Bayesian multivariate logistic regression

Bayesian analyses of multivariate binary or categorical outcomes typically rely on probit or mixed effects logistic regression models that do not have a marginal logistic structure for the individual outcomes. In addition, difficulties arise when simple noninformative priors are chosen for the covariance parameters. Motivated by these problems, we propose a new type of multivariate logistic distribution that can be used to construct a likelihood for multivariate logistic regression analysis of binary and categorical data. The model for individual outcomes has a marginal logistic structure, simplifying interpretation. We follow a Bayesian approach to estimation and inference, developing an efficient data augmentation algorithm for posterior computation. The method is illustrated with application to a neurotoxicology study.     

Mapping Informative Clusters in a Hierarchial Framework of fMRI Multivariate Analysis

Pattern recognition methods have become increasingly popular in fMRI data analysis, which are powerful in discriminating between multi-voxel patterns of brain activities associated with different mental states. However, when they are used in functional brain mapping, the location of discriminative voxels varies significantly, raising difficulties in interpreting the locus of the effect. Here we proposed a hierarchical framework of multivariate approach that maps informative clusters rather than voxels to achieve reliable functional brain mapping without compromising the discriminative power. In particular, we first searched for local homogeneous clusters that consisted of voxels with similar response profiles. Then, a multi-voxel classifier was built for each cluster to extract discriminative information from the multi-voxel patterns. Finally, through multivariate ranking, outputs from the classifiers were served as a multi-cluster pattern to identify informative clusters by examining interactions among clusters. Results from both simulated and real fMRI data demonstrated that this hierarchical approach showed better performance in the robustness of functional brain mapping than traditional voxel-based multivariate methods. In addition, the mapped clusters were highly overlapped for two perceptually equivalent object categories, further confirming the validity of our approach. In short, the hierarchical framework of multivariate approach is suitable for both pattern classification and brain mapping in fMRI studies.     

Mapping informative clusters in a hierarchical [corrected] framework of FMRI multivariate analysis

Pattern recognition methods have become increasingly popular in fMRI data analysis, which are powerful in discriminating between multi-voxel patterns of brain activities associated with different mental states. However, when they are used in functional brain mapping, the location of discriminative voxels varies significantly, raising difficulties in interpreting the locus of the effect. Here we proposed a hierarchical framework of multivariate approach that maps informative clusters rather than voxels to achieve reliable functional brain mapping without compromising the discriminative power. In particular, we first searched for local homogeneous clusters that consisted of voxels with similar response profiles. Then, a multi-voxel classifier was built for each cluster to extract discriminative information from the multi-voxel patterns. Finally, through multivariate ranking, outputs from the classifiers were served as a multi-cluster pattern to identify informative clusters by examining interactions among clusters. Results from both simulated and real fMRI data demonstrated that this hierarchical approach showed better performance in the robustness of functional brain mapping than traditional voxel-based multivariate methods. In addition, the mapped clusters were highly overlapped for two perceptually equivalent object categories, further confirming the validity of our approach. In short, the hierarchical framework of multivariate approach is suitable for both pattern classification and brain mapping in fMRI studies.     

Parsimonious mixtures of multivariate contaminated normal distributions

A mixture of multivariate contaminated normal distributions is developed for model‐based clustering. In addition to the parameters of the classical normal mixture, our contaminated mixture has, for each cluster, a parameter controlling the proportion of mild outliers and one specifying the degree of contamination. Crucially, these parameters do not have to be specified a priori, adding a flexibility to our approach. Parsimony is introduced via eigen‐decomposition of the component covariance matrices, and sufficient conditions for the identifiability of all the members of the resulting family are provided. An expectation‐conditional maximization algorithm is outlined for parameter estimation and various implementation issues are discussed. Using a large‐scale simulation study, the behavior of the proposed approach is investigated and comparison with well‐established finite mixtures is provided. The performance of this novel family of models is also illustrated on artificial and real data.     

Factor analytic models of clustered multivariate data with informative censoring

This article describes a general class of factor analytic models for the analysis of clustered multivariate data in the presence of informative missingness. We assume that there are distinct sets of cluster-level latent variables related to the primary outcomes and to the censoring process, and we account for dependency between these latent variables through a hierarchical model. A linear model is used to relate covariates and latent variables to the primary outcomes for each subunit. A generalized linear model accounts for covariate and latent variable effects on the probability of censoring for subunits within each cluster. The model accounts for correlation within clusters and within subunits through a flexible factor analytic framework that allows multiple latent variables and covariate effects on the latent variables. The structure of the model facilitates implementation of Markov chain Monte Carlo methods for posterior estimation. Data from a spermatotoxicity study are analyzed to illustrate the proposed approach.     

