Ming-Hui Chen

A New Skewed Link Model for Dichotomous Quantal Response Data

Abstract The logit, probit, and student t-links are widely used in modeling dichotomous quantal response data. Most of the commonly used link functions are symmetric, except the complementary log-log link. However, in some applications the overall fit can be significantly improved by the use of an asymmetric link. In this article we propose a new skewed link model for analyzing binary response data with covariates. Introducing a skewed distribution for the underlying latent variable, we develop a class of asymmetric link models for binary response data. Using a Bayesian approach, we first characterize the propriety of the posterior distributions using standard improper priors. We further propose informative priors using historical data from a similar previous study. We examine the proposed method through a large-scale simulation study and use data from a prostate cancer study to demonstrate the use of historical data in Bayesian model fitting and comparison of skewed link models.

Power prior distributions for generalized linear models

In this article, we propose a class of prior distributions called the power prior distributions. The power priors are based on the notion of the availability of historical data, and are of great potential use in this context. We demonstrate how to construct these priors and elicit their hyperparameters. We examine the theoretical properties of these priors in detail and obtain some very general conditions for propriety as well as lower bounds on the normalizing constants. We extensively discuss the normal, binomial, and Poisson regression models. Extensions of the priors are given along with numerical examples to illustrate the methodology.

Propriety of posterior distribution for dichotomous quantal response models

In this article, we investigate the property of posterior distribution for dichotomous quantal response models using a uniform prior distribution on the regression parameters. Sufficient and necessary conditions for the propriety of the posterior distribution with a general link function are established. In addition, the sufficient conditions for the existence of the posterior moments and the posterior moment generating function are also obtained. Finally, the relationship between the propriety of posterior distribution and the existence of the maximum likelihood estimate is examined.

Prior elicitation for model selection and estimation in generalized linear mixed models

Generalized linear models serve as a useful class of regression models for discrete and continuous data. In applications such as longitudinal studies, observations are typically correlated. The correlation structure in the data is induced by introducing a random effect, leading to the generalized linear mixed model (GLMM). In this paper, we propose a class of informative prior distributions for the class of GLMMs and investigate their theoretical as well as computational properties. Specifically, we investigate conditions for propriety of the proposed priors for the class of GLMMs and show that they are proper under some very general conditions. In addition, we examine recent computational methods such as hierarchical centering and semi-hierarchical centering for performing Gibbs sampling for this class of models. One of the main applications of the proposed priors is variable subset selection. Novel computational tools are developed for sampling from the posterior distributions and computing the prior and posterior model probabilities. We demonstrate our methodology with two real longitudinal datasets.

Noninformative priors for one-parameter item response models

Abstract We present a unified Bayesian approach for the analysis of one-parameter item response models. A necessary and sufficient condition is given for the propriety of posteriors under improper priors with nonidentifiable likelihoods. Posterior distributions for item and subject parameters may be improper when the sum of the binary responses for an item or subject takes its minimum or maximum possible value. When the item parameters have a flat prior but the item totals do not fall at a boundary value, we prove the propriety of the Bayesian joint posterior under some sufficient conditions on the joint (proper) distribution of the subject parameters. The methods are implemented using Markov chain Monte Carlo and illustrated with an example from a cross-over study comparing three medical treatments.

Maximum likelihood inference for the Cox regression model with applications to missing covariates

In this paper, we carry out an in-depth theoretical investigation for existence of maximum likelihood estimates for the Cox model (Cox, 1972, 1975) both in the full data setting as well as in the presence of missing covariate data. The main motivation for this work arises from missing data problems, where models can easily become difficult to estimate with certain missing data configurations or large missing data fractions. We establish necessary and sufficient conditions for existence of the maximum partial likelihood estimate (MPLE) for completely observed data (i.e., no missing data) settings as well as sufficient conditions for existence of the maximum likelihood estimate (MLE) for survival data with missing covariates via a profile likelihood method. Several theorems are given to establish these conditions. A real dataset from a cancer clinical trial is presented to further illustrate the proposed methodology.

Bayesian Analysis of Binary Data Using Skewed Logit Models

The logistic regression is one of the most widely used models for binary response data in medical and epidemiologic studies. However, in some applications, the overall fit can be improved significantly by the use of a noncanonical link and in particular by an asymmetric link. In this paper, we consider a skewed logit model for analyzing binary response data with presence of covariates. Using a Bayesian approach, we propose informative priors using historical data from a similar previous study. We use various Bayesian methods for model comparison and model adequacy. More specifically, we use conditional predictive ordinates (CPO) to develop the pseudo-Bayes factor and Bayesian standardized residuals. In addition, we propose Bayesian latent residuals for assessing choices of link functions. Data sets from a prostate cancer study are used to demonstrate the methodology as well as the role of informative priors in model comparison and model adequacy.