Integrative Bayesian models using Post-selective inference: A case study in radiogenomics

Integrative analyses based on statistically relevant associations between genomics and a wealth of intermediary phenotypes (such as imaging) provide vital insights into their clinical relevance in terms of the disease mechanisms. Estimates for uncertainty in the resulting integrative models are however unreliable unless inference accounts for the selection of these associations with accuracy. In this paper, we develop selection-aware Bayesian methods, which (1) counteract the impact of model selection bias through a “selection-aware posterior” in a flexible class of integrative Bayesian models post a selection of promising variables via ℓ₁-regularized algorithms; (2) strike an inevitable trade-off between the quality of model selection and inferential power when the same data set is used for both selection and uncertainty estimation. Central to our methodological development, a carefully constructed conditional likelihood function deployed with a reparameterization mapping provides tractable updates when gradient-based Markov chain Monte Carlo (MCMC) sampling is used for estimating uncertainties from the selection-aware posterior. Applying our methods to a radiogenomic analysis, we successfully recover several important gene pathways and estimate uncertainties for their associations with patient survival times.     

Bayesian Analysis of Composite Quantile Regression

This paper introduces a Bayesian approach for composite quantile regression employing the skewed Laplace distribution for the error distribution. We use a two-level hierarchical Bayesian model for coefficient estimation and future selection which assumes a prior distribution that favors sparseness. An efficient Gibbs sampling algorithm is developed to update the unknown quantities from the posteriors. The proposed approach is illustrated via simulation studies and two real datasets. Results indicate that the proposed approach performs quite good in comparison to the other approaches.     

Latent likelihood ratio tests for assessing spatial kernels in epidemic models

One of the most important issues in the critical assessment of spatio-temporal stochastic models for epidemics is the selection of the transmission kernel used to represent the relationship between infectious challenge and spatial separation of infected and susceptible hosts. As the design of control strategies is often based on an assessment of the distance over which transmission can realistically occur and estimation of this distance is very sensitive to the choice of kernel function, it is important that models used to inform control strategies can be scrutinised in the light of observation in order to elicit possible evidence against the selected kernel function. While a range of approaches to model criticism is in existence, the field remains one in which the need for further research is recognised. In this paper, building on earlier contributions by the authors, we introduce a new approach to assessing the validity of spatial kernels—the latent likelihood ratio tests—which use likelihood-based discrepancy variables that can be used to compare the fit of competing models, and compare the capacity of this approach to detect model mis-specification with that of tests based on the use of infection-link residuals. We demonstrate that the new approach can be used to formulate tests with greater power than infection-link residuals to detect kernel mis-specification particularly when the degree of mis-specification is modest. This new tests avoid the use of a fully Bayesian approach which may introduce undesirable complications related to computational complexity and prior sensitivity.

     

Bayesian covariance selection in generalized linear mixed models

The generalized linear mixed model (GLMM), which extends the generalized linear model (GLM) to incorporate random effects characterizing heterogeneity among subjects, is widely used in analyzing correlated and longitudinal data. Although there is often interest in identifying the subset of predictors that have random effects, random effects selection can be challenging, particularly when outcome distributions are nonnormal. This article proposes a fully Bayesian approach to the problem of simultaneous selection of fixed and random effects in GLMMs. Integrating out the random effects induces a covariance structure on the multivariate outcome data, and an important problem that we also consider is that of covariance selection. Our approach relies on variable selection-type mixture priors for the components in a special Cholesky decomposition of the random effects covariance. A stochastic search MCMC algorithm is developed, which relies on Gibbs sampling, with Taylor series expansions used to approximate intractable integrals. Simulated data examples are presented for different exponential family distributions, and the approach is applied to discrete survival data from a time-to-pregnancy study.     

