A Meta-Analysis Based Hierarchical Variance Model for Powering One and Two-Sample t-Tests

Sample size determination (SSD) is essential in statistical inference and hypothesis testing, as it directly affects the accuracy and power of the analysis. We propose a SSD methodology for one and two-sample t-tests that ensures clinical relevance using a pre-determined unstandardized effect size. Our novel approach leverages Bayesian meta-analysis to account for the uncertainty surrounding the variance, a common issue in SSD. By incorporating prior knowledge from related studies via a Bayesian gamma-inverse gamma model, we obtain an informative posterior predictive distribution for the variance that leads to better decisions about sample size. For efficient posterior sampling, we propose an empirical Bayes approach which is implemented with systematic sampling to facilitate computation. Simulations and empirical studies demonstrate that our methodology outperforms other aggregate approaches (simple average, weighted average, median) in variance estimation for SSD, especially in meta-analyses with large disparity in sample size and moderate variance. Thus, it offers a robust and practical solution for sample size determination in t-tests.