A Bayesian random effects model for analyzing mixed negative binomial and continuous longitudinal responses

ABSTRACT

A general Bayesian random effects model for analyzing longitudinal mixed correlated continuous and negative binomial responses with and without missing data is presented. This Bayesian model, given some random effects, uses a normal distribution for the continuous response and a negative binomial distribution for the count response. A Markov Chain Monte Carlo sampling algorithm is described for estimating the posterior distribution of the parameters. This Bayesian model is illustrated by a simulation study. For sensitivity analysis to investigate the change of parameter estimates with respect to the perturbation from missing at random to not missing at random assumption, the use of posterior curvature is proposed. The model is applied to a medical data, obtained from an observational study on women, where the correlated responses are the negative binomial response of joint damage and continuous response of body mass index. The simultaneous effects of some covariates on both responses are also investigated.     

Bayesian and maximum likelihood estimations of the inverted exponentiated half logistic distribution under progressive Type II censoring

In this paper, the estimation of parameters, reliability and hazard functions of a inverted exponentiated half logistic distribution (IEHLD) from progressive Type II censored data has been considered. The Bayes estimates for progressive Type II censored IEHLD under asymmetric and symmetric loss functions such as squared error, general entropy and linex loss function are provided. The Bayes estimates for progressive Type II censored IEHLD parameters, reliability and hazard functions are also obtained under the balanced loss functions. However, the Bayes estimates cannot be obtained explicitly, Lindley approximation method and importance sampling procedure are considered to obtain the Bayes estimates. Furthermore, the asymptotic normality of the maximum likelihood estimates is used to obtain the approximate confidence intervals. The highest posterior density credible intervals of the parameters based on importance sampling procedure are computed. Simulations are performed to see the performance of the proposed estimates. For illustrative purposes, two data sets have been analyzed.     

A wavelet approach to shape analysis for spinal curves

We present a new method to describe shape change and shape differences in curves, by constructing a deformation function in terms of a wavelet decomposition. Wavelets form an orthonormal basis which allows representations at multiple resolutions. The deformation function is estimated, in a fully Bayesian framework, using a Markov chain Monte Carlo algorithm. This Bayesian formulation incorporates prior information about the wavelets and the deformation function. The flexibility of the MCMC approach allows estimation of complex but clinically important summary statistics, such as curvature in our case, as well as estimates of deformation functions with variance estimates, and allows thorough investigation of the posterior distribution. This work is motivated by multi-disciplinary research involving a large-scale longitudinal study of idiopathic scoliosis in UK children. This paper provides novel statistical tools to study this spinal deformity, from which 5% of UK children suffer. Using the data we consider statistical inference for shape differences between normals, scoliotics and developers of scoliosis, in particular for spinal curvature, and look at longitudinal deformations to describe shape changes with time.     

Bayesian analysis of circular distributions based on non-negative trigonometric sums

ABSTRACT

Fernández-Durán [Circular distributions based on nonnegative trigonometric sums. Biometrics. 2004;60:499–503] developed a new family of circular distributions based on non-negative trigonometric sums that is suitable for modelling data sets that present skewness and/or multimodality. In this paper, a Bayesian approach to deriving estimates of the unknown parameters of this family of distributions is presented. Because the parameter space is the surface of a hypersphere and the dimension of the hypersphere is an unknown parameter of the distribution, the Bayesian inference must be based on transdimensional Markov Chain Monte Carlo (MCMC) algorithms to obtain samples from the high-dimensional posterior distribution. The MCMC algorithm explores the parameter space by moving along great circles on the surface of the hypersphere. The methodology is illustrated with real and simulated data sets.     

Inference on unknown parameters of a Burr distribution under hybrid censoring

Based on hybrid censored data, the problem of making statistical inference on parameters of a two parameter Burr Type XII distribution is taken up. The maximum likelihood estimates are developed for the unknown parameters using the EM algorithm. Fisher information matrix is obtained by applying missing value principle and is further utilized for constructing the approximate confidence intervals. Some Bayes estimates and the corresponding highest posterior density intervals of the unknown parameters are also obtained. Lindley’s approximation method and a Markov Chain Monte Carlo (MCMC) technique have been applied to evaluate these Bayes estimates. Further, MCMC samples are utilized to construct the highest posterior density intervals as well. A numerical comparison is made between proposed estimates in terms of their mean square error values and comments are given. Finally, two data sets are analyzed using proposed methods.     

Bayesian inference for circular distributions with unknown normalising constants

Very often, the likelihoods for circular data sets are of quite complicated forms, and the functional forms of the normalising constants, which depend upon the unknown parameters, are unknown. This latter problem generally precludes rigorous, exact inference (both classical and Bayesian) for circular data.Noting the paucity of literature on Bayesian circular data analysis, and also because realistic data analysis is naturally permitted by the Bayesian paradigm, we address the above problem taking a Bayesian perspective. In particular, we propose a methodology that combines importance sampling and Markov chain Monte Carlo (MCMC) in a very effective manner to sample from the posterior distribution of the parameters, given the circular data. With simulation study and real data analysis, we demonstrate the considerable reliability and flexibility of our proposed methodology in analysing circular data.     