Comparison of approaches to combine species distribution models based on different sets of predictors

Distribution models should take into account the different limiting factors that simultaneously influence species ranges. Species distribution models built with different explanatory variables can be combined into more comprehensive ones, but the resulting models should maximize complementarity and avoid redundancy. Our aim was to compare the different methods available for combining species distribution models. We modelled 19 threatened vertebrate species in mainland Spain, producing models according to three individual explanatory factors: spatial constraints, topography and climate, and human influence. We used five approaches for model combination: Bayesian inference, Akaike weight averaging, stepwise variable selection, updating, and fuzzy logic. We compared the performance of these approaches by assessing different aspects of their classification and discrimination capacity. We demonstrated that different approaches to model combination give rise to disparities in the model outputs. Bayesian integration was systematically affected by an error in the equations that are habitually used in distribution modelling. Akaike weights produced models that were driven by the best single factor and therefore failed at combining the models effectively. The updating and the stepwise approaches shared recalibration as the basic concept for model combination, were very similar in their performance, and showed the highest sensitivity and discrimination capacity. The fuzzy‐logic approach yielded models with the highest classification capacity according to Cohen's kappa. In conclusion: 1) Bayesian integration, employing the currently used equation, and the Akaike weight procedure should be avoided; 2) the updating and stepwise approaches can be considered minor variants of the same recalibrating approach; and 3) there is a trade‐off between this recalibrating approach, which has the highest sensitivity, and fuzzy logic, which has the highest overall classification capacity. Recalibration is better if unfavourable conditions in one environmental factor may be counterbalanced with favourable conditions in a different factor, otherwise fuzzy logic is better.     

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.     

Experimental Design for Bayesian Estimations in the Linear Regression Model Taking Costs into Account

It is assumed that a known, correct, linear regression model (model I) is given. Let the problem be based on a Bayesian estimation of the regression parameter so that any available a priori information regarding this parameter can be used. This Bayesian estimation is, squared loss, an optimal strategy for the overall problem, which is divided into an estimation and a design problem. For practical reasons, the effort involved in performing the experiment will be taken into account as costs. In other words, the experimental design must result in the greatest possible accuracy for a given total cost (restriction of the sample size n). The linear cost function k(x) = 1 + c (x - a)/(b - a) is used to construct costoptimal experimental designs for simple linear regression by means of V = H = [a, b] in a way similar to that used for classical optimality criteria. The complicated structures of these designs and the difficulty in determining them by a direct approach have made it appear advisable to describe an iterative procedure for the construction of cost-optimal designs.     

On the Use of Bayesian Methods for Evaluating Compartmental Neural Models

Computational modeling is being used increasingly in neuroscience. In deriving such models, inference issues such as model selection, model complexity, and model comparison must be addressed constantly. In this article we present briefly the Bayesian approach to inference. Under a simple set of commonsense axioms, there exists essentially a unique way of reasoning under uncertainty by assigning a degree of confidence to any hypothesis or model, given the available data and prior information. Such degrees of confidence must obey all the rules governing probabilities and can be updated accordingly as more data becomes available. While the Bayesian methodology can be applied to any type of model, as an example we outline its use for an important, and increasingly standard, class of models in computational neuroscience—compartmental models of single neurons. Inference issues are particularly relevant for these models: their parameter spaces are typically very large, neurophysiological and neuroanatomical data are still sparse, and probabilistic aspects are often ignored. As a tutorial, we demonstrate the Bayesian approach on a class of one-compartment models with varying numbers of conductances. We then apply Bayesian methods on a compartmental model of a real neuron to determine the optimal amount of noise to add to the model to give it a level of spike time variability comparable to that found in the real cell.     

Bayesian Population Physiologically-Based Pharmacokinetic (PBPK) Approach for a Physiologically Realistic Characterization of Interindividual Variability in Clinically Relevant Populations

Interindividual variability in anatomical and physiological properties results in significant differences in drug pharmacokinetics. The consideration of such pharmacokinetic variability supports optimal drug efficacy and safety for each single individual, e.g. by identification of individual-specific dosings. One clear objective in clinical drug development is therefore a thorough characterization of the physiological sources of interindividual variability. In this work, we present a Bayesian population physiologically-based pharmacokinetic (PBPK) approach for the mechanistically and physiologically realistic identification of interindividual variability. The consideration of a generic and highly detailed mechanistic PBPK model structure enables the integration of large amounts of prior physiological knowledge, which is then updated with new experimental data in a Bayesian framework. A covariate model integrates known relationships of physiological parameters to age, gender and body height. We further provide a framework for estimation of the a posteriori parameter dependency structure at the population level. The approach is demonstrated considering a cohort of healthy individuals and theophylline as an application example. The variability and co-variability of physiological parameters are specified within the population; respectively. Significant correlations are identified between population parameters and are applied for individual- and population-specific visual predictive checks of the pharmacokinetic behavior, which leads to improved results compared to present population approaches. In the future, the integration of a generic PBPK model into an hierarchical approach allows for extrapolations to other populations or drugs, while the Bayesian paradigm allows for an iterative application of the approach and thereby a continuous updating of physiological knowledge with new data. This will facilitate decision making e.g. from preclinical to clinical development or extrapolation of PK behavior from healthy to clinically significant populations.     

