Conservative prior distributions for variance parameters in hierarchical models

Bayesian hierarchical models typically involve specifying prior distributions for one or more variance components. This is rather removed from the observed data, so specification based on expert knowledge can be difficult. While there are suggestions for “default” priors in the literature, often a conditionally conjugate inverse‐gamma specification is used, despite documented drawbacks of this choice. The authors suggest “conservative” prior distributions for variance components, which deliberately give more weight to smaller values. These are appropriate for investigators who are skeptical about the presence of variability in the second‐stage parameters (random effects) and want to particularly guard against inferring more structure than is really present. The suggested priors readily adapt to various hierarchical modelling settings, such as fitting smooth curves, modelling spatial variation and combining data from multiple sites.     

Sensitivity analysis for variance parameters in Bayesian simplex mixed models for proportional data

In this article, we present a Bayesian modeling for response variables restricted to the interval (0, 1), such as proportions and rates, using the simplex distribution for cases in which data have a longitudinal form, taking random effects into account. In order to investigate the stability of posterior distribution, we study through sensitivity analysis, the effect of three different uniparametric prior distributions for variance parameters of random effect on the final estimation. For this purpose, we consider homogeneous and heterogeneous structures for parameters in location and dispersion submodels. Models and results are illustrated with simulated and real data application.     

Bayesian inference for a flexible class of bivariate beta distributions

Several bivariate beta distributions have been proposed in the literature. In particular, Olkin and Liu [A bivariate beta distribution. Statist Probab Lett. 2003;62(4):407–412] proposed a 3 parameter bivariate beta model which Arnold and Ng [Flexible bivariate beta distributions. J Multivariate Anal. 2011;102(8):1194–1202] extend to 5 and 8 parameter models. The 3 parameter model allows for only positive correlation, while the latter models can accommodate both positive and negative correlation. However, these come at the expense of a density that is mathematically intractable. The focus of this research is on Bayesian estimation for the 5 and 8 parameter models. Since the likelihood does not exist in closed form, we apply approximate Bayesian computation, a likelihood free approach. Simulation studies have been carried out for the 5 and 8 parameter cases under various priors and tolerance levels. We apply the 5 parameter model to a real data set by allowing the model to serve as a prior to correlated proportions of a bivariate beta binomial model. Results and comparisons are then discussed.     

A Bayesian method for safety signal detection in ongoing blinded randomised controlled trials

Sponsors have a responsibility to minimise risk to participants in clinical studies through safety monitoring. The FDA Final Rule for IND Safety Reporting requires routine aggregate safety evaluation, including in ongoing blinded studies. We are interested in estimating the probability that the true adverse event rate in the experimental arm exceeds that in the control arm. We developed a Bayesian approach that specifies an informative meta-analytic predictive prior on the event probability in the control arm and an uninformative prior on that in the experimental arm. We combined these priors with a mixture likelihood that considers each patient in the ongoing blinded study may belong to the experimental or control arm. This allowed us to estimate the quantity of interest without unblinding. We evaluated our method by simulation, pairing scenarios that differed only in whether a safety signal was present or missing, and quantifying the ability of our model to discriminate using signal detection theory. Our approach shows benefit. It detects safety signals more reliably with greater sample sizes and for common rather than rare events. Performance does not deteriorate markedly when historical studies exhibit heterogeneous hazards or non-constant hazards. Our method will allow us to monitor safety signals in ongoing blinded studies with the goal of earlier identification and risk mitigation. Our method could be adapted to use informative priors on both arms or predictive covariates where pertinent data exist. We stress that ongoing safety monitoring should involve a multi-disciplinary team where statistical methods are paired with medical judgement.     

Heterogeneity and study size in random-effects meta-analysis

It is well known that heterogeneity between studies in a meta-analysis can be either caused by diversity, for example, variations in populations and interventions, or caused by bias, that is, variations in design quality and conduct of the studies. Heterogeneity that is due to bias is difficult to deal with. On the other hand, heterogeneity that is due to diversity is taken into account by a standard random-effects model. However, such a model generally assumes that heterogeneity does not vary according to study-level variables such as the size of the studies in the meta-analysis and the type of study design used. This paper develops models that allow for this type of variation in heterogeneity and discusses the properties of the resulting methods. The models are fitted using the maximum-likelihood method and by modifying the Paule–Mandel method. Furthermore, a real-world argument is given to support the assumption that the inter-study variance is inversely proportional to study size. Under this assumption, the corresponding random-effects method is shown to be connected with standard fixed-effect meta-analysis in a way that may well appeal to many clinicians. The models and methods that are proposed are applied to data from two large systematic reviews.     

