Covariance regression with random forests.

Capturing the conditional covariances or correlations among the elements of a multivariate response vector based on covariates is important to various fields including neuroscience, epidemiology and biomedicine. We propose a new method called Covariance Regression with Random Forests (CovRegRF) to estimate the covariance matrix of a multivariate response given a set of covariates, using a random forest framework. Random forest trees are built with a splitting rule specially designed to maximize the difference between the sample covariance matrix estimates of the child nodes. We also propose a significance test for the partial effect of a subset of covariates. We evaluate the performance of the proposed method and significance test through a simulation study which shows that the proposed method provides accurate covariance matrix estimates and that the Type-1 error is well controlled. An application of the proposed method to thyroid disease data is also presented. CovRegRF is implemented in a freely available R package on CRAN.

Longitudinal regression of covariance matrix outcomes.

In this study, a longitudinal regression model for covariance matrix outcomes is introduced. The proposal considers a multilevel generalized linear model for regressing covariance matrices on (time-varying) predictors. This model simultaneously identifies covariate-associated components from covariance matrices, estimates regression coefficients, and captures the within-subject variation in the covariance matrices. Optimal estimators are proposed for both low-dimensional and high-dimensional cases by maximizing the (approximated) hierarchical-likelihood function. These estimators are proved to be asymptotically consistent, where the proposed covariance matrix estimator is the most efficient under the low-dimensional case and achieves the uniformly minimum quadratic loss among all linear combinations of the identity matrix and the sample covariance matrix under the high-dimensional case. Through extensive simulation studies, the proposed approach achieves good performance in identifying the covariate-related components and estimating the model parameters. Applying to a longitudinal resting-state functional magnetic resonance imaging data set from the Alzheimer's Disease (AD) Neuroimaging Initiative, the proposed approach identifies brain networks that demonstrate the difference between males and females at different disease stages. The findings are in line with existing knowledge of AD and the method improves the statistical power over the analysis of cross-sectional data.

A combined multilevel factor analysis and covariance regression model with mixed effects in the mean and variance structure.

Li et al developed a multilevel covariance regression (MCR) model as an extension of the covariance regression model of Hoff and Niu. This model assumes a hierarchical structure for the mean and the covariance matrix. Here, we propose the combined multilevel factor analysis and covariance regression model in a Bayesian framework, simultaneously modeling the MCR model and a multilevel factor analysis (MFA) model. The proposed model replaces the responses in the MCR part with the factor scores coming from an MFA model. Via a simulation study and the analysis of real data, we show that the proposed model is quite efficient when the responses of the MCR model are not measured directly but are latent variables such as the patient experience measurements in our motivating dataset.

Reducing subspace models for large-scale covariance regression.

We develop an envelope model for joint mean and covariance regression in the large p, small n setting. In contrast to existing envelope methods, which improve mean estimates by incorporating estimates of the covariance structure, we focus on identifying covariance heterogeneity by incorporating information about mean-level differences. We use a Monte Carlo EM algorithm to identify a low-dimensional subspace that explains differences in both means and covariances as a function of covariates, and then use MCMC to estimate the posterior uncertainty conditional on the inferred low-dimensional subspace. We demonstrate the utility of our model on a motivating application on the metabolomics of aging. We also provide R code that can be used to develop and test other generalizations of the response envelope model.

Regression models for partially localized fMRI connectivity analyses.

Background: Brain functional connectivity analysis of resting-state functional magnetic resonance imaging (fMRI) data is typically performed in a standardized template space assuming consistency of connections across subjects. Analysis methods can come in the form of one-edge-at-a-time analyses or dimension reduction/decomposition methods. Common to these approaches is an assumption that brain regions are functionally aligned across subjects; however, it is known that this functional alignment assumption is often violated. Methods: In this paper, we use subject-level regression models to explain intra-subject variability in connectivity. Covariates can include factors such as geographic distance between two pairs of brain regions, whether the two regions are symmetrically opposite (homotopic), and whether the two regions are members of the same functional network. Additionally, a covariate for each brain region can be included, to account for the possibility that some regions have consistently higher or lower connectivity. This style of analysis allows us to characterize the fraction of variation explained by each type of covariate. Additionally, comparisons across subjects can then be made using the fitted connectivity regression models, offering a more parsimonious alternative to edge-at-a-time approaches. Results: We apply our approach to Human Connectome Project data on 268 regions of interest (ROIs), grouped into eight functional networks. We find that a high proportion of variation is explained by region covariates and network membership covariates, while geographic distance and homotopy have high relative importance after adjusting for the number of predictors. We also find that the degree of data repeatability using our connectivity regression model-which uses only partial location information about pairs of ROI's-is comparably as high as the repeatability obtained using full location information. Discussion: While our analysis uses data that have been transformed into a common template-space, we also envision the method being useful in multi-atlas registration settings, where subject data remains in its own geometry and templates are warped instead. These results suggest the tantalizing possibility that fMRI connectivity analysis can be performed in subject-space, using less aggressive registration, such as simple affine transformations, multi-atlas subject-space registration, or perhaps even no registration whatsoever.

