Network-based models are popular in statistical applications. The main advantage of using a network-based model is that one can understand and evaluate its semantics and properties rather straightforwardly. In this dissertation, we study the statistical properties for network-based models and apply these models to genetic association studies and genetic risk prediction studies.In Chapter 2, we propose a conditional autoregressive (CAR) model to account for possible heterogeneous genetic effects among individuals. In the proposed method, the genetic effect is considered as a random effect and a score test is developed to test the variance component of a genetic random effect. Through simulations, we compare the type I error and power performance of the proposed method with those of the generalized genetic random field (GGRF) and the sequence kernel association test (SKAT) methods under different disease scenarios. We find that our method outperforms the other two methods when (i) the rare variants have the major contribution to the disease, or (ii) the genetic effects vary in different individuals or subgroups of individuals. Finally, we illustrate the new method by applying it to the whole genome sequencing data from the Alzheimer's Disease Neuroimaging Initiative.A kernel-based neural network (KNN) method is proposed in Chapter 3 for genetic risk prediction. KNN inherits the high-dimensional feature from classical kernel methods and the non-linear and non-additive features from neural networks. KNN summarizes a large number of genetic variants into kernel matrices and uses the kernel matrices as input matrices. Based on these kernel matrices, KNN builds a feedforward neural network to model the complex relationship between genetic variants and a disease outcome. Minimum norm quadratic unbiased estimation (MINQUE) is implemented in KNN to make parameter estimation feasible. Through theoretical proof and simulations, we demonstrate that KNN can attain lower average prediction error than LMM. Finally, we illustrate KNN by an application to the sequencing data from the Alzheimer's Disease Neuroimaging Initiate.Nowadays, neural network is one of the most popularly used methods in machine learning and artificial intelligence. However, as a statistical model, few researches focuses on statistical properties for neural networks and these properties will be studied in Chapter 4. A neural network can be classified into a nonlinear regression regression framework. However, if we consider it parametrically, due to the unidentifiability of the parameters, it is difficult to derive its asymptotic properties. Instead, we consider the estimation problem in a nonparametric regression framework and use the results from sieve estimation to establish the consistency, the rates of convergence, and the asymptotic normality of the neural network estimators. We also illustrate the validity of the theories via simulations.
Read
