Bayesian evaluation of informative hypotheses in cluster-randomized trials

Abstract

Researchers often have informative hypotheses in mind when comparing means across treatment groups, such as H₁ : μ_A < μ_B < μ_C and H₂ : μ_B < μ_A < μ_C, and want to compare these hypotheses to each other directly. This can be done by means of Bayesian inference. This article discusses the disadvantages of the frequentist approach to null hypothesis testing and the advantages of the Bayesian approach. It demonstrates how to use the Bayesian approach to hypothesis testing in the setting of cluster-randomized trials. The data from a school-based smoking prevention intervention with four treatment groups are used to illustrate the Bayesian approach. The main advantage of the Bayesian approach is that it provides a degree of evidence from the collected data in favor of an informative hypothesis. Furthermore, a simulation study was conducted to investigate how Bayes factors behave with cluster-randomized trials. The results from the simulation study showed that the Bayes factor increases with increasing number of clusters, cluster size, and effect size, and decreases with increasing intraclass correlation coefficient. The effect of the number of clusters is stronger than the effect of cluster size. With a small number of clusters, the effect of increasing cluster size may be weak, especially when the intraclass correlation coefficient is large. In conclusion, the study showed that the Bayes factor is affected by sample size and intraclass correlation similarly to how these parameters affect statistical power in the frequentist approach of null hypothesis significance testing. Bayesian evaluation may be used as an alternative to null hypotheses testing.

Keywords: Bayes factors; Bayesian inference; Cluster-randomized trial; Informative hypotheses; Null hypothesis testing; Simulation study.

Fig. 1

Two-dimensional representation of a likelihood function for two independent means μ₁ and μ₂ (left) and of the likelihood function for hypothesis H_i : μ₁ > μ₂ (right)

Fig. 2

Error probability (top row), indecision probability (middle row), and median Bayes factor (BF; bottom row) as a function of intraclass correlation coefficient (columns), cluster size (horizontal axis within each plot), and number of clusters (separate lines within each plot). Population with (μ_A, μ_B, μ_C) = (0, 0.2, 0.4). The legend as given in the upper left graph holds for all graphs

Fig. 3

Error probability (top row), indecision probability (middle row), and median Bayes factor (BF; bottom row) as a function of intraclass correlation coefficient (columns), cluster size (horizontal axis within each plot), and number of clusters (separate lines within each plot). Population with (μ_A, μ_B, μ_C) = (0, 0.1, 0.2). The legend given in the upper left graph holds for all graphs