Bayesian regression with multivariate linear splines

We present a Bayesian analysis of a piecewise linear model constructed by using basis functions which generalizes the univariate linear spline to higher dimensions. Prior distributions are adopted on both the number and the locations of the splines, which leads to a model averaging approach to prediction with predictive distributions that take into account model uncertainty. Conditioning on the data produces a Bayes local linear model with distributions on both predictions and local linear parameters. The method is spatially adaptive and covariate selection is achieved by using splines of lower dimension than the data.     

The mode oriented stochastic search (MOSS) algorithm for log-linear models with conjugate priors

We describe a novel stochastic search algorithm for rapidly identifying regions of high posterior probability in the space of decomposable, graphical and hierarchical log-linear models. Our approach is based on the Diaconis–Ylvisaker conjugate prior for log-linear parameters. We discuss the computation of Bayes factors through Laplace approximations and the Bayesian iterative proportional fitting algorithm for sampling model parameters. We use our model determination approach in a sparse eight-way contingency table.     

Estimation in the generalized linear empirical bayes model using the extended quasi-likelihood

A generalized linear empirical Bayes model is developed for empirical Bayes analysis of several means in natural exponential families. A unified approach is presented for all natural exponential families with quadratic variance functions (the Normal, Poisson, Binomial, Gamma, and two others.) The hyperparameters are estimated using the extended quasi-likelihood of Nelder and Pregibon (1987), which is easily implemented via the GLIM package. The accuracy of these estimates is developed by asymptotic approximation of the variance. Two data examples are illustrated.     

Learning from a lot: Empirical Bayes for high‐dimensional model‐based prediction

Empirical Bayes is a versatile approach to “learn from a lot” in two ways: first, from a large number of variables and, second, from a potentially large amount of prior information, for example, stored in public repositories. We review applications of a variety of empirical Bayes methods to several well‐known model‐based prediction methods, including penalized regression, linear discriminant analysis, and Bayesian models with sparse or dense priors. We discuss “formal” empirical Bayes methods that maximize the marginal likelihood but also more informal approaches based on other data summaries. We contrast empirical Bayes to cross‐validation and full Bayes and discuss hybrid approaches. To study the relation between the quality of an empirical Bayes estimator and p, the number of variables, we consider a simple empirical Bayes estimator in a linear model setting. We argue that empirical Bayes is particularly useful when the prior contains multiple parameters, which model a priori information on variables termed “co‐data”. In particular, we present two novel examples that allow for co‐data: first, a Bayesian spike‐and‐slab setting that facilitates inclusion of multiple co‐data sources and types and, second, a hybrid empirical Bayes–full Bayes ridge regression approach for estimation of the posterior predictive interval.     

Building blocks for graphical belief models

The graphical belief model is a versatile tool for modeling complex systems. The graphical structure and its implicit probabilistic and logical independence conditions define the relationships between many of the variables of the problem. The graphical model is composed of a collection of local models:models of both interactions between the variables sharing a common hyperedge and information about single variables. These local models can be constructed with either probability distributions or belief functions. This paper takes the latter approach and describes simple models for univariate and multivariate belief functions. The examples are taken from both reliability and knowledge representation problems.     

Inference in model-based cluster analysis

A new approach to cluster analysis has been introduced based on parsimonious geometric modelling of the within-group covariance matrices in a mixture of multivariate normal distributions, using hierarchical agglomeration and iterative relocation. It works well and is widely used via the MCLUST software available in S-PLUS and StatLib. However, it has several limitations: there is no assessment of the uncertainty about the classification, the partition can be suboptimal, parameter estimates are biased, the shape matrix has to be specified by the user, prior group probabilities are assumed to be equal, the method for choosing the number of groups is based on a crude approximation, and no formal way of choosing between the various possible models is included. Here, we propose a new approach which overcomes all these difficulties. It consists of exact Bayesian inference via Gibbs sampling, and the calculation of Bayes factors (for choosing the model and the number of groups) from the output using the Laplace–Metropolis estimator. It works well in several real and simulated examples.     

