Normal-normal Hierarchical Model Research Articles

Bayesian random-effects meta-analysis with empirical heterogeneity priors for application in health technology assessment with very few studies.

In Bayesian random-effects meta-analysis, the use of weakly informative prior distributions is of particular benefit in cases where only a few studies are included, a situation often encountered in health technology assessment (HTA). Suggestions for empirical prior distributions are available in the literature but it is unknown whether these are adequate in the context of HTA. Therefore, a database of all relevant meta-analyses conducted by the Institute for Quality and Efficiency in Health Care (IQWiG, Germany) was constructed to derive empirical prior distributions for the heterogeneity parameter suitable for HTA. Previously, an extension to the normal-normal hierarchical model had been suggested for this purpose. For different effect measures, this extended model was applied on the database to conservatively derive a prior distribution for the heterogeneity parameter. Comparison of a Bayesian approach using the derived priors with IQWiG's current standard approach for evidence synthesis shows favorable properties. Therefore, these prior distributions are recommended for future meta-analyses in HTA settings and could be embedded into the IQWiG evidence synthesis approach in the case of very few studies.

Quantification of Empirical Determinacy: The Impact of Likelihood Weighting on Posterior Location and Spread in Bayesian Meta-Analysis Estimated with JAGS and INLA

The popular Bayesian meta-analysis expressed by Bayesian normal-normal hierarchical model (NNHM) synthesizes knowledge from several studies and is highly relevant in practice. Moreover, NNHM is the simplest Bayesian hierarchical model (BHM), which illustrates problems typical in more complex BHMs. Until now, it has been unclear to what extent the data determines the marginal posterior distributions of the parameters in NNHM. To address this issue we computed the second derivative of the Bhattacharyya coefficient with respect to the weighted likelihood, defined the total empirical determinacy (TED), the proportion of the empirical determinacy of location to TED (pEDL), and the proportion of the empirical determinacy of spread to TED (pEDS). We implemented this method in the R package \texttt{ed4bhm} and considered two case studies and one simulation study. We quantified TED, pEDL and pEDS under different modeling conditions such as model parametrization, the primary outcome, and the prior. This clarified to what extent the location and spread of the marginal posterior distributions of the parameters are determined by the data. Although these investigations focused on Bayesian NNHM, the method proposed is applicable more generally to complex BHMs.

Bayesian approaches to designing replication studies.

Replication studies are essential for assessing the credibility of claims from original studies. A critical aspect of designing replication studies is determining their sample size; a too-small sample size may lead to inconclusive studies whereas a too-large sample size may waste resources that could be allocated better in other studies. Here, we show how Bayesian approaches can be used for tackling this problem. The Bayesian framework allows researchers to combine the original data and external knowledge in a design prior distribution for the underlying parameters. Based on a design prior, predictions about the replication data can be made, and the replication sample size can be chosen to ensure a sufficiently high probability of replication success. Replication success may be defined by Bayesian or non-Bayesian criteria and different criteria may also be combined to meet distinct stakeholders and enable conclusive inferences based on multiple analysis approaches. We investigate sample size determination in the normal-normal hierarchical model where analytical results are available and traditional sample size determination is a special case where the uncertainty on parameter values is not accounted for. We use data from a multisite replication project of social-behavioral experiments to illustrate how Bayesian approaches can help design informative and cost-effective replication studies. Our methods can be used through the R package BayesRepDesign. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

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.

Sensitivity and identification quantification by a relative latent model complexity perturbation in Bayesian meta-analysis.

In recent years, Bayesian meta‐analysis expressed by a normal–normal hierarchical model (NNHM) has been widely used for combining evidence from multiple studies. Data provided for the NNHM are frequently based on a small number of studies and on uncertain within‐study standard deviation values. Despite the widespread use of Bayesian NNHM, it has always been unclear to what extent the posterior inference is impacted by the heterogeneity prior (sensitivity S) and by the uncertainty in the within‐study standard deviation values (identification I). Thus, to answer this question, we developed a unified method to simultaneously quantify both sensitivity and identification (S‐I) for all model parameters in a Bayesian NNHM, based on derivatives of the Bhattacharyya coefficient with respect to relative latent model complexity (RLMC) perturbations. Three case studies exemplify the applicability of the method proposed: historical data for a conventional therapy, data from which one large study is first included and then excluded, and two subgroup meta‐analyses specified by their randomization status. We analyzed six scenarios, crossing three RLMC targets with two heterogeneity priors (half‐normal, half‐Cauchy). The results show that S‐I explicitly reveals which parameters are affected by the heterogeneity prior and by the uncertainty in the within‐study standard deviation values. In addition, we compare the impact of both heterogeneity priors and quantify how S‐I values are affected by omitting one large study and by the randomization status. Finally, the range of applicability of S‐I is extended to Bayesian NtHM. A dedicated R package facilitates automatic S‐I quantification in applied Bayesian meta‐analyses.

