Index: Roever 2020 - Bayesian Random-Effects Meta-Analysis with bayesmeta

Roever 2020 - Bayesian Random-Effects Meta-Analysis with bayesmeta

Routing Summary

This folder fully captures Röver (2020), Bayesian Random-Effects Meta-Analysis Using the bayesmeta R Package (J. Stat. Softw. 93(6); arXiv:1711.08683): the normal-normal hierarchical model, priors for the effect and heterogeneity, closed-form marginal likelihood and conditional posteriors, the DIRECT normal-mixture computation, prediction, shrinkage, credible intervals, posterior predictive $p$-values, the REML connection and the pediatric liver transplant example. 9 notes; all printed example numbers were recomputed independently in Python.

• Need the paper on one page, headline numbers, or the section-to-note map? → Roever 2020 - Bayesian Random-Effects Meta-Analysis with bayesmeta - Overview
• Need why Bayesian for few studies, Bayesian vs frequentist, or the software landscape? → Roever 2020 - Overview and Motivation of Bayesian Meta-Analysis
• Need the NNHM definition and likelihood? → Roever 2020 - Normal-Normal Hierarchical Model and Likelihood
• Need $\tau$ priors (half-normal, Jeffreys, uniform shrinkage…), unit-information priors, the uniform-prior conservatism theorem, or prior predictive checks? → Roever 2020 - Priors for the Effect and Heterogeneity
• Need the marginal likelihood, conditional normal posterior of $\mu$, and the mixture factorisation (with derivation)? → Roever 2020 - Marginal Likelihood and Conditional Posteriors
• Need prediction for a new study, shrinkage formulas, or credible-interval types? → Roever 2020 - Prediction, Shrinkage and Credible Intervals
• Need posterior predictive $p$-values or the link to REML? → Roever 2020 - Posterior Predictive Checks and Frequentist Connection
• Need the DIRECT algorithm, the divergence formula or the PIT calibration check? → Roever 2020 - DIRECT Algorithm Implementation and Calibration
• Need the six-study IL-2RA example with every printed number? → Roever 2020 - Pediatric Liver Transplant Worked Example

Concept Map

Concept	Note	Type	Depends On	Key Result
Why Bayesian with small $k$	Roever 2020 - Overview and Motivation of Bayesian Meta-Analysis	concept	none	Cochrane median 3 studies; 57% zero $\hat{\tau }$
NNHM	Roever 2020 - Normal-Normal Hierarchical Model and Likelihood	definition	Viechtbauer 2010 - Fixed, Random and Mixed-Effects Models	$y_i\sim N(\mu ,{\sigma }_i^2+{\tau }^2)$
Unit information prior	Roever 2020 - Priors for the Effect and Heterogeneity	definition	NNHM	log-OR: $N(0,4^2)$
$\tau$ prior classes (Table 1)	Roever 2020 - Priors for the Effect and Heterogeneity	definition	NNHM	four classes; Jeffreys proper for $k\ge 2$
Uniform-prior conservatism	Roever 2020 - Priors for the Effect and Heterogeneity	theorem	monotone priors	$F_p\ge F_1$
Marginal likelihood of $\tau$	Roever 2020 - Marginal Likelihood and Conditional Posteriors	theorem	NNHM	Eq. 16-18, 22
Conditional posterior of $\mu$	Roever 2020 - Marginal Likelihood and Conditional Posteriors	theorem	marginal likelihood	normal with $\hat{\mu }(\tau ),\hat{\sigma }(\tau )$
Posterior as normal mixture	Roever 2020 - Marginal Likelihood and Conditional Posteriors	theorem	conditional posterior	$\int N(\mu \mid \tau )p(\tau \mid y)d\tau$
Predictive distribution	Roever 2020 - Prediction, Shrinkage and Credible Intervals	theorem	mixture	variance ${\hat{\sigma }}^2+{\tau }^2$
Shrinkage estimates	Roever 2020 - Prediction, Shrinkage and Credible Intervals	theorem	mixture	precision-weighted pull to $\hat{\mu }(\tau )$
Shortest/HPD intervals	Roever 2020 - Prediction, Shrinkage and Credible Intervals	definition	posterior	default in bayesmeta
Posterior predictive $p$	Roever 2020 - Posterior Predictive Checks and Frequentist Connection	definition	posterior	$p=0.01$ for $\mu \ge 0$ (2 studies)
REML as conditional posterior	Roever 2020 - Posterior Predictive Checks and Frequentist Connection	theorem	uniform $\mu$ prior	identical mean and SE
DIRECT algorithm	Roever 2020 - DIRECT Algorithm Implementation and Calibration	concept	mixture	17 components for $\delta =1\%$
PIT calibration	Roever 2020 - DIRECT Algorithm Implementation and Calibration	theorem	prior-generated data	uniform PIT
IL-2RA example	Roever 2020 - Pediatric Liver Transplant Worked Example	example	all	$\mu$ median -1.57 [-2.23, -0.93]

Notes

• Roever 2020 - Bayesian Random-Effects Meta-Analysis with bayesmeta - Overview — CONTAINS: question, contribution, headline numbers, full Source Map including appendices.
• Roever 2020 - Overview and Motivation of Bayesian Meta-Analysis — CONTAINS: Sec. 1 and Sec. 4: small-$k$ evidence, frequentist plug-in critique, Bayesian advantages, software landscape, interval semantics.
• Roever 2020 - Normal-Normal Hierarchical Model and Likelihood — CONTAINS: Eq. 1-3 and 13-14, definitions, Python log-likelihood.
• Roever 2020 - Priors for the Effect and Heterogeneity — CONTAINS: priors for $\mu$; unit information (App. A.1); informative, proper and improper $\tau$ priors (Eq. 4-12); Table 1; uniform-prior conservatism theorem (App. A.2); prior predictive table; code.
• Roever 2020 - Marginal Likelihood and Conditional Posteriors — CONTAINS: Eq. 15-26 with derivation (App. A.3), normal-prior version, factorisation theorem, grid code.
• Roever 2020 - Prediction, Shrinkage and Credible Intervals — CONTAINS: Eq. 27-29, interval types, transplant shrinkage numbers.
• Roever 2020 - Posterior Predictive Checks and Frequentist Connection — CONTAINS: REML link example, posterior predictive $p$-value procedure, two-study results (Fig. 6-7), code.
• Roever 2020 - DIRECT Algorithm Implementation and Calibration — CONTAINS: internals, divergence formula, PIT calibration protocol (Fig. 8), code.
• Roever 2020 - Pediatric Liver Transplant Worked Example — CONTAINS: Tables 2-3, bayesmeta calls and outputs, posterior queries, risk difference, shrinkage, prior variations, Python reproduction.

Sources

• Röver C (2020). Bayesian Random-Effects Meta-Analysis Using the bayesmeta R Package. Journal of Statistical Software 93(6). PDF (arXiv v2): Roever 2020 - Bayesian Random-Effects Meta-Analysis Using the bayesmeta R Package

See Also

• Viechtbauer 2010 - Conducting Meta-Analyses in R with the metafor Package - Overview, IntHout 2014 - HKSJ vs DerSimonian-Laird Random-Effects Meta-Analysis - Overview
• Hierarchical Models, Constructing Priors for Effect Sizes, Posterior Sampling and Summarization