Bayesian Hierarchical Models

INTRODUCTION

This JAMA Guide to Statistics and Methods discusses the use, limitations, and interpretation of Bayesian hierarchical modeling, a statistical procedure that integrates information across multiple levels and uses prior information about likely treatment effects and their variability to estimate true vs random treatment effects.

Treatment effects will differ from one study to another evaluating similar therapies, both because of random variation between individual patients and owing to true differences that exist because of other differences, including inclusion criteria and temporal trends. The sources of variability have many levels; one level involves the random differences between individual patients, and another level involves the systematic differences that exist between studies. This multilevel or hierarchical information occurs in many research settings, such as in cluster-randomized trials and meta-analyses.^1,2 Sources of variation can be better understood and quantified if treatment effect estimates from each individual study are examined in relation to the totality of information available in all the studies.

Bayesian analysis differs from the usual frequentist approach (eg, use of P values or confidence intervals). Rather than focusing on the probability of different patterns in outcomes assuming specific treatment effects, Bayesian analysis relies on the use of prior information in combination with data from a study to calculate the probabilities of a treatment effect.³ Readers may be familiar with Bayesian analysis when used in randomized clinical trials.^4,5 In this type of Bayesian analysis, patients are considered largely equivalent except with respect to the assigned treatment, and the goal is to estimate the probability of an overall treatment effect in the population.

In contrast, a Bayesian hierarchical model (BHM) is a statistical procedure that integrates information across many levels, so multiple quantities are estimated simultaneously, and explicitly separates the observed variability into parts attributable to random differences and true differences.⁶ The model has 2 key characteristics. First, there is a hierarchical or multilevel structure. For example, if multiple studies were conducted to evaluate diabetes management strategies, the first-level data may be improvements in hemoglobin A_1C values in individual patients, the second-level data may be the mean improvements for patients within each trial, and the third-level data may be the average improvements in trials grouped according to the type of disease management strategy. Second, prior information is used to reflect available information, even if vague, regarding the likely values and variability at each level of the hierarchy (eg, the variability of improvements in patients in a single trial, the variability of average treatment effects between trials using similar disease management strategies, and the variability of treatment effects among groups of trials that use different disease management strategies). Using Bayes theorem, prior information, and the data, the BHM yields estimates of the true effects at each level of the hierarchy.^3,6 Estimates of true treatment effects may be derived for individual patients, patient subgroups, individual trials, or groups of trials. Each of these ...