We're currently running a conjoint experiment in 26 countries with 2000 participants per country and would like to use a multilevel model. We've done up most of the pre analysis plan and run some simulations using LMER and the results look good. However, my prof would like to analyse the random effect for each country and eventually draw inferences from them. I'm opposed to this because AFAIK these aren't actually estimates; these are conditional means/modes/predictions generated from the estimated parameters (see here: Intuition about parameter estimation in mixed models (variance parameters vs. conditional modes) ). They do not have SEs nor confidence intervals. I'm now at an impasse because I too would like to make inferences on these "estimates" but as far as I'm aware, the only way to do so is to switch to a bayesian framework. I'm not a bayesian myself and have very little experience in it. As much as possible I would like to remain in the frequentist world. To my mind the only way I can do this however is generate bootstrapped confidence intervals and SDs for these "estimates", but this would be very computationally expensive. Does anyone have other suggestions?
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