Scale mixtures log-Birnbaum–Saunders regression models with censored data: a Bayesian approach

The main objective of this paper is to develop a full Bayesian analysis for the Birnbaum–Saunders (BS) regression model based on scale mixtures of the normal (SMN) distribution with right-censored survival data. The BS distributions based on SMN models are a very general approach for analysing lifetime data, which has as special cases the Student-t-BS, slash-BS and the contaminated normal-BS distributions, being a flexible alternative to the use of the corresponding BS distribution or any other well-known compatible model, such as the log-normal distribution. A Gibbs sample algorithm with Metropolis–Hastings algorithm is used to obtain the Bayesian estimates of the parameters. Moreover, some discussions on the model selection to compare the fitted models are given and case-deletion influence diagnostics are developed for the joint posterior distribution based on the Kullback–Leibler divergence. The newly developed procedures are illustrated on a real data set previously analysed under BS regression models.     

Influence analysis in skew-Birnbaum–Saunders regression models and applications

In this paper, we propose a method to assess influence in skew-Birnbaum–Saunders regression models, which are an extension based on the skew-normal distribution of the usual Birnbaum–Saunders (BS) regression model. An interesting characteristic that the new regression model has is the capacity of predicting extreme percentiles, which is not possible with the BS model. In addition, since the observed likelihood function associated with the new regression model is more complex than that from the usual model, we facilitate the parameter estimation using a type-EM algorithm. Moreover, we employ influence diagnostic tools that considers this algorithm. Finally, a numerical illustration includes a brief simulation study and an analysis of real data in order to show the proposed methodology.     

Testing for homogeneity in cross-lagged panel studies

Panel studies are statistical studies in which two or more variables are observed for two or more subjects at two or more points In time. Cross- lagged panel studies are those studies in which the variables are continuous and divide naturally into two effects or impacts of each set of variables on the other. If a regression approach is taken5 a regression structure Is formulated for the cross-lagged models This structure may assume that the regression parameters are homogeneous across waves and across subpopulations. Under such assumptions the methods of multivariate regression analysis can be adapted to make inferences about the parameters. These inferences are limited to the degree that homogeneity of the parameters Is 'supported b}T the data. We consider the problem of testing the hypotheses of homogeneity and consider the problem of making statistical inferences about the cross-effects should there be evidence against one of the homogeneity assumptions. We demonstrate the methods developed by applying then to two panel data sets.     

A new class of survival regression models with heavy-tailed errors: robustness and diagnostics

Birnbaum-Saunders models have largely been applied in material fatigue studies and reliability analyses to relate the total time until failure with some type of cumulative damage. In many problems related to the medical field, such as chronic cardiac diseases and different types of cancer, a cumulative damage caused by several risk factors might cause some degradation that leads to a fatigue process. In these cases, BS models can be suitable for describing the propagation lifetime. However, since the cumulative damage is assumed to be normally distributed in the BS distribution, the parameter estimates from this model can be sensitive to outlying observations. In order to attenuate this influence, we present in this paper BS models, in which a Student-t distribution is assumed to explain the cumulative damage. In particular, we show that the maximum likelihood estimates of the Student-t log-BS models attribute smaller weights to outlying observations, which produce robust parameter estimates. Also, some inferential results are presented. In addition, based on local influence and deviance component and martingale-type residuals, a diagnostics analysis is derived. Finally, a motivating example from the medical field is analyzed using log-BS regression models. Since the parameter estimates appear to be very sensitive to outlying and influential observations, the Student-t log-BS regression model should attenuate such influences. The model checking methodologies developed in this paper are used to compare the fitted models.     

Fast and robust bootstrap

In this paper we review recent developments on a bootstrap method for robust estimators which is computationally faster and more resistant to outliers than the classical bootstrap. This fast and robust bootstrap method is, under reasonable regularity conditions, asymptotically consistent. We describe the method in general and then consider its application to perform inference based on robust estimators for the linear regression and multivariate location-scatter models. In particular, we study confidence and prediction intervals and tests of hypotheses for linear regression models, inference for location-scatter parameters and principal components, and classification error estimation for discriminant analysis.     

Semiparametric Regression Estimation in Copula Models

We consider some methods of semiparametric regression estimation in multivariate models when the common distribution function is represented using a copula and the marginals satisfy a generalized regression model using a transfer functional. Sufficient conditions for consistency and joint asymptotic normality of the finite-dimensional parameters are obtained.     

Likelihood-Based Inference for Multivariate Skew-Normal Regression Models

In this article, we present EM algorithms for performing maximum likelihood estimation for three multivariate skew-normal regression models of considerable practical interest. We also consider the restricted estimation of the parameters of certain important special cases of two models. The methodology developed is applied in the analysis of longitudinal data on dental plaque and cholesterol levels.     

