Chris Hans

Gene expression profiling and genetic markers in glioblastoma survival

Despite the strikingly grave prognosis for older patients with glioblastomas, significant variability in patient outcome is experienced. To explore the potential for developing improved prognostic capabilities based on the elucidation of potential biological relationships, we did analyses of genes commonly mutated, amplified, or deleted in glioblastomas and DNA microarray gene expression data from tumors of glioblastoma patients of age >50 for whom survival is known. No prognostic significance was associated with genetic changes in epidermal growth factor receptor (amplified in 17 of 41 patients), TP53 (mutated in 11 of 41 patients), p16INK4A (deleted in 15 of 33 patients), or phosphatase and tensin homologue (mutated in 15 of 41 patients). Statistical analysis of the gene expression data in connection with survival involved exploration of regression models on small subsets of genes, based on computational search over multiple regression models with cross-validation to assess predictive validity. The analysis generated a set of regression models that, when weighted and combined according to posterior probabilities implied by the statistical analysis, identify patterns in expression of a small subset of genes that are associated with survival and have value in assessing survival risks. The dominant genes across such multiple regression models involve three key genes-SPARC (Osteonectin), Doublecortex, and Semaphorin3B-which play key roles in cellular migration processes. Additional analysis, based on statistical graphical association models constructed using similar computational analysis methods, reveals other genes which support the view that multiple mediators of tumor invasion may be important prognostic factor in glioblastomas in older patients.

Experiments in Stochastic Computation for High-Dimensional Graphical Models

We discuss the implementation, development and performance of methods of stochastic computation in Gaussian graphical models. We view these methods from the perspective of high-dimensional model search, with a particular interest in the scalability with dimension of Markov chain Monte Carlo (MCMC) and other stochastic search methods. After reviewing the structure and context of undirected Gaussian graphical models and model uncertainty (covariance selection), we discuss prior specifications, including new priors over models, and then explore a number of examples using various methods of stochastic computation. Traditional MCMC methods are the point of departure for this experimentation; we then develop alternative stochastic search ideas and contrast this new approach with MCMC. Our examples range from low (12–20) to moderate (150) dimension, and combine simple synthetic examples with data analysis from gene expression studies. We conclude with comments about the need and potential for new computational methods in far higher dimensions, including constructive approaches to Gaussian graphical modeling and computation.

Gene Expression Profiles of Multiple Breast Cancer Phenotypes and Response to Neoadjuvant Chemotherapy

Purpose: Breast cancer is a heterogeneous disease, and markers for disease subtypes and therapy response remain poorly defined. For that reason, we employed a prospective neoadjuvant study in locally advanced breast cancer to identify molecular signatures of gene expression correlating with known prognostic clinical phenotypes, such as inflammatory breast cancer or the presence of hypoxia. In addition, we defined molecular signatures that correlate with response to neoadjuvant chemotherapy. Experimental Design: Tissue was collected under ultrasound guidance from patients with stage IIB/III breast cancer before four cycles of neoadjuvant liposomal doxorubicin paclitaxel chemotherapy combined with local whole breast hyperthermia. Gene expression analysis was done using Affymetrix U133 Plus 2.0 GeneChip arrays. Results: Gene expression patterns were identified that defined the phenotypes of inflammatory breast cancer as well as tumor hypoxia. In addition, molecular signatures were identified that predicted the persistence of malignancy in the axillary lymph nodes after neoadjuvant chemotherapy. This persistent lymph node signature significantly correlated with disease-free survival in two separate large populations of breast cancer patients. Conclusions: Gene expression signatures have the capacity to identify clinically significant features of breast cancer and can predict which individual patients are likely to be resistant to neoadjuvant therapy, thus providing the opportunity to guide treatment decisions.

