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Researchers often wish to test a large set of related interventions or approaches to implementation. A factorial experiment accomplishes this by examining not only basic treatment-control comparisons but also the effects of multiple implementation factors such as different dosages or implementation strategies, and the interactions between these factor levels. However, traditional methods of statistical inference may require prohibitively large sample sizes to perform complex factorial experiments. In this paper, we present a Bayesian approach to factorial design. By using hierarchical priors and partial pooling, we show how Bayesian analysis substantially increases the precision of estimates in complex experiments with many factors and factor levels, while controlling the risk of false positives from multiple comparisons. Using an experiment we performed for the U.S. Department of Education as a motivating example, we perform power calculations for both classical and Bayesian methods. We repeatedly simulate factorial experiments with a variety of sample sizes and numbers of treatment arms to estimate the minimum detectable effect (MDE) for each combination. We found that the Bayesian approach yields substantially lower MDEs when compared with classical methods for complex factorial experiments. For example, to test 72 treatment arms (five factors with two or three levels each), a classical experiment requires nearly twice the sample size as a Bayesian experiment to obtain a given MDE. Overall, our study showed that Bayesian methods are a valuable tool for researchers interested in studying complex interventions. They make factorial experiments with many treatment arms vastly more feasible.

This study appeared in the January 2019 edition of Evaluation Review.