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Background Inference using standard linear regression models (LMs) relies on assumptions that are rarely satisfied in practice. Substantial departures, if not addressed, have serious impacts on any inference and conclusions; potentially rendering them invalid and misleading. Count, bounded and skewed outcomes, common in physical activity research, can substantially violate LM assumptions. A common approach to handle these is to transform the outcome and apply a LM. However, a transformation may not suffice. Methods In this paper, we introduce the generalized linear model (GLM), a generalization of the LM, as an approach for the appropriate modelling of count and non-normally distributed (i.e., bounded and skewed) outcomes. Using data from a study of physical activity among older adults, we demonstrate appropriate methods to analyse count, bounded and skewed outcomes. Results We show how fitting an LM when inappropriate, especially for the type of outcomes commonly encountered in physical activity research, substantially impacts the analysis, inference, and conclusions compared to a GLM. Conclusions GLMs which more appropriately model non-normally distributed response variables should be considered as more suitable approaches for managing count, bounded and skewed outcomes rather than simply relying on transformations. We recommend that physical activity researchers add the GLM to their statistical toolboxes and become aware of situations when GLMs are a better method than traditional approaches for modeling count, bounded and skewed outcomes.

In this article, a multivariate extension of the unit-sinh-normal (USHN) distribution is presented. The new distribution, which is obtained from the conditionally specified distributions methodology, is absolutely continuous, and its marginal distributions are univariate USHN. The properties of the multivariate USHN distribution are studied in detail, and statistical inference is carried out from a classical approach using the maximum likelihood method. The new multivariate USHN distribution is suitable for modeling bounded data, especially in the (0,1)p region. In addition, the proposed distribution is extended to the case of the regression model and, for the latter, the Fisher information matrix is derived. The numerical results of a small simulation study and two applications with real data sets allow us to conclude that the proposed distribution, as well as its extension to regression models, are potentially useful to analyze the data of proportions, rates, or indices when modeling them jointly considering different degrees of correlation that may exist in the study variables is of interest.

In this article, we propose a new two-parameter distribution for bounded data such as rates, proportions, or percentages. The density function of the proposed distribution, presenting monotonic, unimodal, and inverse-unimodal shapes, tends to a positive finite value at the lower end of its support, which can lead to a better fit of the lower empirical quantiles. We derive some of the main structural properties of the new distribution. We make a description of the skewness and kurtosis of the distribution. We discuss the parameter estimation under the maximum likelihood method. We developed a simulation study to evaluate the behavior of the estimators. Finally, we present two applications to real data providing evidence that the proposed distribution can perform better than the popular beta and Kumaraswamy distributions.

ilustraciones La distribución Unitaria Birnbaum Saunders (UBS), [Mazucheli et al., 2018a], tiene soporte en el intervalo (0,1), motivo por el cual se emplea con éxito en el modelamiento de tasas e indicadores. Se presentan dos nuevas distribuciones bivariadas, la distribución Bivariada Birnbaum Saunders Unitaria (BVUBS) y la distribución Bivariada Sinh-Normal Birnbaum Saunders Unitaria (BVUSHN), además como efecto natural el modelo de regresión para el caso de covariables en el modelo, empleando para ello el concepto de distribuciones condicionalmente especificadas, dichas distribuciones son capaces de modelar tasas y proporciones en el plano unidad, y presentan un mejor ajuste a datos comparadas con otras distribuciones. Igualmente, se presentan algunas propiedades generales de los modelos, valores esperados e inferencia por máxima verosimilitud y aplicación a datos reales. Conjuntamente al presente trabajo de maestría se publica el artículo The Multivariate Skewed Log-Birnbaum–Saunders Distribution and Its Associated Regression Model, [Martínez-Flórez et al., 2023], el cual se enfocó en la extensión multivariada de la distribución Sinh-Normal Unitaria, estudiando en detalle las propiedades de la distribución e inferencia estadística, se incluye un estudio de simulación asociado al modelo de regresión y dos aplicaciones con datos reales, logrando concluir que son potencialmente útiles para modelar datos de proporciones, tasas o índices. (Texto tomado de la fuente) The unit-Birnbaum-Saunders distribution (UBS), [Mazucheli et al., 2018a], has support in the interval (0,1), which is why it is used successfully in the modeling of rates and indicators. Two new bivariate distributions are presented, the Bivariate Unit-Birnbaum-Saunders distribution (BVUBS) and the Bivariate Unit-Sinh-Normal Birnbaum Saunders distribution (BVUSHN), as well as the natural effect the regression model for the case of covariates in the model, using the concept of conditionally specified distributions, these distributions are capable of modeling rates and proportions in the unit plane, and present a better fit to data compared to other distributions. Likewise, some general properties of the models, expected values and inference by maximum likelihood and application to real data are presented. The article The Multivariate Skewed Log-Birnbaum–Saunders Distribution and Its Associated Regression Model, [Martínez-Flórez et al., 2023], which focused on the multivariate extension of the Unit-Sinh-Normal Distribution, is published together with this master's thesis, studying in detail the properties of the distribution and statistical inference, a simulation study associated with the regression model and two applications with real data are included. We conclude that they are potentially useful for modeling ratio, rate or index data. Maestría Magíster en Ciencias - Estadística Profundización

