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.

Prediction of Brain Connectivity Map in Resting-State fMRI Data Using Shrinkage Estimator

Introduction:

In recent years, brain functional connectivity studies are extended using the advanced statistical methods. Functional connectivity is identified by synchronous activation in a spatially distinct region of the brain in resting-state functional Magnetic Resonance Imaging (MRI) data. For this purpose there are several methods such as seed-based correlation analysis based on temporal correlation between different Regions of Interests (ROIs) or between brain’s voxels of prior seed.

Methods:

In the current study, test-retest Resting State functional MRI (rs-fMRI) data of 21 healthy subjects were analyzed to predict second replication connectivity map using first replication data. A potential estimator is “raw estimator” that uses the first replication data from each subject to predict the second replication connectivity map of the same subject. The second estimator, “mean estimator” uses the average of all sample subjects' connectivity to estimate the correlation map. Shrinkage estimator is made by shrinking raw estimator towards the average connectivity map of all subjects' first replicate. Prediction performance of the second replication correlation map is evaluated by Mean Squared Error (MSE) criteria.

Results:

By the employment of seed-based correlation analysis and choosing precentral gyrus as the ROI over 21 subjects in the study, on average MSE for raw, mean and shrinkage estimator were 0.2169, 0.1118, and 0.1103, respectively. Also, percent reduction of MSE for shrinkage and mean estimator in comparison with raw estimator is 49.14 and 48.45, respectively.

Conclusion:

Shrinkage approach has the positive effect on the prediction of functional connectivity. When data has a large between session variability, prediction of connectivity map can be improved by shrinking towards population mean.

Shrinkage estimation of non-negative mean vector with unknown covariance under balance loss

Parameter estimation in multivariate analysis is important, particularly when parameter space is restricted. Among different methods, the shrinkage estimation is of interest. In this article we consider the problem of estimating the p-dimensional mean vector in spherically symmetric models. A dominant class of Baranchik-type shrinkage estimators is developed that outperforms the natural estimator under the balance loss function, when the mean vector is restricted to lie in a non-negative hyperplane. In our study, the components of the diagonal covariance matrix are assumed to be unknown. The performance evaluation of the proposed class of estimators is checked through a simulation study along with a real data analysis.

Predicting individual brain functional connectivity using a Bayesian hierarchical model

Network-oriented analysis of functional magnetic resonance imaging (fMRI), especially resting-state fMRI, has revealed important association between abnormal connectivity and brain disorders such as schizophrenia, major depression and Alzheimer's disease. Imaging-based brain connectivity measures have become a useful tool for investigating the pathophysiology, progression and treatment response of psychiatric disorders and neurodegenerative diseases. Recent studies have started to explore the possibility of using functional neuroimaging to help predict disease progression and guide treatment selection for individual patients. These studies provide the impetus to develop statistical methodology that would help provide predictive information on disease progression-related or treatment-related changes in neural connectivity. To this end, we propose a prediction method based on Bayesian hierarchical model that uses individual's baseline fMRI scans, coupled with relevant subject characteristics, to predict the individual's future functional connectivity. A key advantage of the proposed method is that it can improve the accuracy of individualized prediction of connectivity by combining information from both group-level connectivity patterns that are common to subjects with similar characteristics as well as individual-level connectivity features that are particular to the specific subject. Furthermore, our method also offers statistical inference tools such as predictive intervals that help quantify the uncertainty or variability of the predicted outcomes. The proposed prediction method could be a useful approach to predict the changes in individual patient's brain connectivity with the progression of a disease. It can also be used to predict a patient's post-treatment brain connectivity after a specified treatment regimen. Another utility of the proposed method is that it can be applied to test-retest imaging data to develop a more reliable estimator for individual functional connectivity. We show there exists a nice connection between our proposed estimator and a recently developed shrinkage estimator of connectivity measures in the neuroimaging community. We develop an expectation-maximization (EM) algorithm for estimation of the proposed Bayesian hierarchical model. Simulations studies are performed to evaluate the accuracy of our proposed prediction methods. We illustrate the application of the methods with two data examples: the longitudinal resting-state fMRI from ADNI2 study and the test-retest fMRI data from Kirby21 study. In both the simulation studies and the fMRI data applications, we demonstrate that the proposed methods provide more accurate prediction and more reliable estimation of individual functional connectivity as compared with alternative methods.

Shrinkage prediction of seed-voxel brain connectivity using resting state fMRI

Resting-state functional magnetic resonance imaging (rs-fMRI) is used to investigate synchronous activations in spatially distinct regions of the brain, which are thought to reflect functional systems supporting cognitive processes. Analyses are often performed using seed-based correlation analysis, allowing researchers to explore functional connectivity between data in a seed region and the rest of the brain. Using scan-rescan rs-fMRI data, we investigate how well the subject-specific seed-based correlation map from the second replication of the study can be predicted using data from the first replication. We show that one can dramatically improve prediction of subject-specific connectivity by borrowing strength from the group correlation map computed using all other subjects in the study. Even more surprisingly, we found that the group correlation map provided a better prediction of a subject's connectivity than the individual's own data. While further discussion and experimentation are required to understand how this can be used in practice, results indicate that shrinkage-based methods that borrow strength from the population mean should play a role in rs-fMRI data analysis.

A new noninferiority test for independent dichotomous variables based on a shrinkage proportion estimator

A new noninferiority test for the difference between two independent proportions is presented. The test is based on a Wald-type statistic in which maximum likelihood estimators and a type of shrinkage estimator are used to estimate proportions. This new test was compared with another Wald-type test that has been shown to behave well in terms of test size and power. For the comparison, the behavior of the new test, in terms of its size and power, was analyzed over several configurations. While the two tests exhibited similar behavior, the new test is easier to implement and thus constitutes a practical alternative.