How vague is vague? How informative is informative? Reference analysis for Bayesian meta-analysis.

Meta‐analysis provides important insights for evidence‐based medicine by synthesizing evidence from multiple studies which address the same research question. Within the Bayesian framework, meta‐analysis is frequently expressed by a Bayesian normal‐normal hierarchical model (NNHM). Recently, several publications have discussed the choice of the prior distribution for the between‐study heterogeneity in the Bayesian NNHM and used several “vague” priors. However, no approach exists to quantify the informativeness of such priors, and thus, we develop a principled reference analysis framework for the Bayesian NNHM acting at the posterior level. The posterior reference analysis (post‐RA) is based on two posterior benchmarks: one induced by the improper reference prior, which is minimally informative for the data, and the other induced by a highly anticonservative proper prior. This approach applies the Hellinger distance to quantify the informativeness of a heterogeneity prior of interest by comparing the corresponding marginal posteriors with both posterior benchmarks. The post‐RA is implemented in the freely accessible R package ra4bayesmeta and is applied to two medical case studies. Our findings show that anticonservative heterogeneity priors produce platykurtic posteriors compared with the reference posterior, and they produce shorter 95% credible intervals (CrI) and optimistic inference compared with the reference prior. Conservative heterogeneity priors produce leptokurtic posteriors, longer 95% CrI and cautious inference. The novel post‐RA framework could support numerous Bayesian meta‐analyses in many research fields, as it determines how informative a heterogeneity prior is for the actual data as compared with the minimally informative reference prior.

3 Estimating the Burden of Dementia in Latin America and the Caribbean: A Systematic Review and Meta-Analysis

Abstract Background and Objectives Rapid increase in life expectancy has resulted in an increase in the global burden of dementia that is expected to become a leading cause of morbidity in the future. Low- and middle-income countries are expected to bear an increasing majority of the burden, but lack data for accurate burden estimates that are key for informing policy and planning. Bayesian methods have recently gained recognition over traditional frequentist approaches for modelling disease burden for their superiority in dealing with severely limited data. This study provides updated estimates of dementia prevalence in Latin America and the Caribbean (LAC) for the years 2015, 2020 and 2030. Given the paucity of data, estimates were developed using a Bayesian methodology and confirmed by the traditional frequentist approach, with the aim of providing methodological insights for future disease burden estimates. Methods A comprehensive systematic literature search was conducted to identify all relevant primary studies published between the years 2010–2018. The quality of the included studies was critically assessed. A random-effects model (REM) and a Bayesian normal-normal hierarchical model (NNHM) were used to obtain the pooled prevalence estimate of dementia for people aged 60 and above. The latter model was also developed to estimate age-specific dementia prevalence. Using UN population estimates, total and age-specific projections of the burden of dementia were calculated. Results The prevalence of dementia in LAC was found to be 14% (10–21%) in those above age 60 based on REM, and 8% (5–11.5%) based on NNHM. The prevalence increased from 2% (1–4%) in people aged 60–69 to 29% (20–37%) in people above the age of 80. The number of people living with dementia in LAC in 2015 was estimated at 5.68 million, with future projections of 6.86 million in 2020 and 9.94 million in 2030. Conclusions The findings of this review found that burden of dementia in LAC is substantial and continues to rapidly grow. The projected rise in dementia cases in the future should prompt urgent governmental response to address this growing public health issue. We were also able to demonstrate that given the overall paucity of data, a Bayesian approach was superior for estimating disease prevalence and burden.

On weakly informative prior distributions for the heterogeneity parameter in Bayesian random-effects meta-analysis.

The normal-normal hierarchical model (NNHM) constitutes a simple and widely used framework for meta-analysis. In the common case of only few studies contributing to the meta-analysis, standard approaches to inference tend to perform poorly, and Bayesian meta-analysis has been suggested as a potential solution. The Bayesian approach, however, requires the sensible specification of prior distributions. While noninformative priors are commonly used for the overall mean effect, the use of weakly informative priors has been suggested for the heterogeneity parameter, in particular in the setting of (very) few studies. To date, however, a consensus on how to generally specify a weakly informative heterogeneity prior is lacking. Here we investigate the problem more closely and provide some guidance on prior specification.

