Bayesian longitudinal data analysis with mixed models and thick-tailed distributions using MCMC

Linear mixed effects models are frequently used to analyse longitudinal data, due to their flexibility in modelling the covariance structure between and within observations. Further, it is easy to deal with unbalanced data, either with respect to the number of observations per subject or per time period, and with varying time intervals between observations. In most applications of mixed models to biological sciences, a normal distribution is assumed both for the random effects and for the residuals. This, however, makes inferences vulnerable to the presence of outliers. Here, linear mixed models employing thick-tailed distributions for robust inferences in longitudinal data analysis are described. Specific distributions discussed include the Student-t, the slash and the contaminated normal. A Bayesian framework is adopted, and the Gibbs sampler and the Metropolis-Hastings algorithms are used to carry out the posterior analyses. An example with data on orthodontic distance growth in children is discussed to illustrate the methodology. Analyses based on either the Student-t distribution or on the usual Gaussian assumption are contrasted. The thick-tailed distributions provide an appealing robust alternative to the Gaussian process for modelling distributions of the random effects and of residuals in linear mixed models, and the MCMC implementation allows the computations to be performed in a flexible manner.

Discrete coherent states and probability distributions in finite-dimensional spaces

Operator bases are discussed in connection with the construction of phase space representatives of operators in finite-dimensional spaces, and their properties are presented. It is also shown how these operator bases allow for the construction of a finite harmonic oscillator-like coherent state. Creation and annihilation operators for the Fock finite-dimensional space are discussed and their expressions in terms of the operator bases are explicitly written. The relevant finite-dimensional probability distributions are obtained and their limiting behavior for an infinite-dimensional space are calculated which agree with the well known results. (C) 1996 Academic Press, Inc.

SOME CONSEQUENCES OF A SYMMETRY IN STRONG DISTRIBUTIONS

Some polynomials and interpolatory quadrature rules associated with strong Stieltjes distributions are considered, especially when the distributions satisfy a Certain symmetric property. (C) 1995 Academic Press, Inc.

Interval training at 95% and 100% of the velocity at VO2 max: effects on aerobic physiological indexes and running performance

The objective of this study was to analyze the effect of two different high-intensity interval training (HIT) programs on selected aerobic physiological indices and 1500 and 5000 m running performance in well-trained runners. The following tests were completed (n = 17): (i) incremental treadmill test to determine maximal oxygen uptake (VO2max), running velocity associated with VO2 max (VVO2max), and the velocity corresponding to 3.5 mmol/L of blood lactate concentration (vOBLA); (ii) submaximal constant-intensity test to determine running economy (RE); and (iii) 1500 and 5000 m time trials on a 400 m track. Runners were then randomized into 95% vVO(2max) or 100% vVO(2max) groups, and undertook a 4 week training program consisting of 2 HIT sessions (performed at 95% or 100% vVO(2max), respectively) and 4 submaximal run sessions per week. Runners were retested on all parameters at the completion of the training program. The VO2 max values were not different after training for both groups. There was a significant increase in post-training vVO(2 max), RE, and 1500 in running performance in the 100% vVO(2 max) group. The vOBLA and 5000 m running performance were significantly higher after the training period for both groups. We conclude that vOBLA and 5000 m running performance can be significantly improved in well-trained runners using a 4 week training program consisting of 2 HIT sessions (performed at 95% or 100% vVO(2max)) and 4 submaximal run sessions per week. However, the improvement in vVO(2 max), RE, and 1500 in running performance seems to be dependent on the HIT program at 100% vVO(2 max).