Stochastic functional data analysis: a diffusion model-based approach

This article presents a new modeling strategy in functional data analysis. We consider the problem of estimating an unknown smooth function given functional data with noise. The unknown function is treated as the realization of a stochastic process, which is incorporated into a diffusion model. The method of smoothing spline estimation is connected to a special case of this approach. The resulting models offer great flexibility to capture the dynamic features of functional data, and allow straightforward and meaningful interpretation. The likelihood of the models is derived with Euler approximation and data augmentation. A unified Bayesian inference method is carried out via a Markov chain Monte Carlo algorithm including a simulation smoother. The proposed models and methods are illustrated on some prostate-specific antigen data, where we also show how the models can be used for forecasting.     

Bayesian variable selection for latent class models

In this article, we develop a latent class model with class probabilities that depend on subject-specific covariates. One of our major goals is to identify important predictors of latent classes. We consider methodology that allows estimation of latent classes while allowing for variable selection uncertainty. We propose a Bayesian variable selection approach and implement a stochastic search Gibbs sampler for posterior computation to obtain model-averaged estimates of quantities of interest such as marginal inclusion probabilities of predictors. Our methods are illustrated through simulation studies and application to data on weight gain during pregnancy, where it is of interest to identify important predictors of latent weight gain classes.     

Coestimation of recombination,substitution and molecular adaptation rates by approximate Bayesian computation

The estimation of parameters in molecular evolution may be biased when some processes are not considered. For example, the estimation of selection at the molecular level using codon-substitution models can have an upward bias when recombination is ignored. Here we address the joint estimation of recombination, molecular adaptation and substitution rates from coding sequences using approximate Bayesian computation (ABC). We describe the implementation of a regression-based strategy for choosing subsets of summary statistics for coding data, and show that this approach can accurately infer recombination allowing for intracodon recombination breakpoints, molecular adaptation and codon substitution rates. We demonstrate that our ABC approach can outperform other analytical methods under a variety of evolutionary scenarios. We also show that although the choice of the codon-substitution model is important, our inferences are robust to a moderate degree of model misspecification. In addition, we demonstrate that our approach can accurately choose the evolutionary model that best fits the data, providing an alternative for when the use of full-likelihood methods is impracticable. Finally, we applied our ABC method to co-estimate recombination, substitution and molecular adaptation rates from 24 published human immunodeficiency virus 1 coding data sets.     

Performance-based selection of likelihood models for phylogeny estimation

Phylogenetic estimation has largely come to rely on explicitly model-based methods. This approach requires that a model be chosen and that that choice be justified. To date, justification has largely been accomplished through use of likelihood-ratio tests (LRTs) to assess the relative fit of a nested series of reversible models. While this approach certainly represents an important advance over arbitrary model selection, the best fit of a series of models may not always provide the most reliable phylogenetic estimates for finite real data sets, where all available models are surely incorrect. Here, we develop a novel approach to model selection, which is based on the Bayesian information criterion, but incorporates relative branch-length error as a performance measure in a decision theory (DT) framework. This DT method includes a penalty for overfitting, is applicable prior to running extensive analyses, and simultaneously compares all models being considered and thus does not rely on a series of pairwise comparisons of models to traverse model space. We evaluate this method by examining four real data sets and by using those data sets to define simulation conditions. In the real data sets, the DT method selects the same or simpler models than conventional LRTs. In order to lend generality to the simulations, codon-based models (with parameters estimated from the real data sets) were used to generate simulated data sets, which are therefore more complex than any of the models we evaluate. On average, the DT method selects models that are simpler than those chosen by conventional LRTs. Nevertheless, these simpler models provide estimates of branch lengths that are more accurate both in terms of relative error and absolute error than those derived using the more complex (yet still wrong) models chosen by conventional LRTs. This method is available in a program called DT-ModSel.     

