Nonconjugate Bayesian analysis of variance component models

We consider the usual normal linear mixed model for variance components from a Bayesian viewpoint. With conjugate priors and balanced data, Gibbs sampling is easy to implement; however, simulating from full conditionals can become difficult for the analysis of unbalanced data with possibly nonconjugate priors, thus leading one to consider alternative Markov chain Monte Carlo schemes. We propose and investigate a method for posterior simulation based on an independence chain. The method is customized to exploit the structure of the variance component model, and it works with arbitrary prior distributions. As a default reference prior, we use a version of Jeffreys' prior based on the integrated (restricted) likelihood. We demonstrate the ease of application and flexibility of this approach in familiar settings involving both balanced and unbalanced data.     

Bayesian Approach to Proportional Odds Models for Survival Analysis

In this paper Bayesian approach is adopted to develop inferences about parameters in proportional odds models. Bayesian posterior intervals for coefficients in proportional odds models are derived by using approximation given in Pregibon (1981). The results are illustrated by using the lung cancer survival data reported by Prentice (1973).     

Bayes estimates as an approximation to maximum likelihood estimates

Ronald A. Fisher, who is the founder of maximum likelihood estimation (ML estimation), criticized the Bayes estimation of using a uniform prior distribution, because we can create estimates arbitrarily if we use Bayes estimation by changing the transformation used before the analysis. Thus, the Bayes estimates lack the scientific objectivity, especially when the amount of data is small. However, we can use the Bayes estimates as an approximation to the objective ML estimates if we use an appropriate transformation that makes the posterior distribution close to a normal distribution. One-to-one correspondence exists between a uniform prior distribution under a transformed scale and a non-uniform prior distribution under the original scale. For this reason, the Bayes estimation of ML estimates is essentially identical to the estimation using Jeffreys prior.     

A closer look at the distribution of number needed to treat (NNT): a Bayesian approach

In this paper, I present a Bayesian approach to estimation of the number needed to treat (NNT). The use of NNT as a measure of clinical benefit is now becoming commonplace. Various methods of estimation have been proposed, but none of them seem to provide entirely good estimates. Very little has been done to understand the statistical properties of NNT. Here, I derive the posterior distribution of NNT and use simulations to investigate the general behaviour of the distribution. The posterior mode of the distribution is proposed as a point estimate and results are compared with the conventional method of estimation of NNT done by inversion.     

A hierarchical Bayesian approach to modeling embryo implantation following in vitro fertilization

In vitro fertilization and embryo transfer (IVF-ET) is considered a method of last resort for treating infertility. Oocytes taken from a woman are fertilized in vitro, and one or more resulting embryos are transferred into the uterus, with the hope that at least one will implant and result in pregnancy. Successful implantation depends on both embryo viability and uterine receptivity. This has led to the development of the EU model for embryo implantation, wherein uterine receptivity is characterized by a latent binary variable U and embryo viability is characterized by a latent binomial variable E representing the number of viable embryos among those selected for transfer. The observed number of implantations is the product of E and U. Zhou and Weinberg (1998) developed a regression formulation of the EU model in which embryo viabilities are independent within patients. We extend their methodology to a Bayesian hierarchical framework that allows for correlation between the embryo viabilities and gives explicit characterization of patient-level heterogeneity. When some subjects have zero implantations, the likelihood for the hierarchical EU model is relatively flat and therefore using prior information for key parametersis needed. This provides a key motivation for adopting a Bayesian approach. The model is used to assess the effect of hydrosalpinx on embryo implantation in a cohort of 288 women undergoing IVF-ET because of tubal disease. Hydrosalpinx is a build-up of fluid in the Fallopian tubes, which sometimes leaks to the uterus and may reduce the likelihood of implantation. The EU model is well suited to this question because hydrosalpinx is thought to affect implantation by reducing uterine receptivity only. Our analysis indicates substantial subject-level heterogeneity with respect to embryo viability, suggesting the utility of a multi-level model.     

Bayesian semiparametric proportional odds models

Methodology for implementing the proportional odds regression model for survival data assuming a mixture of finite Polya trees (MPT) prior on baseline survival is presented. Extensions to frailties and generalized odds rates are discussed. Although all manner of censoring and truncation can be accommodated, we discuss model implementation, regression diagnostics, and model comparison for right-censored data. An advantage of the MPT model is the relative ease with which predictive densities, survival, and hazard curves are generated. Much discussion is devoted to practical implementation of the proposed models, and a novel MCMC algorithm based on an approximating parametric normal model is developed. A modest simulation study comparing the small sample behavior of the MPT model to a rank-based estimator and a real data example is presented.     

