Type of Sums of Squares
Hello,
JASP supports different Types of Sums of Squares, which correspond to different ways to define marginal means and hence main effects in classic Frequentist multi-way ANOVA. Is there a correspondence to this in the Bayesian ANOVA? Does any of the Bayes Factors in a factorial ANOVA actually map onto what would be called a main effect in the Frequentist ANOVA? And if so, under what kind of definition of marginal means (unweighted, aka Type III; weighted by harmonic mean of sample size, aka Type II)? I am currently thinking the resulting BFs do not map onto main effects, but I am curious to learn more.
Comments
OK, just read Richard's slides from ICPS2015, where he talks about the difference between model-based and effects-based approaches. The former being the Bayesian ANOVA approach, the latter being the Frequentist approach. That makes some sense now.
Any additional comments still appreciated.
Well, we'd have to solicit comments from Richard I guess!
Cheers,
E.J.
Also, for the benefit of others in this forum, can you link to those slides from Richard?
Cheers,
E.J.
Sure thing EJ - here are the slides:
https://static.jasp-stats.org/presentations/SARMAC2015/RM/talk.html#1
The relevant one is #15.
From my understanding, the Frequentist main effects allow different weighting (commonly known as Type of Sums of Squares) - or to be more precise the marginal means are weighted differently. Manually, these Type of Sums of Squares can also be obtained by using different types of coded variables (e.g., dummy codes, effect codes, centered dummy codes). The Bayesian approach (as implemented in the BayesFactor package) remains invariant to these different coding schemes.
Yes, I recall a few papers critiquing Type III sums of squares because of issues with differences due to scale transformations. And there's also a new paper out by Rouder, Morey and colleagues (in PBR) about model-based ANOVA approaches.
Yeah, none of the sums of squares types (I,II,III) really captures what one would call an average main effect in unbalanced factorial designs. Rolf Steyer wrote a paper about this a while ago. I believe that the Bayesian ANOVA as implemented in BayesFactor avoids this issue altogether (the different weighting of the marginal means), as it focuses on model comparison, and is not effects-based. I think I have seen this paper by Rouder - was that the one with plot of the multivariate Cauchy distribution?
It is this one: http://pcl.missouri.edu/sites/default/files/Rouder.etal_.pbr_.2016.pdf
Do you have a reference for the Steyer paper in return? :-)
E.J.
Thanks for sharing - I did NOT see this paper. I remembered a different one. Looks interesting. I just emailed Steyer and he told me that his paper is not in press yet, in fact it is currently a doctoral dissertation and unfinished. Sorry, E.J. - no reference, but hopefully soonish.
no worries, thanks for asking