Systematic review estimating the burden of dementia in the WHO Southeast Asia Region using Bayesian and frequentist approaches

BackgroundRapid increase in life expectancy in low- and middle-income countries including the World Health Organization’s Southeast Asia Region (SEAR) has resulted in an increase in the global burden of dementia, which is expected to become a leading cause of morbidity. Accurate burden estimates are key for informing policy and planning. Given the paucity of data, estimates were developed using both a Bayesian methodology and as well as a traditional frequentist approach to gain better insights into methodological approaches for disease burden estimates.MethodsSeven databases were searched for studies published between 2010-2018 regarding dementia prevalence in SEAR, generating 8 relevant articles. A random-effects model (REM) and a Bayesian normal-normal hierarchical model (NNHM) were used to obtain the pooled prevalence estimate of dementia for people aged 60 and above in SEAR. The latter model was also developed to estimate age-specific dementia prevalence. Using UN population estimates for SEAR, total and age-specific projections of the burden of dementia in 2015, 2020 and 2030 were calculated.ResultsThe prevalence of dementia in SEAR was found to be 3% (95% confidence interval (CI) = 2-6%) in those above age 60 based on REM, and 3.1% (95% credible interval = 1.5-5.0%) based on the NNHM. The estimated prevalence varies with age, increasing from 1.6% (95% credible interval = 0.8-2.5%) in people aged 60-69 to 12.4% (95% credible interval = 5.6-20%) in people above the age of 80. The risk of developing dementia increased exponentially with age. The number of people living with dementia in SEAR in 2015 was estimated at 5.51 million (95% credible interval = 2.66-8.82), with projections of 6.66 million (95% credible interval = 3.21-10.7) in 2020 and 9.6 million (95% credible interval = 4.62-15.36) in 2030.ConclusionThe burden of dementia in SEAR is substantial and will continue to increase rapidly by 2030. The lack of research focusing on dementia in SEAR points to a significant under-recognition of this disease. The projected rise in dementia cases in the future should prompt urgent governmental response to address this growing public health issue. We also argue that given the overall paucity of data for the region, the Bayesian approach offers a promising methodology for improved estimates of disease prevalence and burden and should continue to be explored.

A Comparison Between a Meta-analytic Approach and Power Prior Approach to Using Historical Control Information in Clinical Trials With Binary Endpoints.

In the process of research and development of a new treatment, clinical trials are conducted to evaluate its safety and efficacy. Key to streamlining the process is to utilize appropriate historical information on an outcome of a control treatment when designing and analyzing a clinical trial. For the use of such historical control information, there exist a meta-analytic approach and power prior approach. In this article, we evaluate their performance with regard to the type I error (TIE) rate and power through a simulation study where we analyze the data on a binary outcome of an experimental treatment and a control treatment from a new small-scale trial, along with the corresponding data of the control treatment from multiple historical trials. The reason is that the difference in the performance between the 2 approaches has not been clear. When historical trials were homogeneous, the power was higher in the power prior approach and the meta-analytic approach using a beta-binomial model with a less noninformative prior than the other approaches. However, when heterogeneous historical trials were mixed, the power was lower, or the TIE rates got inflated. To make use of historical control data, if importance is attached to control of the TIE rate, the meta-analytic approach using a normal-normal hierarchical model may be preferable to the power prior approach, whereas if attached to improvement of the power, this preference be reversed. Anyway, the best approach should be chosen by comparing the operational characteristics of the approaches.

Bayesian Random-Effects Meta-Analysis Using the bayesmeta R Package

The random-effects or normal-normal hierarchical model is commonly utilized in a wide range of meta-analysis applications. A Bayesian approach to inference is very attractive in this context, especially when a meta-analysis is based only on few studies. The bayesmeta R package provides readily accessible tools to perform Bayesian meta-analyses and generate plots and summaries, without having to worry about computational details. It allows for flexible prior specification and instant access to the resulting posterior distributions, including prediction and shrinkage estimation, and facilitating for example quick sensitivity checks. The present paper introduces the underlying theory and showcases its usage.

A Comparison Between a Meta-analytic Approach and Power Prior Approach to Using Historical Control Information in Clinical Trials With Binary Endpoints.

In the process of research and development of a new treatment, clinical trials are conducted to evaluate its safety and efficacy. Key to streamlining the process is to utilize appropriate historical information on an outcome of a control treatment when designing and analyzing a clinical trial. For the use of such historical control information, there exist a meta-analytic approach and power prior approach. In this article, we evaluate their performance with regard to the type I error (TIE) rate and power through a simulation study where we analyze the data on a binary outcome of an experimental treatment and a control treatment from a new small-scale trial, along with the corresponding data of the control treatment from multiple historical trials. The reason is that the difference in the performance between the 2 approaches has not been clear. When historical trials were homogeneous, the power was higher in the power prior approach and the meta-analytic approach using a beta-binomial model with a less noninformative prior than the other approaches. However, when heterogeneous historical trials were mixed, the power was lower, or the TIE rates got inflated. To make use of historical control data, if importance is attached to control of the TIE rate, the meta-analytic approach using a normal-normal hierarchical model may be preferable to the power prior approach, whereas if attached to improvement of the power, this preference be reversed. Anyway, the best approach should be chosen by comparing the operational characteristics of the approaches.