Effect of Prior Specification on Bayesian Design for Two-Sample Comparison of a Binary Outcome

Closed forms of the optimal Bayesian design for the estimation of the log odds-ratio from a two-sample experiment are obtained. The effect of various specifications of prior distributions are considered and plausible circumstances where unequal allocation is optimal are presented.     

Calculation of marginal densities for parameters of multinomial distributions

The full Bayesian analysis of multinomial data using informative and flexible prior distributions has, in the past, been restricted by the technical problems involved in performing the numerical integrations required to obtain marginal densities for parameters and other functions thereof. In this paper it is shown that Gibbs sampling is suitable for obtaining accurate approximations to marginal densities for a large and flexible family of posterior distributions—the family. The method is illustrated with a three-way contingency table. Two alternative Monte Carlo strategies are also discussed.     

Subroutines for computing minimum variance unbiased estimators of functions of the parameters in the normal and gamma distributions

In this paper subroutines are given which calculate the uniformly minimum variance unbiased estimators (UMVUE’ s) of a broad class of functions of the parameters of the normal and gamma distributions. These subroutines employ the new expressions for the UMVUE’ s given recently by Gray, Watkins, and Schucany (1973), Woodward and Gray (1975), and Gray, Schucany, and Woodward (1976). In order to employ the subroutines here the user need only be able to provide a FORTRAN function subprogram to calculate derivatives of the function, either analytically or numerically.     

On the Choice of a Prior for Bayesian D-Optimal Designs for the Logistic Regression Model with a Single Predictor

The Bayesian design approach accounts for uncertainty of the parameter values on which optimal design depends, but Bayesian designs themselves depend on the choice of a prior distribution for the parameter values. This article investigates Bayesian D-optimal designs for two-parameter logistic models, using numerical search. We show three things: (1) a prior with large variance leads to a design that remains highly efficient under other priors, (2) uniform and normal priors lead to equally efficient designs, and (3) designs with four or five equidistant equally weighted design points are highly efficient relative to the Bayesian D-optimal designs.     

The conditional variance for gamma mixtures of normal distributions

This paper is concerned with obtaining an expression for the conditional variance-covariance matrix when the random vector is gamma scaled of a multivariate normal distribution. We show that the conditional variance is not degenerate as in the multivariate normal distribution, but depends upon a positive function for which various asymptotic properties are derived. A discussion section is included commenting on the usefulness of these results     

Robust exchangeability designs for early phase clinical trials with multiple strata

Clinical trials with multiple strata are increasingly used in drug development. They may sometimes be the only option to study a new treatment, for example in small populations and rare diseases. In early phase trials, where data are often sparse, good statistical inference and subsequent decision‐making can be challenging. Inferences from simple pooling or stratification are known to be inferior to hierarchical modeling methods, which build on exchangeable strata parameters and allow borrowing information across strata. However, the standard exchangeability (EX) assumption bears the risk of too much shrinkage and excessive borrowing for extreme strata. We propose the exchangeability–nonexchangeability (EXNEX) approach as a robust mixture extension of the standard EX approach. It allows each stratum‐specific parameter to be exchangeable with other similar strata parameters or nonexchangeable with any of them. While EXNEX computations can be performed easily with standard Bayesian software, model specifications and prior distributions are more demanding and require a good understanding of the context. Two case studies from phases I and II (with three and four strata) show promising results for EXNEX. Data scenarios reveal tempered degrees of borrowing for extreme strata, and frequentist operating characteristics perform well for estimation (bias, mean‐squared error) and testing (less type‐I error inflation). Copyright © 2015 John Wiley & Sons, Ltd.     

Bayesian analysis for mixtures of discrete distributions with a non-parametric component

Bayesian finite mixture modelling is a flexible parametric modelling approach for classification and density fitting. Many areas of application require distinguishing a signal from a noise component. In practice, it is often difficult to justify a specific distribution for the signal component; therefore, the signal distribution is usually further modelled via a mixture of distributions. However, modelling the signal as a mixture of distributions is computationally non-trivial due to the difficulties in justifying the exact number of components to be used and due to the label switching problem. This paper proposes the use of a non-parametric distribution to model the signal component. We consider the case of discrete data and show how this new methodology leads to more accurate parameter estimation and smaller false non-discovery rate. Moreover, it does not incur the label switching problem. We show an application of the method to data generated by ChIP-sequencing experiments.     

Minimum variance unbiased estimation in doubly type II censored samples from families of distributions involving two truncation parameters

In this paper we construct uniformly minimum variance unbiased estimators for U-estimable functions when the underlying family of distributions involves two unknown truncation parameters and the sample is doubly Type II censored. Previous relevant results for the complete sample case are obtained as special cases of our results.     

