Augmented mixed models for clustered proportion data
Dipankar Bandyopadhyay, Diana M. Galvis, Víctor H. Lachos
Often in biomedical research, we deal with continuous (clustered) proportion responses ranging between zero and one quantifying the disease status of the cluster units. Interestingly, the study population might also consist of relatively disease-free as well as highly diseased subjects, contributing to proportion values in the interval [0, 1]. Regression on a variety of parametric densities with support lying in (0, 1), such as beta regression, can assess important covariate effects. However, they are deemed inappropriate due the presence of zeros and/or ones. To evade this, we introduce a class of general proportion density (GPD), and further augment the probabilities of zero and one to this GPD, controlling for the clustering. Our approach is Bayesian, and presents a computationally convenient framework amenable to available freeware.Bayesian case-deletion influence diagnostics based on q-divergence measures are automatic from the MCMC output. The methodology is illustrated using both simulation studies and application to a real dataset from a clinical periodontology study.
Censored Linear Regression Models for Irregularly Observed Longitudinal Data using the Multivariate-t Distribution
Aldo M. Garay, Luis M. Castro, Jacek Leskow, Víctor H. Lachos
In AIDS studies it is quite common to observe viral load measurements collected irregularlyover time. Moreover, these measurements can be subjected to some upper and/or lower detection limitsdepending on the quantification assays. A complication arises when these continuous repeated measureshave a heavy-tailed behavior. For such data structures, we propose a robust structure for a censoredlinear model based on the multivariate Student-t distribution. To compensate for the autocorrelationexisting among irregularly observed measures, a damped exponential correlation structure is employed.An efficient EM-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-t distribution. The methodology is illustrated through an application to an HIV-AIDS study and several simulation studies.
Further examples of maximal curves which cannot be covered by the Hermitian curve
Saeed Tafazolian, Arnoldo Teheran-Herrera, Fernando Torres
We construct examples of curves defined over the finite field Fq6 which are covered by the GK-curve. Thus such curves are maximal over Fq6 although they cannot be covered by the Hermitian curve for q > 2. We also give examples of maximal curvesthat cannot be Galois covered by the Hermitian curve over the finite field Fq2n with n > 3 odd and q > 2. We point out some applications to codes related to an array coming from telescopic semigroups.
Improving Shewhart-type Generalized Variance Control Charts for Multivariate Process Variability Monitoring using Cornish-Fisher Quantile Correction, Meijer-G Function and Other Tools
Emanuel P. Barbosa, Mario A. Gneri, Ariane Meneguetti
This paper presents an improved version of the Shewhart-type generalized variance |S| control chart for multivariate Gaussian process dispersion monitoring, based on the Cornish-Fisher quantile formula for non-normality correction of the traditional nor-mal based 3-sigma chart limits.Also, the exact sample distribution of |S| and its quantiles (chart exact limits) are obtained through the Meijer-G function (inverse Mellin-Barnes integral transform), and an auxiliary control chart based on the trace of the standardized S matrix is introduced in order to avoid non detection of certain changes in the process variance-covariance Σ matrix.The performance of the proposed CF-corrected control chart is compared, considering false alarm risk (using analytical and simulation tools), with the traditional normal based chart and with the exact distributed based chart (for dimensions d = 2 and d = 3). This study shows that the proposed control limit corrections do remove the drawback of excess of false alarm associated with the traditional normal based |S| control chart.The proposed new chart (with its corresponding auxiliary chart) is illustrated with two numerical examples.
On the curve Y n = X m + X over finite fields
Saeed Tafazolian, Fernando Torres
We show that a maximal curve over Fq2 defined by the affine equation y n = f (x), where f (x) ∈ Fq2 [x] has degree coprime to n, is such that n is a divisor of q + 1 if and only if f (x) has a root in Fq2 . In this case, all the roots of f (x) belong to Fq2 ;cf. Thm. 1.2, Thm. 4.3 in [J. Pure Appl. Algebra 212 (2008), 2513–2521]. In particular, we characterize certain maximal curves defined by equations of type y n = xm + x over finite fields.
