysis of data with different dimensions, such as scalars, vectors, functions and images. This is of high practical and methodological relevance, as in the course of the technical progress, data with increasing complexity and dimensionality becomes available, requiring the extension of statistical models to new types of data and leading to the development of completely new statistical methods. In the first part of the thesis, multivariate functional principal component analysis (MFPCA) is developed for functional data on different dimensional domains. This is a novel method, as existing approaches for MFPCA are restricted to multivariate functional data on the same, one-dimensional interval. Using the new approach, principal components for data consisting e.g. of functions and images (i.e. functions on a two-dimensional domain) can be obtained, taking potential covariation in the elements into account. The thesis constructs a thorough theoretical basis for multivariate functional data on different dimensional domains and derives a theoretical relationship between univariate and multivariate functional principal component analysis for finite sample sizes. The results can be used to estimate multivariate functional principal components, eigenvalues and scores based on their univariate counterparts. It is shown how the method can be extended to univariate elements in general basis representations and to a weighted version of MFPCA to correct for differences in domain, range or variation of the elements. The approach is also applicable for sparse data or data with measurement error. The finite sample performance of the new method is evaluated in a simulation study with different levels of complexity. Moreover, asymptotic properties for large sample sizes are derived in two theorems, using results from perturbation theory and showing consistency of the proposed estimators. The estimation algorithm has been implemented in a publicly available R-package MFPCA, together with another R-package funData for representing functional data in an object-oriented manner. The thesis provides an introduction to the software and the underlying concepts. The new approach is illustrated in an application to a neuroimaging dataset. The aim here is to examine the relationship between trajectories of a neuropsychological test score over time and FDG-PET brain scans at baseline, that can be interpreted as functions on a three-dimensional domain, as the latter might be predictive of subsequent cognitive decline. The results show that estimates obtained from the new MFPCA method are meaningful from a medical point of view and provide new insights into the data. The second part of the thesis is concerned with scalar-on-image regression. This class of statistical methods models the relation of a scalar outcome and an image predictor, hence data with different dimensions and a complex dependence structure. It is representative for a broad class of statistical models for complex data, which intrinsically is unidentifiable, as in general the number of observations will be low compared to the number of pixels in the image. Strong model assumptions are thus required to obtain a unique solution, which is of course conditional on the hypotheses made on the true coefficient image. In the thesis, different models for scalar-on-image regression with different assumptions are compared with respect to their ability to give reliable and interpretable estimates. To this end, new measures for quantifying the influence of model assumptions are developed and analyzed in a simulation study for nine different scalar-on-image models. The relevance of the topic is illustrated in a practical neuroimaging application. It is shown that different models with different assumptions can lead to results that share common patterns, but can differ substantially in their details, as model assumptions can have a strong influence on the estimates. This can entail the risk of over-interpreting effects that are mainly driven by the model assumptions.