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Birnbaum-Saunders frailty regression models: Diagnostics and application to medical data: Birnbaum-Saunders frailty regression models

In survival models, some covariates affecting the lifetime could not be observed or measured. These covariates may correspond to environmental or genetic factors and be considered as a random effect related to a frailty of the individuals explaining their survival times. We propose a methodology based on a Birnbaum-Saunders frailty regression model, which can be applied to censored or uncensored data. Maximum-likelihood methods are used to estimate the model parameters and to derive local influence techniques. Diagnostic tools are important in regression to detect anomalies, as departures from error assumptions and presence of outliers and influential cases. Normal curvatures for local influence under different perturbations are computed and two types of residuals are introduced. Two examples with uncensored and censored real-world data illustrate the proposed methodology. Comparison with classical frailty models is carried out in these examples, which shows the superiority of the proposed model.

Robust multivariate control charts based on Birnbaum–Saunders distributions

ABSTRACT Multivariate control charts are powerful and simple visual tools for monitoring the quality of a process. This multivariate monitoring is carried out by considering simultaneously several correlated quality characteristics and by determining whether these characteristics are in control or out of control. In this paper, we propose a robust methodology using multivariate quality control charts for subgroups based on generalized Birnbaum–Saunders distributions and an adapted Hotelling statistic. This methodology is constructed for Phases I and II of control charts. We estimate the corresponding parameters with the maximum likelihood method and use parametric bootstrapping to obtain the distribution of the adapted Hotelling statistic. In addition, we consider the Mahalanobis distance to detect multivariate outliers and use it to assess the adequacy of the distributional assumption. A Monte Carlo simulation study is conducted to evaluate the proposed methodology and to compare it with a standard methodology. This study reports the good performance of our methodology. An illustration with real-world air quality data of Santiago, Chile, is provided. This illustration shows that the methodology is useful for alerting early episodes of extreme air pollution, thus preventing adverse effects on human health.

On a tobit–Birnbaum–Saunders model with an application to medical data

ABSTRACT The tobit model allows a censored response variable to be described by covariates. Its applications cover different areas such as economics, engineering, environment and medicine. A strong assumption of the standard tobit model is that its errors follow a normal distribution. However, not all applications are well modeled by this distribution. Some efforts have relaxed the normality assumption by considering more flexible distributions. Nevertheless, the presence of asymmetry could not be well described by these flexible distributions. A real-world data application of measles vaccine in Haiti is explored, which confirms this asymmetry. We propose a tobit model with errors following a Birnbaum–Saunders (BS) distribution, which is asymmetrical and has shown to be a good alternative for describing medical data. Inference based on the maximum likelihood method and a type of residual are derived for the tobit–BS model. We perform global and local influence diagnostics to assess the sensitivity of the maximum likelihood estimators to atypical cases. A Monte Carlo simulation study is carried out to empirically evaluate the performance of these estimators. We conduct a data analysis for the mentioned application of measles vaccine based on the proposed model with the help of the R software. The results show the good performance of the tobit–BS model.

Kriging with external drift in a Birnbaum–Saunders geostatistical model

Spatial models to describe dependent georeferenced data are applied in different fields and, particularly, are used to analyze earth and environmentalxa0data. Most of these applications are carried out under Gaussian spatial models. The Birnbaum–Saunders distribution is a unimodal and positively skewed model which has received considerable attention in several areas, including earth and environmental sciences. In addition, theoretical arguments have been provided to justify its usage in the data modeling from these sciences, at least in the same settings where the lognormal distribution can be employed. This paper presents kriging with external drift based on a Birnbaum–Saunders spatial model. The maximum likelihood method is considered to estimate its parameters. The results obtained in the paper are illustrated by an experimental data set related to agricultural management. Specifically, in this illustration, the spatial variability of magnesium content in the soil as a function of calcium content is analyzed.

Environmental Applications Based on Birnbaum–Saunders Models

We discuss some environmental applications of methodologies based on the Birnbaum–Saunders model, which is an asymmetrical statistical distribution that is being widely considered to describe data collected in earth sciences. We present a formal justification, by means of the proportionate effect law, to use the Birnbaum–Saunders model as a useful distribution for environmental and regional variables. The methodologies discussed in this work include exceedance probabilities, X-bar control charts, np control charts, and spatial models. Applications with real-world environmental data sets are carried out for each discussed methodology.

