New Methods for Discovering Hidden Dependence and for Assessing the Possible Influence of Unobserved Variables.

Abstract

The biological interpretation of neuroimaging data often depends on changes in the dependence structure between locations in the brain. A major challenge in neuroscience is uncovering the relationship between consciousness and brain activity. Electroencephalography (EEG) recordings made on human subjects who are given anesthesia for surgery provide an opportunity to directly study this relationship. The main focus in this area has been on changes in the connectivity between brain regions that occur as the consciousness state changes. Connectivity can be assessed in terms of the statistical dependence between EEG measurements from different recording sites on the scalp.

In this thesis, we consider two approaches for capturing changes in the dependence structure among several time series. We first consider the possibility that dependence between two series may be localized to a specific frequency band, and hence cannot be uncovered using global measures dependence. We propose methods to characterize the frequency-specific dependence in such data. We then consider the possibility that the dependence between two series can be revealed by applying a local transformation. We optimize over a class of such transformations to maximize a simple association measure, leading to a new measure of dependence for serially observed data. These two new methods are used to analyze a data that consists of multi-channel EEG recordings of multiple subjects under several consciousness states.

Another question that arises in analyzing complex biological data sets is whether there exists an unobserved variable responsible for all apparent relationships between a given set of observed variables and the outcome. In the last chapter, we propose an approach to understanding under what circumstances a single unmeasured variable could explain the entire observed relationship between an outcome and several observed predictors. The unobservable regression of interest is characterized in terms of three quantities: the distribution of the unobserved covariate, the effect size of the unobserved covariate, and the net dependence between the unobserved and the observed covariates. We derive an explicit functional relationship among these quantities, and how this in turn can be used to learn about possible alternative explanations for an observed multiple regression relationship.