Dynamically borrowing strength from another study through shrinkage estimation.

Meta-analytic methods may be used to combine evidence from different sources of information. Quite commonly, the normal-normal hierarchical model (NNHM) including a random-effect to account for between-study heterogeneity is utilized for such analyses. The same modeling framework may also be used to not only derive a combined estimate, but also to borrow strength for a particular study from another by deriving a shrinkage estimate. For instance, a small-scale randomized controlled trial could be supported by a non-randomized study, e.g. a clinical registry. This would be particularly attractive in the context of rare diseases. We demonstrate that a meta-analysis still makes sense in this extreme case, effectively based on a synthesis of only two studies, as illustrated using a recent trial and a clinical registry in Creutzfeld-Jakob disease. Derivation of a shrinkage estimate within a Bayesian random-effects meta-analysis may substantially improve a given estimate even based on only a single additional estimate while accounting for potential effect heterogeneity between the studies. Alternatively, inference may equivalently be motivated via a model specification that does not require a common overall mean parameter but considers the treatment effect in one study, and the difference in effects between the studies. The proposed approach is quite generally applicable to combine different types of evidence originating, e.g. from meta-analyses or individual studies. An application of this more general setup is provided in immunosuppression following liver transplantation in children.

Bayesian hierarchical distributed lag models for summer ozone exposure and cardio‐respiratory mortality

In this article we develop Bayesian hierarchical distributed lag models for estimating associations between daily variations in summer ozone levels and daily variations in cardiovascular and respiratory (CVDRESP) mortality counts for 19 large U.S. cities included in the National Morbidity, Mortality and Air Pollution Study (NMMAPS) for the summers of 1987-1994. In the first stage, we define a semi-parametric distributed lag Poisson regression model to estimate city-specific relative rates of CVDRESP mortality associated with short-term exposure to summer ozone. In the second stage, we specify a class of distributions for the true city-specific relative rates to estimate an overall effect by taking into account the variability within and across cities. We perform the calculations with respect to several random effects distributions (normal, t-student, and mixture of normal), thus relaxing the common assumption of a two-stage normal-normal hierarchical model. We assess the sensitivity of the results to: (i) lag structure for ozone exposure; (ii) degree of adjustment for long-term trends; (iii) inclusion of other pollutants in the model; (iv) heat waves; (v) random effects distributions; and (vi) prior hyperparameters. On average across cities, we found that a 10ppb increase in summer ozone level over the previous week is associated with a 1.25 per cent increase in CVDRESP mortality (95 per cent posterior regions: 0.47, 2.03). The relative rate estimates are also positive and statistically significant at lags 0, 1 and 2. We found that associations between summer ozone and CVDRESP mortality are sensitive to the confounding adjustment for PM10, but are robust to: (i) the adjustment for long-term trends, other gaseous pollutants (NO2, SO2 and CO); (ii) the distributional assumptions at the second stage of the hierarchical model; and (iii) the prior distributions on all unknown parameters. Bayesian hierarchical distributed lag models and their application to the NMMAPS data allow us to estimate of an acute health effect associated with exposure to ambient air pollution in the last few days on average across several locations. The application of these methods and the systematic assessment of the sensitivity of findings to model assumptions provide important epidemiological evidence for future air quality regulations.

Underestimation of standard errors in multi-site time series studies.

Multi-site time series studies of the association of air pollution with mortality and morbidity have figured prominently in the literature as comprehensive approaches for estimating short-term effects of air pollution on health. Hierarchical models are generally used to combine site-specific information and to estimate pooled air pollution effects while taking into account both within-site statistical uncertainty and across-site heterogeneity. Within a site, characteristics of time series data of air pollution and health (small pollution effects, missing data, and highly correlated predictors) make the modeling of all sources of uncertainty challenging. One potential consequence is underestimation of the statistical variance of the site-specific effects to be combined.In this paper, we investigate the impact of variance underestimation on the pooled relative rate estimate. We focused on two-stage normal-normal hierarchical models and on underestimation of the statistical variance at the first stage. By mathematical considerations and simulation studies, we found that variance underestimation did not affect the pooled estimate substantially. However, the pooled estimate was somewhat sensitive to variance underestimation when the number of sites was small and underestimation was severe. These simulation results are applicable to any two-stage normal-normal hierarchical model for combining information of site-specific results (including meta-analyses), and they can easily be extended to more general hierarchical formulations. We also examined the impact of variance underestimation on the national average relative rate estimate from the National Morbidity, Mortality and Air Pollution Study. We found that variance underestimation as large as 40% had little effect on the national average.