Merlise A. Clyde

Mixtures of g Priors for Bayesian Variable Selection

Zellners g prior remains a popular conventional prior for use in Bayesian variable selection, despite several undesirable consistency issues. In this article we study mixtures of g priors as an alternative to default g priors that resolve many of the problems with the original formulation while maintaining the computational tractability that has made the g prior so popular. We present theoretical properties of the mixture g priors and provide real and simulated examples to compare the mixture formulation with fixed g priors, empirical Bayes approaches, and other default procedures. Please see Arnold Zellners letter and the authors response.

Flexible empirical Bayes estimation for wavelets

Wavelet shrinkage estimation is an increasingly popular method for signal denoising and compression. Although Bayes estimators can provide excellent mean‐squared error (MSE) properties, the selection of an effective prior is a difficult task. To address this problem, we propose empirical Bayes (EB) prior selection methods for various error distributions including the normal and the heavier‐tailed Student t‐distributions. Under such EB prior distributions, we obtain threshold shrinkage estimators based on model selection, and multiple‐shrinkage estimators based on model averaging. These EB estimators are seen to be computationally competitive with standard classical thresholding methods, and to be robust to outliers in both the data and wavelet domains. Simulated and real examples are used to illustrate the flexibility and improved MSE performance of these methods in a wide variety of settings.

Redefine Statistical Significance

We propose to change the default P-value threshold for statistical significance from 0.05 to 0.005 for claims of new discoveries.

Prediction via Orthogonalized Model Mixing

Abstract We introduce an approach and algorithms for model mixing in large prediction problems with correlated predictors. We focus on the choice of predictors in linear models, and mix over possible subsets of candidate predictors. Our approach is based on expressing the space of models in terms of an orthogonalization of the design matrix. Advantages are both statistical and computational. Statistically, orthogonalization often leads to a reduction in the number of competing models by eliminating correlations. Computationally, large model spaces cannot be enumerated; recent approaches are based on sampling models with high posterior probability via Markov chains. Based on orthogonalization of the space of candidate predictors, we can approximate the posterior probabilities of models by products of predictor-specific terms. This leads to an importance sampling function for sampling directly from the joint distribution over the model space, without resorting to Markov chains. Compared to the latter, ortho...

Model uncertainty and health effect studies for particulate matter

There are many aspects of model choice that are involved in health effect studies of particulate matter and other pollutants. Some of these choices concern which pollutants and confounding variables should be included in the model, what type of lag structure for the covariates should be used, which interactions need to be considered, and how to model nonlinear trends. Because of the large number of potential variables, model selection is often used to find a parsimonious model. Different model selection strategies may lead to very different models and conclusions for the same set of data. As variable selection may involve numerous test of hypotheses, the resulting significance levels may be called into question, and there is the concern that the positive associations are a result of multiple testing. Bayesian Model Averaging is an alternative that can be used to combine inferences from multiple models and incorporate model uncertainty. This paper presents objective prior distributions for Bayesian Model Averaging in generalized linear models so that Bayesian model selection corresponds to standard methods of model selection, such as the Akaike Information Criterion (AIC) or Bayes Information Criterion (BIC), and inferences within a model are based on standard maximum likelihood estimation. These methods allow non-Bayesians to describe the level of uncertainty due to model selection, and can be used to combine inferences by averaging over a wider class of models using readily available summary statistics from standard model fitting programs. Using Bayesian Model Averaging and objective prior distributions, we re-analyze data from Birmingham, AL and illustrate the role of model uncertainty in inferences about the effect of particulate matter on elderly mortality. Copyright

Clinical response to varying the stimulus parameters in deep brain stimulation for essential tremor

Deep brain stimulation (DBS) of the ventral intermediate nucleus of the thalamus for essential tremor is sometimes limited by side effects. The mechanisms by which DBS alleviates tremor or causes side effects are unclear; thus, it is difficult to select stimulus parameters that maximize the width of the therapeutic window. The goal of this study was to quantify the impact on side effect intensity (SE), tremor amplitude, and the therapeutic window of varying stimulus parameters. Tremor amplitude and SE were recorded at 40 to 90 combinations of pulse width, frequency, and voltage across 14 thalami. Posterior variable inclusion probabilities indicated that frequency and voltage were the most important predictors of both SE and tremor amplitude. The amount of tremor suppression achieved at frequencies of 90 to 100 Hz was not different from that at 160 to 170 Hz. However, the width of the therapeutic window decreased significantly and power consumption increased as frequency was increased above 90 to 100 Hz. Improved understanding of the relationships between stimulus parameters and clinical responses may lead to improved techniques of stimulus parameter adjustment.