Multifractal characterization of saprolite particle-size distributions after topsoil removal

Multifractal analysis is now increasingly used to characterize soil properties as it may provide more information than a single fractal model. During the building of a large reservoir on the Parana River (Brazil), a highly weathered soil profile was excavated to a depth between 5 and 8 m. Excavation resulted in an abandoned area with saprolite materials and, in this area, an experimental field was established to assess the effectiveness of different soil rehabilitation treatments. The experimental design consisted of randomized blocks. The aim of this work was to characterize particle-size distributions of the saprolite material and use the information obtained to assess between-block variability. Particle-size distributions of the experimental plots were characterized by multifractal techniques. Ninety-six soil samples were analyzed routinely for particle-size distribution by laser diffractometry in a range of scales, varying from 0.390 to 2000 mu m. Six different textural classes (USDA) were identified with a clay content ranging from 16.9% to 58.4%. Multifractal models described reasonably well the scaling properties of particle-size distributions of the saprolite material. This material exhibits a high entropy dimension, D-1. Parameters derived from the left side (q > 0) of the f(alpha) spectra, D-1, the correlation dimension (D-2) and the range (alpha(0)-alpha(q+)), as well as the total width of the spectra (alpha(max) - alpha(min)) all showed dependence on the clay content. Sand, silt and clay contents were significantly different among treatments as a consequence of soil intrinsic variability. The D, and the Holder exponent of order zero, alpha(0), were not significantly different between treatments; in contrast, D-2 and several fractal attributes describing the width of the f(alpha) spectra were significantly different between treatments. The only parameter showing significant differences between sampling depths was (alpha(0) - alpha(q+)). Scale independent fractal attributes may be useful for characterizing intrinsic particle-size distribution variability. (c) 2006 Elsevier B.V. All rights reserved.

Robust Bayesian analysis of heavy-tailed stochastic volatility models using scale mixtures of normal distributions

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

A Note on the Prior Distributions of Weibull Parameters for the Reliability Function

In Bayesian Inference it is often desirable to have a posterior density reflecting mainly the information from sample data. To achieve this purpose it is important to employ prior densities which add little information to the sample. We have in the literature many such prior densities, for example, Jeffreys (1967), Lindley (1956); (1961), Hartigan (1964), Bernardo (1979), Zellner (1984), Tibshirani (1989), etc. In the present article, we compare the posterior densities of the reliability function by using Jeffreys, the maximal data information (Zellner, 1984), Tibshirani's, and reference priors for the reliability function R(t) in a Weibull distribution.

The frog Dermatonotus muelleri (Boettger 1885) (Anura Microhylidae) shifts its search tactics in response to two different prey distributions

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Elicitation of multivariate prior distributions: A nonparametric Bayesian approach

In the context of Bayesian statistical analysis, elicitation is the process of formulating a prior density f(.) about one or more uncertain quantities to represent a person's knowledge and beliefs. Several different methods of eliciting prior distributions for one unknown parameter have been proposed. However, there are relatively few methods for specifying a multivariate prior distribution and most are just applicable to specific classes of problems and/or based on restrictive conditions, such as independence of variables. Besides, many of these procedures require the elicitation of variances and correlations, and sometimes elicitation of hyperparameters which are difficult for experts to specify in practice. Garthwaite et al. (2005) discuss the different methods proposed in the literature and the difficulties of eliciting multivariate prior distributions. We describe a flexible method of eliciting multivariate prior distributions applicable to a wide class of practical problems. Our approach does not assume a parametric form for the unknown prior density f(.), instead we use nonparametric Bayesian inference, modelling f(.) by a Gaussian process prior distribution. The expert is then asked to specify certain summaries of his/her distribution, such as the mean, mode, marginal quantiles and a small number of joint probabilities. The analyst receives that information, treating it as a data set D with which to update his/her prior beliefs to obtain the posterior distribution for f(.). Theoretical properties of joint and marginal priors are derived and numerical illustrations to demonstrate our approach are given. (C) 2010 Elsevier B.V. All rights reserved.

Quantum and semiclassical Husimi distributions for a one-dimensional resonant system

We compare exact and semiclassical Husimi distributions for the single eigenstates of a one-dimensional resonant Hamiltonian. We find that both distributions concentrate near the unstable fixed points even when these points are made complex by suitably varying a parameter. © 1992 The American Physical Society.