Bayesian validation assessment of multivariate computational models

Multivariate model validation is a complex decision-making problem involving comparison of multiple correlated quantities, based upon the available information and prior knowledge. This paper presents a Bayesian risk-based decision method for validation assessment of multivariate predictive models under uncertainty. A generalized likelihood ratio is derived as a quantitative validation metric based on Bayes’ theorem and Gaussian distribution assumption of errors between validation data and model prediction. The multivariate model is then assessed based on the comparison of the likelihood ratio with a Bayesian decision threshold, a function of the decision costs and prior of each hypothesis. The probability density function of the likelihood ratio is constructed using the statistics of multiple response quantities and Monte Carlo simulation. The proposed methodology is implemented in the validation of a transient heat conduction model, using a multivariate data set from experiments. The Bayesian methodology provides a quantitative approach to facilitate rational decisions in multivariate model assessment under uncertainty.     

Bayesian inference method for model validation and confidence extrapolation

This paper presents a Bayesian-hypothesis-testing-based methodology for model validation and confidence extrapolation under uncertainty, using limited test data. An explicit expression of the Bayes factor is derived for the interval hypothesis testing. The interval method is compared with the Bayesian point null hypothesis testing approach. The Bayesian network with Markov Chain Monte Carlo simulation and Gibbs sampling is explored for extrapolating the inference from the validated domain at the component level to the untested domain at the system level. The effect of the number of experiments on the confidence in the model validation decision is investigated. The probabilities of Type I and Type II errors in decision-making during the model validation and confidence extrapolation are quantified. The proposed methodologies are applied to a structural mechanics problem. Numerical results demonstrate that the Bayesian methodology provides a quantitative approach to facilitate rational decisions in model validation and confidence extrapolation under uncertainty.     

Robust confidence interval for a residual standard deviation

The residual standard deviation of a general linear model provides information about predictive accuracy that is not revealed by the multiple correlation or regression coefficients. The classic confidence interval for a residual standard deviation is hypersensitive to minor violations of the normality assumption and its robustness does not improve with increasing sample size. An approximate confidence interval for the residual standard deviation is proposed and shown to be robust to moderate violations of the normality assumption with robustness to extreme non-normality that improves with increasing sample size.     

A Semiparametric Bayesian Approach to Multivariate Longitudinal Data

We extend the standard multivariate mixed model by incorporating a smooth time effect and relaxing distributional assumptions. We propose a semiparametric Bayesian approach to multivariate longitudinal data using a mixture of Polya trees prior distribution. Usually, the distribution of random effects in a longitudinal data model is assumed to be Gaussian. However, the normality assumption may be suspect, particularly if the estimated longitudinal trajectory parameters exhibit multimodality and skewness. In this paper we propose a mixture of Polya trees prior density to address the limitations of the parametric random effects distribution. We illustrate the methodology by analyzing data from a recent HIV-AIDS study.     

Bayesian local influence analysis of skew-normal spatial dynamic panel data models

The existing studies on spatial dynamic panel data model (SDPDM) mainly focus on the normality assumption of response variables and random effects. This assumption may be inappropriate in some applications. This paper proposes a new SDPDM by assuming that response variables and random effects follow the multivariate skew-normal distribution. A Markov chain Monte Carlo algorithm is developed to evaluate Bayesian estimates of unknown parameters and random effects in skew-normal SDPDM by combining the Gibbs sampler and the Metropolis–Hastings algorithm. A Bayesian local influence analysis method is developed to simultaneously assess the effect of minor perturbations to the data, priors and sampling distributions. Simulation studies are conducted to investigate the finite-sample performance of the proposed methodologies. An example is illustrated by the proposed methodologies.     

Dynamic Bayesian analysis of generalized odds ratios assuming multivariate skew-normal distribution for the error terms in the system equation

In this paper, we develop a methodology for the dynamic Bayesian analysis of generalized odds ratios in contingency tables. It is a standard practice to assume a normal distribution for the random effects in the dynamic system equations. Nevertheless, the normality assumption may be unrealistic in some applications and hence the validity of inferences can be dubious. Therefore, we assume a multivariate skew-normal distribution for the error terms in the system equation at each step. Moreover, we introduce a moving average approach to elicit the hyperparameters. Both simulated data and real data are analyzed to illustrate the application of this methodology.     

A decision theoretic approach to a distribution–free compound classification problem

A compound decision problem with component decision problem being the classification of a random sample as having come from one of the finite number of univariate populations is investigated. The Bayesian approach is discussed. A distribution–free decision rule is presented which has asymptotic risk equal to zero. The asymptotic efficiencies of these rules are discussed.

The results of a compter simulation are presented which compares the Bayes rule to the distribution–free rule under the assumption of normality. It is found that the distribution–free rule can be recommended in situations where certain key lo cation parameters are not known precisely and/or when certain distributional assumptions are not satisfied.     