A semiparametric Bayesian approach for structural equation models

In the development of structural equation models (SEMs), observed variables are usually assumed to be normally distributed. However, this assumption is likely to be violated in many practical researches. As the non‐normality of observed variables in an SEM can be obtained from either non‐normal latent variables or non‐normal residuals or both, semiparametric modeling with unknown distribution of latent variables or unknown distribution of residuals is needed. In this article, we find that an SEM becomes nonidentifiable when both the latent variable distribution and the residual distribution are unknown. Hence, it is impossible to estimate reliably both the latent variable distribution and the residual distribution without parametric assumptions on one or the other. We also find that the residuals in the measurement equation are more sensitive to the normality assumption than the latent variables, and the negative impact on the estimation of parameters and distributions due to the non‐normality of residuals is more serious. Therefore, when there is no prior knowledge about parametric distributions for either the latent variables or the residuals, we recommend making parametric assumption on latent variables, and modeling residuals nonparametrically. We propose a semiparametric Bayesian approach using the truncated Dirichlet process with a stick breaking prior to tackle the non‐normality of residuals in the measurement equation. Simulation studies and a real data analysis demonstrate our findings, and reveal the empirical performance of the proposed methodology. A free WinBUGS code to perform the analysis is available in Supporting Information.     

A Bayesian approach to DNA sequence segmentation

Many deoxyribonucleic acid (DNA) sequences display compositional heterogeneity in the form of segments of similar structure. This article describes a Bayesian method that identifies such segments by using a Markov chain governed by a hidden Markov model. Markov chain Monte Carlo (MCMC) techniques are employed to compute all posterior quantities of interest and, in particular, allow inferences to be made regarding the number of segment types and the order of Markov dependence in the DNA sequence. The method is applied to the segmentation of the bacteriophage lambda genome, a common benchmark sequence used for the comparison of statistical segmentation algorithms.     

Bayesian design of noninferiority trials for medical devices using historical data

Summary We develop a new Bayesian approach of sample size determination (SSD) for the design of noninferiority clinical trials. We extend the fitting and sampling priors of Wang and Gelfand (2002, Statistical Science 17 , 193–208) to Bayesian SSD with a focus on controlling the type I error and power. Historical data are incorporated via a hierarchical modeling approach as well as the power prior approach of Ibrahim and Chen (2000, Statistical Science 15 , 46–60). Various properties of the proposed Bayesian SSD methodology are examined and a simulation‐based computational algorithm is developed. The proposed methodology is applied to the design of a noninferiority medical device clinical trial with historical data from previous trials.     

A Bayesian approach for the analysis of panel-count data with dependent termination

We consider modeling and Bayesian analysis for panel-count data when the termination time for each subject may depend on its history of the recurrent events. We propose a fully specified semiparametric model for the joint distribution of the recurrent events and the termination time. For this model, we provide a natural motivation, derive several novel properties, and develop a Bayesian analysis based on a Markov chain Monte Carlo algorithm. Comparisons are made to other existing models and methods for panel-count data. We demonstrate the usefulness of our new models and methodologies through the reanalysis of a data set from a clinical trial.     

Insights from a general,full-likelihood Bayesian approach to inferring shared evolutionary events from genomic data: Inferring shared demographic events is challenging*

Factors that influence the distribution, abundance, and diversification of species can simultaneously affect multiple evolutionary lineages within or across communities. These include changes to the environment or inter-specific ecological interactions that cause ranges of multiple species to contract, expand, or fragment. Such processes predict temporally clustered evolutionary events across species, such as synchronous population divergences and/or changes in population size. There have been a number of methods developed to infer shared divergences or changes in population size, but not both, and the latter has been limited to approximate methods. We introduce a full-likelihood Bayesian method that uses genomic data to estimate temporal clustering of an arbitrary mix of population divergences and population-size changes across taxa. Using simulated data, we find that estimating the timing and sharing of demographic changes tends to be inaccurate and sensitive to prior assumptions, which is in contrast to accurate, precise, and robust estimates of shared divergence times. We also show that previous estimates of co-expansion among five Alaskan populations of three-spine sticklebacks (Gasterosteus aculeatus) were likely driven by prior assumptions and ignoring invariant characters. We conclude by discussing potential avenues to improve the estimation of synchronous demographic changes across populations.     

A Bayesian model comparison approach to inferring positive selection

A popular approach to detecting positive selection is to estimate the parameters of a probabilistic model of codon evolution and perform inference based on its maximum likelihood parameter values. This approach has been evaluated intensively in a number of simulation studies and found to be robust when the available data set is large. However, uncertainties in the estimated parameter values can lead to errors in the inference, especially when the data set is small or there is insufficient divergence between the sequences. We introduce a Bayesian model comparison approach to infer whether the sequence as a whole contains sites at which the rate of nonsynonymous substitution is greater than the rate of synonymous substitution. We incorporated this probabilistic model comparison into a Bayesian approach to site-specific inference of positive selection. Using simulated sequences, we compared this approach to the commonly used empirical Bayes approach and investigated the effect of tree length on the performance of both methods. We found that the Bayesian approach outperforms the empirical Bayes method when the amount of sequence divergence is small and is less prone to false-positive inference when the sequences are saturated, while the results are indistinguishable for intermediate levels of sequence divergence.