Multivariate measurement error models based on Student-t distribution under censored responses

Measurement error models constitute a wide class of models that include linear and nonlinear regression models. They are very useful to model many real-life phenomena, particularly in the medical and biological areas. The great advantage of these models is that, in some sense, they can be represented as mixed effects models, allowing us to implement well-known techniques, like the EM-algorithm for the parameter estimation. In this paper, we consider a class of multivariate measurement error models where the observed response and/or covariate are not fully observed, i.e., the observations are subject to certain threshold values below or above which the measurements are not quantifiable. Consequently, these observations are considered censored. We assume a Student-t distribution for the unobserved true values of the mismeasured covariate and the error term of the model, providing a robust alternative for parameter estimation. Our approach relies on a likelihood-based inference using an EM-type algorithm. The proposed method is illustrated through some simulation studies and the analysis of an AIDS clinical trial dataset.     

Bayesian analysis in multivariate regression models with conjugate priors

In this paper, we consider the full rank multivariate regression model with matrix elliptically contoured distributed errors. We formulate a conjugate prior distribution for matrix elliptical models and derive the posterior distributions of mean and scale matrices. In the sequel, some characteristics of regression matrix parameters are also proposed.     

On multivariate nonlinear regression models with stationary correlated errors

In this paper we consider the statistical analysis of multivariate multiple nonlinear regression models with correlated errors, using Finite Fourier Transforms. Consistency and asymptotic normality of the weighted least squares estimates are established under various conditions on the regressor variables. These conditions involve different types of scalings, and the scaling factors are obtained explicitly for various types of nonlinear regression models including an interesting model which requires the estimation of unknown frequencies. The estimation of frequencies is a classical problem occurring in many areas like signal processing, environmental time series, astronomy and other areas of physical sciences. We illustrate our methodology using two real data sets taken from geophysics and environmental sciences. The data we consider from geophysics are polar motion (which is now widely known as “Chandlers Wobble”), where one has to estimate the drift parameters, the offset parameters and the two periodicities associated with elliptical motion. The data were first analyzed by Arato, Kolmogorov and Sinai who treat it as a bivariate time series satisfying a finite order time series model. They estimate the periodicities using the coefficients of the fitted models. Our analysis shows that the two dominant frequencies are 12 h and 410 days. The second example, we consider is the minimum/maximum monthly temperatures observed at the Antarctic Peninsula (Faraday/Vernadsky station). It is now widely believed that over the past 50 years there is a steady warming in this region, and if this is true, the warming has serious consequences on ecology, marine life, etc. as it can result in melting of ice shelves and glaciers. Our objective here is to estimate any existing temperature trend in the data, and we use the nonlinear regression methodology developed here to achieve that goal.     

Estimates of regression coefficients based on the sign covariance matrix

Summary. A new estimator of the regression parameters is introduced in a multivariate multiple-regression model in which both the vector of explanatory variables and the vector of response variables are assumed to be random. The affine equivariant estimate matrix is constructed using the sign covariance matrix (SCM) where the sign concept is based on Oja's criterion function. The influence function and asymptotic theory are developed to consider robustness and limiting efficiencies of the SCM regression estimate. The estimate is shown to be consistent with a limiting multinormal distribution. The influence function, as a function of the length of the contamination vector, is shown to be linear in elliptic cases; for the least squares (LS) estimate it is quadratic. The asymptotic relative efficiencies with respect to the LS estimate are given in the multivariate normal as well as the t -distribution cases. The SCM regression estimate is highly efficient in the multivariate normal case and, for heavy-tailed distributions, it performs better than the LS estimate. Simulations are used to consider finite sample efficiencies with similar results. The theory is illustrated with an example.     

Multivariate Dispersion Models Generated From Gaussian Copula

In this paper a class of multivariate dispersion models generated from the multivariate Gaussian copula is presented. Being a multivariate extension of Jørgensen's (1987a) dispersion models, this class of multivariate models is parametrized by marginal position, dispersion and dependence parameters, producing a large variety of multivariate discrete and continuous models including the multivariate normal as a special case. Properties of the multivariate distributions are investigated, some of which are similar to those of the multivariate normal distribution, which makes these models potentially useful for the analysis of correlated non-normal data in a way analogous to that of multivariate normal data. As an example, we illustrate an application of the models to the regression analysis of longitudinal data, and establish an asymptotic relationship between the likelihood equation and the generalized estimating equation of Liang & Zeger (1986).     

State-space models for multivariate longitudinal data of mixed types

We propose a class of state-space models for multivariate longitudinal data where the components of the response vector may have different distributions. The approach is based on the class of Tweedie exponential dispersion models, which accommodates a wide variety of discrete, continuous and mixed data. The latent process is assumed to be a Markov process, and the observations are conditionally independent given the latent process, over time as well as over the components of the response vector. This provides a fully parametric alternative to the quasilikelihood approach of Liang and Zeger. We estimate the regression parameters for time-varying covariates entering either via the observation model or via the latent process, based on an estimating equation derived from the Kalman smoother. We also consider analysis of residuals from both the observation model and the latent process.     

