My guess is that the optimum isn’t well defined and that we’re seeing two local optima. I don’t have time at the moment to check the profiles (the code for this isn’t finalized in MixedModels.jl and I – and I suspect @dmbates too – have spent too much time in Julia lately to remember how to produce a profile plot quickly there). Because looking at the random effects – which is where the magic happens – we’re seeing approximately the same two answers again and again. Given that there are only 4 groups, this isn’t too surprising, but I would have to double check.