Summarizing empirical information on between-study heterogeneity for Bayesian random-effects meta-analysis.

In Bayesian meta-analysis, the specification of prior probabilities for the between-study heterogeneity is commonly required, and is of particular benefit in situations where only few studies are included. Among the considerations in the set-up of such prior distributions, the consultation of available empirical data on a set of relevant past analyses sometimes plays a role. How exactly to summarize historical data sensibly is not immediately obvious; in particular, the investigation of an empirical collection of heterogeneity estimates will not target the actual problem and will usually only be of limited use. The commonly used normal-normal hierarchical model for random-effects meta-analysis is extended to infer a heterogeneity prior. Using an example data set, we demonstrate how to fit a distribution to empirically observed heterogeneity data from a set of meta-analyses. Considerations also include the choice of a parametric distribution family. Here, we focus on simple and readily applicable approaches to then translate these into (prior) probability distributions.

State-space prior distribution for parameter of nonhomogeneous Poisson spatiotemporal models.

This article proposes a new class of nonhomogeneous Poisson spatiotemporal model. In this approach, we use a state-space model-based prior distribution to handle the scale and shape parameters of the Weibull intensity function. The proposed prior distribution enables the inclusion of changes in the behavior of the intensity function over time. In defining the spatial correlation function of the model, we include anisotropy via spatial deformation. We estimate the model parameters from a Bayesian perspective, employ the Markov chain Monte Carlo approach, and validate this estimation procedure through a simulation exercise. Finally, the extreme rainfall in the southern semiarid region in northeastern Brazil is analyzed using the R10mm index. The proposed model showed better fit and prediction ability than did other nonhomogeneous Poisson spatiotemporal models available in the literature. This improvement in performance is mainly due to the flexibility of the intensity function that is achieved by allowing the incorporation, in time, of the climatic characteristics of this region.

Prior Choices of Between-Study Heterogeneity in Contemporary Bayesian Network Meta-analyses: an Empirical Study.

BACKGROUND: Network meta-analysis (NMA) is a popular tool to compare multiple treatments in medical research. It is frequently implemented via Bayesian methods. The prior choice of between-study heterogeneity is critical in Bayesian NMAs. This study evaluates the impact of different priors for heterogeneity on NMA results. METHODS: We identified all NMAs with binary outcomes published in The BMJ, JAMA, and The Lancet during 2010-2018, and extracted information about their prior choices for heterogeneity. Our primary analyses focused on those with publicly available full data. We re-analyzed the NMAs using 3 commonly-used non-informative priors and empirical informative log-normal priors. We obtained the posterior median odds ratios and 95% credible intervals of all comparisons, assessed the correlation among different priors, and used Bland-Altman plots to evaluate their agreement. The kappa statistic was also used to evaluate the agreement among these priors regarding statistical significance. RESULTS: Among the selected Bayesian NMAs, 52.3% did not specify the prior choice for heterogeneity, and 84.1% did not provide rationales. We re-analyzed 19 NMAs with full data available, involving 894 studies, 173 treatments, and 395,429 patients. The correlation among posterior median (log) odds ratios using different priors were generally very strong for NMAs with over 20 studies. The informative priors produced substantially narrower credible intervals than non-informative priors, especially for NMAs with few studies. Bland-Altman plots and kappa statistics indicated strong overall agreement, but this was not always the case for a specific NMA. CONCLUSIONS: Priors should be routinely reported in Bayesian NMAs. Sensitivity analyses are recommended to examine the impact of priors, especially for NMAs with relatively small sample sizes. Informative priors may produce substantially narrower credible intervals for such NMAs.

Bayesian model-averaged meta-analysis in medicine.

We outline a Bayesian model-averaged (BMA) meta-analysis for standardized mean differences in order to quantify evidence for both treatment effectiveness δ and across-study heterogeneity τ . We construct four competing models by orthogonally combining two present-absent assumptions, one for the treatment effect and one for across-study heterogeneity. To inform the choice of prior distributions for the model parameters, we used 50% of the Cochrane Database of Systematic Reviews to specify rival prior distributions for δ and τ . The relative predictive performance of the competing models and rival prior distributions was assessed using the remaining 50% of the Cochrane Database. On average, ℋ1r -the model that assumes the presence of a treatment effect as well as across-study heterogeneity-outpredicted the other models, but not by a large margin. Within ℋ1r , predictive adequacy was relatively constant across the rival prior distributions. We propose specific empirical prior distributions, both for the field in general and for each of 46 specific medical subdisciplines. An example from oral health demonstrates how the proposed prior distributions can be used to conduct a BMA meta-analysis in the open-source software R and JASP. The preregistered analysis plan is available at https://osf.io/zs3df/.

