10 Bayesian Factorial Design
In many scenarios, your goal is to use data to design the optimal treatment. This is where factorial designs come into play. A factorial design is an experimental setup where multiple factors are manipulated or varied simultaneously. Each factor can have multiple levels, and the experiment involves testing all possible combinations of these factor levels. This allows us to not only assess the main effects of each factor but also to investigate the interactions between factors.
Bayesian analysis offers a powerful approach to factorial designs, especially when dealing with complex experiments with many factors and levels. By incorporating prior knowledge and using hierarchical models, Bayesian methods can improve the precision of estimates and control the risk of false positives from multiple comparisons.
In a Bayesian factorial design, we start with prior distributions for the effects of each factor level and the interactions between them. These priors can be based on previous research, expert opinion, or simply reflect our uncertainty about the effects. As we collect data from the experiment, we update these priors using Bayes’ theorem, resulting in posterior distributions that reflect our updated beliefs about the effects.
One of the key advantages of Bayesian factorial designs is the ability to “borrow strength” across different factor levels and interactions. This means that if the data for one factor level are limited, the model can use information from other factor levels to improve the estimate for that level. This is particularly useful in complex experiments where some factor combinations might have smaller sample sizes.
Kassler, Nichols-Barrer, and Finucane (2018) Beyond “Treatment versus Control”: How Bayesian Analysis Makes Factorial Experiments Feasible in Education Research.
Example with code
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