Multisensory Oddity Detection as Bayesian Inference

A key goal for the perceptual system is to optimally combine information from all the senses that may be available in order to develop the most accurate and unified picture possible of the outside world. The contemporary theoretical framework of ideal observer maximum likelihood integration (MLI) has been highly successful in modelling how the human brain combines information from a variety of different sensory modalities. However, in various recent experiments involving multisensory stimuli of uncertain correspondence, MLI breaks down as a successful model of sensory combination. Within the paradigm of direct stimulus estimation, perceptual models which use Bayesian inference to resolve correspondence have recently been shown to generalize successfully to these cases where MLI fails. This approach has been known variously as model inference, causal inference or structure inference. In this paper, we examine causal uncertainty in another important class of multi-sensory perception paradigm – that of oddity detection and demonstrate how a Bayesian ideal observer also treats oddity detection as a structure inference problem. We validate this approach by showing that it provides an intuitive and quantitative explanation of an important pair of multi-sensory oddity detection experiments – involving cues across and within modalities – for which MLI previously failed dramatically, allowing a novel unifying treatment of within and cross modal multisensory perception. Our successful application of structure inference models to the new ‘oddity detection’ paradigm, and the resultant unified explanation of across and within modality cases provide further evidence to suggest that structure inference may be a commonly evolved principle for combining perceptual information in the brain.     

Variable selection for nonparametric additive Cox model with interval-censored data

The standard Cox model is perhaps the most commonly used model for regression analysis of failure time data but it has some limitations such as the assumption on linear covariate effects. To relax this, the nonparametric additive Cox model, which allows for nonlinear covariate effects, is often employed, and this paper will discuss variable selection and structure estimation for this general model. For the problem, we propose a penalized sieve maximum likelihood approach with the use of Bernstein polynomials approximation and group penalization. To implement the proposed method, an efficient group coordinate descent algorithm is developed and can be easily carried out for both low- and high-dimensional scenarios. Furthermore, a simulation study is performed to assess the performance of the presented approach and suggests that it works well in practice. The proposed method is applied to an Alzheimer's disease study for identifying important and relevant genetic factors.     

Applications of the Dirichlet distribution to forensic match probabilities

The Dirichlet distribution provides a convenient conjugate prior for Bayesian analyses involving multinomial proportions. In particular, allele frequency estimation can be carried out with a Dirichlet prior. If data from several distinct populations are available, then the parameters characterizing the Dirichlet prior can be estimated by maximum likelihood and then used for allele frequency estimation in each of the separate populations. This empirical Bayes procedure tends to moderate extreme multinomial estimates based on sample proportions. The Dirichlet distribution can also be employed to model the contributions from different ancestral populations in computing forensic match probabilities. If the ancestral populations are in genetic equilibrium, then the product rule for computing match probabilities is valid conditional on the ancestral contributions to a typical person of the reference population. This fact facilitates computation of match probabilities and tight upper bounds to match probabilities.Editor's commentsThe author continues the formal Bayesian analysis introduced by Gjertson & Morris in this voluem. He invokes Dirichlet distributions, and so brings rigor to the discussion of the effects of population structure on match probabilities. The increased computational burden this approach entails should not be regarded as a hindrance.     

Statistical Evaluation of Toxicological Experimental Design for Bayesian Model Averaged Benchmark Dose Estimation with Dichotomous Data

A method is presented to statistically evaluate toxicity study design for dose– response assessment aimed at minimizing the uncertainty in resulting Benchmark dose (BMD) estimates. Although the BMD method has been accepted as a valuable tool for risk assessment, the traditional no observed adverse effect level (NOAEL)/lowest observed adverse effective level (LOAEL) approach is still the principal basis for toxicological study design. To develop similar protocols for experimental design for BMD estimation, methods are needed that account for variability in experimental outcomes, and uncertainty in dose–response model selection and model parameter estimates. Based on Bayesian model averaging (BMA) BMD estimation, this study focuses on identifying the study design criteria that can reduce the uncertainty in BMA BMD estimates by using a Monte Carlo pre-posterior analysis on BMA BMD predictions. The results suggest that (1) as more animals are tested there is less uncertainty in BMD estimates; (2) one relatively high dose is needed and other doses can then be appropriately spread over the resulting dose scale; (3) placing different numbers of animals in different dose groups has very limited influence on improving BMD estimation; and (4) when the total number of animals is fixed, using more (but smaller) dose groups is a preferred strategy.     

