Maria DeYoreo

A Fully Nonparametric Modeling Approach to Binary Regression

© 2015 International Society for Bayesian Analysis. We propose a general nonparametric Bayesian framework for binary regression, which is built from modeling for the joint response-covariate distribution. The observed binary responses are assumed to arise from underlying continuous random variables through discretization, and we model the joint distribution of these latent responses and the covariates using a Dirichlet process mixture of multivariate normals. We show that the kernel of the induced mixture model for the observed data is identifiable upon a restriction on the latent variables. To allow for appropriate dependence structure while facilitating identifiability, we use a square-root-free Cholesky decomposition of the covariance matrix in the normal mixture kernel. In addition to allowing for the necessary restriction, this modeling strategy provides substantial simplifications in implementation of Markov chain Monte Carlo posterior simulation. We present two data examples taken from areas for which the methodology is especially well suited. In particular, the first example involves estimation of relationships between environmental variables, and the second develops inference for natural selection surfaces in evolutionary biology. Finally, we discuss extensions to regression settings with ordinal responses.

Bayesian Nonparametric Modeling for Multivariate Ordinal Regression

ABSTRACT Univariate or multivariate ordinal responses are often assumed to arise from a latent continuous parametric distribution, with covariate effects that enter linearly. We introduce a Bayesian nonparametric modeling approach for univariate and multivariate ordinal regression, which is based on mixture modeling for the joint distribution of latent responses and covariates. The modeling framework enables highly flexible inference for ordinal regression relationships, avoiding assumptions of linearity or additivity in the covariate effects. In standard parametric ordinal regression models, computational challenges arise from identifiability constraints and estimation of parameters requiring nonstandard inferential techniques. A key feature of the nonparametric model is that it achieves inferential flexibility, while avoiding these difficulties. In particular, we establish full support of the nonparametric mixture model under fixed cut-off points that relate through discretization the latent continuous responses with the ordinal responses. The practical utility of the modeling approach is illustrated through application to two datasets from econometrics, an example involving regression relationships for ozone concentration, and a multirater agreement problem. Supplementary materials with technical details on theoretical results and on computation are available online.

Categorical data fusion using auxiliary information

In data fusion analysts seek to combine information from two databases comprised of disjoint sets of individuals, in which some variables appear in both databases and other variables appear in only one database. Most data fusion techniques rely on variants of conditional independence assumptions. When inappropriate, these assumptions can result in unreliable inferences. We propose a data fusion technique that allows analysts to easily incorporate auxiliary information on the dependence structure of variables not observed jointly; we refer to this auxiliary information as glue. With this technique, we fuse two marketing surveys from the book publisher HarperCollins using glue from the online, rapid-response polling company CivicScience. The fused data enable estimation of associations between peoples preferences for authors and for learning about new books. The analysis also serves as a case study on the potential for using online surveys to aid data fusion.

Bayesian Mixture Models with Focused Clustering for Mixed Ordinal and Nominal Data

In some contexts, mixture models can fit certain variables well at the expense of others in ways beyond the analysts control. For example, when the data include some variables with non-trivial amounts of missing values, the mixture model may fit the marginal distributions of the nearly and fully complete variables at the expense of the variables with high fractions of missing data. Motivated by this setting, we present a mixture model for mixed ordinal and nominal data that splits variables into two groups, focus variables and remainder variables. The model allows the analyst to specify a rich sub-model for the focus variables and a simpler sub-model for remainder variables, yet still capture associations among the variables. Using simulations, we illustrate advantages and limitations of focused clustering compared to mixture models that do not distinguish variables. We apply the model to handle missing values in an analysis of the 2012 American National Election Study, estimating relationships among voting behavior, ideology, and political party affiliation.

Modeling for Dynamic Ordinal Regression Relationships: An Application to Estimating Maturity of Rockfish in California

ABSTRACT We develop a Bayesian nonparametric framework for modeling ordinal regression relationships, which evolve in discrete time. The motivating application involves a key problem in fisheries research on estimating dynamically evolving relationships between age, length, and maturity, the latter recorded on an ordinal scale. The methodology builds from nonparametric mixture modeling for the joint stochastic mechanism of covariates and latent continuous responses. This approach yields highly flexible inference for ordinal regression functions while at the same time avoiding the computational challenges of parametric models that arise from estimation of cut-off points relating the latent continuous and ordinal responses. A novel-dependent Dirichlet process prior for time-dependent mixing distributions extends the model to the dynamic setting. The methodology is used for a detailed study of relationships between maturity, age, and length for Chilipepper rockfish, using data collected over 15 years along the coast of California. Supplementary materials for this article are available online.

Reducing Costs and Improving Fit for Clinical Trials that Have Positive-Valued Data

ABSTRACT In many fields of research variables are often both continuous and restricted to be positive. We analyzed 70 endpoints that contained continuous and positive dependent variables from 6 clinical and 3 preclinical trials. The impact of data transformation and adjustment for baseline on the fit of the model was studied. On average, including baseline as a covariate decreases the necessary sample size to achieve a particular precision in the treatment effect estimate by about 70% as compared to a model that ignores baseline, or by 20% to 33% as compared to models that only adjust for baseline in the response. Additionally, log transformation of the endpoint (either the direct outcome or a baseline-adjusted outcome) appears to decrease the sample size needed on average by 20% to 35%. We draw three conclusions from this work. First, if a baseline is available, use of baseline as a covariate should always be undertaken. Second, although we recommend exploration of data from previous studies, percent change from baseline analyses should not be undertaken unless there is strong empirical evidence that for that endpoint it is preferred. Finally, and again with the caveat that nothing replaces exploration of data from previous studies, log transformation ought to be the default analysis of positive data unless exploration of previous data provides convincing evidence that the natural scale is preferred.

A Bayesian nonparametric Markovian model for non-stationary time series

Stationary time series models built from parametric distributions are, in general, limited in scope due to the assumptions imposed on the residual distribution and autoregression relationship. We present a modeling approach for univariate time series data, which makes no assumptions of stationarity, and can accommodate complex dynamics and capture non-standard distributions. The model for the transition density arises from the conditional distribution implied by a Bayesian nonparametric mixture of bivariate normals. This results in a flexible autoregressive form for the conditional transition density, defining a time-homogeneous, non-stationary Markovian model for real-valued data indexed in discrete time. To obtain a computationally tractable algorithm for posterior inference, we utilize a square-root-free Cholesky decomposition of the mixture kernel covariance matrix. Results from simulated data suggest that the model is able to recover challenging transition densities and non-linear dynamic relationships. We also illustrate the model on time intervals between eruptions of the Old Faithful geyser. Extensions to accommodate higher order structure and to develop a state-space model are also discussed.