Multivariate thin plate spline estimates for the posterior probabilities in the classification problem

A nonparametric estimate for the posterior probabilities in the classification problem using multivariate thin plate splines is proposed. This method presents a nonpararnetric alternative to logistic discrimination as well as to survival curve estimation. The degree of smoothness of the estimate is determined from the data using generalized crossvalidation.     

Subjective priors for the dirichlet process

This paper reviews difficulties with the interpretation and use of the prior parameter u required in the Dirichlet approach to nonpararnetric Bayesian statistics. Two subjective prior distributions are introduced and studied. These priors are obtained computationally by requiring that the experimenter specify certain constraints.     

Analyzing space-time sensor network data under suppression and failure in transmission

In this paper we present a fully model-based analysis of the effects of suppression and failure in data transmission with sensor networks. Sensor networks are becoming an increasingly common data collection mechanism in a variety of fields. Sensors can be created to collect data at very high temporal resolution. However, during periods when the process is following a stable path, transmission of such high resolution data would carry little additional information with regard to the process model, i.e., all of the data that is collected need not be transmitted. In particular, when there is cost to transmission, we find ourselves moving to consideration of suppression in transmission. Additionally, for many sensor networks, in practice, we will experience failures in transmission—messages sent by a sensor but not received at the gateway, messages sent but arriving corrupted. Evidently, both suppression and failure lead to information loss which will be reflected in inference associated with our process model. Our effort here is to assess the impact of such information loss under varying extents of suppression and varying incidence of failure. We consider two illustrative process models, presenting fully model-based analyses of suppression and failure using hierarchical models. Such models naturally facilitate borrowing strength across nodes, leveraging all available data to learn about local process behavior.     

Some comments on efficiency gains from auxiliary information for right-censored data

Heavily right-censored time to event, or survival, data arise frequently in research areas such as medicine and industrial reliability. Recently, there have been suggestions that auxiliary outcomes which are more fully observed may be used to “enhance” or increase the efficiency of inferences for a primary survival time variable. However, efficiency gains from this approach have mostly been very small. Most of the situations considered have involved semiparametric models, so in this note we consider two very simple fully parametric models. In the one case involving a correlated auxiliary variable that is always observed, we find that efficiency gains are small unless the response and auxiliary variable are very highly correlated and the response is heavily censored. In the second case, which involves an intermediate stage in a three-stage model of failure, the efficiency gains can be more substantial. We suggest that careful study of specific situations is needed to identify opportunities for “enhanced” inferences, but that substantial gains seem more likely when auxiliary information involves structural information about the failure process.     

Using auxiliary data for parameter estimation with non-ignorably missing outcomes

We propose a method for estimating parameters in generalized linear models when the outcome variable is missing for some subjects and the missing data mechanism is non-ignorable. We assume throughout that the covariates are fully observed. One possible method for estimating the parameters is maximum likelihood with a non-ignorable missing data model. However, caution must be used when fitting non-ignorable missing data models because certain parameters may be inestimable for some models. Instead of fitting a non-ignorable model, we propose the use of auxiliary information in a likelihood approach to reduce the bias, without having to specify a non-ignorable model. The method is applied to a mental health study.     

Conjugacy class prior distributions on metric-based ranking models

Summary. A new class of prior distributions for metric-based models in the analysis of fully and partially ranked data is developed. This class is attractive because it provides a meaningful way to encapsulate prior information about the parameters of the model. Three examples illustrate the ideas developed in the paper.     

Bayesian inference of the fully specified subdistribution model for survival data with competing risks

Competing risks data are routinely encountered in various medical applications due to the fact that patients may die from different causes. Recently, several models have been proposed for fitting such survival data. In this paper, we develop a fully specified subdistribution model for survival data in the presence of competing risks via a subdistribution model for the primary cause of death and conditional distributions for other causes of death. Various properties of this fully specified subdistribution model have been examined. An efficient Gibbs sampling algorithm via latent variables is developed to carry out posterior computations. Deviance information criterion (DIC) and logarithm of the pseudomarginal likelihood (LPML) are used for model comparison. An extensive simulation study is carried out to examine the performance of DIC and LPML in comparing the cause-specific hazards model, the mixture model, and the fully specified subdistribution model. The proposed methodology is applied to analyze a real dataset from a prostate cancer study in detail.     