Estimating multiple-membership logit models with mixed effects: indirect inference versus data cloning

Multiple-membership logit models with random effects are models for clustered binary data, where each statistical unit can belong to more than one group. The likelihood function of these models is analytically intractable. We propose two different approaches for parameter estimation: indirect inference and data cloning (DC). The former is a non-likelihood-based method which uses an auxiliary model to select reasonable estimates. We propose an auxiliary model with the same dimension of parameter space as the target model, which is particularly convenient to reach good estimates very fast. The latter method computes maximum likelihood estimates through the posterior distribution of an adequate Bayesian model, fitted to cloned data. We implement a DC algorithm specifically for multiple-membership models. A Monte Carlo experiment compares the two methods on simulated data. For further comparison, we also report Bayesian posterior mean and Integrated Nested Laplace Approximation hybrid DC estimates. Simulations show a negligible loss of efficiency for the indirect inference estimator, compensated by a relevant computational gain. The approaches are then illustrated with two real examples on matched paired data.     

Bayes estimates in multivariate semiparametric linear models

Bayes estimates are derived in multivariate linear models with unknown distribution. The prior distribution is defined using a Dirichlet prior for the unknown error distribution and a normal-Wishart distribution for the parameters. The posterior distribution is determined and explicit expressions are given in the special cases of location-scale and two-sample models. The calculation of self-informative limits of Bayes estimates yields standard estimates.     

Noninformative nonparametric quantile estimation for simple random samples

For noninformative nonparametric estimation of finite population quantiles under simple random sampling, estimation based on the Polya posterior is similar to estimation based on the Bayesian approach developed by Ericson (J. Roy. Statist. Soc. Ser. B 31 (1969) 195) in that the Polya posterior distribution is the limit of Ericson's posterior distributions as the weight placed on the prior distribution diminishes. Furthermore, Polya posterior quantile estimates can be shown to be admissible under certain conditions. We demonstrate the admissibility of the sample median as an estimate of the population median under such a set of conditions. As with Ericson's Bayesian approach, Polya posterior-based interval estimates for population quantiles are asymptotically equivalent to the interval estimates obtained from standard frequentist approaches. In addition, for small to moderate sized populations, Polya posterior-based interval estimates for quantiles of a continuous characteristic of interest tend to agree with the standard frequentist interval estimates.     

On estimating parameters of a progressively censored lognormal distribution

We consider the problem of making statistical inference on unknown parameters of a lognormal distribution under the assumption that samples are progressively censored. The maximum likelihood estimates (MLEs) are obtained by using the expectation-maximization algorithm. The observed and expected Fisher information matrices are provided as well. Approximate MLEs of unknown parameters are also obtained. Bayes and generalized estimates are derived under squared error loss function. We compute these estimates using Lindley's method as well as importance sampling method. Highest posterior density interval and asymptotic interval estimates are constructed for unknown parameters. A simulation study is conducted to compare proposed estimates. Further, a data set is analysed for illustrative purposes. Finally, optimal progressive censoring plans are discussed under different optimality criteria and results are presented.     

Approximate Bayesian computation with composite score functions

Both approximate Bayesian computation (ABC) and composite likelihood methods are useful for Bayesian and frequentist inference, respectively, when the likelihood function is intractable. We propose to use composite likelihood score functions as summary statistics in ABC in order to obtain accurate approximations to the posterior distribution. This is motivated by the use of the score function of the full likelihood, and extended to general unbiased estimating functions in complex models. Moreover, we show that if the composite score is suitably standardised, the resulting ABC procedure is invariant to reparameterisations and automatically adjusts the curvature of the composite likelihood, and of the corresponding posterior distribution. The method is illustrated through examples with simulated data, and an application to modelling of spatial extreme rainfall data is discussed.     

Posterior probability intervals for wavelet thresholding

Summary. We use cumulants to derive Bayesian credible intervals for wavelet regression estimates. The first four cumulants of the posterior distribution of the estimates are expressed in terms of the observed data and integer powers of the mother wavelet functions. These powers are closely approximated by linear combinations of wavelet scaling functions at an appropriate finer scale. Hence, a suitable modification of the discrete wavelet transform allows the posterior cumulants to be found efficiently for any given data set. Johnson transformations then yield the credible intervals themselves. Simulations show that these intervals have good coverage rates, even when the underlying function is inhomogeneous, where standard methods fail. In the case where the curve is smooth, the performance of our intervals remains competitive with established nonparametric regression methods.     

Properties of the beta regression model for small area estimation of proportions and application to estimation of poverty rates

Abstract

Linear mixed effects models have been popular in small area estimation problems for modeling survey data when the sample size in one or more areas is too small for reliable inference. However, when the data are restricted to a bounded interval, the linear model may be inappropriate, particularly if the data are near the boundary. Nonlinear sampling models are becoming increasingly popular for small area estimation problems when the normal model is inadequate. This paper studies the use of a beta distribution as an alternative to the normal distribution as a sampling model for survey estimates of proportions which take values in (0, 1). Inference for small area proportions based on the posterior distribution of a beta regression model ensures that point estimates and credible intervals take values in (0, 1). Properties of a hierarchical Bayesian small area model with a beta sampling distribution and logistic link function are presented and compared to those of the linear mixed effect model. Propriety of the posterior distribution using certain noninformative priors is shown, and behavior of the posterior mean as a function of the sampling variance and the model variance is described. An example using 2010 Small Area Income and Poverty Estimates (SAIPE) data is given, and a numerical example studying small sample properties of the model is presented.     

