Framework for Skew-Probit Links in Binary Regression

We review several asymmetrical links for binary regression models and present a unified approach for two skew-probit links proposed in the literature. Moreover, under skew-probit link, conditions for the existence of the ML estimators and the posterior distribution under improper priors are established. The framework proposed here considers two sets of latent variables which are helpful to implement the Bayesian MCMC approach. A simulation study to criteria for models comparison is conducted and two applications are made. Using different Bayesian criteria we show that, for these data sets, the skew-probit links are better than alternative links proposed in the literature.

Bayesian binary regression model: an application to in-hospital death after AMI prediction

Um modelo bayesiano de regressão binária é desenvolvido para predizer óbito hospitalar em pacientes acometidos por infarto agudo do miocárdio. Métodos de Monte Carlo via Cadeias de Markov (MCMC) são usados para fazer inferência e validação. Uma estratégia para construção de modelos, baseada no uso do fator de Bayes, é proposta e aspectos de validação são extensivamente discutidos neste artigo, incluindo a distribuição a posteriori para o índice de concordância e análise de resíduos. A determinação de fatores de risco, baseados em variáveis disponíveis na chegada do paciente ao hospital, é muito importante para a tomada de decisão sobre o curso do tratamento. O modelo identificado se revela fortemente confiável e acurado, com uma taxa de classificação correta de 88% e um índice de concordância de 83%.

Bayesian outlier analysis in binary regression

We propose alternative approaches to analyze residuals in binary regression models based on random effect components. Our preferred model does not depend upon any tuning parameter, being completely automatic. Although the focus is mainly on accommodation of outliers, the proposed methodology is also able to detect them. Our approach consists of evaluating the posterior distribution of random effects included in the linear predictor. The evaluation of the posterior distributions of interest involves cumbersome integration, which is easily dealt with through stochastic simulation methods. We also discuss different specifications of prior distributions for the random effects. The potential of these strategies is compared in a real data set. The main finding is that the inclusion of extra variability accommodates the outliers, improving the adjustment of the model substantially, besides correctly indicating the possible outliers.

On Graphically Checking Goodness-of-fit of Binary Logistic Regression Models

OBJECTIVES: This paper is concerned with checking goodness-of-fit of binary logistic regression models. For the practitioners of data analysis, the broad classes of procedures for checking goodness-of-fit available in the literature are described. The challenges of model checking in the context of binary logistic regression are reviewed. As a viable solution, a simple graphical procedure for checking goodness-of-fit is proposed. METHODS: The graphical procedure proposed relies on pieces of information available from any logistic analysis; the focus is on combining and presenting these in an informative way. RESULTS: The information gained using this approach is presented with three examples. In the discussion, the proposed method is put into context and compared with other graphical procedures for checking goodness-of-fit of binary logistic models available in the literature. CONCLUSION: A simple graphical method can significantly improve the understanding of any logistic regression analysis and help to prevent faulty conclusions.

Validation of Logistic Regression Models for Landslide Susceptibility Maps

A wide range of numerical models and tools have been developed over the last decades to support the decision making process in environmental applications, ranging from physical models to a variety of statistically-based methods. In this study, a landslide susceptibility map of a part of Three Gorges Reservoir region of China was produced, employing binary logistic regression analyses. The available information includes the digital elevation model of the region, geological map and different GIS layers including land cover data obtained from satellite imagery. The landslides were observed and documented during the field studies. The validation analysis is exploited to investigate the quality of mapping.

Latent residual analysis in binary regression with skewed link

Model diagnostics is an integral part of model determination and an important part of the model diagnostics is residual analysis. We adapt and implement residuals considered in the literature for the probit, logistic and skew-probit links under binary regression. New latent residuals for the skew-probit link are proposed here. We have detected the presence of outliers using the residuals proposed here for different models in a simulated dataset and a real medical dataset.

Evaluating Prediction Rules for t-Year Survivors With Censored Regression Models

Suppose that we are interested in establishing simple, but reliable rules for predicting future t-year survivors via censored regression models. In this article, we present inference procedures for evaluating such binary classification rules based on various prediction precision measures quantified by the overall misclassification rate, sensitivity and specificity, and positive and negative predictive values. Specifically, under various working models we derive consistent estimators for the above measures via substitution and cross validation estimation procedures. Furthermore, we provide large sample approximations to the distributions of these nonsmooth estimators without assuming that the working model is correctly specified. Confidence intervals, for example, for the difference of the precision measures between two competing rules can then be constructed. All the proposals are illustrated with two real examples and their finite sample properties are evaluated via a simulation study.

Optimal crossover designs for logistic regression models in pharmacodynamics

Pharmacodynamics (PD) is the study of the biochemical and physiological effects of drugs. The construction of optimal designs for dose-ranging trials with multiple periods is considered in this paper, where the outcome of the trial (the effect of the drug) is considered to be a binary response: the success or failure of a drug to bring about a particular change in the subject after a given amount of time. The carryover effect of each dose from one period to the next is assumed to be proportional to the direct effect. It is shown for a logistic regression model that the efficiency of optimal parallel (single-period) or crossover (two-period) design is substantially greater than a balanced design. The optimal designs are also shown to be robust to misspecification of the value of the parameters. Finally, the parallel and crossover designs are combined to provide the experimenter with greater flexibility.

Censored Linear Regression Models For Irregularly Observed Longitudinal Data Using The Multivariate-t Distribution.

