Evaluation of an average numerator relationship matrix model and a Bayesian hierarchical model for growth traits in Nellore cattle with uncertain paternity

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

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.

Bayesian L-optimal exact design of experiments for biological kinetic models

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

Robust linear mixed models with normal/independent distributions and Bayesian MCMC implementation

Linear mixed effects models have been widely used in analysis of data where responses are clustered around some random effects, so it is not reasonable to assume independence between observations in the same cluster. In most biological applications, it is assumed that the distributions of the random effects and of the residuals are Gaussian. This makes inferences vulnerable to the presence of outliers. Here, linear mixed effects models with normal/independent residual distributions for robust inferences are described. Specific distributions examined include univariate and multivariate versions of the Student-t, the slash and the contaminated normal. A Bayesian framework is adopted and Markov chain Monte Carlo is used to carry out the posterior analysis. The procedures are illustrated using birth weight data on rats in a texicological experiment. Results from the Gaussian and robust models are contrasted, and it is shown how the implementation can be used for outlier detection. The thick-tailed distributions provide an appealing robust alternative to the Gaussian process in linear mixed models, and they are easily implemented using data augmentation and MCMC techniques.

The zero-inflated Conway-Maxwell-Poisson distribution: Bayesian inference, regression modeling and influence diagnostic

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Computational tools for comparing asymmetric GARCH models via Bayes factors

In this paper we use Markov chain Monte Carlo (MCMC) methods in order to estimate and compare GARCH models from a Bayesian perspective. We allow for possibly heavy tailed and asymmetric distributions in the error term. We use a general method proposed in the literature to introduce skewness into a continuous unimodal and symmetric distribution. For each model we compute an approximation to the marginal likelihood, based on the MCMC output. From these approximations we compute Bayes factors and posterior model probabilities. (C) 2012 IMACS. Published by Elsevier B.V. All rights reserved.

Parameter recovery for a skew-normal IRT model under a Bayesian approach: hierarchical framework, prior and kernel sensitivity and sample size

Item response theory (IRT) comprises a set of statistical models which are useful in many fields, especially when there is an interest in studying latent variables (or latent traits). Usually such latent traits are assumed to be random variables and a convenient distribution is assigned to them. A very common choice for such a distribution has been the standard normal. Recently, Azevedo et al. [Bayesian inference for a skew-normal IRT model under the centred parameterization, Comput. Stat. Data Anal. 55 (2011), pp. 353-365] proposed a skew-normal distribution under the centred parameterization (SNCP) as had been studied in [R. B. Arellano-Valle and A. Azzalini, The centred parametrization for the multivariate skew-normal distribution, J. Multivariate Anal. 99(7) (2008), pp. 1362-1382], to model the latent trait distribution. This approach allows one to represent any asymmetric behaviour concerning the latent trait distribution. Also, they developed a Metropolis-Hastings within the Gibbs sampling (MHWGS) algorithm based on the density of the SNCP. They showed that the algorithm recovers all parameters properly. Their results indicated that, in the presence of asymmetry, the proposed model and the estimation algorithm perform better than the usual model and estimation methods. Our main goal in this paper is to propose another type of MHWGS algorithm based on a stochastic representation (hierarchical structure) of the SNCP studied in [N. Henze, A probabilistic representation of the skew-normal distribution, Scand. J. Statist. 13 (1986), pp. 271-275]. Our algorithm has only one Metropolis-Hastings step, in opposition to the algorithm developed by Azevedo et al., which has two such steps. This not only makes the implementation easier but also reduces the number of proposal densities to be used, which can be a problem in the implementation of MHWGS algorithms, as can be seen in [R.J. Patz and B.W. Junker, A straightforward approach to Markov Chain Monte Carlo methods for item response models, J. Educ. Behav. Stat. 24(2) (1999), pp. 146-178; R. J. Patz and B. W. Junker, The applications and extensions of MCMC in IRT: Multiple item types, missing data, and rated responses, J. Educ. Behav. Stat. 24(4) (1999), pp. 342-366; A. Gelman, G.O. Roberts, and W.R. Gilks, Efficient Metropolis jumping rules, Bayesian Stat. 5 (1996), pp. 599-607]. Moreover, we consider a modified beta prior (which generalizes the one considered in [3]) and a Jeffreys prior for the asymmetry parameter. Furthermore, we study the sensitivity of such priors as well as the use of different kernel densities for this parameter. Finally, we assess the impact of the number of examinees, number of items and the asymmetry level on the parameter recovery. Results of the simulation study indicated that our approach performed equally as well as that in [3], in terms of parameter recovery, mainly using the Jeffreys prior. Also, they indicated that the asymmetry level has the highest impact on parameter recovery, even though it is relatively small. A real data analysis is considered jointly with the development of model fitting assessment tools. The results are compared with the ones obtained by Azevedo et al. The results indicate that using the hierarchical approach allows us to implement MCMC algorithms more easily, it facilitates diagnosis of the convergence and also it can be very useful to fit more complex skew IRT models.

