emmeans - interaction contrasts After reading about interactions contrasts in emmeans, I just wanted to make sure I understood it correctly. Say I have a model with a group*time interaction effect, and I set up emmeans as follows:
emm <- emmeans(lme, ~ Group * Session)

And then use
contrast(emm, interaction = TRUE, "pairwise", adjust="mvt")

It outputs something like
Group_pairwise        Session_pairwise   estimate    SE     df   t.ratio p.value
Group_A - Group_B     Session1 - Session2  x.xxx       x.xxxx xx.x x.xxx   0.001
Group_A - Group_B     Session1 - Session3 ...
...

Does the first line for example then say that the difference between Session 1 and 2 within Group_A is significantly different to the difference between Session 1 and 2 within Group_B?
So it is basically a contrast of contrasts?

I guess if a Cohen's d makes sense, you can do something like
CON <- contrast(emm, interaction = TRUE, "pairwise", adjust="mvt")
eff_size(CON, sigma = ???, edf = ???, method = "identity")

(see the examples in the help page for eff_size)
but you need to replace the ???s with reasonable values -- and be able to explain and defend them. You apparently have a mixed model, and I'm not sure Cohen's d is even defined because there are multiple sigma values involved. But I suppose that for edf you can use #groups - 1 for the coarsest grouping. For sigma, possibly you can do VarCorr(model) to see all the variance estimates; and then combine them. For instance, if there are three SDs sd1, sd2, sd3 you might use
sigma <- sqrt(sd1^2 + sd2^2 + sd3^2)

I am very leery of all of this because you can't find an example where this was actually ever done with interaction contrasts, and because in any but the simplest situations, the whole idea of effect size is flaky; people improvise something such as what I suggest, and then nobody really knows what you're talking about. You wind up with numbers, but they answer a question that can't be stated clearly.
