A tutorial on selecting, translating, and reporting variance priors in Bayesian multilevel models

Bayesian multilevel models are increasingly used in psychological, educational, and behavioral research, yet applied guidance on specifying priors for scalar variance components remains limited. This tutorial provides a practical workflow for identifying target variance components, selecting benchmark values, calibrating priors on variance and standard-deviation scales, translating them into software-compatible prior specifications, evaluating their implications through prior predictive checks, posterior comparison, sensitivity analysis, and transparent reporting. We review several commonly used prior families, including gamma, inverse-gamma, half-t, and half-Cauchy distributions, and clarify how their parameterizations relate across variance, precision (the reciprocal of variance), and standard-deviation formulations. The tutorial focuses on helping researchers align prior choice with substantive knowledge, model parameterization, software constraints, and sensitivity goals. As a worked example, we use a two-level analysis of TIMSS 2019 data to show how benchmark information from a non-overlapping reference sample can be calibrated into alternative variance-prior specifications and then evaluated in a smaller worked example sample. The empirical illustration is used as a pedagogical example to demonstrate the workflow and its associated checking procedures. We emphasize practical safeguards, including prior-density inspection, prior predictive checking, posterior comparison, convergence assessment, sensitivity analysis, and transparent reporting of prior assumptions. The tutorial concludes by summarizing practical, literature-grounded considerations for documenting prior rationale, implementation decisions, and robustness checks in Bayesian multilevel applications.