In this paper, we recommend a mechanism for determining whether to logit or not to logit data in the unit interval which is based on quantile estimation of data between 0 and 1. By using a simulated dataset generated from a Beta regression model, the estimated quantile for this model perform better than those based on the linear quantile regression with logit transformation. Further, we investigate the performance of the quantile regression estimators based on the LQR and we conclude that it is better than those based on the Beta regression when the distribution is contaminated with 10% uniform numbers between 0 and 1. The proposed recommendation is that we can use logit transformation LQR if (1) we are dealing with quantile estimation in data between 0 and 1 (2) we ascertain that the data fit well to the contemplated bounded data regressions (whether Beta Regression or otherwise) and (3) if the fit of the model is suspected.

In patient follow-up studies, events of interest may take place between periodic clinical assessments and so the exact time of onset is not observed. Such events are known as ‘bounded’ or ‘interval-censored’. Methods for handling such events can be categorised as either (i) applying multiple imputation (MI) strategies or (ii) taking a full likelihood-based (LB) approach. We focused on MI strategies, rather than LB methods, because of their flexibility. We evaluated MI strategies for bounded event times in a competing risks analysis, examining the extent to which interval boundaries, features of the data distribution and substantive analysis model are accounted for in the imputation model. Candidate imputation models were predictive mean matching (PMM); log-normal regression with postimputation back-transformation; normal regression with and without restrictions on the imputed values and Delord and Genin’s method1 based on sampling from the cumulative incidence function. We used a simulation study to compare MI methods and one LB method when data were missing at random and missing not at random, also varying the proportion of missing data, and then applied the methods to a haematopoietic stem cell transplantation dataset. We found that cumulative incidence and median event time estimation were sensitive to model misspecification. In a competing risks analysis, we found that it is more important to account for features of the data distribution than to restrict imputed values based on interval boundaries or to ensure compatibility with the substantive analysis by sampling from the cumulative incidence function. We recommend MI by type 1 PMM.

Considering the impact of the number of potential new coronavirus infections in each city, this paper explores the relationship between temperature and cumulative confirmed cases of COVID-19 in mainland China through the non-parametric method. In this paper, the floating population of each city in Wuhan is taken as a proxy variable for the number of potential new coronavirus infections. Firstly, to use the non-parametric method correctly, the symmetric Gauss kernel and asymmetric Gamma kernel are applied to estimate the density of cumulative confirmed cases of COVID-19 in China. The result confirms that the Gamma kernel provides a more reasonable density estimation of bounded data than the Gauss kernel. Then, through the non-parametric method based on the Gamma kernel estimation, this paper finds a positive relationship between Wuhan’s mobile population and cumulative confirmed cases, while the relationship between temperature and cumulative confirmed cases is inconclusive in China when the impact of the number of potential new coronavirus infections in each city is considered. Compared with the weather, the potentially infected population plays a more critical role in spreading the virus. Therefore, the role of prevention and control measures is more important than weather factors. Even in summer, we should also pay attention to the prevention and control of the epidemic.

