Quantifying uncertainty: a practical comparison of Bayesian and frequentist meta-analytic frameworks applied to lead exposure and children’s IQ

Abstract

Uncertainty quantification is central to evidence synthesis for risk assessment and policy-making. Frequentist meta-analysis - currently predominant in epidemiology - quantifies uncertainty through confidence intervals, while Bayesian meta-analysis provides posterior distributions. We performed frequentist and Bayesian meta-analyses to compare uncertainty quantification and applicability for risk assessment using lead exposure and children’s IQ as a case study. We systematically searched PubMed and Scopus for epidemiological studies on blood lead level (BLL) and IQ score in children. After screening, risk of bias (RoB) in the identified studies was assessed and regression coefficients were harmonised to the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\text{l}\text{o}\text{g}}_{e}$$\end{document} scale. In both meta-analyses, we included the same effect estimates, assumed a multilevel structure and used random-effects models. The Bayesian meta-analysis employed a hierarchical model with weakly-informative priors. Sensitivity analyses examined robustness to RoB and prior specification. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\tau\:}^{2}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{I}^{2}$$\end{document} were estimated as measures of between-study heterogeneity. Twelve studies were included in the meta-analysis. The frequentist model yielded a pooled estimate of -2.45 (95% confidence interval: -3.82 to -1.08) IQ points per unit increase in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\text{l}\text{o}\text{g}}_{e}$$\end{document}-transformed BLL (µg/dL) and a 95% predictive interval of -6.53 to 1.53. The Bayesian model yielded a mean pooled estimate of -2.28 (95% credible interval: -3.54 to -1.04) and a 95% predictive interval of -6.28 to 1.69. Between-study heterogeneity was substantial (frequentist: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\tau\:}^{2}$$\end{document} = 3.32, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\text{I}}^{2}$$\end{document} = 85.0%; Bayesian: posterior median \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\tau\:}^{2}$$\end{document} = 3.26, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:{\text{I}}^{2}$$\end{document} = 84.7%). The Bayesian approach provided a full posterior distribution for heterogeneity, directly quantifying uncertainty from between-study variation. Sensitivity analyses showed both meta-analyses were robust to RoB, and Bayesian estimates were insensitive to tested priors. Depending on the specified prior configuration, the Bayesian model was less sensitive to studies reporting substantial uncertainty, resulting in a lower pooled effect compared to the frequentist model. This regularisation is particularly valuable in small meta-analyses, where frequentist heterogeneity estimates can be unstable. We provide updated pooled estimates for the effect of lead on children’s IQ. The Bayesian framework provides posterior distributions, enabling direct probability statements, e.g. about exceeding risk thresholds, and comparative assessments between pollutants. These features are particularly valuable for risk assessment and policy-making.