Network-based genomic discovery: application and comparison of Markov random field models

Summary.  As biological knowledge accumulates rapidly, gene networks encoding genomewide gene–gene interactions have been constructed. As an improvement over the standard mixture model that tests all the genes identically and independently distributed a priori , Wei and co-workers have proposed modelling a gene network as a discrete or Gaussian Markov random field (MRF) in a mixture model to analyse genomic data. However, how these methods compare in practical applications is not well understood and this is the aim here. We also propose two novel constraints in prior specifications for the Gaussian MRF model and a fully Bayesian approach to the discrete MRF model. We assess the accuracy of estimating the false discovery rate by posterior probabilities in the context of MRF models. Applications to a chromatin immuno-precipitation–chip data set and simulated data show that the modified Gaussian MRF models have superior performance compared with other models, and both MRF-based mixture models, with reasonable robustness to misspecified gene networks, outperform the standard mixture model.     

Accounting for uncertainty in health economic decision models by using model averaging

Summary.  Health economic decision models are subject to considerable uncertainty, much of which arises from choices between several plausible model structures, e.g. choices of covariates in a regression model. Such structural uncertainty is rarely accounted for formally in decision models but can be addressed by model averaging. We discuss the most common methods of averaging models and the principles underlying them. We apply them to a comparison of two surgical techniques for repairing abdominal aortic aneurysms. In model averaging, competing models are usually either weighted by using an asymptotically consistent model assessment criterion, such as the Bayesian information criterion, or a measure of predictive ability, such as Akaike's information criterion. We argue that the predictive approach is more suitable when modelling the complex underlying processes of interest in health economics, such as individual disease progression and response to treatment.     

An extended random-effects approach to modeling repeated, overdispersed count data

Non-Gaussian outcomes are often modeled using members of the so-called exponential family. The Poisson model for count data falls within this tradition. The family in general, and the Poisson model in particular, are at the same time convenient since mathematically elegant, but in need of extension since often somewhat restrictive. Two of the main rationales for existing extensions are (1) the occurrence of overdispersion, in the sense that the variability in the data is not adequately captured by the model's prescribed mean-variance link, and (2) the accommodation of data hierarchies owing to, for example, repeatedly measuring the outcome on the same subject, recording information from various members of the same family, etc. There is a variety of overdispersion models for count data, such as, for example, the negative-binomial model. Hierarchies are often accommodated through the inclusion of subject-specific, random effects. Though not always, one conventionally assumes such random effects to be normally distributed. While both of these issues may occur simultaneously, models accommodating them at once are less than common. This paper proposes a generalized linear model, accommodating overdispersion and clustering through two separate sets of random effects, of gamma and normal type, respectively. This is in line with the proposal by Booth et al. (Stat Model 3:179-181, 2003). The model extends both classical overdispersion models for count data (Breslow, Appl Stat 33:38-44, 1984), in particular the negative binomial model, as well as the generalized linear mixed model (Breslow and Clayton, J Am Stat Assoc 88:9-25, 1993). Apart from model formulation, we briefly discuss several estimation options, and then settle for maximum likelihood estimation with both fully analytic integration as well as hybrid between analytic and numerical integration. The latter is implemented in the SAS procedure NLMIXED. The methodology is applied to data from a study in epileptic seizures.     

