Gilberto A. Paula

Influence diagnostics in log-Birnbaum-Saunders regression models with censored data

In this paper we discuss log-Birnbaum-Saunders regression models with censored observations. This kind of model has been largely applied to study material lifetime subject to failure or stress. The score functions and observed Fisher information matrix are given as well as the process for estimating the regression coefficients and shape parameter is discussed. The normal curvatures of local influence are derived under various perturbation schemes and two deviance-type residuals are proposed to assess departures from the log-Birnbaum-Saunders error assumption as well as to detect outlying observations. Finally, a data set from the medical area is analyzed under log-Birnbaum-Saunders regression models. A diagnostic analysis is performed in order to select an appropriate model.

Influence Diagnostics in log-Birnbaum-Saunders Regression Models

In this paper we present various diagnostic methods for a linear regression model under a logarithmic Birnbaum-Saunders distribution for the errors, which may be applied for accelerated life testing or to compare the median lives of several populations. Some influence methods, such as the local influence, total local influence of an individual and generalized leverage are derived, analysed and discussed. We also present a connection between the local influence and generalized leverage methods. A discussion of the computation of the likelihood displacement as well as the normal curvature in the local influence method are presented. Finally, an example with real data is given for illustration.

Local influence in elliptical linear regression models

Influence diagnostic methods are extended in this paper to elliptical linear models. These include several symmetric multivariate distributions such as the normal, Student t-, Cauchy and logistic distributions, among others. For a particular perturbation scheme and for the likelihood displacement the diagnostics agree with those developed for the normal linear regression model by Cook when the coefficients and the scale parameter are treated separately. This result shows the invariance of the diagnostics with respect to the induced model in the elliptical linear family. However, if the coefficients and the scale parameter are treated jointly we have a different diagnostic for each induced model, which makes this approach helpful for selecting the less sensitive model in the elliptical linear family. An example on the salinity of water is given for illustration.

Assessment of local influence in elliptical linear models with longitudinal structure

The aim of this paper is to derive local influence curvatures under various perturbation schemes for elliptical linear models with longitudinal structure. The elliptical class provides a useful generalization of the normal model since it covers both light- and heavy-tailed distributions for the errors, such as Student-t, power exponential, contaminated normal, among others. It is well known that elliptical models with longer-than-normal tails may present robust parameter estimates against outlying observations. However, little has been investigated on the robustness aspects of the parameter estimates against perturbation schemes. We use appropriate derivative operators to express the normal curvatures in tractable forms for any correlation structure. Estimation procedures for the position and variance-covariance parameters are also presented. A data set previously analyzed under a normal linear mixed model is reanalyzed under elliptical models. Local influence graphics are used to select less sensitive models with respect to some perturbation schemes.

Influence diagnostics in generalized log-gamma regression models

We discuss in this paper application of influence diagnostics in generalized log-gamma regression models considering the possibility of censored observations. We derive the appropriate matrices for assessing the local influence on the parameter estimates as well as the predictions from the fitted model under different perturbation schemes. The effect of censoring on local influence is also investigated. An example, for which we apply the diagnostic methods, is given as illustration.

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.

Generalized log-gamma regression models with cure fraction

In this paper, the generalized log-gamma regression model is modified to allow the possibility that long-term survivors may be present in the data. This modification leads to a generalized log-gamma regression model with a cure rate, encompassing, as special cases, the log-exponential, log-Weibull and log-normal regression models with a cure rate typically used to model such data. The models attempt to simultaneously estimate the effects of explanatory variables on the timing acceleration/deceleration of a given event and the surviving fraction, that is, the proportion of the population for which the event never occurs. The normal curvatures of local influence are derived under some usual perturbation schemes and two martingale-type residuals are proposed to assess departures from the generalized log-gamma error assumption as well as to detect outlying observations. Finally, a data set from the medical area is analyzed.

Assessing local influence in restricted regression models

Abstract In this paper we propose a score distance in order to detect influential observations on the removal of the ordinary least-squares estimate outside the restricted parametric region in inequality constrained linear regression models. We apply the local influence method (Cook, J. Roy. Statist. Soc. Ser. B. 48, 1986) to assess the effect of small perturbations on this displacement. The results are extended to generalized linear models and two illustrative examples are given.

An R implementation for generalized Birnbaum-Saunders distributions

The Birnbaum-Saunders (BS) model is a positively skewed statistical distribution that has received great attention in recent decades. A generalized version of this model was derived based on symmetrical distributions in the real line named the generalized BS (GBS) distribution. The R package named gbs was developed to analyze data from GBS models. This package contains probabilistic and reliability indicators and random number generators from GBS distributions. Parameter estimates for censored and uncensored data can also be obtained by means of likelihood methods from the gbs package. Goodness-of-fit and diagnostic methods were also implemented in this package in order to check the suitability of the GBS models. In this article, the capabilities and features of the gbs package are illustrated by using simulated and real data sets. Shape and reliability analyses for GBS models are presented. A simulation study for evaluating the quality and sensitivity of the estimation method developed in the package is provided and discussed.

Restricted methods in symmetrical linear regression models

In this paper we discuss the problem of testing equality and inequality constraints in symmetrical linear regression models. This class of models includes all symmetric continuous distributions, such as normal, Student-t, Pearson VII, power exponential and logistic, among others. It is commonly used for the analysis of data containing influential or outlying observations with responses supposedly normal. Iterative processes for evaluating the parameters under equality and inequality constraints are presented. The asymptotic null distribution of three asymptotically equivalent one-sided tests is showed to be invariant with the symmetrical error. A sensitivity study to investigate the robustness of the maximum likelihood estimates from some symmetrical models against high leverage and influential observations is presented. An illustrative example with presence of influential observations on the decisions from the statistical tests of different symmetrical models is given. The robustness aspects of such models are also discussed.