Constrained Adjusted Maximum a Posteriori Estimation of Bayesian Network Parameters.

Maximum a posteriori estimation (MAP) with Dirichlet prior has been shown to be effective in improving the parameter learning of Bayesian networks when the available data are insufficient. Given no extra domain knowledge, uniform prior is often considered for regularization. However, when the underlying parameter distribution is non-uniform or skewed, uniform prior does not work well, and a more informative prior is required. In reality, unless the domain experts are extremely unfamiliar with the network, they would be able to provide some reliable knowledge on the studied network. With that knowledge, we can automatically refine informative priors and select reasonable equivalent sample size (ESS). In this paper, considering the parameter constraints that are transformed from the domain knowledge, we propose a Constrained adjusted Maximum a Posteriori (CaMAP) estimation method, which is featured by two novel techniques. First, to draw an informative prior distribution (or prior shape), we present a novel sampling method that can construct the prior distribution from the constraints. Then, to find the optimal ESS (or prior strength), we derive constraints on the ESS from the parameter constraints and select the optimal ESS by cross-validation. Numerical experiments show that the proposed method is superior to other learning algorithms.

Predictively consistent prior effective sample sizes.

Determining the sample size of an experiment can be challenging, even more so when incorporating external information via a prior distribution. Such information is increasingly used to reduce the size of the control group in randomized clinical trials. Knowing the amount of prior information, expressed as an equivalent prior effective sample size (ESS), clearly facilitates trial designs. Various methods to obtain a prior's ESS have been proposed recently. They have been justified by the fact that they give the standard ESS for one-parameter exponential families. However, despite being based on similar information-based metrics, they may lead to surprisingly different ESS for nonconjugate settings, which complicates many designs with prior information. We show that current methods fail a basic predictive consistency criterion, which requires the expected posterior-predictive ESS for a sample of size N to be the sum of the prior ESS and N. The expected local-information-ratio ESS is introduced and shown to be predictively consistent. It corrects the ESS of current methods, as shown for normally distributed data with a heavy-tailed Student-t prior and exponential data with a generalized Gamma prior. Finally, two applications are discussed: the prior ESS for the control group derived from historical data and the posterior ESS for hierarchical subgroup analyses.

Bayesian leveraging of historical control data for a clinical trial with time-to-event endpoint.

The recent 21st Century Cures Act propagates innovations to accelerate the discovery, development, and delivery of 21st century cures. It includes the broader application of Bayesian statistics and the use of evidence from clinical expertise. An example of the latter is the use of trial-external (or historical) data, which promises more efficient or ethical trial designs. We propose a Bayesian meta-analytic approach to leverage historical data for time-to-event endpoints, which are common in oncology and cardiovascular diseases. The approach is based on a robust hierarchical model for piecewise exponential data. It allows for various degrees of between trial-heterogeneity and for leveraging individual as well as aggregate data. An ovarian carcinoma trial and a non-small cell cancer trial illustrate methodological and practical aspects of leveraging historical data for the analysis and design of time-to-event trials.

Ensuring identifiability in hierarchical mixed effects Bayesian models.

Ecologists are increasingly familiar with Bayesian statistical modeling and its associated Markov chain Monte Carlo (MCMC) methodology to infer about or to discover interesting effects in data. The complexity of ecological data often suggests implementation of (statistical) models with a commensurately rich structure of effects, including crossed or nested (i.e., hierarchical or multi-level) structures of fixed and/or random effects. Yet, our experience suggests that most ecologists are not familiar with subtle but important problems that often arise with such models and with their implementation in popular software. Of foremost consideration for us is the notion of effect identifiability, which generally concerns how well data, models, or implementation approaches inform about, i.e., identify, quantities of interest. In this paper, we focus on implementation pitfalls that potentially misinform subsequent inference, despite otherwise informative data and models. We illustrate the aforementioned issues using random effects regressions on synthetic data. We show how to diagnose identifiability issues and how to remediate these issues with model reparameterization and computational and/or coding practices in popular software, with a focus on JAGS, OpenBUGS, and Stan. We also show how these solutions can be extended to more complex models involving multiple groups of nested, crossed, additive, or multiplicative effects, for models involving random and/or fixed effects. Finally, we provide example code (JAGS/OpenBUGS and Stan) that practitioners can modify and use for their own applications.

