Thanks for your thoughts, Tobias, it helped me clarify for myself how I would think about this problem.  I especially appreciated the way you worded the following, as it's a good reminder to remember what we are doing here:

When you take out the fixed Days effect, you model your data with the assumption that there is no effect of Days common to all subjects (= fixed effect is 0).

This also reminded me that some of the confusion for this may be caused by folks switching between likelihood (fixed and random effects) and MCMC (everything random) methods.  This is understandable.  In a Bayesian statistical model for this same problem, I could define the distribution of the slopes with an overall slope mean instead of the distribution of the differences between each slope and the population-level slope with a 0 mean.

You are right that if the fixed effect isn't 0, the model without it will generally fit worse (this is very easy to see in simulation).  In that case, comparing models with and without the fixed slope might help someone realize that a random slopes only model isn't what they are looking for.

I would guess that your overall fit is worse when you leave out the fixed effect, reflecting that there IS a common effect.

My question of statistical theory now seems too hard to word clearly.  I was mostly mulling over the possibility of maximizing the likelihood with a different definition for the mean of the distribution of the random effects.  I'm sure I can research that for myself if I want to take the time.

Ariel


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