Recalibrating single-study effect sizes using hierarchical Bayesian models

There are growing concerns about commonly inflated effect sizes in small neuroimaging studies, yet no study has addressed recalibrating effect size estimates for small samples. To tackle this issue, we propose a hierarchical Bayesian model to adjust the magnitude of single-study effect sizes while incorporating a tailored estimation of sampling variance.We estimated the effect sizes of case-control differences on brain structural features between individuals who were dependent on alcohol, nicotine, cocaine, methamphetamine, or cannabis and non-dependent participants for 21 individual studies (Total cases: 903; Total controls: 996). Then, the study-specific effect sizes were modeled using a hierarchical Bayesian approach in which the parameters of the study-specific effect size distributions were sampled from a higher-order overarching distribution. The posterior distribution of the overarching and study-specific parameters was approximated using the Gibbs sampling method.The results showed shrinkage of the posterior distribution of the study-specific estimates toward the overarching estimates given the original effect sizes observed in individual studies. Differences between the original effect sizes (i.e., Cohen's d) and the point estimate of the posterior distribution ranged from 0 to 0.97. The magnitude of adjustment was negatively correlated with the sample size (r = −0.27, p < 0.001) and positively correlated with empirically estimated sampling variance (r = 0.40, p < 0.001), suggesting studies with smaller samples and larger sampling variance tended to have greater adjustments.Our findings demonstrate the utility of the hierarchical Bayesian model in recalibrating single-study effect sizes using information from similar studies. This suggests that Bayesian utilization of existing knowledge can be an effective alternative approach to improve the effect size estimation in individual studies, particularly for those with smaller samples.