In my understanding, the idea of âeffective number of parametersâ is not so much one about counting number of parmaters, but more one about estimating model-complexity. (Sorry, if my phrasing above was misleading).

The general idea is that the number of parameters is an indicator of model complexity, for which one penalizes if one compares models with information criteria like BIC, AIC, DIC, WAIC. The authors of these information criteria estimate model complexity (number of effective parameters) differently, and more spohisticated estimators of model complexity as used in the WAIC imply that model complexity is not necessarily a monotone (linear?) function of the number of random effect parameters.

One way to see this is to consider Equation 13 in the above linked paper and to realize that the variance over mcmc samples in the log predictive density is higher in a model in which ârandom effectsâ* are not constrained by an additional hyper-parameter, compared to a model with hyper-parameter (e.g. variance of random effects) for the random effects. That is, the model with more parameters would have lower estimated complexity than the model with fewer parameters.

So, one could say that the model complexity of a hierachical/multilevel model is higher than that of a fixed effects model, but lower than the complexity of a model that estimates independent random effects.

*putting this in parantheses, because Iâm not sure one would still call this a random effects model, if one estimated group-specific effects independently.