Incorporation of frailties into a cure rate regression model and its diagnostics and application to melanoma data: A cure rate model with frailties and its diagnostics

Cure rate models have been widely studied to analyze time-to-event data with a cured fraction of patients. Our proposal consists of incorporating frailty into a cure rate model, as an alternative to the existing models to describe this type of data, based on the Birnbaum-Saunders distribution. Such a distribution has theoretical arguments to model medical data and has shown empirically to be a good option for their analysis. An advantage of the proposed model is the possibility to jointly consider the heterogeneity among patients by their frailties and the presence of a cured fraction of them. In addition, the number of competing causes is described by the negative binomial distribution, which absorbs several particular cases. We consider likelihood-based methods to estimate the model parameters and to derive influence diagnostics for this model. We assess local influence on the parameter estimates under different perturbation schemes. Deriving diagnostic tools is needed in all statistical modeling, which is another novel aspect of our proposal. Numerical evaluation of the considered model is performed by Monte Carlo simulations and by an illustration with melanoma data, both of which show its good performance and its potential applications. Particularly, the illustration confirms the importance of statistical diagnostics in the modeling.

Multivariate Generalized Birnbaum-Saunders Models Applied to Case Studies in Bio-Engineering and Industry

Birnbaum-Saunders models are receiving considerable attention in the literature. Multivariate regression models are a useful tool in the multivariate analysis, which takes into account the correlation between variables. Diagnostic analysis is an important aspect to be considered in the statistical modeling. In this work, we formulate a statistical methodology based on multivariate generalized Birnbaum-Saunders regression models and their diagnostics. We implement the obtained results in the R software, which are illustrated with two real-world multivariate data sets related to case studies in bio-engineering and industry to show their potential applications.

Cumulative damage and times of occurrence for a multicomponent system: A discrete time approach

Abstract A discrete time stochastic model for a multicomponent system is presented, which consists of two random vectors representing a multivariate cumulative damage and their corresponding failure times. The times of occurrence of some events, for the system components, are correlated and their associate cumulative damages are assumed to be additive. Since, in general, it is not possible to obtain a closed form for the distribution of these random vectors, their asymptotic distribution is studied. A central limit theorem and a large deviation principle for the multivariate cumulative damage are derived. An application to neurophysiology is presented. Parameters associated with the mean and covariance matrix of the shocks are assumed known. Otherwise, they can be estimated through well-known methods. However, the critical levels (thresholds) of resistance for the components of the system are assumed to be unknown parameters. One of the objectives of this work is to carry out asymptotic statistical inference on these parameters. To this end, the asymptotic distribution of certain Mahalanobis type distances is studied, which enables us to estimate the parameters of interest and to test hypotheses concerning their values. Numerical results complete the analysis.

On a Business Confidence Index and Its Data Analytics: A Chilean Case

In this work, we present a methodology based on a Chilean business confidence index, which allows us to describe aspects of the market at a global level, as well as at industrial and sector levels of Chilean great brands. We introduce some issues related to business intelligence, customer surveys, market variables, and the confidence index mentioned. In addition, we carry out analytics of real-world data using this index, whose results show the competitiveness of some Chilean great brands.

A new estimator for the covariance of the PLS coefficients estimator with applications to chemical data: An estimator for the covariance of the PLS coefficients estimator

Partial least squares (PLS) regression is a multivariate technique developed to solve the problem of multicollinearity and high dimensionality in explanatory variables. Several efforts have been made to improve the estimation of the covariance matrix of the PLS coefficients estimator. We propose a new estimator for this covariance matrix and prove its unbiasedness and consistency. We conduct a Monte Carlo simulation study to compare the proposed estimator and one based on the modified jackknife method, showing the advantages of the new estimator in terms of accuracy and computational efficiency. We illustrate the proposed method with three univariate and multivariate real‐world chemical data sets. In these illustrations, important findings are discovered because the conclusions of the studies change drastically when using the proposed estimation method in relation to the standard method, implying a change in the decisions to be made by the chemical practitioners.