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.

[1]  J. Berger.  Statistical Decision Theory and Bayesian Analysis , 1988 .

[2]  D. V. Lindley.  The distinction between inference and decision , 2004, Synthese.

[3]  Franz J. Neyer,et al.  A Gentle Introduction to Bayesian Analysis: Applications to Developmental Research , 2013, Child development.

[4]  Haitao Chu,et al.  mmeta: An R Package for Multivariate Meta-Analysis. , 2014, Journal of statistical software.

[5]  D. Bartholomew,et al.  A comparison of some Bayesian and frequentist inferences. II. , 1965, Biometrika.

[6]  B. Sinha,et al.  Statistical Meta-Analysis with Applications , 2008 .

[7]  David J. Lunn,et al.  Fully Bayesian hierarchical modelling in two stages, with application to meta-analysis , 2013, Journal of the Royal Statistical Society. Series C, Applied statistics.

[8]  Rebecca M. Turner,et al.  Predictive distributions were developed for the extent of heterogeneity in meta-analyses of continuous outcome data , 2015, Journal of clinical epidemiology.

[9]  L. Wasserman,et al.  The Selection of Prior Distributions by Formal Rules , 1996 .

[10]  L. Hedges,et al.  Statistical Methods for Meta-Analysis , 1987 .

[11]  Steff Lewis,et al.  Forest plots: trying to see the wood and the trees , 2001, BMJ : British Medical Journal.

[12]  Stephen Senn,et al.  (www.interscience.wiley.com) DOI: 10.1002/sim.2639 Trying to be precise about vagueness , 2022 .

[13]  Iven Van Mechelen,et al.  Posterior predictive checks: Principles and discussion , 2000, Comput. Stat..

[14]  C. D. Litton,et al.  Theory of Probability (3rd Edition) , 1984 .

[15]  James O. Berger,et al.  Objective Bayesian Methods for Model Selection: Introduction and Comparison , 2001 .

[16]  D. Spiegelhalter,et al.  Disease Mapping With WinBUGS and MLwiN, Bayesian Approaches to Clinical Trials and Health Care Evaluation , 2004 .

[17]  I. J. P. Howard.  Meta-Analysis withR , 2015 .

[18]  N. L. Johnson,et al.  Continuous Univariate Distributions. , 1995 .

[19]  A. Raftery,et al.  Probabilistic forecasts, calibration and sharpness , 2007 .

[20]  H. Jeffreys.  An invariant form for the prior probability in estimation problems , 1946, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[21]  Dan Jackson,et al.  Predictive distributions for between-study heterogeneity and simple methods for their application in Bayesian meta-analysis , 2014, Statistics in medicine.

[22]  Donald B. Rubin,et al.  Validation of Software for Bayesian Models Using Posterior Quantiles , 2006 .

[23]  R. Tibshirani.  Noninformative priors for one parameter of many , 1989 .

[24]  Andrew Gelman,et al.  Two simple examples for understanding posterior p-values whose distributions are far from uniform , 2013 .

[25]  Jonathan J Deeks,et al.  Issues in the selection of a summary statistic for meta‐analysis of clinical trials with binary outcomes , 2002, Statistics in medicine.

[26]  Wolfgang Viechtbauer,et al.  Conducting Meta-Analyses in R with the metafor Package , 2010 .

[27]  L. Hedges,et al.  Introduction to Meta‐Analysis , 2009, International Coaching Psychology Review.

[28]  Xiao-Li Meng,et al.  Posterior Predictive $p$-Values , 1994 .

[29]  L. Wasserman,et al.  A Reference Bayesian Test for Nested Hypotheses and its Relationship to the Schwarz Criterion , 1995 .

[30]  D. Spiegelhalter.  Incorporating Bayesian Ideas into Health-Care Evaluation , 2004 .

[31]  Rebecca M Turner,et al.  Characteristics of meta-analyses and their component studies in the Cochrane Database of Systematic Reviews: a cross-sectional, descriptive analysis , 2011, BMC medical research methodology.

[32]  Tim Friede,et al.  Using phase II data for the analysis of phase III studies: An application in rare diseases , 2016, Clinical trials.

