Title: Bounding the Effects of Unobserved Covariates in Regression Analysis
Advisors: Professor Vijay Nair, Professor Kerby Shedden
Committee Member: Assistant Professor Xuanlong Nguyen
Abstract: Multiple regression analysis is used in applied research to understand the relationship between an outcome and several observed factors. One question that may be relevant in some settings is whether one, as opposed to several of the factors are associated with the outcome. For example in neuroscience, a researcher may wish to know whether a certain task involves coordination of several brain regions. In many scientific settings, only a subset of the factors of interest can be directly measured, hence this type of question cannot be directly answered based on a regression analysis involving observable quantities. We propose an approach to understanding under what circumstances a single unmeasured factor could explain the entire regression 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 factor, the effect size of the unobserved factor, and the net dependence between the unobserved factor and the observed factors. We show how these three quantities can be bounded in terms of each other, and how this in turn can be used to learn about possible alternative explanations for an observed multiple regression relationship.