Miguel de Carvalho

Bayesian Nonparametric ROC Regression Modeling

The receiver operating characteristic (ROC) curve is the most widely used measure for evaluating the discriminatory performance of a continuous biomarker. Incorporating covariates in the analysis can potentially enhance information gath- ered from the biomarker, as its discriminatory ability may depend on these. In this paper we propose a dependent Bayesian nonparametric model for conditional ROC estimation. Our model is based on dependent Dirichlet processes, where the covariate-dependent ROC curves are indirectly modeled using probability models for related probability distributions in the diseased and healthy groups. Our ap- proach allows for the entire distribution in each group to change as a function of the covariates, provides exact posterior inference up to a Monte Carlo error, and can easily accommodate multiple continuous and categorical predictors. Simula- tion results suggest that, regarding the mean squared error, our approach performs better than its competitors for small sample sizes and nonlinear scenarios. The proposed model is applied to data concerning diagnosis of diabetes.

Spectral Density Ratio Models for Multivariate Extremes

The modeling of multivariate extremes has received increasing recent attention because of its importance in risk assessment. In classical statistics of extremes, the joint distribution of two or more extremes has a nonparametric form, subject to moment constraints. This article develops a semiparametric model for the situation where several multivariate extremal distributions are linked through the action of a covariate on an unspecified baseline distribution, through a so-called density ratio model. Theoretical and numerical aspects of empirical likelihood inference for this model are discussed, and an application is given to pairs of extreme forest temperatures. Supplementary materials for this article are available online.

A Euclidean likelihood estimator for bivariate tail dependence

The spectral measure plays a key role in the statistical modeling of multivariate extremes. Estimation of the spectral measure is a complex issue, given the need to obey a certain moment condition. We propose a Euclidean likelihood-based estimator for the spectral measure which is simple and explicitly defined, with its expression being free of Lagrange multipliers. Our estimator is shown to have the same limit distribution as the maximum empirical likelihood estimator of Einmahl and Segers (2009). Numerical experiments suggest an overall good performance and identical behavior to the maximum empirical likelihood estimator. We illustrate the method in an extreme temperature data analysis.

Spectral density regression for bivariate extremes

We introduce a density regression model for the spectral density of a bivariate extreme value distribution, that allows us to assess how extremal dependence can change over a covariate. Inference is performed through a double kernel estimator, which can be seen as an extension of the Nadaraya–Watson estimator where the usual scalar responses are replaced by mean constrained densities on the unit interval. Numerical experiments with the methods illustrate their resilience in a variety of contexts of practical interest. An extreme temperature dataset is used to illustrate our methods.

Bayesian Nonparametric Approaches for ROC Curve Inference

The development of medical diagnostic tests is of great importance in clinical practice, public health, and medical research. The receiver operating characteristic (ROC) curve is a popular tool for evaluating the accuracy of such tests. We review Bayesian nonparametric methods based on Dirichlet process mixtures and the Bayesian bootstrap for ROC curve estimation and regression. The methods are illustrated by means of data concerning diagnosis of lung cancer in women.

Bayesian Nonparametric Biostatistics

We discuss some typical applications of Bayesian nonparametrics in biostatistics. The chosen applications highlight how Bayesian nonparametrics can contribute to addressing some fundamental questions that arise in biomedical research. In particular, we review some modern Bayesian semi- and nonparametric approaches for modeling longitudinal, survival, and medical diagnostic outcome data. Our discussion includes methods for longitudinal data analysis, non-proportional hazards survival analysis, joint modeling of longitudinal and survival data, longitudinal diagnostic test outcome data, and receiver operating characteristic curves. Throughout, we make comparisons among competing BNP models for the various data types considered.

Confidence intervals for the minimum of a function using extreme value statistics

Stochastic search algorithms are becoming an increasingly popular tool in the optimisation community. The random structure of these methods allows us to sample from the range of a function and to obtain estimates of its global minimum. However, a major advantage of stochastic search algorithms over deterministic algorithms, which is frequently unexplored, is that they also allow us to obtain interval estimates. In this paper, we put forward such advantage by providing guidance on how to combine stochastic search and optimisation methods with extreme value theory. To illustrate this approach we use several well-known objective functions. The obtained results are encouraging, suggesting that the interval estimates yield by this approach can be helpful for supplementing point estimates produced by other sophisticated optimisation methods.

Time-Varying Extreme Value Dependence with Application to Leading European Stock Markets

Extremal dependence between international stock markets is of particular interest in todays global financial landscape. However, previous studies have shown this dependence is not necessarily stationary over time. We concern ourselves with modeling extreme value dependence when that dependence is changing over time, or other suitable covariate. Working within a framework of asymptotic dependence, we introduce a regression model for the angular density of a bivariate extreme value distribution that allows us to assess how extremal dependence evolves over a covariate. We apply the proposed model to assess the dynamics governing extremal dependence of some leading European stock markets over the last three decades, and find evidence of an increase in extremal dependence over recent years.

Nonparametric Bayesian covariate-adjusted estimation of the Youden index

A novel nonparametric regression model is developed for evaluating the covariate-specific accuracy of a continuous biological marker. Accurately screening diseased from nondiseased individuals and correctly diagnosing disease stage are critically important to health care on several fronts, including guiding recommendations about combinations of treatments and their intensities. The accuracy of a continuous medical test or biomarker varies by the cutoff threshold (c) used to infer disease status. Accuracy can be measured by the probability of testing positive for diseased individuals (the true positive probability or sensitivity, Se(c), of the test), and the true negative probability (specificity, Sp(c)) of the test. A commonly used summary measure of test accuracy is the Youden index, YI=max{Se(c)+Sp(c)-1:c∈ℝ}, which is popular due in part to its ease of interpretation and relevance to population health research. In addition, clinical practitioners benefit from having an estimate of the optimal cutoff that maximizes sensitivity plus specificity available as a byproduct of estimating YI. We develop a highly flexible nonparametric model to estimate YI and its associated optimal cutoff that can respond to unanticipated skewness, multimodality, and other complexities because data distributions are modeled using dependent Dirichlet process mixtures. Important theoretical results on the support properties of the model are detailed. Inferences are available for the covariate-specific Youden index and its corresponding optimal cutoff threshold. The value of our nonparametric regression model is illustrated using multiple simulation studies and data on the age-specific accuracy of glucose as a biomarker of diabetes.

Functional covariate-adjusted partial area under the specificity-ROC curve with an application to metabolic syndrome diagnosis

Partially funded by Fondecyt Grants 11130541 (first author) and 11121186 (second author). Supported in part by the Spanish Ministry of Science and Innovation through project MTM2008-03010