Diversity in substance use behaviour among street children of Delhi under Bayesian paradigm.

BACKGROUND: Shannon's index is one of the measures of biodiversity, which is intended to quantify both richness and evenness of the species/individuals in the ecosystem or community. However, application of Shannon's index in the field of substance use among the street children has not been done till date. METHODS: This paper is concerned with methods of estimating Shannon's diversity index (SDI), which can be used to capture the variation in the population due to certain characteristics. Under the consideration that the probability of abundance, based on certain characteristics in the population, is a random phenomenon, we derive a Bayesian estimate in connection with Shannon's information measure and their properties (mean and variance), by using a probability matching prior, through simulation and compared it with those of the classical estimates of Shannon. The theoretical framework has been applied to the primary survey data of substance use among the street children in Delhi, collected during 2015. The measure of diversity was estimated across different age profiles and districts. RESULTS: The results unrevealing the diversity estimate for street children corresponding to each region of Delhi, under both the classical and Bayesian paradigms. Although the estimates were close to one another, a striking difference was noted in the age profile of children. CONCLUSIONS: The Bayesian methodology provided evidence for a greater likelihood of finding substance-using street children, belonging to the lower age group (7-10, maximum Bayesian entropy-3.73), followed by the middle (11-14) and upper age group (15-18). Moreover, the estimated variance under the Bayesian paradigm was lesser than that of the classical estimate. There is ample scope for further refinement in these estimates, by considering more covariates that may have a possible role in initiating substance use among street children in developing countries like India.

Bayesian Extended Redundancy Analysis: A Bayesian Approach to Component-based Regression with Dimension Reduction.

Extended redundancy analysis (ERA) combines linear regression with dimension reduction to explore the directional relationships between multiple sets of predictors and outcome variables in a parsimonious manner. It aims to extract a component from each set of predictors in such a way that it accounts for the maximum variance of outcome variables. In this article, we extend ERA into the Bayesian framework, called Bayesian ERA (BERA). The advantages of BERA are threefold. First, BERA enables to make statistical inferences based on samples drawn from the joint posterior distribution of parameters obtained from a Markov chain Monte Carlo algorithm. As such, it does not necessitate any resampling method, which is on the other hand required for (frequentist's) ordinary ERA to test the statistical significance of parameter estimates. Second, it formally incorporates relevant information obtained from previous research into analyses by specifying informative power prior distributions. Third, BERA handles missing data by implementing multiple imputation using a Markov Chain Monte Carlo algorithm, avoiding the potential bias of parameter estimates due to missing data. We assess the performance of BERA through simulation studies and apply BERA to real data regarding academic achievement.

A group-specific prior distribution for effect-size heterogeneity in meta-analysis.

While both methodological and applied work on Bayesian meta-analysis have flourished, Bayesian modeling of differences between groups of studies remains scarce in meta-analyses in psychology, education, and the social sciences. On rare occasions when Bayesian approaches have been used, non-informative prior distributions have been chosen. However, more informative prior distributions have recently garnered popularity. We propose a group-specific weakly informative prior distribution for the between-studies standard-deviation parameter in meta-analysis. The proposed prior distribution incorporates a frequentist estimate of the between-studies standard deviation as the noncentrality parameter in a folded noncentral t distribution. This prior distribution is then separately modeled for each subgroup of studies, as determined by a categorical factor. Use of the new prior distribution is shown in two extensive examples based on a published meta-analysis on psychological interventions aimed at increasing optimism. We compare the folded noncentral t prior distribution to several non-informative prior distributions. We conclude with discussion, limitations, and avenues for further development of Bayesian meta-analysis in psychology and the social sciences.

Impact of prior distributions and central tendency measures on Bayesian intravoxel incoherent motion model fitting.

