Graduate Student Seminar

Speaker 1: Young-Geun Choi

Title

Positive definite modification of regularized covariance matrix estimators via linear shrinkage

Abstract

Covariance matrix plays an important role in many multivariate statistical procedures. Examples are principal component analysis (PCA), linear discriminant analysis (LDA), mean-vector homogeneity testing, and so forth. In this work, we are interested in positive definiteness (PDness) of covariance matrices. In the recent high-dimension literature, the estimation of covariance matrix involves a 'regularization' of sample covariance matrix such as 'thresholding', 'banding', or 'l1-penalty based optimization', and these techniques easily violate the PDness of covariance matrix. To solve this difficulty, we propose a linear-shrinkage type correction, which is simply a linear combination of an original non-PD covariance matrix estimator and the identity matrix. Our main contribution is to inspect the choice of linear coefficients for preserving the same asymptotic error rate. This result is also generically applicable for other non-PD estimation problems, including regularized inverse covariance matrix and covariance matrix estimation from missing data. Numerical study shows that the proposed correction is much faster than other optimization-based competitives, while empirical errors are still comparable. This project is the main topic of my Ph.D dissertation, under the supervision of Dr. Johan Lim.

Speaker 2: Gregory Haber

Title

Design and Analysis of Group Randomized Treatment Trials with Random Group Switching: A Similarity Based Approach

Abstract

When individual treatment is not possible or desirable, group-randomized trials (GRTs), in which interventions take place at a group level, have become the standard in many diverse fields of research. The defining characteristic of GRTs, in terms of analysis, is a positive correlation between members of the same group, often denoted as the intra-class correlation. While this correlation may be quite small (< .01), ignoring groups and treating all observations as independent is known to lead to heavily inflated type I error rates. Ways of addressing this issue have been the focus of much of the GRT literature over the past 30 years. Despite this, one area that has received little, if any, attention is the impact of individuals switching groups (within a single treatment arm) over the course of a study. This situation may arise for a variety of reasons in a given trial and is likely to lead to a much more complicated correlation structure than what is accounted for by current methodology. In this talk, we present simulation studies designed to look at the impact of such group switching on type I error rates, as well as new methods for modelling the correlation among participants which we demonstrate to be robust across a wider variety of practical situations.