Statistical Methods for Constructing Heterogeneous Biomarker Networks
The theme of this dissertation is to construct heterogeneous biomarker networks using graphical models for understanding disease progression and prognosis. Biomarkers may organize into networks of connected regions. Substantial heterogeneity in networks between individuals and subgroups of individuals is observed. The strengths of network connections may vary across subjects depending on subject-specific covariates (e.g., genetic variants, age). In addition, the connectivities between biomarkers, as subject-specific network features, have been found to predict disease clinical outcomes. Thus, it is important to accurately identify biomarker network structure and estimate the strength of connections. Graphical models have been extensively used to construct complex networks. However, the estimated networks are at the population level, not accounting for subjects’ covariates. More flexible covariate-dependent graphical models are needed to capture the heterogeneity in subjects and further create new network features to improve prediction of disease clinical outcomes and stratify subjects into clinically meaningful groups. A large number of parameters are required in covariate-dependent graphical models. Regularization needs to be imposed to handle the high-dimensional parameter space. Furthermore, personalized clinical symptom networks can be constructed to investigate co-occurrence of clinical symptoms. When there are multiple biomarker modalities, the estimation of a target biomarker network can be improved by incorporating prior network information from the external modality. This dissertation contains four parts to achieve these goals: (1) An efficient l0-norm feature selection method based on augmented and penalized minimization to tackle the high-dimensional parameter space involved in covariate-dependent graphical models; (2) A two-stage approach to identify disease-associated biomarker network features; (3) An application to construct personalized symptom networks; (4) A node-wise biomarker graphical model to leverage the shared mechanism between multi-modality data when external modality data is available. In the first part of the dissertation, we propose a two-stage procedure to regularize l0-norm as close as possible and solve it by a highly efficient and simple computational algorithm. Advances in high-throughput technologies in genomics and imaging yield unprecedentedly large numbers of prognostic biomarkers. To accommodate the scale of biomarkers and study their association with disease outcomes, penalized regression is often used to identify important biomarkers. The ideal variable selection procedure would search for the best subset of predictors, which is equivalent to imposing an l0-penalty on the regression coefficients. Since this optimization is a non-deterministic polynomial-time hard (NP-hard) problem that does not scale with number of biomarkers, alternative methods mostly place smooth penalties on the regression parameters, which lead to computationally feasible optimization problems. However, empirical studies and theoretical analyses show that convex approximation of l0-norm (e.g., l1) does not outperform their l0 counterpart. The progress for l0-norm feature selection is relatively slower, where the main methods are greedy algorithms such as stepwise regression or orthogonal matching pursuit. Penalized regression based on regularizing l0-norm remains much less explored in the literature. In this work, inspired by the recently popular augmenting and data splitting algorithms including alternating direction method of multipliers, we propose a two-stage procedure for l0-penalty variable selection, referred to as augmented penalized minimization-L0 (APM-L0). APM-L0 targets l0-norm as closely as possible while keeping computation tractable, efficient, and simple, which is achieved by iterating between a convex regularized regression and a simple hard-thresholding estimation. The procedure can be viewed as arising from regularized optimization with truncated l1 norm. Thus, we propose to treat regularization parameter and thresholding parameter as tuning parameters and select based on cross-validation. A one-step coordinate descent algorithm is used in the first stage to significantly improve computational efficiency. Through extensive simulation studies and real data application, we demonstrate superior performance of the proposed method in terms of selection accuracy and computational speed as compared to existing methods. The proposed APM-L0 procedure is implemented in the R-package APML0. In the second part of the dissertation, we develop a two-stage method to estimate biomarker networks that account for heterogeneity among subjects and evaluate the network’s association with disease clinical outcome. In the first stage, we propose a conditional Gaussian graphical model with mean and precision matrix depending on covariates to obtain subject- or subgroup-specific networks. In the s