PURPOSE: Bayesian model fitting has been proposed as a robust alternative for intravoxel incoherent motion (IVIM) model-fitting parameter estimation. However, consensus regarding choice of prior distribution and posterior distribution central tendency measure is needed. The aim of this study was to compare the quality of IVIM parameter estimates produced by different prior distributions and central tendency measures, and to gain knowledge about the effect of these choices. METHODS: Three prior distributions (uniform, reciprocal, and lognormal) and two measures of central tendency (mean and mode) found in the literature were studied using simulations and in vivo data from a tumor mouse model. RESULTS: Simulations showed that the uniform and lognormal priors were superior to the reciprocal prior, especially for the parameters D and f and clinically relevant SNR levels. The choice of central tendency measure had less effect on the results, but had some effects on estimation bias. Results based on simulations and in vivo data agreed well, indicating high validity of the simulations. CONCLUSIONS: Choice of prior distribution and central tendency measure affects the results of Bayesian IVIM parameter estimates. This must be considered when comparing results from different studies. The best overall quality of IVIM parameter estimates was obtained using the lognormal prior. Magn Reson Med 79:1674-1683, 2018. © 2017 International Society for Magnetic Resonance in Medicine.

Genomic prediction using an iterative conditional expectation algorithm for a fast BayesC-like model.

Genomic prediction is feasible for estimating genomic breeding values because of dense genome-wide markers and credible statistical methods, such as Genomic Best Linear Unbiased Prediction (GBLUP) and various Bayesian methods. Compared with GBLUP, Bayesian methods propose more flexible assumptions for the distributions of SNP effects. However, most Bayesian methods are performed based on Markov chain Monte Carlo (MCMC) algorithms, leading to computational efficiency challenges. Hence, some fast Bayesian approaches, such as fast BayesB (fBayesB), were proposed to speed up the calculation. This study proposed another fast Bayesian method termed fast BayesC (fBayesC). The prior distribution of fBayesC assumes that a SNP with probability γ has a non-zero effect which comes from a normal density with a common variance. The simulated data from QTLMAS XII workshop and actual data on large yellow croaker were used to compare the predictive results of fBayesB, fBayesC and (MCMC-based) BayesC. The results showed that when γ was set as a small value, such as 0.01 in the simulated data or 0.001 in the actual data, fBayesB and fBayesC yielded lower prediction accuracies (abilities) than BayesC. In the actual data, fBayesC could yield very similar predictive abilities as BayesC when γ ≥ 0.01. When γ = 0.01, fBayesB could also yield similar results as fBayesC and BayesC. However, fBayesB could not yield an explicit result when γ ≥ 0.1, but a similar situation was not observed for fBayesC. Moreover, the computational speed of fBayesC was significantly faster than that of BayesC, making fBayesC a promising method for genomic prediction.

Bayesian survival analysis in clinical trials: What methods are used in practice?

Background Bayesian statistics are an appealing alternative to the traditional frequentist approach to designing, analysing, and reporting of clinical trials, especially in rare diseases. Time-to-event endpoints are widely used in many medical fields. There are additional complexities to designing Bayesian survival trials which arise from the need to specify a model for the survival distribution. The objective of this article was to critically review the use and reporting of Bayesian methods in survival trials. Methods A systematic review of clinical trials using Bayesian survival analyses was performed through PubMed and Web of Science databases. This was complemented by a full text search of the online repositories of pre-selected journals. Cost-effectiveness, dose-finding studies, meta-analyses, and methodological papers using clinical trials were excluded. Results In total, 28 articles met the inclusion criteria, 25 were original reports of clinical trials and 3 were re-analyses of a clinical trial. Most trials were in oncology (n = 25), were randomised controlled (n = 21) phase III trials (n = 13), and half considered a rare disease (n = 13). Bayesian approaches were used for monitoring in 14 trials and for the final analysis only in 14 trials. In the latter case, Bayesian survival analyses were used for the primary analysis in four cases, for the secondary analysis in seven cases, and for the trial re-analysis in three cases. Overall, 12 articles reported fitting Bayesian regression models (semi-parametric, n = 3; parametric, n = 9). Prior distributions were often incompletely reported: 20 articles did not define the prior distribution used for the parameter of interest. Over half of the trials used only non-informative priors for monitoring and the final analysis (n = 12) when it was specified. Indeed, no articles fitting Bayesian regression models placed informative priors on the parameter of interest. The prior for the treatment effect was based on historical data in only four trials. Decision rules were pre-defined in eight cases when trials used Bayesian monitoring, and in only one case when trials adopted a Bayesian approach to the final analysis. Conclusion Few trials implemented a Bayesian survival analysis and few incorporated external data into priors. There is scope to improve the quality of reporting of Bayesian methods in survival trials. Extension of the Consolidated Standards of Reporting Trials statement for reporting Bayesian clinical trials is recommended.

