total (pooled) variance: examples [Power / Sample Size]

Dear BEQool.

❝ So if we have e.g. 3x6x3 Williams crossover design (with 3 treatments), the number in the denominator depends on the approach we use?

No, for both approaches the denominator is 3. It's not affected by whether the model is complete or incomplete.
Nevertheless someone else could check it. I read about it many years ago (mentioned here and statistical background somewhere else, I don't remember where). And "a long time ago in a galaxy far, far away" I also recalculated the estimate (obtained from ANOVA - GLM and equation with 3 as denominator) using the mixed model where the inter-subject variability is obtained more directly.

❝ If we use "Two at a Time" approach with incomplete block designs, we are only evaluating 2 treatments and therefore we should use number 2 in the denominator; but if we use "All at Once" approach, we are evaluating all 3 treatments and we have to use number 3 in the denominator?

As Helmut mentioned, the "Two at a Time" is preffered approach. In the "All at Once" analysis, 90% confidence intervals in comparison of T and R1 are (slightly) affected by values of R2 and vice versa. It is not desirable in any of comparisons (e.g. when R1 is from EU and R2 from USA).
So: "..., the analysis for each comparison should be conducted excluding the data from the treatments that are not relevant for the comparison in question." (EMA Guideline 1401)

❝ Additional question (just to be sure): we also use number 2 in the denominator (to get inter-subject variance) when we have replicate designs with 2 treatments (regradless of the partial- or full-replicate design)?

Yes. At least I hope so, I have experience only with full replicate designs so far.