A Mixed-Effect Model for Positive Responses Augmented by Zeros
Mariana R. Motta, Diana M. Galvis, Víctor H. Lachos, Filidor E. Vilca-Labra, Valéria Troncoso Baltar, Eliseu Verly Junior, Regina Mara Fisberg, Dirce Maria Lobo Marchioni
In this work we propose a model for positive and zero responses by means of a zero augmented mixed regression model. Under this class, we are particularly interested in studying positive responses whose distribution accommodates skewness. At the same time, responses can be zero and therefore we justify the use of a zero-augmented mixture model. We model the mean of the positive response in a logarithm scale and the mixture probability in a logit scale, both as afunction of fixed and random effects. Here, the random effects link the two random components through their joint distribution and incorporate within subject correlation due to repeated measurements and between-subject heterogeneity.An MCMC algorithm is tailored to obtain Bayesian posterior distributions of the unknown quantities of interest and Bayesian case-deletion influence diagnostics based on the q-divergence measure is performed. We motivate and illustrate the proposed methodology by means of a data set from a 24 hours dietary recall study obtained in the city of S ̃o Paulo, Brazil, and present a simulation study a to evaluate the performance of the proposed methods.Bayesian inference, gamma distribution, log-normal distribution, mixed models, random effects, usual intake, zero-augmented data.
An Ergodic Description of Ground States
Eduardo Garibaldi, Philippe Thieullen
Given a translation-invariant Hamiltonian H, a ground state on the lattice Zd is a configuration whose energy, calculated with respect to H, cannot be lowered by altering its states on a finite number of sites. The set formed by these configurations is translation-invariant. Given an observable defined on the space of configurations, a minimizing measure is a translation-invariantprobability which minimizes the average of. If 0 is the mean contribution of all interactions to the site 0, we show that any configuration of the support of a minimizing measure is necessarily a ground state.
Discrete weak-KAM methods for stationary uniquely ergodic setting
Eduardo Garibaldi, Samuel Petite, Philippe Thieullen
The Frenkel-Kontorova model describes how an infinite chain of atoms minimizes the total energy of the system when the energy takes into account the interaction of nearest neighbors as well as the interaction with an exterior environment. An almost-periodic environment leads to consider a family of interaction energies which is stationary with respect to a minimal topological dynamical system. We introduce, in this context, the notion of calibrated configuration (stronger than the standard minimizing condition) and, for continuous superlinear interaction energies, we prove its existence for some environment of the dynamical system. Furthermore, in one dimension, we give sufficient conditions on the family of interaction energies to ensure the existence of calibrated configurations for any environment when the underlying dynamics is uniquely ergodic. The main mathematical tools for this study are developed in the frameworks of discrete weak KAM theory, Aubry-Mather theory and spaces of Delone sets.
Diagnostics for censored mixed-effects models using the multivariate t-distribution
Dipankar Bandyopadhyay, Larissa A. Matos, Luis M. Castro, Víctor H. Lachos
In biomedical studies on HIV RNA dynamics, the viral loads generate repeated measures that are often subjected to (upper and lower) detection limits, and hence these responses are either left- or right-censored. Linear and non-linear mixed-effects censored (LMEC/NLMEC) models are routinely used to analyze these longitudinal data, with normality assumptions for the random effects and residual errors. However, the derived inference may not be robust when these underlying normality assumptions are questionable, specially presence of outliers and thick-tails. Motivated by this, Matos et al. (2013b) recently proposed an exact EM-type algorithm for LMEC/NLMEC models using a multivariate Student’s-t distribution, with closed-form expressions at the E-step. In this paper, we develop influence diagnostics for LMEC/NLMEC models using multivariate Student’s-t density, based on the conditional expectation of the complete data log-likelihood which eliminates the complexity associated with the approach of Cook (1977, 1986) for censored mixed-effects models. The new methodology is illustrated through an application to a longitudinal HIV dataset using the NLMEC framework. In addition, a simulation study is presented, which explores the accuracy of the proposed measures in detecting influential observations in heavy-tailed censored data under different perturbation schemes.
Estimation Methods for Multivariate Tobit Confirmatory Factor Analysis
D. R. Costa, Víctor H. Lachos, J. L. Bazan, Caio L. N. Azevedo
We propose two methods for estimating multivariate Tobit Confirmatory Factor Analysis (TCFA) with covariates, one from Bayesian and another from likelihood based perspectives. TCFA is particularly useful in analysis of multivariate data with censored information. In contrast with previous developments that utilize Monte Carlo simulations for maximum likelihood estimation, an exact EM-type algorithm is proposed, which uses closed form expressions at the E-step that rely on the mean and variance of a truncated multinormal distribution and can be computed using available software. Through simulation studies, we compare the performance of the proposed algorithm when the censored pattern is ignored for different levels of censoring. Our results suggest that this algorithm has excellent performance, since it recovered the true parameters of the TCFA model much better than did the traditional CFA model. In addition, by considering a hierarchical formulation of the models, we also explore the estimation of the parameters via MCMC techniques by using proper priors. A Bayesian case deletion influence diagnostic based on the q-divergence measure and model selection criteria is also developed and applied to analyze a real dataset from an education assessment. In addition, a simulation study is conducted to compare the performance of the proposed method with the traditional CFA model.
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