Bayesian estimation in multiple comparisons

Abstract

Traditional regression models typically estimate parameters for a factor F by designating one level as a reference (intercept) and calculating slopes for other levels of F. While this approach often aligns with our research question(s), it limits direct comparisons between all pairs of levels within F and requires additional procedures for generating these comparisons. Moreover, Frequentist methods often rely on corrections (e.g., Bonferroni or Tukey), which can reduce statistical power and inflate uncertainty by mechanically widening confidence intervals. This paper demonstrates how Bayesian hierarchical models provide a robust framework for parameter estimation in the context of multiple comparisons. By leveraging entire posterior distributions, these models produce estimates for all pairwise comparisons without requiring post hoc adjustments. The hierarchical structure, combined with the use of priors, naturally incorporates shrinkage, pulling extreme estimates toward the overall mean. This regularization improves the stability and reliability of estimates, particularly in the presence of sparse or noisy data, and leads to more conservative comparisons. Bayesian models also offer a flexible framework for addressing heteroscedasticity by directly modeling variance structures and incorporating them into the posterior distribution. The result is a coherent approach to exploring differences between levels of F, where parameter estimates reflect the full uncertainty of the data.