Hi Jacob,

I think there is a way to explain this phenomenon that could be helpful. 

Imagine a dataset where the *true* data model is that there is 0 variation in the random cluster slopes. But there is, of course, some residual error variance. Now as we generate the observed data from this data model, we start by giving every cluster the same slope. Then we add in the perturbations due to the random residual effects. It is clear that after adding in the random residual effects, the clusters will no longer have identical slopes, even though the clusters "truly" have identical slopes.

So in your dataset, if you calculate the simple "no pooling" estimates of the cluster slopes (i.e., from separate within-cluster regressions), there will of course be some variation in these slope estimates. But if we consider the magnitude of the estimated residual error variance, it is clear that the amount of variation in these observed slopes is completely consistent with a data model where *all* of the variation in these slopes is due simply to the random residual effects. We do not need to posit any variation in the latent cluster slopes in order to account for the amount of variation that we observed in the simple cluster slopes. So we end up with an estimate of 0 slope variance.

Hope this helps to give some intuition about it. I would be happy to hear comments from others about this interpretation of the situation. 

Jake 

> Date: Tue, 17 Sep 2013 10:06:10 -0400
> From: bbolker@gmail.com
> To: segerfan83@gmail.com; r-sig-mixed-models@r-project.org
> Subject: Re: [R-sig-ME] Lack of Variation in Slopes
> 
> On 13-09-17 09:18 AM, AvianResearchDivision wrote:
> > Ben,
> > 
> > What are the chances that there is absolutely no variation in slopes?  This
> > seems beyond unlikely, but is probably rooted in my limited understanding
> > of what the model is actually doing.  It seems you would always expect some
> > sort of variation, even if it is the smallest amount.  Maybe I am
> > misinterpreting what this means.  Anyways, thank you very much for taking
> > the time to answer my many questions up to this point.  I do very much
> > appreciate it.
> > 
> > Jacob
> 
>  It's not that there is no variation; it's that the maximum likelihood
> estimate is that there is no variation.  This happens a lot in mixed
> models, especially when the variance is small or the data set is small
> or noisy.  See e.g. http://rpubs.com/bbolker/4187
> 
>  [cc'ing back to r-sig-mixed-models ]
> 
> 
> > 
> > 
> > On Tue, Sep 17, 2013 at 9:06 AM, Ben Bolker <bbolker@gmail.com> wrote:
> > 
> >> AvianResearchDivision <segerfan83@...> writes:
> >>
> >>>
> >>
> >>   [snip]
> >>
> >>> You were spot on with your suggestion.  Thank you very much for your
> >> help.
> >>> I had mentioned before that I probably should keep random slopes within
> >> my
> >>> models since that is a primary interest of my study, regardless of
> >> whether
> >>> a LRT suggests they are needed.  Would you still recommend keeping them
> >>> within the model?
> >>>
> >>> Jacob
> >>
> >>   I don't think it will hurt.
> >>
> >> _______________________________________________
> >> R-sig-mixed-models@r-project.org mailing list
> >> https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models
> >>
> >
> 
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