In acquired immunodeficiency syndrome (AIDS) studies it is quite common to observe viral load measurements collected irregularly over time. Moreover, these measurements can be subjected to some upper and/or lower detection limits depending on the quantification assays. A complication arises when these continuous repeated measures have a heavy-tailed behavior. For such data structures, we propose a robust structure for a censored linear model based on the multivariate Student's t-distribution. To compensate for the autocorrelation existing among irregularly observed measures, a damped exponential correlation structure is employed. An efficient expectation maximization type algorithm is developed for computing the maximum likelihood estimates, obtaining as a by-product the standard errors of the fixed effects and the log-likelihood function. The proposed algorithm uses closed-form expressions at the E-step that rely on formulas for the mean and variance of a truncated multivariate Student's t-distribution. The methodology is illustrated through an application to an Human Immunodeficiency Virus-AIDS (HIV-AIDS) study and several simulation studies.

Residuals for log-Burr XII regression models in survival analysis

In this paper, we compare three residuals to assess departures from the error assumptions as well as to detect outlying observations in log-Burr XII regression models with censored observations. These residuals can also be used for the log-logistic regression model, which is a special case of the log-Burr XII regression model. For different parameter settings, sample sizes and censoring percentages, various simulation studies are performed and the empirical distribution of each residual is displayed and compared with the standard normal distribution. These studies suggest that the residual analysis usually performed in normal linear regression models can be straightforwardly extended to the modified martingale-type residual in log-Burr XII regression models with censored data.

Log-modified Weibull regression models with censored data: Sensitivity and residual analysis

This paper proposes a regression model considering the modified Weibull distribution. This distribution can be used to model bathtub-shaped failure rate functions. Assuming censored data, we consider maximum likelihood and Jackknife estimators for the parameters of the model. We derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and we also present some ways to perform global influence. Besides, for different parameter settings, sample sizes and censoring percentages, various simulations are performed and the empirical distribution of the modified deviance residual is displayed and compared with the standard normal distribution. These studies suggest that the residual analysis usually performed in normal linear regression models can be straightforwardly extended for a martingale-type residual in log-modified Weibull regression models with censored data. Finally, we analyze a real data set under log-modified Weibull regression models. A diagnostic analysis and a model checking based on the modified deviance residual are performed to select appropriate models. (c) 2008 Elsevier B.V. All rights reserved.

On estimation and influence diagnostics for zero-inflated negative binomial regression models

The zero-inflated negative binomial model is used to account for overdispersion detected in data that are initially analyzed under the zero-Inflated Poisson model A frequentist analysis a jackknife estimator and a non-parametric bootstrap for parameter estimation of zero-inflated negative binomial regression models are considered In addition an EM-type algorithm is developed for performing maximum likelihood estimation Then the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and some ways to perform global influence analysis are derived In order to study departures from the error assumption as well as the presence of outliers residual analysis based on the standardized Pearson residuals is discussed The relevance of the approach is illustrated with a real data set where It is shown that zero-inflated negative binomial regression models seems to fit the data better than the Poisson counterpart (C) 2010 Elsevier B V All rights reserved

Regression models for grouped survival data: Estimation and sensitivity analysis

In this study, regression models are evaluated for grouped survival data when the effect of censoring time is considered in the model and the regression structure is modeled through four link functions. The methodology for grouped survival data is based on life tables, and the times are grouped in k intervals so that ties are eliminated. Thus, the data modeling is performed by considering the discrete models of lifetime regression. The model parameters are estimated by using the maximum likelihood and jackknife methods. To detect influential observations in the proposed models, diagnostic measures based on case deletion, which are denominated global influence, and influence measures based on small perturbations in the data or in the model, referred to as local influence, are used. In addition to those measures, the local influence and the total influential estimate are also employed. Various simulation studies are performed and compared to the performance of the four link functions of the regression models for grouped survival data for different parameter settings, sample sizes and numbers of intervals. Finally, a data set is analyzed by using the proposed regression models. (C) 2010 Elsevier B.V. All rights reserved.

Parametric correlation functions to model the structure of permanent environmental (co)variances in milk yield random regression models

The objective of the present study was to estimate milk yield genetic parameters applying random regression models and parametric correlation functions combined with a variance function to model animal permanent environmental effects. A total of 152,145 test-day milk yields from 7,317 first lactations of Holstein cows belonging to herds located in the southeastern region of Brazil were analyzed. Test-day milk yields were divided into 44 weekly classes of days in milk. Contemporary groups were defined by herd-test-day comprising a total of 2,539 classes. The model included direct additive genetic, permanent environmental, and residual random effects. The following fixed effects were considered: contemporary group, age of cow at calving (linear and quadratic regressions), and the population average lactation curve modeled by fourth-order orthogonal Legendre polynomial. Additive genetic effects were modeled by random regression on orthogonal Legendre polynomials of days in milk, whereas permanent environmental effects were estimated using a stationary or nonstationary parametric correlation function combined with a variance function of different orders. The structure of residual variances was modeled using a step function containing 6 variance classes. The genetic parameter estimates obtained with the model using a stationary correlation function associated with a variance function to model permanent environmental effects were similar to those obtained with models employing orthogonal Legendre polynomials for the same effect. A model using a sixth-order polynomial for additive effects and a stationary parametric correlation function associated with a seventh-order variance function to model permanent environmental effects would be sufficient for data fitting.