Moving beyond noninformative priors: why and how to choose weakly informative priors in Bayesian analyses

Throughout the last two decades, Bayesian statistical methods have proliferated throughout ecology and evolution. Numerous previous references established both philosophical and computational guidelines for implementing Bayesian methods. However, protocols for incorporating prior information, the defining characteristic of Bayesian philosophy, are nearly nonexistent in the ecological literature. Here, I hope to encourage the use of weakly informative priors in ecology and evolution by providing a ‘consumer's guide’ to weakly informative priors. The first section outlines three reasons why ecologists should abandon noninformative priors: 1) common flat priors are not always noninformative, 2) noninformative priors provide the same result as simpler frequentist methods, and 3) noninformative priors suffer from the same high type I and type M error rates as frequentist methods. The second section provides a guide for implementing informative priors, wherein I detail convenient ‘reference’ prior distributions for common statistical models (i.e. regression, ANOVA, hierarchical models). I then use simulations to visually demonstrate how informative priors influence posterior parameter estimates. With the guidelines provided here, I hope to encourage the use of weakly informative priors for Bayesian analyses in ecology. Ecologists can and should debate the appropriate form of prior information, but should consider weakly informative priors as the new ‘default’ prior for any Bayesian model.     

Random effects selection in linear mixed models

We address the important practical problem of how to select the random effects component in a linear mixed model. A hierarchical Bayesian model is used to identify any random effect with zero variance. The proposed approach reparameterizes the mixed model so that functions of the covariance parameters of the random effects distribution are incorporated as regression coefficients on standard normal latent variables. We allow random effects to effectively drop out of the model by choosing mixture priors with point mass at zero for the random effects variances. Due to the reparameterization, the model enjoys a conditionally linear structure that facilitates the use of normal conjugate priors. We demonstrate that posterior computation can proceed via a simple and efficient Markov chain Monte Carlo algorithm. The methods are illustrated using simulated data and real data from a study relating prenatal exposure to polychlorinated biphenyls and psychomotor development of children.     

Robust modeling of additive and nonadditive variation with intuitive inclusion of expert knowledge

We propose a novel Bayesian approach that robustifies genomic modeling by leveraging expert knowledge (EK) through prior distributions. The central component is the hierarchical decomposition of phenotypic variation into additive and nonadditive genetic variation, which leads to an intuitive model parameterization that can be visualized as a tree. The edges of the tree represent ratios of variances, for example broad-sense heritability, which are quantities for which EK is natural to exist. Penalized complexity priors are defined for all edges of the tree in a bottom-up procedure that respects the model structure and incorporates EK through all levels. We investigate models with different sources of variation and compare the performance of different priors implementing varying amounts of EK in the context of plant breeding. A simulation study shows that the proposed priors implementing EK improve the robustness of genomic modeling and the selection of the genetically best individuals in a breeding program. We observe this improvement in both variety selection on genetic values and parent selection on additive values; the variety selection benefited the most. In a real case study, EK increases phenotype prediction accuracy for cases in which the standard maximum likelihood approach did not find optimal estimates for the variance components. Finally, we discuss the importance of EK priors for genomic modeling and breeding, and point to future research areas of easy-to-use and parsimonious priors in genomic modeling.     

Statistical multivariate metabolite profiling for aiding biomarker pattern detection and mechanistic interpretations in GC/MS based metabolomics

A strategy for robust and reliable mechanistic statistical modelling of metabolic responses in relation to drug induced toxicity is presented. The suggested approach addresses two cases commonly occurring within metabonomic toxicology studies, namely; 1) A pre-defined hypothesis about the biological mechanism exists and 2) No such hypothesis exists. GC/MS data from a liver toxicity study consisting of rat urine from control rats and rats exposed to a proprietary AstraZeneca compound were resolved by means of hierarchical multivariate curve resolution (H-MCR) generating 287 resolved chromatographic profiles with corresponding mass spectra. Filtering according to significance in relation to drug exposure rendered in 210 compound profiles, which were subjected to further statistical analysis following correction to account for the control variation over time. These dose related metabolite traces were then used as new observations in the subsequent analyses. For case 1, a multivariate approach, named Target Batch Analysis, based on OPLS regression was applied to correlate all metabolite traces to one or more key metabolites involved in the pre-defined hypothesis. For case 2, principal component analysis (PCA) was combined with hierarchical cluster analysis (HCA) to create a robust and interpretable framework for unbiased mechanistic screening. Both the Target Batch Analysis and the unbiased approach were cross-verified using the other method to ensure that the results did match in terms of detected metabolite traces. This was also the case, implying that this is a working concept for clustering of metabolites in relation to their toxicity induced dynamic profiles regardless if there is a pre-existing hypothesis or not. For each of the methods the detected metabolites were subjected to identification by means of data base comparison as well as verification in the raw data. The proposed strategy should be seen as a general approach for facilitating mechanistic modelling and interpretations in metabolomic studies.     