Models and methods for power monolithic microwave integrated circuits

Computer aided design of Monolithic Microwave Integrated Circuits (MMICs) depends critically on active device models that are accurate, computationally efficient, and easily extracted from measurements or device simulators. Empirical models of active electron devices, which are based on actual device measurements, do not provide a detailed description of the electron device physics. However they are numerically efficient and quite accurate. These characteristics make them very suitable for MMIC design in the framework of commercially available CAD tools. In the empirical model formulation it is very important to separate linear memory effects (parasitic effects) from the nonlinear effects (intrinsic effects). Thus an empirical active device model is generally described by an extrinsic linear part which accounts for the parasitic passive structures connecting the nonlinear intrinsic electron device to the external world. An important task circuit designers deal with is evaluating the ultimate potential of a device for specific applications. In fact once the technology has been selected, the designer would choose the best device for the particular application and the best device for the different blocks composing the overall MMIC. Thus in order to accurately reproducing the behaviour of different-in-size devices, good scalability properties of the model are necessarily required. Another important aspect of empirical modelling of electron devices is the mathematical (or equivalent circuit) description of the nonlinearities inherently associated with the intrinsic device. Once the model has been defined, the proper measurements for the characterization of the device are performed in order to identify the model. Hence, the correct measurement of the device nonlinear characteristics (in the device characterization phase) and their reconstruction (in the identification or even simulation phase) are two of the more important aspects of empirical modelling. This thesis presents an original contribution to nonlinear electron device empirical modelling treating the issues of model scalability and reconstruction of the device nonlinear characteristics. The scalability of an empirical model strictly depends on the scalability of the linear extrinsic parasitic network, which should possibly maintain the link between technological process parameters and the corresponding device electrical response. Since lumped parasitic networks, together with simple linear scaling rules, cannot provide accurate scalable models, either complicate technology-dependent scaling rules or computationally inefficient distributed models are available in literature. This thesis shows how the above mentioned problems can be avoided through the use of commercially available electromagnetic (EM) simulators. They enable the actual device geometry and material stratification, as well as losses in the dielectrics and electrodes, to be taken into account for any given device structure and size, providing an accurate description of the parasitic effects which occur in the device passive structure. It is shown how the electron device behaviour can be described as an equivalent two-port intrinsic nonlinear block connected to a linear distributed four-port passive parasitic network, which is identified by means of the EM simulation of the device layout, allowing for better frequency extrapolation and scalability properties than conventional empirical models. Concerning the issue of the reconstruction of the nonlinear electron device characteristics, a data approximation algorithm has been developed for the exploitation in the framework of empirical table look-up nonlinear models. Such an approach is based on the strong analogy between timedomain signal reconstruction from a set of samples and the continuous approximation of device nonlinear characteristics on the basis of a finite grid of measurements. According to this criterion, nonlinear empirical device modelling can be carried out by using, in the sampled voltage domain, typical methods of the time-domain sampling theory.