Beyond massive univariate tests: Covariance regression reveals complex patterns of functional connectivity related to attention-deficit/hyperactivity disorder, age, sex, and response control.

Background: Studies of brain functional connectivity (FC) typically involve massive univariate tests, performing statistical analysis on each individual connection. In this study we apply a novel whole-matrix regression approach referred to as Covariate Assisted Principal (CAP) regression to identify resting-state FC brain networks associated with attention-deficit/hyperactivity disorder (ADHD) and response control. Methods: Participants included 8-12 year-old children with ADHD (n=115, 29 girls) and typically developing controls (n=102, 35 girls) who completed a resting-state fMRI scan and a go/no-go task (GNG). We modeled three sets of covariates to identify resting-state networks associated with an ADHD diagnosis, sex, and response inhibition (commission errors) and variability (ex-Gaussian parameter tau). Results: The first network includes FC between striatal-cognitive control (CC) network subregions and thalamic-default mode network (DMN) subregions and is positively related to age. The second consists of FC between CC-visual-somatomotor regions and between CC-DMN subregions and is positively associated with response variability in boys with ADHD. The third consists of FC within the DMN and between DMN-CC-visual regions and differs between boys with and without ADHD. The fourth consists of FC between visual-somatomotor regions and between visual-DMN regions and differs between girls and boys with ADHD and is associated with response inhibition and variability in boys with ADHD. Unique networks were also identified in each of the three models suggesting some specificity to the covariates of interest. Conclusions: These findings demonstrate the utility of our novel covariance regression approach to studying functional brain networks relevant for development, behavior, and psychopathology.

Model averaging estimation for high-dimensional covariance matrices with a network structure.

In this paper, we develop a model averaging method to estimate a high-dimensional covariance matrix, where the candidate models are constructed by different orders of polynomial functions. We propose a Mallows-type model averaging criterion and select the weights by minimizing this criterion, which is an unbiased estimator of the expected in-sample squared error plus a constant. Then, we prove the asymptotic optimality of the resulting model average covariance estimators. Finally, we conduct numerical simulations and a case study on Chinese airport network structure data to demonstrate the usefulness of the proposed approaches.

Bayesian hierarchical factor regression models to infer cause of death from verbal autopsy data.

In low-resource settings where vital registration of death is not routine it is often of critical interest to determine and study the cause of death (COD) for individuals and the cause-specific mortality fraction (CSMF) for populations. Post-mortem autopsies, considered the gold standard for COD assignment, are often difficult or impossible to implement due to deaths occurring outside the hospital, expense, and/or cultural norms. For this reason, Verbal Autopsies (VAs) are commonly conducted, consisting of a questionnaire administered to next of kin recording demographic information, known medical conditions, symptoms, and other factors for the decedent. This article proposes a novel class of hierarchical factor regression models that avoid restrictive assumptions of standard methods, allow both the mean and covariance to vary with COD category, and can include covariate information on the decedent, region, or events surrounding death. Taking a Bayesian approach to inference, this work develops an MCMC algorithm and validates the FActor Regression for Verbal Autopsy (FARVA) model in simulation experiments. An application of FARVA to real VA data shows improved goodness-of-fit and better predictive performance in inferring COD and CSMF over competing methods. Code and a user manual are made available at https://github.com/kelrenmor/farva.

JOINT MEAN AND COVARIANCE MODELING OF MULTIPLE HEALTH OUTCOME MEASURES.

Health exams determine a patient's health status by comparing the patient's measurement with a population reference range, a 95% interval derived from a homogeneous reference population. Similarly, most of the established relation among health problems are assumed to hold for the entire population. We use data from the 2009-2010 National Health and Nutrition Examination Survey (NHANES) on four major health problems in the U.S. and apply a joint mean and covariance model to study how the reference ranges and associations of those health outcomes could vary among subpopulations. We discuss guidelines for model selection and evaluation, using standard criteria such as AIC in conjunction with posterior predictive checks. The results from the proposed model can help identify subpopulations in which more data need to be collected to refine the reference range and to study the specific associations among those health problems.

Multivariate Heteroscedasticity Models for Functional Brain Connectivity.

Functional brain connectivity is the co-occurrence of brain activity in different areas during resting and while doing tasks. The data of interest are multivariate timeseries measured simultaneously across brain parcels using resting-state fMRI (rfMRI). We analyze functional connectivity using two heteroscedasticity models. Our first model is low-dimensional and scales linearly in the number of brain parcels. Our second model scales quadratically. We apply both models to data from the Human Connectome Project (HCP) comparing connectivity between short and conventional sleepers. We find stronger functional connectivity in short than conventional sleepers in brain areas consistent with previous findings. This might be due to subjects falling asleep in the scanner. Consequently, we recommend the inclusion of average sleep duration as a covariate to remove unwanted variation in rfMRI studies. A power analysis using the HCP data shows that a sample size of 40 detects 50% of the connectivity at a false discovery rate of 20%. We provide implementations using R and the probabilistic programming language Stan.