Multistate recapture models: Modelling incomplete individual histories

Multistate capture-recapture models are a natural generalization of the usual one-site recapture models. Similarly, individuals are sampled on discrete occasions, at which they may be captured or not. However, contrary to the one-site case, the individuals can move within a finite set of states between occasions. The growing interest in spatial aspects of population dynamics presently contributes to making multistate models a very promising tool for population biology. We review first the interest and the potential of multistate models, in particular when they are used with individual states as well as geographical sites. Multistate models indeed constitute canonical capture-recapture models for individual categorical covariates changing over time, and can be linked to longitudinal studies with missing data and models such as hidden Markov chains. Multistate models also provide a promising tool for handling heterogeneity of capture, provided states related to capturability can be defined and used. Such an approach could be relevant for population size estimation in closed populations. Multistate models also constitute a natural framework for mixtures of information in individual history data. Presently, most models can be fit using program MARK. As an example, we present a canonical model for multisite accession to reproduction, which fully generalizes a classical one-site model. In the generalization proposed, one can estimate simultaneously age-dependent rates of accession to reproduction, natal and breeding dispersal. Finally, we discuss further generalizations - such as a multistate generalization of growth rate models and models for data where the state in which an individual is detected is known with uncertainty - and prospects for software development.     

Multistate recapture models: modelling incomplete individual histories

Multistate capture-recapture models are a natural generalization of the usual one-site recapture models. Similarly, individuals are sampled on discrete occasions, at which they may be captured or not. However, contrary to the one-site case, the individuals can move within a finite set of states between occasions. The growing interest in spatial aspects of population dynamics presently contributes to making multistate models a very promising tool for population biology. We review first the interest and the potential of multistate models, in particular when they are used with individual states as well as geographical sites. Multistate models indeed constitute canonical capture-recapture models for individual categorical covariates changing over time, and can be linked to longitudinal studies with missing data and models such as hidden Markov chains. Multistate models also provide a promising tool for handling heterogeneity of capture, provided states related to capturability can be defined and used. Such an approach could be relevant for population size estimation in closed populations. Multistate models also constitute a natural framework for mixtures of information in individual history data. Presently, most models can be fit using program MARK. As an example, we present a canonical model for multisite accession to reproduction, which fully generalizes a classical one-site model. In the generalization proposed, one can estimate simultaneously age-dependent rates of accession to reproduction, natal and breeding dispersal. Finally, we discuss further generalizations - such as a multistate generalization of growth rate models and models for data where the state in which an individual is detected is known with uncertainty - and prospects for software development.     

General multivariate linear models for longitudional studies

Longitudinal studies occcur frequently in many different disciplines. To fully utilize the potential value of the information contained in a longitudinal data, various multivariate linear models have been proposed. The methodology and analysis are somewhat unique in their own ways and their relationships are not well understood and presented. This article describes a general multivaritate linear model for longitudinal data and attempts to provide a constructive formulation of the components in the mean response profile. The objective is to point out the extension and connections of some well-known models that have been obscured by different areas of application. More imporiantly, the model is expressed in a unified regression form from the subject matter considerations. Such an approach is simpler and more intuitive than other ways to modeling and parameter estimation. As a cmsequeace the analyses of the general class cf models for longitudional data can be casily implemented with standard software.     

Prior Distributions for Item Parameters in IRT Models

The focus of this article is on the choice of suitable prior distributions for item parameters within item response theory (IRT) models. In particular, the use of empirical prior distributions for item parameters is proposed. Firstly, regression trees are implemented in order to build informative empirical prior distributions. Secondly, model estimation is conducted within a fully Bayesian approach through the Gibbs sampler, which makes estimation feasible also with increasingly complex models. The main results show that item parameter recovery is improved with the introduction of empirical prior information about item parameters, also when only a small sample is available.     