Multiple testing in spatial epidemiology: a Bayesian approach

In this work we aim to propose a new approach for preliminary epidemiological studies on Standardized Mortality Ratios (SMR) collected in many spatial regions. A preliminary study on SMRs aims to formulate hypotheses to be investigated via individual epidemiological studies that avoid bias carried on by aggregated analyses. Starting from collecting disease counts and calculating expected disease counts by means of reference population disease rates, in each area an SMR is derived as the MLE under the Poisson assumption on each observation. Such estimators have high standard errors in small areas, i.e. where the expected count is low either because of the low population underlying the area or the rarity of the disease under study. Disease mapping models and other techniques for screening disease rates among the map aiming to detect anomalies and possible high-risk areas have been proposed in literature according to the classic and the Bayesian paradigm. Our proposal is approaching this issue by a decision-oriented method, which focus on multiple testing control, without however leaving the preliminary study perspective that an analysis on SMR indicators is asked to. We implement the control of the FDR, a quantity largely used to address multiple comparisons problems in the eld of microarray data analysis but which is not usually employed in disease mapping. Controlling the FDR means providing an estimate of the FDR for a set of rejected null hypotheses. The small areas issue arises diculties in applying traditional methods for FDR estimation, that are usually based only on the p-values knowledge (Benjamini and Hochberg, 1995; Storey, 2003). Tests evaluated by a traditional p-value provide weak power in small areas, where the expected number of disease cases is small. Moreover tests cannot be assumed as independent when spatial correlation between SMRs is expected, neither they are identical distributed when population underlying the map is heterogeneous. The Bayesian paradigm oers a way to overcome the inappropriateness of p-values based methods. Another peculiarity of the present work is to propose a hierarchical full Bayesian model for FDR estimation in testing many null hypothesis of absence of risk.We will use concepts of Bayesian models for disease mapping, referring in particular to the Besag York and Mollié model (1991) often used in practice for its exible prior assumption on the risks distribution across regions. The borrowing of strength between prior and likelihood typical of a hierarchical Bayesian model takes the advantage of evaluating a singular test (i.e. a test in a singular area) by means of all observations in the map under study, rather than just by means of the singular observation. This allows to improve the power test in small areas and addressing more appropriately the spatial correlation issue that suggests that relative risks are closer in spatially contiguous regions. The proposed model aims to estimate the FDR by means of the MCMC estimated posterior probabilities b i's of the null hypothesis (absence of risk) for each area. An estimate of the expected FDR conditional on data (\FDR) can be calculated in any set of b i's relative to areas declared at high-risk (where thenull hypothesis is rejected) by averaging the b i's themselves. The\FDR can be used to provide an easy decision rule for selecting high-risk areas, i.e. selecting as many as possible areas such that the\FDR is non-lower than a prexed value; we call them\FDR based decision (or selection) rules. The sensitivity and specicity of such rule depend on the accuracy of the FDR estimate, the over-estimation of FDR causing a loss of power and the under-estimation of FDR producing a loss of specicity. Moreover, our model has the interesting feature of still being able to provide an estimate of relative risk values as in the Besag York and Mollié model (1991). A simulation study to evaluate the model performance in FDR estimation accuracy, sensitivity and specificity of the decision rule, and goodness of estimation of relative risks, was set up. We chose a real map from which we generated several spatial scenarios whose counts of disease vary according to the spatial correlation degree, the size areas, the number of areas where the null hypothesis is true and the risk level in the latter areas. In summarizing simulation results we will always consider the FDR estimation in sets constituted by all b i's selected lower than a threshold t. We will show graphs of the\FDR and the true FDR (known by simulation) plotted against a threshold t to assess the FDR estimation. Varying the threshold we can learn which FDR values can be accurately estimated by the practitioner willing to apply the model (by the closeness between\FDR and true FDR). By plotting the calculated sensitivity and specicity (both known by simulation) vs the\FDR we can check the sensitivity and specicity of the corresponding\FDR based decision rules. For investigating the over-smoothing level of relative risk estimates we will compare box-plots of such estimates in high-risk areas (known by simulation), obtained by both our model and the classic Besag York Mollié model. All the summary tools are worked out for all simulated scenarios (in total 54 scenarios). Results show that FDR is well estimated (in the worst case we get an overestimation, hence a conservative FDR control) in small areas, low risk levels and spatially correlated risks scenarios, that are our primary aims. In such scenarios we have good estimates of the FDR for all values less or equal than 0.10. The sensitivity of\FDR based decision rules is generally low but specicity is high. In such scenario the use of\FDR = 0:05 or\FDR = 0:10 based selection rule can be suggested. In cases where the number of true alternative hypotheses (number of true high-risk areas) is small, also FDR = 0:15 values are well estimated, and \FDR = 0:15 based decision rules gains power maintaining an high specicity. On the other hand, in non-small areas and non-small risk level scenarios the FDR is under-estimated unless for very small values of it (much lower than 0.05); this resulting in a loss of specicity of a\FDR = 0:05 based decision rule. In such scenario\FDR = 0:05 or, even worse,\FDR = 0:1 based decision rules cannot be suggested because the true FDR is actually much higher. As regards the relative risk estimation, our model achieves almost the same results of the classic Besag York Molliè model. For this reason, our model is interesting for its ability to perform both the estimation of relative risk values and the FDR control, except for non-small areas and large risk level scenarios. A case of study is nally presented to show how the method can be used in epidemiology.

