Bayesian Time-Varying Meta-Analysis via Hierarchical Mean-Variance Random-effects Models

TL;DR

Meta-analysis often faces heteroscedastic and time-varying effects that limit standard pooling methods. We introduce STREAM, a Bayesian framework that performs shrinkage on both mean effects $\\theta_i$ and sampling variances $\\sigma_i^2$ within a hierarchical structure and models time effects $\\theta_c$ with a Gaussian process, enabling borrowing strength across experiments and flexible time trends. Posterior inference via MCMC in Stan yields predictive distributions for future experiments, including $\\tilde{\\theta}_j$ and $\\tilde{\\sigma}_j^2$, enabling accurate point and interval estimates. Across simulations with varying time patterns and heterogeneity, STREAM consistently outperforms baselines, and in a real promotions dataset STREAM achieves superior predictive performance, illustrating its practical utility for marketing analytics and potentially other domains.

Abstract

Meta-analysis is widely used to integrate results from multiple experiments to obtain generalized insights. Since meta-analysis datasets are often heteroscedastic due to varying subgroups and temporal heterogeneity arising from experiments conducted at different time points, the typical meta-analysis approach, which assumes homoscedasticity, fails to adequately address this heteroscedasticity among experiments. This paper proposes a new Bayesian estimation method that simultaneously shrinks estimates of the means and variances of experiments using a hierarchical Bayesian approach while accounting for time effects through a Gaussian process. This method connects experiments via the hierarchical framework, enabling "borrowing strength" between experiments to achieve high-precision estimates of each experiment's mean. The method can flexibly capture potential time trends in datasets by modeling time effects with the Gaussian process. We demonstrate the effectiveness of the proposed method through simulation studies and illustrate its practical utility using a real marketing promotions dataset.

Figures (3)

• Figure 1: Comparison of the distribution of the original variable $y$ and the log-transformed variable $\log y$.
• Figure 2: Estimated merchant and business type effects.
• Figure 3: Estimated time effect.