Criteria for Linear Model Selection Based on Kullback's Symmetric Divergence

Model selection criteria are frequently developed by constructing estimators of discrepancy measures that assess the disparity between the 'true' model and a fitted approximating model. The Akaike information criterion (AIC) and its variants result from utilizing Kullback's directed divergence as the targeted discrepancy. The directed divergence is an asymmetric measure of separation between two statistical models, meaning that an alternative directed divergence can be obtained by reversing the roles of the two models in the definition of the measure. The sum of the two directed divergences is Kullback's symmetric divergence. In the framework of linear models, a comparison of the two directed divergences reveals an important distinction between the measures. When used to evaluate fitted approximating models that are improperly specified, the directed divergence which serves as the basis for AIC is more sensitive towards detecting overfitted models, whereas its counterpart is more sensitive towards detecting underfitted models. Since the symmetric divergence combines the information in both measures, it functions as a gauge of model disparity which is arguably more balanced than either of its individual components. With this motivation, the paper proposes a new class of criteria for linear model selection based on targeting the symmetric divergence. The criteria can be regarded as analogues of AIC and two of its variants: 'corrected' AIC or AICc and 'modified' AIC or MAIC. The paper examines the selection tendencies of the new criteria in a simulation study and the results indicate that they perform favourably when compared to their AIC analogues.     

Fitting E(max) models to clinical trial dose-response data

We consider fitting Emax models to the primary endpoint for a parallel group dose–response clinical trial. Such models can be difficult to fit using Maximum Likelihood if the data give little information about the maximum possible response. Consequently, we consider alternative models that can be derived as limiting cases, which can usually be fitted. Furthermore we propose two model selection procedures for choosing between the different models. These model selection procedures are compared with two model selection procedures which have previously been used. In a simulation study we find that the model selection procedure that performs best depends on the underlying true situation. One of the new model selection procedures gives what may be regarded as the most robust of the procedures.     

Bayesian estimation of the multidimensional graded response model with nonignorable missing data

A Bayesian approach is developed for analysing item response models with nonignorable missing data. The relevant model for the observed data is estimated concurrently in conjunction with the item response model for the missing-data process. Since the approach is fully Bayesian, it can be easily generalized to more complicated and realistic models, such as those models with covariates. Furthermore, the proposed approach is illustrated with item response data modelled as the multidimensional graded response models. Finally, a simulation study is conducted to assess the extent to which the bias caused by ignoring the missing-data mechanism can be reduced.     

Count Data Models with Correlated Unobserved Heterogeneity

Abstract. As previously argued, the correlation between included and omitted regressors generally causes inconsistency of standard estimators for count data models. Non‐linear instrumental variables estimation of an exponential model under conditional moment restrictions is one of the proposed remedies. This approach is extended here by fully exploiting the model assumptions and thereby improving efficiency of the resulting estimator. Empirical likelihood in particular has favourable properties in this setting compared with the two‐step generalized method of moments, as demonstrated in a Monte Carlo experiment. The proposed method is applied to the estimation of a cigarette demand function.     

A copula-based Markov chain model for the analysis of binary longitudinal data

A fully parametric first-order autoregressive (AR(1)) model is proposed to analyse binary longitudinal data. By using a discretized version of a copula, the modelling approach allows one to construct separate models for the marginal response and for the dependence between adjacent responses. In particular, the transition model that is focused on discretizes the Gaussian copula in such a way that the marginal is a Bernoulli distribution. A probit link is used to take into account concomitant information in the behaviour of the underlying marginal distribution. Fixed and time-varying covariates can be included in the model. The method is simple and is a natural extension of the AR(1) model for Gaussian series. Since the approach put forward is likelihood-based, it allows interpretations and inferences to be made that are not possible with semi-parametric approaches such as those based on generalized estimating equations. Data from a study designed to reduce the exposure of children to the sun are used to illustrate the methods.     

Bayesian analyses of an exponential-Poisson and related zero augmented type models

We consider several alternatives to the continuous exponential-Poisson distribution in order to accommodate the occurrence of zeros. Three of these are modifications of the exponential-Poisson model. One of these remains a fully continuous model. The other models we consider are all semi-continuous models, each with a discrete point mass at zero and a continuous density on the positive values. All of the models are applied to two environmental data sets concerning precipitation, and their Bayesian analyses using MCMC are discussed. This discussion covers convergence of the MCMC simulations and model selection procedures and considerations.