On the prior distribution of extinction time.

Bayesian inference about the extinction of a species based on a record of its sightings requires the specification of a prior distribution for extinction time. Here, I critically review some specifications in the context of a specific model of the sighting record. The practical implication of the choice of prior distribution is illustrated through an application to the sighting record of the Caribbean monk seal.

Sequentially updating the likelihood of success of a Phase 3 pivotal time-to-event trial based on interim analyses or external information.

When performing a pivotal clinical trial, it may be of interest to assess the probability of success (PoS) of that trial. Initially evaluated when the trial is designed, PoS can be updated as the trial progresses and new information about the drug effect becomes available. Such information can be external to the trial, such as results from trials conducted in parallel, or internal, such as continuing after an interim analysis. We develop a framework to update PoS based on such internal and external information for a time-to-event endpoint and illustrate it using a realistic development program for a new molecule.

Bayes Factors Have Frequency Properties-This Should Not Be Ignored: A Rejoinder to Morey, Wagenmakers, and Rouder.

Hoijtink, van Kooten, and Hulsker ( 2016 ) outline a research agenda for Bayesian psychologists: evaluate and use the frequency properties of Bayes factors. Morey, Wagenmakers, and Rouder ( 2016 ) respond that Bayes factors calibrated using frequency properties should not be used. This paper contains the response of Hoijtink, van Kooten, and Hulsker to the criticism of Morey, Wagenmakers, and Rouder ( 2016 ).

Why Bayesian Psychologists Should Change the Way They Use the Bayes Factor.

The discussion following Bem's ( 2011 ) psi research highlights that applications of the Bayes factor in psychological research are not without problems. The first problem is the omission to translate subjective prior knowledge into subjective prior distributions. In the words of Savage ( 1961 ): "they make the Bayesian omelet without breaking the Bayesian egg." The second problem occurs if the Bayesian egg is not broken: the omission to choose default prior distributions such that the ensuing inferences are well calibrated. The third problem is the adherence to inadequate rules for the interpretation of the size of the Bayes factor. The current paper will elaborate these problems and show how to avoid them using the basic hypotheses and statistical model used in the first experiment described in Bem ( 2011 ). It will be argued that a thorough investigation of these problems in the context of more encompassing hypotheses and statistical models is called for if Bayesian psychologists want to add a well-founded Bayes factor to the tool kit of psychological researchers.

Bayesian predictive power: choice of prior and some recommendations for its use as probability of success in drug development.

Bayesian predictive power, the expectation of the power function with respect to a prior distribution for the true underlying effect size, is routinely used in drug development to quantify the probability of success of a clinical trial. Choosing the prior is crucial for the properties and interpretability of Bayesian predictive power. We review recommendations on the choice of prior for Bayesian predictive power and explore its features as a function of the prior. The density of power values induced by a given prior is derived analytically and its shape characterized. We find that for a typical clinical trial scenario, this density has a u-shape very similar, but not equal, to a ß-distribution. Alternative priors are discussed, and practical recommendations to assess the sensitivity of Bayesian predictive power to its input parameters are provided. Copyright © 2016 John Wiley & Sons, Ltd.

A counterfactual p-value approach for benefit-risk assessment in clinical trials.

Clinical trials generally allow various efficacy and safety outcomes to be collected for health interventions. Benefit-risk assessment is an important issue when evaluating a new drug. Currently, there is a lack of standardized and validated benefit-risk assessment approaches in drug development due to various challenges. To quantify benefits and risks, we propose a counterfactual p-value (CP) approach. Our approach considers a spectrum of weights for weighting benefit-risk values and computes the extreme probabilities of observing the weighted benefit-risk value in one treatment group as if patients were treated in the other treatment group. The proposed approach is applicable to single benefit and single risk outcome as well as multiple benefit and risk outcomes assessment. In addition, the prior information in the weight schemes relevant to the importance of outcomes can be incorporated in the approach. The proposed CPs plot is intuitive with a visualized weight pattern. The average area under CP and preferred probability over time are used for overall treatment comparison and a bootstrap approach is applied for statistical inference. We assess the proposed approach using simulated data with multiple efficacy and safety endpoints and compare its performance with a stochastic multi-criteria acceptability analysis approach.