Please see the attached screenshot repeat the Google search for full details Best wishes David Booth

Can you tell us how many units are at each level?

The reason I ask is that if there are few units (say) at level 3 there will be a difficulty estimating the variance. If that is the case I would fit a two level model (leave out level 3) and the variance will then accrue to the lowest included level, that is level 2. You could also then include dummies (fixed effects) for the few level 3 units.

The reason for using Bayesian MCMC estimation is shown by Article How Many Countries for Multilevel Modeling? A Comparison of ...

Non -Bayesian estimation can give poor estimates as the simulation study shows.

I have written a guide to understanding MCMC in Book Developing multilevel models for analysing contextuality, he...

That guide is for continuous response ; I attach something on binary response and multilevel logit model ; you have to be careful about interpreting these models as the level 1 variance can never decrease as you include predictor variables in the fixed part; this is covered in the document.

To answer two of you questions directly:

I would use a model designed for binary outcomes eg a logit rather than assume continuous data.

If you are not wanting to use (sampling) weights then a lack of facilities with Bayesian estimation in Mplus should not worry you!

This model/type of analysis is very complex, and you mention that the results appear to be "fragile" or unstable. I would contact the Mplus support directly. They will be happy to look over your output to see if the results can be trusted. They can also give you further advice on how to best approach this type of analysis in Mplus.

Thank you all for your help!

Kelvyn Jones In fact, there are only 24 units at level 3, which worries me a little. However, I want to investigate a cross-level interaction, so that I need the three-level model. I will have a look at your recommended links and literature, thank you! Christian Geiser And I will also contact the Mplus support, you are right, they are in fact very helpful. I will see then if I can conduct my analyses as planned or need to find another (continuous) operationalization for my outcome.

I would not be too worried abut 24 units to estimate the highest level variance/covariances ( I was worried that you may have had only 5 or so). The MCMC approach is useful because it is full uncertainty modelling that is both the fixed and random parts are estimated taking account of their uncertainty.

You may want to have a look at this:

Article Understanding and misunderstanding Group mean centering: a c...

and the answers to this

https://www.researchgate.net/post/Cross-level-interaction-without-specifying-a-random-slope-for-the-Level-1-variable#:~:text=A%20cross%2Dlevel%20interaction%20is,with%20the%20cross%2Dlevel%20interaction.