A method for combining p-values in meta-analysis by gamma distributions

Combining p-values from statistical tests across different studies is the most commonly used approach in meta-analysis for evolutionary biology. The most commonly used p-value combination methods mainly incorporate the z-transform tests (e.g., the un-weighted z-test and the weighted z-test) and the gamma-transform tests (e.g., the CZ method [Z. Chen, W. Yang, Q. Liu, J.Y. Yang, J. Li, and M.Q. Yang, A new statistical approach to combining p-values using gamma distribution and its application to genomewide association study, Bioinformatics 15 (2014), p. S3]). However, among these existing p-value combination methods, no method is uniformly most powerful in all situations [Chen et al. 2014]. In this paper, we propose a meta-analysis method based on the gamma distribution, MAGD, by pooling the p-values from independent studies. The newly proposed test, MAGD, allows for flexible accommodating of the different levels of heterogeneity of effect sizes across individual studies. The MAGD simultaneously retains all the characters of the z-transform tests and the gamma-transform tests. We also propose an easy-to-implement resampling approach for estimating the empirical p-values of MAGD for the finite sample size. Simulation studies and two data applications show that the proposed method MAGD is essentially as powerful as the z-transform tests (the gamma-transform tests) under the circumstance with the homogeneous (heterogeneous) effect sizes across studies.     

Efficient simulation of tail probabilities in a queueing model with heterogeneous servers

This paper considers a multi-server queue with Markov-modulated Poisson input and server-dependent phase-type service times. We develop an efficient rare-event simulation technique to estimate the probability that the number of customers in this system reaches a high value. Relying on explicit bounds on the probability under consideration as well as the associated likelihood ratio, we succeed in proving that the proposed estimator is of bounded relative error. Simulation experiments illustrate the significant speed-up that can be achieved by the proposed algorithm.     

Modeling with a large class of unimodal multivariate distributions

In this paper we introduce a new class of multivariate unimodal distributions, motivated by Khintchine's representation for unimodal densities on the real line. We start by introducing a new class of unimodal distributions which can then be naturally extended to higher dimensions, using the multivariate Gaussian copula. Under both univariate and multivariate settings, we provide MCMC algorithms to perform inference about the model parameters and predictive densities. The methodology is illustrated with univariate and bivariate examples, and with variables taken from a real data set.     

Testing for variance changes in autoregressive models with unknown order

The problem of change point in autoregressive process is studied in this article. We propose a Bayesian information criterion-iterated cumulative sums of squares algorithm to detect the variance changes in an autoregressive series with unknown order. Simulation results and two examples are presented, where it is shown to have good performances when the sample size is relatively small.     

Confidence intervals for variance components in unbalanced one-way random effects model using non-normal distributions

In scenarios where the variance of a response variable can be attributed to two sources of variation, a confidence interval for a ratio of variance components gives information about the relative importance of the two sources. For example, if measurements taken from different laboratories are nine times more variable than the measurements taken from within the laboratories, then 90% of the variance in the responses is due to the variability amongst the laboratories and 10% of the variance in the responses is due to the variability within the laboratories. Assuming normally distributed sources of variation, confidence intervals for variance components are readily available. In this paper, however, simulation studies are conducted to evaluate the performance of confidence intervals under non-normal distribution assumptions. Confidence intervals based on the pivotal quantity method, fiducial inference, and the large-sample properties of the restricted maximum likelihood (REML) estimator are considered. Simulation results and an empirical example suggest that the REML-based confidence interval is favored over the other two procedures in unbalanced one-way random effects model.     

Estimation of scale parameters in mixture distributions

Simultaneous estimation of scale parameters is considered in mixture distributions under squared-error loss. A general class of estimators is obtained which dominates the componentwise best multiple estimators and the moment estimators. As special cases, improved estimators are obtained for the multivariate t-distribution and the p-variate Lomax distribution.     

Inferences from logistic regression models in the presence of small samples,rare events,nonlinearity, and multicollinearity with observational data

The logistic regression model has been widely used in the social and natural sciences and results from studies using this model can have significant policy impacts. Thus, confidence in the reliability of inferences drawn from these models is essential. The robustness of such inferences is dependent on sample size. The purpose of this article is to examine the impact of alternative data sets on the mean estimated bias and efficiency of parameter estimation and inference for the logistic regression model with observational data. A number of simulations are conducted examining the impact of sample size, nonlinear predictors, and multicollinearity on substantive inferences (e.g. odds ratios, marginal effects) when using logistic regression models. Findings suggest that small sample size can negatively affect the quality of parameter estimates and inferences in the presence of rare events, multicollinearity, and nonlinear predictor functions, but marginal effects estimates are relatively more robust to sample size.