Simulation is definitely the way to go in my view. More generally, though, I find that power analysis often has to rely on so many unverified a priori assumptions that it probably won't be of much more use than general experience-based rules of thumb (cf. Marc’s response).

Hello Nina.  Jos Twisk's nice (introductory) book on multilevel models includes a chapter on sample size estimation.  HTH.

UPDATE:  I just noticed that PASS uses simulation to perform sample size estimation for "mixed effects models".  See the second link below.

http://www.cambridge.org/ca/academic/subjects/medicine/epidemiology-public-health-and-medical-statistics/applied-multilevel-analysis-practical-guide-medical-researchers?format=PB&isbn=9780521614986

https://ncss-wpengine.netdna-ssl.com/wp-content/themes/ncss/pdf/Procedures/PASS/Mixed_Models-Simulation.pdf

https://www.ncss.com/software/pass/pass-documentation/#Simulation

I vote for the suggestion to use simulation. R does this nicely, but many other programs will too.

Thank you very much, everyone!

Jake Westfall has a calculator for mixed effect (and a couple of good papers together with Judd). Go on this site: http://jakewestfall.org/ (Web apps).

You also find simulations on a paper I submitted as a project. Personally, I've come to conclusion that I won't do much research anymore with less than 1,600 observations per cell (40 participants 40 stimuli, or 20 participants 80 stimuli, or 80 participants 20 stimuli). Such a design allows you to find anything that is interesting, even though it may be slightly overpower for large effects).

Marc Brysbaert Do you have publish this paper yet? I would like to apply your rule of 1600 observations per cell. I am also wondering hoy the type of factor (within versus between) may influence this number. For instance in my case I have an interaction between a within (2 types of stimuli) and a between (2 groups of subject). Thanks in advance Boris

I found this one: Article Power Analysis and Effect Size in Mixed Effects Models: A Tutorial

Yes, that's the reference. It works for a pairwise comparison. Unfortunately, you will need more observations (participants) for the interaction you are after, as you can read here:

https://psyarxiv.com/2v4yx/

Will be published shortly in the Journal of Cognition

Thank you very much, Boris

Marc Brysbaert. With some colleagues we are running a study examining the evolution of performance (correct vs. incorrect response) across task duration using logistic mixed-effect models. We have 2 conditions. One with low difficulty (Less inhibiting trials) and the second with high (more inhibiting trials). Our main test will be to assess the moderating effect of the condition on the evolution of the performance across the task: i.e., a trials number (continuous) x conditions (easy vs. difficult)  on the performance (0 vs 1). We are struggling to estimate the correct sample size as the main test is no comparing the absolute performance difference between group but the evolution, that is we want to have the accurate level of power to test for this evolution of time, and we are not sure to know the minimum number of observation by group and time needed,

Another point: we will not have the same number of observations between group (150 observations in the low difficulty group vers 450 in the other; this is because our dependent variable is directly related to the manipulation). We are wondering how manage such unbalanced data observation (1/3 differences).

If you have some time, your help will be very much appreciated. Boris

Probably the best is to run a simulation of what you intend to do (and hope to find). You can adapt one of the programs I used for the latest power analysis paper.

Thank you very much. I will do it.

Simulation is definitely the way to go in my view. More generally, though, I find that power analysis often has to rely on so many unverified a priori assumptions that it probably won't be of much more use than general experience-based rules of thumb (cf. Marc’s response).

Are there any papers in support of this view, i.e.，general expeirence-based rules of thumb？

SimR may be useful for you to conduct power analysis ( Article SIMR: An R package for power analysis of generalized linear ...

I agree with most people here that a simulation-based method is the best approach. However, as someone said, you need to make many assumptions that might not hold in the end. I would like to add that a possible way around this is to use a kind of semi-simulation where you extrapolate from that from a pilot study. See for instance our paper in BMC psychology for an example:

Article A protocol for a three-arm cluster randomized controlled sup...

Further, it might be worth conducting Bayesian design analysis, which is sort of the Bayesian equvalent of power analysis. Here is an article on this. This framework can be expanded to mixed effects models. Bare in mind that the simulations will take a long time in this case and that you might need a powerful computer.

Article A tutorial on Bayes Factor Design Analysis using an informed prior

https://www.surveysystem.com/sscalc.htm

A very nice stand-alone free software for power analysis is G*Power, with a very good and intuitive visual interface, check it out:

http://www.psychologie.hhu.de/arbeitsgruppen/allgemeine-psychologie-und-arbeitspsychologie/gpower.html

To my knowledge doesn't work for mixed effects designs with participants and stimuli as random variables. In addition, does not give useful information about repeated measures design in an intuitive way. Have a look here:

Article How Many Participants Do We Have to Include in Properly Powe...

If you know a good way to get the information out of G*Power for such designs that would be very welcome.

Baker, D. H., Vilidaite, G., Lygo, F. A., Smith, A. K., Flack, T. R., Gouws, A. D., & Andrews, T. J. (2020, July 16). Power Contours: Optimising Sample Size and Precision in Experimental Psychology and

Human Neuroscience. Psychological Methods. Advance online publication.

http://dx.doi.org/10.1037/met0000337