Objective Bayes Factors for Gaussian Directed Acyclic Graphical Models

Abstract. We propose an objective Bayesian method for the comparison of all Gaussian directed acyclic graphical models defined on a given set of variables. The method, which is based on the notion of fractional Bayes factor (BF), requires a single default (typically improper) prior on the space of unconstrained covariance matrices, together with a prior sample size hyper‐parameter, which can be set to its minimal value. We show that our approach produces genuine BFs. The implied prior on the concentration matrix of any complete graph is a data‐dependent Wishart distribution, and this in turn guarantees that Markov equivalent graphs are scored with the same marginal likelihood. We specialize our results to the smaller class of Gaussian decomposable undirected graphical models and show that in this case they coincide with those recently obtained using limiting versions of hyper‐inverse Wishart distributions as priors on the graph‐constrained covariance matrices.     

On MCMC sampling in hierarchical longitudinal models

Markov chain Monte Carlo (MCMC) algorithms have revolutionized Bayesian practice. In their simplest form (i.e., when parameters are updated one at a time) they are, however, often slow to converge when applied to high-dimensional statistical models. A remedy for this problem is to block the parameters into groups, which are then updated simultaneously using either a Gibbs or Metropolis-Hastings step. In this paper we construct several (partially and fully blocked) MCMC algorithms for minimizing the autocorrelation in MCMC samples arising from important classes of longitudinal data models. We exploit an identity used by Chib (1995) in the context of Bayes factor computation to show how the parameters in a general linear mixed model may be updated in a single block, improving convergence and producing essentially independent draws from the posterior of the parameters of interest. We also investigate the value of blocking in non-Gaussian mixed models, as well as in a class of binary response data longitudinal models. We illustrate the approaches in detail with three real-data examples.     

Extended linear empirical bayes estimation

In this paper, the linear empirical Bayes estimation method, which is based on approximation of the Bayes estimator by a linear function, is generalized to an extended linear empirical Bayes estimation technique which represents the Bayes estimator by a series of algebraic polynomials. The extended linear empirical Bayes estimators are elaborated in the case of a location or a scale parameter. The theory is illustrated by examples of its application to the normal distribution with a location parameter and the gamma distribution with a scale parameter. The linear and the extended linear empirical Bayes estimators are constructed in these two cases and, then, studied numerically via Monte Carlo simulations. The simulations show that the extended linear empirical Bayes estimators have better convergence rates than the traditional linear empirical Bayes estimators.     

Local Influence Analysis in AB–BA Crossover Designs

The aim of this article is to develop methodology for detecting influential observations in crossover models with random individual effects. Various case‐weighted perturbations are performed. We obtain the influence of the perturbations on each parameter estimator and on their dispersion matrices. The obtained results exhibit the possibility to obtain closed‐form expressions of the influence using the residuals in mixed linear models. Some graphical tools are also presented.     

Reference priors for linear models with general covariance structures

We develop a new class of reference priors for linear models with general covariance structures. A general Markov chain Monte Carlo algorithm is also proposed for implementing the computation. We present several examples to demonstrate the results: Bayesian penalized spline smoothing, a Bayesian approach to bivariate smoothing for a spatial model, and prior specification for structural equation models.     

Influence on tests with focus on linear models

To assess the influence of single observations on the parameter estimates, case-deletion diagnostics are commonly used in linear regression models; one example is Cook's distance. For nested parametric models we consider a deletion diagnostic for evaluating the influence of a single observation on the likelihood ratio (LR) test. In order to have a common scale as reference, the asymptotic distribution of the diagnostic is derived and the values of the diagnostic are converted to percentiles. We focus on linear models and general linear models, and in these cases explicit results are derived. The performance of the diagnostic is explored in two small bench mark examples from linear regression and in a larger linear mixed model example.     

Bayes estimates in multivariate semiparametric linear models

Bayes estimates are derived in multivariate linear models with unknown distribution. The prior distribution is defined using a Dirichlet prior for the unknown error distribution and a normal-Wishart distribution for the parameters. The posterior distribution is determined and explicit expressions are given in the special cases of location-scale and two-sample models. The calculation of self-informative limits of Bayes estimates yields standard estimates.     