Skew-normal factor analysis models with incomplete data

Traditional factor analysis (FA) rests on the assumption of multivariate normality. However, in some practical situations, the data do not meet this assumption; thus, the statistical inference made from such data may be misleading. This paper aims at providing some new tools for the skew-normal (SN) FA model when missing values occur in the data. In such a model, the latent factors are assumed to follow a restricted version of multivariate SN distribution with additional shape parameters for accommodating skewness. We develop an analytically feasible expectation conditional maximization algorithm for carrying out parameter estimation and imputation of missing values under missing at random mechanisms. The practical utility of the proposed methodology is illustrated with two real data examples and the results are compared with those obtained from the traditional FA counterparts.     

Multivariate extension of chi-squared univariate normality test

We propose a multivariate extension of the univariate chi-squared normality test. Using a known result for the distribution of quadratic forms in normal variables, we show that the proposed test statistic has an approximated chi-squared distribution under the null hypothesis of multivariate normality. As in the univariate case, the new test statistic is based on a comparison of observed and expected frequencies for specified events in sample space. In the univariate case, these events are the standard class intervals, but in the multivariate extension we propose these become hyper-ellipsoidal annuli in multivariate sample space. We assess the performance of the new test using Monte Carlo simulation. Keeping the type I error rate fixed, we show that the new test has power that compares favourably with other standard normality tests, though no uniformly most powerful test has been found. We recommend the new test due to its competitive advantages.     

A Bayesian approach to assessing population bioequivalence in a 2 ×2 crossover design

A Bayesian testing procedure is proposed for assessment of the bioequivalence in both mean and variance, which ensures population bioequivalence under the normality assumption. We derive the joint posterior distribution of the means and variances in a standard 2 ×2 crossover experimental design and propose a Bayesian testing procedure for bioequivalence based on a Markov chain Monte Carlo method. The proposed method is applied to a real data set.     

Bayesian analysis of multivariate t linear mixed models with missing responses at random

The multivariate t linear mixed model (MtLMM) has been recently proposed as a robust tool for analysing multivariate longitudinal data with atypical observations. Missing outcomes frequently occur in longitudinal research even in well controlled situations. As a powerful alternative to the traditional expectation maximization based algorithm employing single imputation, we consider a Bayesian analysis of the MtLMM to account for the uncertainties of model parameters and missing outcomes through multiple imputation. An inverse Bayes formulas sampler coupled with Metropolis-within-Gibbs scheme is used to effectively draw the posterior distributions of latent data and model parameters. The techniques for multiple imputation of missing values, estimation of random effects, prediction of future responses, and diagnostics of potential outliers are investigated as well. The proposed methodology is illustrated through a simulation study and an application to AIDS/HIV data.     

Bayesian inference for the randomly censored Burr-type XII distribution

The article presents the Bayesian inference for the parameters of randomly censored Burr-type XII distribution with proportional hazards. The joint conjugate prior of the proposed model parameters does not exist; we consider two different systems of priors for Bayesian estimation. The explicit forms of the Bayes estimators are not possible; we use Lindley's method to obtain the Bayes estimates. However, it is not possible to obtain the Bayesian credible intervals with Lindley's method; we suggest the Gibbs sampling procedure for this purpose. Numerical experiments are performed to check the properties of the different estimators. The proposed methodology is applied to a real-life data for illustrative purposes. The Bayes estimators are compared with the Maximum likelihood estimators via numerical experiments and real data analysis. The model is validated using posterior predictive simulation in order to ascertain its appropriateness.     

A shared parameter model of longitudinal measurements and survival time with heterogeneous random-effects distribution

Typical joint modeling of longitudinal measurements and time to event data assumes that two models share a common set of random effects with a normal distribution assumption. But, sometimes the underlying population that the sample is extracted from is a heterogeneous population and detecting homogeneous subsamples of it is an important scientific question. In this paper, a finite mixture of normal distributions for the shared random effects is proposed for considering the heterogeneity in the population. For detecting whether the unobserved heterogeneity exits or not, we use a simple graphical exploratory diagnostic tool proposed by Verbeke and Molenberghs [34] to assess whether the traditional normality assumption for the random effects in the mixed model is adequate. In the joint modeling setting, in the case of evidence against normality (homogeneity), a finite mixture of normals is used for the shared random-effects distribution. A Bayesian MCMC procedure is developed for parameter estimation and inference. The methodology is illustrated using some simulation studies. Also, the proposed approach is used for analyzing a real HIV data set, using the heterogeneous joint model for this data set, the individuals are classified into two groups: a group with high risk and a group with moderate risk.     