Shotgun stochastic search for "large p" regression

Model search in regression with very large numbers of candidate predictors raises challenges for both model specification and computation, for which standard approaches such as Markov chain Monte Carlo (MCMC) methods are often infeasible or ineffective. We describe a novel shotgun stochastic search (SSS) approach that explores “interesting” regions of the resulting high-dimensional model spaces and quickly identifies regions of high posterior probability over models. We describe algorithmic and modeling aspects, priors over the model space that induce sparsity and parsimony over and above the traditional dimension penalization implicit in Bayesian and likelihood analyses, and parallel computation using cluster computers. We discuss an example from gene expression cancer genomics, comparisons with MCMC and other methods, and theoretical and simulation-based aspects of performance characteristics in large-scale regression model searches. We also provide software implementing the methods.

Model uncertainty and variable selection in Bayesian lasso regression

While Bayesian analogues of lasso regression have become popular, comparatively little has been said about formal treatments of model uncertainty in such settings. This paper describes methods that can be used to evaluate the posterior distribution over the space of all possible regression models for Bayesian lasso regression. Access to the model space posterior distribution is necessary if model-averaged inference—e.g., model-averaged prediction and calculation of posterior variable inclusion probabilities—is desired. The key element of all such inference is the ability to evaluate the marginal likelihood of the data under a given regression model, which has so far proved difficult for the Bayesian lasso. This paper describes how the marginal likelihood can be accurately computed when the number of predictors in the model is not too large, allowing for model space enumeration when the total number of possible predictors is modest. In cases where the total number of possible predictors is large, a simple Markov chain Monte Carlo approach for sampling the model space posterior is provided. This Gibbs sampling approach is similar in spirit to the stochastic search variable selection methods that have become one of the main tools for addressing Bayesian regression model uncertainty, and the adaption of these methods to the Bayesian lasso is shown to be straightforward.

Elastic Net Regression Modeling With the Orthant Normal Prior

The elastic net procedure is a form of regularized optimization for linear regression that provides a bridge between ridge regression and the lasso. The estimate that it produces can be viewed as a Bayesian posterior mode under a prior distribution implied by the form of the elastic net penalty. This article broadens the scope of the Bayesian connection by providing a complete characterization of a class of prior distributions that generate the elastic net estimate as the posterior mode. The resulting model-based framework allows for methodology that moves beyond exclusive use of the posterior mode by considering inference based on the full posterior distribution. Two characterizations of the class of prior distributions are introduced: a properly normalized, direct characterization, which is shown to be conjugate for linear regression models, and an alternate representation as a scale mixture of normal distributions. Prior distributions are proposed for the regularization parameters, resulting in an infinite mixture of elastic net regression models that allows for adaptive, data-based shrinkage of the regression coefficients. Posterior inference is easily achieved using Markov chain Monte Carlo (MCMC) methods. Uncertainty about model specification is addressed from a Bayesian perspective by assigning prior probabilities to all possible models. Corresponding computational approaches are described. Software for implementing the MCMC methods described in this article, written in C++ with an R package interface, is available at http://www.stat.osu.edu/~hans/software/.

Covariance Decompositions for Accurate Computation in Bayesian Scale-Usage Models

Analyses of multivariate ordinal probit models typically use data augmentation to link the observed (discrete) data to latent (continuous) data via a censoring mechanism defined by a collection of “cutpoints.” Most standard models, for which effective Markov chain Monte Carlo (MCMC) sampling algorithms have been developed, use a separate (and independent) set of cutpoints for each element of the multivariate response. Motivated by the analysis of ratings data, we describe a particular class of multivariate ordinal probit models where it is desirable to use a common set of cutpoints. While this approach is attractive from a data-analytic perspective, we show that the existing efficient MCMC algorithms can no longer be accurately applied. Moreover, we show that attempts to implement these algorithms by numerically approximating required multivariate normal integrals over high-dimensional rectangular regions can result in severely degraded estimates of the posterior distribution. We propose a new data augmentation that is based on a covariance decomposition and that admits a simple and accurate MCMC algorithm. Our data augmentation requires only that univariate normal integrals be evaluated, which can be done quickly and with high accuracy. We provide theoretical results that suggest optimal decompositions within this class of data augmentations, and, based on the theory, recommend default decompositions that we demonstrate work well in practice. This article has supplementary material online.