Thanks Ben, if Rune Haubo is reading this list, it would be nice to see the example that you mentioned. I wonder if certain coding schemes for the categorical variables could minimize this phenomenon. So it would be nice to toy around with the example.

Jake

> To: r-sig-mixed-models@r-project.org
> From: bbolker@gmail.com
> Date: Tue, 3 Dec 2013 21:10:07 +0000
> Subject: Re: [R-sig-ME] Always allow for correlation of random effects?
> 
> AvianResearchDivision <segerfan83@...> writes:
> 
> > 
> > Hi all,
> > 
> > I am working with mixed models and have the following general model
> > structure:
> > 
> > model<-lmer(X~Y*Z+(A|B), where Y is a continuous mean centered
> > environmental variable, Z is Year (2012 or 2013).
> > 
> > Part of my interest in my study is to explore variation in plasticity
> > between individuals, not just population level plasticity.  When settling
> > on a final model, I first check for significance of random 
> > effects and then
> > I worry about the fixed effects structure.  For the random effects
> > structure, even in there is not significant variation between individuals
> > in their slope, I leave this term in the model because it is a primary
> > interest of mine.  My question is, if I am leaving random slopes in all of
> > final models along with random intercepts, should I also allow for the
> > correlation of the random effects, i.e. (A|B) or should I not allow for
> > this, i.e. (A+0|B) if there is no significant correlation when checked LRT
> > fitted by REML?
> > 
> > I initially thought that I would remove the ability for correlated random
> > effects if there was no significant correlation, but then I read the
> > following: "In models with both individual-specifi… 
> > elevations and slopes we
> > allowed for the potential correlation between these, to ensure that BLUP
> > estimates produced by the models were not affected by the method used to
> > centre covariates." from 'Phenotypic plasticity in a maternal trait in red
> > deer" by Nussey et al. 2005.  Do you understand their reasoning?
> 
>   Very briefly (if Gmane lets me): if the correlation between slopes
> and intercepts is suppressed, then you can change the results by
> linear transformation of the 'A' variable (you didn't tell us what
> it was -- it might be helpful); for example, if 'A' is a continuous
> predictor, then centering it will change the answers.  If 'A' is
> a categorical predictor, then changing the contrasts (e.g. from the
> default treatment contrasts to sum-to-zero contrasts) will change the
> answers.  If you allow for the correlations, then the results will be
> invariant to linear transformations/combinations of the 'A' variable.
> Rune Haubo has some nice little examples of this phenomenon suggesting
> that the common practice of dropping the correlations for parsimony
> can have misleading effects -- I don't know if they're publicly available
> anywhere.
> 
>   By the way,
> by dropping the correlations do you mean the difference between
> 
> (1|B) + (0+A|B)
> 
> and
> 
> (A|B)
> 
> ?  This works for a continuous variable, but *not* for a categorical
> variable -- at present if you want to do this for a categorical
> variable you need to generate your own dummy variables (e.g. see
> https://github.com/lme4/lme4/issues/139 )
> 
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