Prediction in multilevel generalized linear models

Summary.  We discuss prediction of random effects and of expected responses in multilevel generalized linear models. Prediction of random effects is useful for instance in small area estimation and disease mapping, effectiveness studies and model diagnostics. Prediction of expected responses is useful for planning, model interpretation and diagnostics. For prediction of random effects, we concentrate on empirical Bayes prediction and discuss three different kinds of standard errors; the posterior standard deviation and the marginal prediction error standard deviation (comparative standard errors) and the marginal sampling standard deviation (diagnostic standard error). Analytical expressions are available only for linear models and are provided in an appendix . For other multilevel generalized linear models we present approximations and suggest using parametric bootstrapping to obtain standard errors. We also discuss prediction of expectations of responses or probabilities for a new unit in a hypothetical cluster, or in a new (randomly sampled) cluster or in an existing cluster. The methods are implemented in gllamm and illustrated by applying them to survey data on reading proficiency of children nested in schools. Simulations are used to assess the performance of various predictions and associated standard errors for logistic random-intercept models under a range of conditions.     

Local influence for incomplete data models

This paper proposes a method to assess the local influence in a minor perturbation of a statistical model with incomplete data. The idea is to utilize Cook's approach to the conditional expectation of the complete-data log-likelihood function in the EM algorithm. It is shown that the method proposed produces analytic results that are very similar to those obtained from a classical local influence approach based on the observed data likelihood function and has the potential to assess a variety of complicated models that cannot be handled by existing methods. An application to the generalized linear mixed model is investigated. Some illustrative artificial and real examples are presented.     

Choosing the link function and accounting for link uncertainty in generalized linear models using Bayes factors

One important component of model selection using generalized linear models (GLM) is the choice of a link function. We propose using approximate Bayes factors to assess the improvement in fit over a GLM with canonical link when a parametric link family is used. The approximate Bayes factors are calculated using the Laplace approximations given in [32], together with a reference set of prior distributions. This methodology can be used to differentiate between different parametric link families, as well as allowing one to jointly select the link family and the independent variables. This involves comparing nonnested models and so standard significance tests cannot be used. The approach also accounts explicitly for uncertainty about the link function. The methods are illustrated using parametric link families studied in [12] for two data sets involving binomial responses. The first author was supported by Sonderforschungsbereich 386 Statistische Analyse Diskreter Strukturen, and the second author by NIH Grant 1R01CA094212-01 and ONR Grant N00014-01-10745.     

Robustness measures for Bayes linear analyses

We consider the role of global robustness measures in Bayes linear analysis. We suggest two such measures, one for expectation comparisons and one for variance comparisons. Geometric interpretations of the measures are presented. The approach is illustrated by considering the robustness of certain multiplicative models to assumptions of independence, with particular application to a problem arising in an asset management model for water resources.     

Influence diagnostics for stratified ordinal contingency tables

Influence diagnostics are investigated in this study. In particular, an approach based on the generalized linear mixed model setting is presented for formulating ordered categorical counts in stratified contingency tables. Deletion diagnostics and their first-order approximations are developed for assessing the stratum-specific influence on parameter estimates in the models. To illustrate the proposed model diagnostic technique, the method is applied to analyze two sets of data: a clinical trial and a survey study. The two examples demonstrate that the presence of influential strata may substantially change the results in ordinal contingency table analysis.     

Comparing two clinical measurements: a linear mixed model approach

In this article, we extended the widely used Bland-Altman graphical technique of comparing two measurements in clinical studies to include an analytical approach using a linear mixed model. The proposed statistical inferences can be conducted easily by commercially available statistical software such as SAS. The linear mixed model approach was illustrated using a real example in a clinical nursing study of oxygen saturation measurements, when functional oxygen saturation was compared against fractional oxy-hemoglobin.     

Projectors and linear estimation in general linear models

The paper gives a self-contained account of minimum disper­sion linear unbiased estimation of the expectation vector in a linear model with the dispersion matrix belonging to some, rather arbitrary, set of nonnegative definite matrices. The approach to linear estimation in general linear models recommended here is a direct generalization of some ideas and results presented by Rao (1973, 19 74) for the case of a general Gauss-Markov model

A new insight into the nature of some estimation problems originaly arising in the context of a general Gauss-Markov model as well as the correspondence of results known in the literature to those obtained in the present paper for general linear models are also given. As preliminary results the theory of projectors defined by Rao (1973) is extended.