A full Bayesian analysis of circular data using the von Mises distribution

This paper presents a full Bayesian analysis of circular data, paying special attention to the von Mises distribution. We obtain samples from the posterior distribution using the Gibbs sampler which, after the introduction of strategic latent variables, has all full conditional distributions of known type.     

Statistical analysis for competing risks model from a Weibull distribution under progressively hybrid censoring

This paper considers the statistical analysis for competing risks model under the Type-I progressively hybrid censoring from a Weibull distribution. We derive the maximum likelihood estimates and the approximate maximum likelihood estimates of the unknown parameters. We then use the bootstrap method to construct the confidence intervals. Based on the non informative prior, a sampling algorithm using the acceptance–rejection sampling method is presented to obtain the Bayes estimates, and Monte Carlo method is employed to construct the highest posterior density credible intervals. The simulation results are provided to show the effectiveness of all the methods discussed here and one data set is analyzed.     

Bayesian inference and prediction of the Pareto distribution based on ordered ranked set sampling

In this paper, order statistics from independent and non identically distributed random variables is used to obtain ordered ranked set sampling (ORSS). Bayesian inference of unknown parameters under a squared error loss function of the Pareto distribution is determined. We compute the minimum posterior expected loss (the posterior risk) of the derived estimates and compare them with those based on the corresponding simple random sample (SRS) to assess the efficiency of the obtained estimates. Two-sample Bayesian prediction for future observations is introduced by using SRS and ORSS for one- and m-cycle. A simulation study and real data are applied to show the proposed results.     

Bayesian estimation and prediction based on lognormal record values

In this paper we consider the problems of estimation and prediction when observed data from a lognormal distribution are based on lower record values and lower record values with inter-record times. We compute maximum likelihood estimates and asymptotic confidence intervals for model parameters. We also obtain Bayes estimates and the highest posterior density (HPD) intervals using noninformative and informative priors under square error and LINEX loss functions. Furthermore, for the problem of Bayesian prediction under one-sample and two-sample framework, we obtain predictive estimates and the associated predictive equal-tail and HPD intervals. Finally for illustration purpose a real data set is analyzed and simulation study is conducted to compare the methods of estimation and prediction.     

Estimation and prediction of the Burr type XII distribution based on record values and inter-record times

The maximum likelihood and Bayesian approaches for parameter estimations and prediction of future record values have been considered for the two-parameter Burr Type XII distribution based on record values with the number of trials following the record values (inter-record times). Firstly, the Bayes estimates are obtained based on a joint bivariate prior for the shape parameters. In this case, the Bayes estimates of the parameters have been developed by using Lindley's approximation and the Markov Chain Monte Carlo (MCMC) method due to the lack of explicit forms under the squared error and the linear-exponential loss functions. The MCMC method has been also used to construct the highest posterior density credible intervals. Secondly, the Bayes estimates are obtained with respect to a discrete prior for the first shape parameter and a conjugate prior for other shape parameter. The Bayes and the maximum likelihood estimates are compared in terms of the estimated risk by the Monte Carlo simulations. We further consider the non-Bayesian and Bayesian prediction for future lower record arising from the Burr Type XII distribution based on record data. The comparison of the derived predictors is carried out by using Monte Carlo simulations. A real data are analysed for illustration purposes.     

Bayesian latent variable model for mixed continuous and ordinal responses with possibility of missing responses

A general framework is proposed for joint modelling of mixed correlated ordinal and continuous responses with missing values for responses, where the missing mechanism for both kinds of responses is also considered. Considering the posterior distribution of unknowns given all available information, a Markov Chain Monte Carlo sampling algorithm via winBUGS is used for estimating the posterior distribution of the parameters. For sensitivity analysis to investigate the perturbation from missing at random to not missing at random, it is shown how one can use some elements of covariance structure. These elements associate responses and their missing mechanisms. Influence of small perturbation of these elements on posterior displacement and posterior estimates is also studied. The model is illustrated using data from a foreign language achievement study.     

Estimation using hybrid censored data from a generalized inverted exponential distribution

ABSTRACT

We consider point and interval estimation of the unknown parameters of a generalized inverted exponential distribution in the presence of hybrid censoring. The maximum likelihood estimates are obtained using EM algorithm. We then compute Fisher information matrix using the missing value principle. Bayes estimates are derived under squared error and general entropy loss functions. Furthermore, approximate Bayes estimates are obtained using Tierney and Kadane method as well as using importance sampling approach. Asymptotic and highest posterior density intervals are also constructed. Proposed estimates are compared numerically using Monte Carlo simulations and a real data set is analyzed for illustrative purposes.