[33]  James G. Scott,et al.  On the half-cauchy prior for a global scale parameter , 2011, 1104.4937.

[34]  D. Handscomb.  THE MONTE CARLO METHOD , 2013 .

[35]  Torrin M. Liddell,et al.  The Bayesian New Statistics: Hypothesis testing, estimation, meta-analysis, and power analysis from a Bayesian perspective , 2016, Psychonomic bulletin & review.

[36]  M. Giovannelli,et al.  Randomized Trial of Basiliximab Induction versus Steroid Therapy in Pediatric Liver Allograft Recipients Under Tacrolimus Immunosuppression , 2006, American journal of transplantation : official journal of the American Society of Transplantation and the American Society of Transplant Surgeons.

[37]  J. Hartung,et al.  A refined method for the meta‐analysis of controlled clinical trials with binary outcome , 2001, Statistics in medicine.

[38]  Gregory Smallwood,et al.  Pediatric liver transplantation with daclizumab induction1 , 2003 .

[39]  Heinz Schmidli,et al.  Meta-analytic-predictive use of historical variance data for the design and analysis of clinical trials , 2017, Comput. Stat. Data Anal..

[40]  Simon G Thompson,et al.  Predicting the extent of heterogeneity in meta-analysis, using empirical data from the Cochrane Database of Systematic Reviews , 2012, International journal of epidemiology.

[41]  T. Friede,et al.  Discrete Approximation of a Mixture Distribution via Restricted Divergence , 2016, 1602.04060.

[42]  Taekyun Kim,et al.  A Note on the -Euler Measures , 2009 .

[43]  A. Obed,et al.  Interleukin 2 receptor antagonists for liver transplant recipients: A systematic review and meta‐analysis of controlled studies , 2011, Hepatology.

[44]  Martyn Plummer,et al.  JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling , 2003 .

[45]  Xavier Rogiers,et al.  Long‐term results of basiliximab induction immunosuppression in pediatric liver transplant recipients , 2005, Pediatric transplantation.

[46]  Edwin T. Jaynes.  Prior Probabilities , 2010, Encyclopedia of Machine Learning.

[47]  A. Dawid.  The Well-Calibrated Bayesian , 1982 .

[48]  Evangelos Kontopantelis,et al.  A Re-Analysis of the Cochrane Library Data: The Dangers of Unobserved Heterogeneity in Meta-Analyses , 2013, PloS one.

[49]  Hal S. Stern,et al.  Posterior Predictive Assessment of Item Response Theory Models , 2006 .

[50]  Eleanor M Pullenayegum,et al.  An informed reference prior for between‐study heterogeneity in meta‐analyses of binary outcomes , 2011, Statistics in medicine.

[51]  Philip C. Gregory,et al.  Bayesian Logical Data Analysis for the Physical Sciences: Acknowledgements , 2005 .

[52]  J. Ioannidis,et al.  The PRISMA Statement for Reporting Systematic Reviews and Meta-Analyses of Studies That Evaluate Health Care Interventions: Explanation and Elaboration , 2009, Annals of Internal Medicine [serial online].

[53]  Thierry Chambert,et al.  Use of posterior predictive checks as an inferential tool for investigating individual heterogeneity in animal population vital rates , 2014, Ecology and evolution.

[54]  S. Thompson,et al.  Quantifying heterogeneity in a meta‐analysis , 2002, Statistics in medicine.

[55]  E. Jaynes.  Probability theory : the logic of science , 2003 .

[56]  Thomas A. Severini,et al.  On the Relationship between Bayesian and Non-Bayesian Interval Estimates , 1991 .

[57]  J. Berger,et al.  A Bayesian Approach to Ranking and Selection of Related Means with Alternatives to Analysis-of-Variance Methodology , 1988 .

[58]  Xiao-Li Meng,et al.  POSTERIOR PREDICTIVE ASSESSMENT OF MODEL FITNESS VIA REALIZED DISCREPANCIES , 1996 .

[59]  M. Daniels.  A prior for the variance in hierarchical models , 1999 .