International audience; We study the estimation, in L p-norm, of density functions dened on [0, 1] d. We construct a new family of kernel density estimators that do not suer from the so-called boundary bias problem and we propose a data-driven procedure based on the Goldenshluger and Lepski approach that jointly selects a kernel and a bandwidth. We derive two estimators that satisfy oracle-type inequalities. They are also proved to be adaptive over a scale of anisotropic or isotropic Sobolev-Slobodetskii classes (which are particular cases of Besov or Sobolev classical classes). The main interest of the isotropic procedure is to obtain adaptive results without any restriction on the smoothness parameter. Abstract Nous étudions l'estimation, en norme L p , d'une densité de probabilté dénie sur [0, 1] d. Nous construisons une nouvelle famille d'estimateurs à noyaux qui ne sont pas biaisés au bord du domaine de dénition et nous proposons une procédure de sélection simultanée d'un noyau et d'une fenêtre de lissage en adaptant la méthode développée par Goldenshluger et Lepski. Deux estimateurs diérents, déduits de cette procédure générale, sont proposés et des inégalités oracles sont établies pour chacun d'eux. Ces inégalités permettent de prouver que les-dits estimateurs sont adapatatifs par rapport à des familles de classes de Sobolev-Slobodetskii anisotropes ou isotropes. Dans cette dernière situation aucune borne supérieure sur le paramètre de régularité n'est imposée.

Observed data often belong to some specific intervals of values (for instance in case of percentages or proportions) or are higher (lower) than pre-specified values (for instance, chemical concentrations are higher than zero). The use of classical principal component analysis (PCA) may lead to extract components such that the reconstructed data take unfeasible values. In order to cope with this problem, a Constrained generalization of PCA is proposed. The new technique, called bounded principal component analysis (B-PCA), detects components such that the reconstructed data are constrained to belong to some pre-specified bounds. This is done by implementing a row-wise alternating least squares (ALS) algorithm, which exploits the potentialities of the least squares with inequality (LSI) algorithm. The results of a simulation study and two applications to bounded data are discussed for evaluating how the method and the algorithm for solving it work in practice. Copyright (C) 2007 John Wiley & Sons, Ltd.

Understanding and quantifying the determinants of the number of sectors or firms exporting in a given country is of relevance for the assessment of trade policies. Estimation of models for the number of sectors, however, poses a challenge because the dependent variable has both a lower and an upper bound, implying that the partial effects of the explanatory variables on the conditional mean of the dependent variable cannot be constant and must approach zero as the conditional mean approaches its bounds. We argue that ignoring these bounds can lead to erroneous conclusions due to the model's mispecification, and propose a flexible specification that accounts for the doubly-bounded nature of the dependent variable. We empirically investigate the problem and the proposed solution, finding significant differences between estimates obtained with the proposed estimator and those obtained with standard approaches.

Objectives The aim of this study was to show the potential of Bayesian analysis in statistical modelling of dental caries data. Because of the bounded nature of the dmft (DMFT) index, zero-inflated binomial (ZIB) and beta-binomial (ZIBB) models were considered. The effects of incorporating prior information available about the parameters of models were also shown. Methods The data set used in this study was the Belo Horizonte Caries Prevention (BELCAP) study (Bohning et al. (1999)), consisting of five variables collected among 797 Brazilian school children designed to evaluate four programmes for reducing caries. Only the eight primary molar teeth were considered in the data set. A data augmentation algorithm was used for estimation. Firstly, noninformative priors were used to express our lack of knowledge about the regression parameters. Secondly, prior information about the probability of being a structural zero dmft and the probability of being caries affected in the subpopulation of susceptible children was incorporated. Results With noninformative priors, the best fitting model was the ZIBB. Education (OR = 0.76, 95% CrI: 0.59, 0.99), all interventions (OR = 0.46, 95% CrI: 0.35, 0.62), rinsing (OR = 0.61, 95% CrI: 0.47, 0.80) and hygiene (OR = 0.65, 95% CrI: 0.49, 0.86) were demonstrated to be factors protecting children from being caries affected. Being male increased the probability of being caries diseased (OR = 1.19, 95% CrI: 1.01, 1.42). However, after incorporating informative priors, ZIB models' estimates were not influenced, while ZIBB models reduced deviance and confirmed the association with all interventions and rinsing only. Discussion In our application, Bayesian estimates showed a similar accuracy and precision than likelihood-based estimates, although they offered many computational advantages and the possibility of expressing all forms of uncertainty in terms of probability. The overdispersion parameter could expound why the introduction of prior information had significant effects on the parameters of the ZIBB model, while ZIB estimates remained unchanged. Finally, the best performance of ZIBB compared to the ZIB model was shown to catch overdispersion in data.