Ethics of randomised controlled trials – not yet time to give up on equipoise

In this commentary on Fries and Krishnan's argument that 'design bias' undermines the status of equipoise as the ethical justification for randomised controlled trials, it is argued that their argument is analogous to Bayesian arguments for the use of informative priors in trial design, but that this does not undermine the importance of equipoise. In particular, mismatches between the outcomes of interest to industrial sponsors of research and outcomes of interest to patients and clinicians ensure that in many cases industry-sponsored trials can fail to reflect the reasonable equipoise of working clinicians.     

Proper multivariate conditional autoregressive models for spatial data analysis

In the past decade conditional autoregressive modelling specifications have found considerable application for the analysis of spatial data. Nearly all of this work is done in the univariate case and employs an improper specification. Our contribution here is to move to multivariate conditional autoregressive models and to provide rich, flexible classes which yield proper distributions. Our approach is to introduce spatial autoregression parameters. We first clarify what classes can be developed from the family of Mardia (1988) and contrast with recent work of Kim et al. (2000). We then present a novel parametric linear transformation which provides an extension with attractive interpretation. We propose to employ these models as specifications for second-stage spatial effects in hierarchical models. Two applications are discussed; one for the two-dimensional case modelling spatial patterns of child growth, the other for a four-dimensional situation modelling spatial variation in HLA-B allele frequencies. In each case, full Bayesian inference is carried out using Markov chain Monte Carlo simulation.     

Analytic approaches to twin data using structural equation models

The classical twin study is the most popular design in behavioural genetics. It has strong roots in biometrical genetic theory, which allows predictions to be made about the correlations between observed traits of identical and fraternal twins in terms of underlying genetic and environmental components. One can infer the relative importance of these 'latent' factors (model parameters) by structural equation modelling (SEM) of observed covariances of both twin types. SEM programs estimate model parameters by minimising a goodness-of-fit function between observed and predicted covariance matrices, usually by the maximum-likelihood criterion. Likelihood ratio statistics also allow the comparison of fit of different competing models. The program Mx, specifically developed to model genetically sensitive data, is now widely used in twin analyses. The flexibility of Mx allows the modelling of multivariate data to examine the genetic and environmental relations between two or more phenotypes and the modelling to categorical traits under liability-threshold models.     

Proportionality of Diagonal Blocks with Blockwise Independence

In this paper, the tests of similarities among group covariance matrices and the differences among block covariance matrices within a group under repeated measurement model are studied. There are nine hierarchical nested structures of covariance matrices which have been tested. The likelihood ratio tests have been derived for these nine hierarchically structured models. An algorithm for determining the numerical solution of the corresponding maximum likelihood equations is also given.     

Tree Biomass Estimation of Chinese fir (Cunninghamia lanceolata) Based on Bayesian Method

Chinese fir (Cunninghamia lanceolata (Lamb.) Hook.) is the most important conifer species for timber production with huge distribution area in southern China. Accurate estimation of biomass is required for accounting and monitoring Chinese forest carbon stocking. In the study, allometric equation was used to analyze tree biomass of Chinese fir. The common methods for estimating allometric model have taken the classical approach based on the frequency interpretation of probability. However, many different biotic and abiotic factors introduce variability in Chinese fir biomass model, suggesting that parameters of biomass model are better represented by probability distributions rather than fixed values as classical method. To deal with the problem, Bayesian method was used for estimating Chinese fir biomass model. In the Bayesian framework, two priors were introduced: non-informative priors and informative priors. For informative priors, 32 biomass equations of Chinese fir were collected from published literature in the paper. The parameter distributions from published literature were regarded as prior distributions in Bayesian model for estimating Chinese fir biomass. Therefore, the Bayesian method with informative priors was better than non-informative priors and classical method, which provides a reasonable method for estimating Chinese fir biomass.     

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.     