Sliced inverse regression for multivariate response regression

We consider a regression analysis of multivariate response on a vector of predictors. In this article, we develop a sliced inverse regression-based method for reducing the dimension of predictors without requiring a prespecified parametric model. Our proposed method preserves as much regression information as possible. We derive the asymptotic weighted chi-squared test for dimension. Simulation results are reported and comparisons are made with three methods—most predictable variates, k-means inverse regression and canonical correlation approach.     

Likelihood-based approaches for multivariate linear models under inequality constraints for incomplete data

In this paper, we consider a multivariate linear model with complete/incomplete data, where the regression coefficients are subject to a set of linear inequality restrictions. We first develop an expectation/conditional maximization (ECM) algorithm for calculating restricted maximum likelihood estimates of parameters of interest. We then establish the corresponding convergence properties for the proposed ECM algorithm. Applications to growth curve models and linear mixed models are presented. Confidence interval construction via the double-bootstrap method is provided. Some simulation studies are performed and a real example is used to illustrate the proposed methods.     

Inference in multivariate linear regression models with elliptically distributed errors

In this study we investigate the problem of estimation and testing of hypotheses in multivariate linear regression models when the errors involved are assumed to be non-normally distributed. We consider the class of heavy-tailed distributions for this purpose. Although our method is applicable for any distribution in this class, we take the multivariate t-distribution for illustration. This distribution has applications in many fields of applied research such as Economics, Business, and Finance. For estimation purpose, we use the modified maximum likelihood method in order to get the so-called modified maximum likelihood estimates that are obtained in a closed form. We show that these estimates are substantially more efficient than least-square estimates. They are also found to be robust to reasonable deviations from the assumed distribution and also many data anomalies such as the presence of outliers in the sample, etc. We further provide test statistics for testing the relevant hypothesis regarding the regression coefficients.     

Estimation and inference of the joint conditional distribution for multivariate longitudinal data using nonparametric copulas

In this paper we study estimating the joint conditional distributions of multivariate longitudinal outcomes using regression models and copulas. For the estimation of marginal models, we consider a class of time-varying transformation models and combine the two marginal models using nonparametric empirical copulas. Our models and estimation method can be applied in many situations where the conditional mean-based models are not good enough. Empirical copulas combined with time-varying transformation models may allow quite flexible modelling for the joint conditional distributions for multivariate longitudinal data. We derive the asymptotic properties for the copula-based estimators of the joint conditional distribution functions. For illustration we apply our estimation method to an epidemiological study of childhood growth and blood pressure.     

On estimation and diagnostics analysis in log-generalized gamma regression model for interval-censored data

The interval-censored survival data appear very frequently, where the event of interest is not observed exactly but it is only known to occur within some time interval. In this paper, we propose a location-scale regression model based on the log-generalized gamma distribution for modelling interval-censored data. We shall be concerned only with parametric forms. The proposed model for interval-censored data represents a parametric family of models that has, as special submodels, other regression models which are broadly used in lifetime data analysis. Assuming interval-censored data, we consider a frequentist analysis, a Jackknife estimator and a non-parametric bootstrap for the model parameters. We derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and present some techniques to perform global influence.     

Gaussian Scale Mixture Models for Robust Linear Multivariate Regression with Missing Data

We present an algorithm for multivariate robust Bayesian linear regression with missing data. The iterative algorithm computes an approximative posterior for the model parameters based on the variational Bayes (VB) method. Compared to the EM algorithm, the VB method has the advantage that the variance for the model parameters is also computed directly by the algorithm. We consider three families of Gaussian scale mixture models for the measurements, which include as special cases the multivariate t distribution, the multivariate Laplace distribution, and the contaminated normal model. The observations can contain missing values, assuming that the missing data mechanism can be ignored. A Matlab/Octave implementation of the algorithm is presented and applied to solve three reference examples from the literature.     

Bivariate location–scale models for regression analysis, with applications to lifetime data

Summary.  The literature on multivariate linear regression includes multivariate normal models, models that are used in survival analysis and a variety of models that are used in other areas such as econometrics. The paper considers the class of location–scale models, which includes a large proportion of the preceding models. It is shown that, for complete data, the maximum likelihood estimators for regression coefficients in a linear location–scale framework are consistent even when the joint distribution is misspecified. In addition, gains in efficiency arising from the use of a bivariate model, as opposed to separate univariate models, are studied. A major area of application for multivariate regression models is to clustered, 'parallel' lifetime data, so we also study the case of censored responses. Estimators of regression coefficients are no longer consistent under model misspecification, but we give simulation results that show that the bias is small in many practical situations. Gains in efficiency from bivariate models are also examined in the censored data setting. The methodology in the paper is illustrated by using lifetime data from the Diabetic Retinopathy Study.