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19.1 Bayesian Multifactor ANOVA 519 In those last equations, the symbol "" means "for all." In words, the last two equations simply mean that the interaction deflections sum to zero along every level of the two predictors. A graphic example of this was presented in the left panel of Figure 19.1, which shows that the heights of the arrows sum to zero along every edge of the parallelogram. Our goal is to estimate the additive and interactive deflections, based on the observed data. It is important to understand that the observed data are not the bars in Figure 19.1; instead, the data are swarms of points at various heights near the heights of the bars. The bars represent the central tendency of the data at each combination of the predictors. Thus, what the equations above actually predict is the central tendency µ at each combination of predictors, and the data are typically modeled as being normally distributed around µ. 19.1.2 The Hierarchical Prior The complete generative model of the data is shown in Figure 19.2. It might look daunting, but it really is merely the diagram for oneway ANOVA, in Figure 18.1, with the hyperprior replicated for each predictor and interaction.