Bayesian Adaptive Sampling for Variable Selection and Model Averaging

For the problem of model choice in linear regression, we introduce a Bayesian adaptive sampling algorithm (BAS), that samples models without replacement from the space of models. For problems that permit enumeration of all models, BAS is guaranteed to enumerate the model space in 2p iterations where p is the number of potential variables under consideration. For larger problems where sampling is required, we provide conditions under which BAS provides perfect samples without replacement. When the sampling probabilities in the algorithm are the marginal variable inclusion probabilities, BAS may be viewed as sampling models “near” the median probability model of Barbieri and Berger. As marginal inclusion probabilities are not known in advance, we discuss several strategies to estimate adaptively the marginal inclusion probabilities within BAS. We illustrate the performance of the algorithm using simulated and real data and show that BAS can outperform Markov chain Monte Carlo methods. The algorithm is implemented in the R package BAS available at CRAN. This article has supplementary material online.

The Equivalence of Constrained and Weighted Designs in Multiple Objective Design Problems

Abstract Several competing objectives may be relevant in the design of an experiment. The competing objectives may not be easy to characterize in a single optimality criterion. One approach to these design problems has been to weight each criterion and find the design that optimizes the weighted average of the criteria. An alternative approach has been to optimize one criterion subject to constraints on the other criteria. An equivalence theorem is presented for the Bayesian constrained design problem. Equivalence theorems are essential in verifying optimality of proposed designs, especially when (as in most nonlinear design problems) numerical optimization is required. This theorem is used to show that the results of Cook and Wong on the equivalence of the weighted and constrained problems apply much more generally. The results are applied to Bayesian nonlinear design problems with several objectives.

Effects of avian seed dispersal on the genetic structure of whitebark pine populations

We used allozyme analysis to examine family structure, the spatial patterning of related individuals, in two populations of whitebark pine (Pinus albicaulis), a subalpine conifer that commonly displays a multistem form. The individual stems within clumps are genetically distinct individuals, having arisen from separate seeds. Individuals within a clump are genetically more similar than individuals in different clumps, but individuals in neighboring clumps do not appear to be more similar than individuals in distant clumps. This family structure appears to be a direct result of the seed‐caching behavior of Clarks nutcrackers (Nucifraga columbiana), the primary dispersal agent for whitebark pine seeds.

Single Nucleotide Polymorphisms in the TP53 Region and Susceptibility to Invasive Epithelial Ovarian Cancer

The p53 protein is critical for multiple cellular functions including cell growth and DNA repair. We assessed whether polymorphisms in the region encoding TP53 were associated with risk of invasive ovarian cancer. The study population includes a total of 5,206 invasive ovarian cancer cases (2,829 of which were serous) and 8,790 controls from 13 case-control or nested case-control studies participating in the Ovarian Cancer Association Consortium (OCAC). Three of the studies performed independent discovery investigations involving genotyping of up to 23 single nucleotide polymorphisms (SNP) in the TP53 region. Significant findings from this discovery phase were followed up for replication in the other OCAC studies. Mixed effects logistic regression was used to generate posterior median per allele odds ratios (OR), 95% probability intervals (PI), and Bayes factors (BF) for genotype associations. Five SNPs showed significant associations with risk in one or more of the discovery investigations and were followed up by OCAC. Mixed effects analysis confirmed associations with serous invasive cancers for two correlated (r(2) = 0.62) SNPs: rs2287498 (median per allele OR, 1.30; 95% PI, 1.07-1.57) and rs12951053 (median per allele OR, 1.19; 95% PI, 1.01-1.38). Analyses of other histologic subtypes suggested similar associations with endometrioid but not with mucinous or clear cell cancers. This large study provides statistical evidence for a small increase in risk of ovarian cancer associated with common variants in the TP53 region.