Effects of selection and drift on G matrix evolution in a heterogeneous environment: a multivariate Qst-Fst Test with the freshwater snail Galba truncatula

Unraveling the effect of selection vs. drift on the evolution of quantitative traits is commonly achieved by one of two methods. Either one contrasts population differentiation estimates for genetic markers and quantitative traits (the Q(st)-F(st) contrast) or multivariate methods are used to study the covariance between sets of traits. In particular, many studies have focused on the genetic variance-covariance matrix (the G matrix). However, both drift and selection can cause changes in G. To understand their joint effects, we recently combined the two methods into a single test (accompanying article by Martin et al.), which we apply here to a network of 16 natural populations of the freshwater snail Galba truncatula. Using this new neutrality test, extended to hierarchical population structures, we studied the multivariate equivalent of the Q(st)-F(st) contrast for several life-history traits of G. truncatula. We found strong evidence of selection acting on multivariate phenotypes. Selection was homogeneous among populations within each habitat and heterogeneous between habitats. We found that the G matrices were relatively stable within each habitat, with proportionality between the among-populations (D) and the within-populations (G) covariance matrices. The effect of habitat heterogeneity is to break this proportionality because of selection for habitat-dependent optima. Individual-based simulations mimicking our empirical system confirmed that these patterns are expected under the selective regime inferred. We show that homogenizing selection can mimic some effect of drift on the G matrix (G and D almost proportional), but that incorporating information from molecular markers (multivariate Q(st)-F(st)) allows disentangling the two effects.     

Hierarchical models facilitate spatial analysis of large data sets: a case study on invasive plant species in the northeastern United States

Many critical ecological issues require the analysis of large spatial point data sets – for example, modelling species distributions, abundance and spread from survey data. But modelling spatial relationships, especially in large point data sets, presents major computational challenges. We use a novel Bayesian hierarchical statistical approach, 'spatial predictive process' modelling, to predict the distribution of a major invasive plant species, Celastrus orbiculatus , in the northeastern USA. The model runs orders of magnitude faster than traditional geostatistical models on a large data set of c . 4000 points, and performs better than generalized linear models, generalized additive models and geographically weighted regression in cross-validation. We also use this approach to model simultaneously the distributions of a set of four major invasive species in a spatially explicit multivariate model. This multispecies analysis demonstrates that some pairs of species exhibit negative residual spatial covariation, suggesting potential competitive interaction or divergent responses to unmeasured factors.     

A nonparametric approach to the discrimination of two cell populations

An approach to the statistical testing of differences between two cell populations the elements of which are characterized by multivariate data is described. The approach is based on the Fisher discriminant and the Kruskal-Wallis test. The method, which is sufficient but not necessary, makes no assumptions about the normality of the data or about the equality of the covariance matrices for each population.     

A predictive model of avian natal dispersal distance provides prior information for investigating response to landscape change

1. Informative Bayesian priors can improve the precision of estimates in ecological studies or estimate parameters for which little or no information is available. While Bayesian analyses are becoming more popular in ecology, the use of strongly informative priors remains rare, perhaps because examples of informative priors are not readily available in the published literature. 2. Dispersal distance is an important ecological parameter, but is difficult to measure and estimates are scarce. General models that provide informative prior estimates of dispersal distances will therefore be valuable. 3. Using a world-wide data set on birds, we develop a predictive model of median natal dispersal distance that includes body mass, wingspan, sex and feeding guild. This model predicts median dispersal distance well when using the fitted data and an independent test data set, explaining up to 53% of the variation. 4. Using this model, we predict a priori estimates of median dispersal distance for 57 woodland-dependent bird species in northern Victoria, Australia. These estimates are then used to investigate the relationship between dispersal ability and vulnerability to landscape-scale changes in habitat cover and fragmentation. 5. We find evidence that woodland bird species with poor predicted dispersal ability are more vulnerable to habitat fragmentation than those species with longer predicted dispersal distances, thus improving the understanding of this important phenomenon. 6. The value of constructing informative priors from existing information is also demonstrated. When used as informative priors for four example species, predicted dispersal distances reduced the 95% credible intervals of posterior estimates of dispersal distance by 8-19%. Further, should we have wished to collect information on avian dispersal distances and relate it to species' responses to habitat loss and fragmentation, data from 221 individuals across 57 species would have been required to obtain estimates with the same precision as those provided by the general model.     

Towards more meaningful hierarchical classification of amino acid scoring matrices.

Hierarchical classifications of the 20 amino acids according to residue relationships within scoring matrices have not hitherto been tested for reliability. In fact, testing here of the residue groupings obtained thus from 18 published matrices shows that they vary considerably in reliability. This behaviour gives a new insight then into the matrices with respect to the relationships between the amino acid scores contained therein. For example, other than the trivial grouping of the 20 amino acids, no reliable residue groupings are present in all 18 matrix amino acid hierarchical classifications. Hierarchical classification of the 18 scoring matrices themselves is investigated in terms of matrix representation and choice of similarity and dissimilarity measures for matrix comparison. There is no absolute standard against which to compare a matrix clustering, of course, but it is possible to assess the usefulness of a measure for the purpose in terms of the reliability of the calculated tree. Matrix representation is shown to be important. Finally, a novel two-step approach for hierarchical classification of the 18 amino acid scoring matrices is described.