Fixed Interaction Effects Projection
Hello, BayesFactor users.
I am writing to inquire about implementing fixed interaction effects in ANOVA models.
The fixed main effects in one-way ANOVA are discussed in Rouder et al. (2012, p.363) and Rouder et al. (2017, p. 312), where a Q matrix of size a x (a - 1) projects a sef of a effects into (a - 1) parameters with the property that the marginal prior on all a effects is identical.
$\mathbf{I} - \mathbf{J} / a = \mathbf{Q} \cdot \mathbf{Q}^\top$
Q is an a x (a - 1) matrix of the (a - 1) eigenvectors of unit length corresponding to the nonzero eigenvalues of the left side term.
For example, the projected effect t* = (t1 - t2) / sqrt{2}, when a = 2.
My interest is in a multiway ANOVA. The four interaction effects will be projected into one term for a 2 x 2 factorial ANOVA, considering all fixed. In this case, how is the projection matrix computed?
Thanks for any comments.
Rouder, J. N., Morey, R. D., Speckman, P. L., & Province, J. M. (2012). Default Bayes factors for ANOVA designs. Journal of Mathematical Psychology, 56, 356-374. https://doi.org/10.1016/j.jmp.2012.08.001
Rouder, J. N., Morey, R. D., Verhagen, J., Swagman, A. R., & Wagenmakers, E.-J. (2017). Bayesian analysis of factorial designs. Psychological Methods, 22, 304-321. https://doi.org/10.1037/met0000057
Comments
I may figure it out: ab* = (ab11 - ab21 - ab12 + ab22) / 2 when projecting four interaction effects into one term, given a 2 x 2 design.
Thanks, I attended the experts to this question.
Yes, Sherloconan, that's correct. You can get the design matrix from a Bayes factor test that will tell you more with the model.matrix, , e.g.:
Best,
Richard
Thank you very much, Dr. Morey and Dr. Wagenmakers.