Behaviour of general dental practitioners in Germany regarding posterior restorations with flowable composites

Because the recommendation to use flowables for posterior restorations is still a matter of debate, the objective of this study was to determine in a nationwide survey in Germany how frequently, for what indications, and for what reasons, German dentists use flowable composites in posterior teeth. In addition, the acceptance of a simplified filling technique for posterior restorations using a low stress flowable composite was evaluated. Completed questionnaires from all over Germany were returned by 1,449 dentists resulting in a response rate of 48.5%; 78.6% of whom regularly used flowable composites for posterior restorations. The most frequent indications were cavity lining (80.1%) and small Class I fillings (74.2%). Flowables were less frequently used for small Class II fillings (22.7%) or other indications (13.6%). Most frequent reasons given for the use of flowables in posterior teeth were the prevention of voids (71.7%) and superior adaptation to cavity walls (72.9%), whereas saving time was considered less important (13.8%). Based on the subjective opinion of the dentists the simplified filling technique seemed to deliver advantages compared to the methods used to date particularly with regard to good cavity adaptation and ease of use. In conclusion, resin composites are the standard material type used for posterior restorations by general dental practitioners in Germany and most dentists use flowable composites as liners.

Posterior Simulation in the Generalized Linear Model with Semiparmetric Random Effects

Generalized linear mixed models with semiparametric random effects are useful in a wide variety of Bayesian applications. When the random effects arise from a mixture of Dirichlet process (MDP) model, normal base measures and Gibbs sampling procedures based on the Pólya urn scheme are often used to simulate posterior draws. These algorithms are applicable in the conjugate case when (for a normal base measure) the likelihood is normal. In the non-conjugate case, the algorithms proposed by MacEachern and Müller (1998) and Neal (2000) are often applied to generate posterior samples. Some common problems associated with simulation algorithms for non-conjugate MDP models include convergence and mixing difficulties. This paper proposes an algorithm based on the Pólya urn scheme that extends the Gibbs sampling algorithms to non-conjugate models with normal base measures and exponential family likelihoods. The algorithm proceeds by making Laplace approximations to the likelihood function, thereby reducing the procedure to that of conjugate normal MDP models. To ensure the validity of the stationary distribution in the non-conjugate case, the proposals are accepted or rejected by a Metropolis-Hastings step. In the special case where the data are normally distributed, the algorithm is identical to the Gibbs sampler.

Checking Assumptions in Latent Class Regression Models via a Markov Chain Monte Carlo Estimation Approach: An Application to Depression and Socio-Economic Status

Latent class regression models are useful tools for assessing associations between covariates and latent variables. However, evaluation of key model assumptions cannot be performed using methods from standard regression models due to the unobserved nature of latent outcome variables. This paper presents graphical diagnostic tools to evaluate whether or not latent class regression models adhere to standard assumptions of the model: conditional independence and non-differential measurement. An integral part of these methods is the use of a Markov Chain Monte Carlo estimation procedure. Unlike standard maximum likelihood implementations for latent class regression model estimation, the MCMC approach allows us to calculate posterior distributions and point estimates of any functions of parameters. It is this convenience that allows us to provide the diagnostic methods that we introduce. As a motivating example we present an analysis focusing on the association between depression and socioeconomic status, using data from the Epidemiologic Catchment Area study. We consider a latent class regression analysis investigating the association between depression and socioeconomic status measures, where the latent variable depression is regressed on education and income indicators, in addition to age, gender, and marital status variables. While the fitted latent class regression model yields interesting results, the model parameters are found to be invalid due to the violation of model assumptions. The violation of these assumptions is clearly identified by the presented diagnostic plots. These methods can be applied to standard latent class and latent class regression models, and the general principle can be extended to evaluate model assumptions in other types of models.