Nonparametric hierarchical Bayes analysis of binomial data via Bernstein polynomial priors

For binomial data analysis, many methods based on empirical Bayes interpretations have been developed, in which a variance‐stabilizing transformation and a normality assumption are usually required. To achieve the greatest model flexibility, we conduct nonparametric Bayesian inference for binomial data and employ a special nonparametric Bayesian prior—the Bernstein–Dirichlet process (BDP)—in the hierarchical Bayes model for the data. The BDP is a special Dirichlet process (DP) mixture based on beta distributions, and the posterior distribution resulting from it has a smooth density defined on [0, 1]. We examine two Markov chain Monte Carlo procedures for simulating from the resulting posterior distribution, and compare their convergence rates and computational efficiency. In contrast to existing results for posterior consistency based on direct observations, the posterior consistency of the BDP, given indirect binomial data, is established. We study shrinkage effects and the robustness of the BDP‐based posterior estimators in comparison with several other empirical and hierarchical Bayes estimators, and we illustrate through examples that the BDP‐based nonparametric Bayesian estimate is more robust to the sample variation and tends to have a smaller estimation error than those based on the DP prior. In certain settings, the new estimator can also beat Stein's estimator, Efron and Morris's limited‐translation estimator, and many other existing empirical Bayes estimators. The Canadian Journal of Statistics 40: 328–344; 2012 © 2012 Statistical Society of Canada     

On implementation of a test for Kronecker product covariance structure for multivariate repeated measures data

Under the assumption of multivariate normality the likelihood ratio test is derived to test a hypothesis for Kronecker product structure on a covariance matrix in the context of multivariate repeated measures data. Although the proposed hypothesis testing can be computationally performed by indirect use of Proc Mixed of SAS, the Proc Mixed algorithm often fails to converge. We provide an alternative algorithm. The algorithm is illustrated with two real data sets. A simulation study is also conducted for the purpose of sample size consideration.     

Model determination for the variance component model using reference priors

For the balanced variance component model when the inference concerning intraclass correlation coefficient is of interest, Bayesian analysis is often appropriate. However, the question remains is to choose the appropriate prior. In this paper, we consider testing of the intraclass correlation coefficient under a default prior specification. Berger and Bernardo's (1992) On the development of the reference prior method. In: Bernardo, J.M., Berger, J.O., Dawid, A.P., Smith, A.F.M. (Eds.), Bayesian Statist. Vol. 4. Oxford University Press, London, pp. 35–60 reference priors are developed and are used to obtain the intrinsic Bayes factor (Berger and Pericchi, 1996) The intrinsic Bayes factor for model selection and prediction. J. Amer. statist. Assoc. 91, 109–122 for the nested models. Influence diagnostics using intrinsic Bayes factors are also developed. Finally, one simulated data is provided which illustrates the proposed methodology with appropriate simulation based on computational formulas. Then in order to overcome the difficulty in Bayesian computation, MCMC method, such as Gibbs sampler and Metropolis–Hastings algorithm, is employed.     

Heteroscedastic regression models with non-normally distributed errors

Although regression estimates are quite robust to slight departure from normality, symmetric prediction intervals assuming normality can be highly unsatisfactory and problematic if the residuals have a skewed distribution. For data with distributions outside the class covered by the Generalized Linear Model, a common way to handle non-normality is to transform the response variable. Unfortunately, transforming the response variable often destroys the theoretical or empirical functional relationship connecting the mean of the response variable to the explanatory variables established on the original scale. Further complication arises if a single transformation cannot both stabilize variance and attain normality. Furthermore, practitioners also find the interpretation of highly transformed data not obvious and often prefer an analysis on the original scale. The present paper presents an alternative approach for handling simultaneously heteroscedasticity and non-normality without resorting to data transformation. Unlike classical approaches, the proposed modeling allows practitioners to formulate the mean and variance relationships directly on the original scale, making data interpretation considerably easier. The modeled variance relationship and form of non-normality in the proposed approach can be easily examined through a certain function of the standardized residuals. The proposed method is seen to remain consistent for estimating the regression parameters even if the variance function is misspecified. The method along with some model checking techniques is illustrated with a real example.     

A Method for Simulating Multivariate Non Normal Distributions with Specified Standardized Cumulants and Intraclass Correlation Coefficients

Intraclass correlation coefficients (ICCs) are commonly used indices in subject areas such as biometrics, longitudinal data analysis, measurement theory, quality control, and survey research. The properties of the ICCs most often used are derived under the assumption of normality. However, real-world data often violate the normality assumption. In view of this, a computationally efficient procedure is developed for simulating multivariate non normal continuous distributions with specified (a) standardized cumulants, (b) Pearson intercorrelations, and (c) ICCs. The linear model specified is a two-factor design with either fixed or random effects. A numerical example is worked and the results of a Monte Carlo simulation are provided to demonstrate and confirm the methodology.     

Comparison of Within-Subject Covariance Matrices in 2 × 2 Crossover Trials with Multivariate Response

Abstract.  In a multivariate framework, it is shown how the two treatments in a 2 × 2 crossover trial with multivariate response can be compared with respect to mean vectors, i.e. fixed treatment effects, as well as within-subject covariance matrices, marginally and simultaneously. No distributional assumption is made about the between-subject variability, whereas multivariate normality is assumed for the within-subject variability. The proposed exact statistical inferences are valid even with few subjects. Data from a crossover trial with bivariate response are analysed with the proposed multivariate methods as well as with univariate methods.