Jeremy Gaskins (Oct. 11)

In the modeling of longitudinal data from several groups, appropriate handling of the dependence structure is of central importance. In many cases, one assumes that the covariance (or correlation) structure is the same for all groups. However, this assumption, if it fails to hold, can have an adverse effect on inference for mean effects. Conversely, if one specifies each of the covariance matrices without regard to the other groups, this can lead to a loss of efficiency if there is information to be gained across groups. It is desirable to develop methods to simultaneously estimate covariance matrices for each group that will borrow strength across groups in a way that is ultimately informed by the data. In addition, for several groups with covariance matrices of even medium dimension, it is difficult to ‘manually’ select a single best parametric model given a huge number of possibilities (e.g., structural zeros and/or commonality of individual parameters across groups). In this seminar we will develop a family of nonparametric priors using the Matrix Stick-Breaking Process of Dunson et al. (2008) that seek to accomplish this task by parameterizing the covariance matrices in terms of the parameters of their modified Cholesky decomposition (Pourahmadi, 1999). We establish some theoretic properties of these priors, examine their effectiveness via a simulation study, and illustrate the priors using data from a longitudinal clinical trial. (Joint work with Dr. Michael Daniels)

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Jorge Carlos Román (Oct. 18)

In recent years, Markov chain Monte Carlo (MCMC) algorithms have proven to be useful for obtaining draws from complex, high-dimensional probability distributions. In this seminar, I will discuss Bayesian general linear mixed models with proper priors as well as "non-informative" improper priors. The posterior densities for these models are intractable in the sense that the integrals required for making inferences cannot be computed in closed form. I will describe block Gibbs samplers that can be used to explore the resulting intractable posterior densities and provide easily-checked conditions under which their underlying Markov chains are geometrically ergodic; that is, they converge to the corresponding posterior in total variation norm at a geometric rate. There are well known advantages to using an MCMC algorithm that is driven by a geometrically ergodic Markov chain. In particular, when the chain is geometrically ergodic, sample averages satisfy central limit theorems, and these allow for the computation of asymptotically valid standard errors for MCMC-based estimates. (Joint work with Dr. Jim Hobert.)

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Professor Kshitij Khare (Nov. 1)

We consider the problem of sparse covariance estimation in high dimensional settings using graphical models. These models can be represented in terms of a graph, where the nodes represent random variables and edges represent their interactions. When the random variables are jointly Gaussian distributed, the lack of edges in such graphs can be interpreted as conditional and/or marginal independencies between these variables. We present a computationally efficient approach for high dimensional sparse covariance estimation in graphical models based on the Cholesky decomposition of the covariance matrix or its inverse. The proposed method is illustrated on both simulated and real data.

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Kenneth Lopiano (Nov. 29)

In environmental studies, relationships among variables that are misaligned in space are routinely assessed. Because the data are misaligned, kriging is often used to predict the covariate at the locations where the response is observed. Using kriging predictions to estimate regression parameters in linear regression models introduces both Berkson and classical measurement error. As a result, the Berkson error induces a covariance structure that is challenging to estimate. We characterize the measurement error as part of a broader class of Berkson error models and develop an estimated generalized least squares estimator using estimated covariance parameters. In working with the induced model, we fully account for the error structure and estimate the covariance parameters using restricted maximum likelihood and method of moments. We assess the performance of the estimators using simulation and illustrate the methodology using publicly available data from the Environmental Protection Agency. We extend the results to another change-of-support problem where the response is observed at the areal unit level and the covariate is observed at the point level using an example from the Centers for Disease Control's Environmental Public Health Tracking Program. Finally, we discuss our current research of spatial misalignment in generalized linear models.

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