Bayesian Updating of Model-Based Risk Estimates Using Imperfect Public Health Surveillance Data

Model-based estimation of the human health risks resulting from exposure to environmental contaminants can be an important tool for structuring public health policy. Due to uncertainties in the modeling process, the outcomes of these assessments are usually probabilistic representations of a range of possible risks. In some cases, health surveillance data are available for the assessment population over all or a subset of the risk projection period and this additional information can be used to augment the model-based estimates. We use a Bayesian approach to update model-based estimates of health risks based on available health outcome data. Updated uncertainty distributions for risk estimates are derived using Monte Carlo sampling, which allows flexibility to model realistic situations including measurement error in the observable outcomes. We illustrate the approach by using imperfect public health surveillance data on lung cancer deaths to update model-based lung cancer mortality risk estimates in a population exposed to ionizing radiation from a uranium processing facility.     

Combined linkage disequilibrium and linkage mapping: Bayesian multilocus approach

Quantitative trait loci (QTL) affecting the phenotype of interest can be detected using linkage analysis (LA), linkage disequilibrium (LD) mapping or a combination of both (LDLA). The LA approach uses information from recombination events within the observed pedigree and LD mapping from the historical recombinations within the unobserved pedigree. We propose the Bayesian variable selection approach for combined LDLA analysis for single-nucleotide polymorphism (SNP) data. The novel approach uses both sources of information simultaneously as is commonly done in plant and animal genetics, but it makes fewer assumptions about population demography than previous LDLA methods. This differs from approaches in human genetics, where LDLA methods use LA information conditional on LD information or the other way round. We argue that the multilocus LDLA model is more powerful for the detection of phenotype–genotype associations than single-locus LDLA analysis. To illustrate the performance of the Bayesian multilocus LDLA method, we analyzed simulation replicates based on real SNP genotype data from small three-generational CEPH families and compared the results with commonly used quantitative transmission disequilibrium test (QTDT). This paper is intended to be conceptual in the sense that it is not meant to be a practical method for analyzing high-density SNP data, which is more common. Our aim was to test whether this approach can function in principle.     

Bayesian infinite mixture model based clustering of gene expression profiles

MOTIVATION: The biologic significance of results obtained through cluster analyses of gene expression data generated in microarray experiments have been demonstrated in many studies. In this article we focus on the development of a clustering procedure based on the concept of Bayesian model-averaging and a precise statistical model of expression data. RESULTS: We developed a clustering procedure based on the Bayesian infinite mixture model and applied it to clustering gene expression profiles. Clusters of genes with similar expression patterns are identified from the posterior distribution of clusterings defined implicitly by the stochastic data-generation model. The posterior distribution of clusterings is estimated by a Gibbs sampler. We summarized the posterior distribution of clusterings by calculating posterior pairwise probabilities of co-expression and used the complete linkage principle to create clusters. This approach has several advantages over usual clustering procedures. The analysis allows for incorporation of a reasonable probabilistic model for generating data. The method does not require specifying the number of clusters and resulting optimal clustering is obtained by averaging over models with all possible numbers of clusters. Expression profiles that are not similar to any other profile are automatically detected, the method incorporates experimental replicates, and it can be extended to accommodate missing data. This approach represents a qualitative shift in the model-based cluster analysis of expression data because it allows for incorporation of uncertainties involved in the model selection in the final assessment of confidence in similarities of expression profiles. We also demonstrated the importance of incorporating the information on experimental variability into the clustering model. AVAILABILITY: The MS Windows(TM) based program implementing the Gibbs sampler and supplemental material is available at http://homepages.uc.edu/~medvedm/BioinformaticsSupplement.htm CONTACT: medvedm@email.uc.edu