Reduced Bayesian Hierarchical Models: Estimating Health Effects of Simultaneous Exposure to Multiple Pollutants

Quantifying the health effects associated with simultaneous exposure to many air pollutants is now a research priority of the US EPA. Bayesian hierarchical models (BHM) have been extensively used in multisite time series studies of air pollution and health to estimate health effects of a single pollutant adjusted for potential confounding of other pollutants and other time-varying factors. However, when the scientific goal is to estimate the impacts of many pollutants jointly, a straightforward application of BHM is challenged by the need to specify a random-effect distribution on a high-dimensional vector of nuisance parameters, which often do not have an easy interpretation. In this paper we introduce a new BHM formulation, which we call "reduced BHM", aimed at analyzing clustered data sets in the presence of a large number of random effects that are not of primary scientific interest. At the first stage of the reduced BHM, we calculate the integrated likelihood of the parameter of interest (e.g. excess number of deaths attributed to simultaneous exposure to high levels of many pollutants). At the second stage, we specify a flexible random-effect distribution directly on the parameter of interest. The reduced BHM overcomes many of the challenges in the specification and implementation of full BHM in the context of a large number of nuisance parameters. In simulation studies we show that the reduced BHM performs comparably to the full BHM in many scenarios, and even performs better in some cases. Methods are applied to estimate location-specific and overall relative risks of cardiovascular hospital admissions associated with simultaneous exposure to elevated levels of particulate matter and ozone in 51 US counties during the period 1999-2005.

Seasonal Analyses of Air Pollution and Mortality in 100 U.S. Cities

Time series models relating short-term changes in air pollution levels to daily mortality counts typically assume that the effects of air pollution on the log relative rate of mortality do not vary with time. However, these short-term effects might plausibly vary by season. Changes in the sources of air pollution and meteorology can result in changes in characteristics of the air pollution mixture across seasons. The authors develop Bayesian semi-parametric hierarchical models for estimating time-varying effects of pollution on mortality in multi-site time series studies. The methods are applied to the updated National Morbidity and Mortality Air Pollution Study database for the period 1987--2000, which includes data for 100 U.S. cities. At the national level, a 10 micro-gram/m3 increase in PM(10) at lag 1 is associated with a 0.15 (95% posterior interval: -0.08, 0.39),0.14 (-0.14, 0.42), 0.36 (0.11, 0.61), and 0.14 (-0.06, 0.34) percent increase in mortality for winter, spring, summer, and fall, respectively. An analysis by geographical regions finds a strong seasonal pattern in the northeast (with a peak in summer) and little seasonal variation in the southern regions of the country. These results provide useful information for understanding particle toxicity and guiding future analyses of particle constituent data.

Bayesian models for the analysis of genetic structure when populations are correlated

Motivation: Population allele frequencies are correlated when populations have a shared history or when they exchange genes. Unfortunately, most models for allele frequency and inference about population structure ignore this correlation. Recent analytical results show that among populations, correlations can be very high, which could affect estimates of population genetic structure. In this study, we propose a mixture beta model to characterize the allele frequency distribution among populations. This formulation incorporates the correlation among populations as well as extending the model to data with different clusters of populations. Results: Using simulated data, we show that in general, the mixture model provides a good approximation of the among-population allele frequency distribution and a good estimate of correlation among populations. Results from fitting the mixture model to a dataset of genotypes at 377 autosomal microsatellite loci from human populations indicate high correlation among populations, which may not be appropriate to neglect. Traditional measures of population structure tend to over-estimate the amount of genetic differentiation when correlation is neglected. Inference is performed in a Bayesian framework.