[R-sig-ME] Always allow for correlation of random effects?

[Viechtbauer Wolfgang (STAT)]

This can happen when the model fitting routine is hitting boundaries of the parameter space. For example, suppose the slope variance of Y is so close to 0 as to be essentially indistinguishable numerically from 0. Whether rho is then -1, 0, +1, or any other value is essentially irrelevant then (since the covariance is then also essentially 0). The routine will give you *some* value (and different routines may give you rather different values), but that parameter is not really estimable. This is what Thierry was alluding to when he said:

> This can be caused by having more complexity than the data allows. Note that the variance of the random slope is small.

Thierry, please correct me if I am misinterpreting your statement.

The profiled log likelihood for parameters that are not identifiable is essentially flat. Roughly speaking, this translates into a large SE for the corresponding parameter estimate. So, even with an 'estimated' correlation of -1, the test of that correlation may not be significant.

Best,
Wolfgang

--   
Wolfgang Viechtbauer, Ph.D., Statistician   
Department of Psychiatry and Psychology   
School for Mental Health and Neuroscience   
Faculty of Health, Medicine, and Life Sciences   
Maastricht University, P.O. Box 616 (VIJV1)   
6200 MD Maastricht, The Netherlands   
+31 (43) 388-4170 | http://www.wvbauer.com   

> -----Original Message-----
> From: r-sig-mixed-models-bounces at r-project.org [mailto:r-sig-mixed-models-
> bounces at r-project.org] On Behalf Of AvianResearchDivision
> Sent: Wednesday, December 04, 2013 14:45
> To: Ben Bolker; r-sig-mixed-models at r-project.org
> Subject: Re: [R-sig-ME] Always allow for correlation of random effects?
> 
> Hi Ben,
> 
> Thank you for the response.  My model actually was this:
> model<-lmer(X~Y*Z+(Y|B), where Y is a continuous grand mean-centered
> environmental variable (where 0 in this case does not have a 'strong
> theoretical meaning' other than the average environment that males (B)
> experience) and B is a factor with 59 levels.  The interaction Y*Z (Z =
> year) is irrelevant I suppose for two reasons.  1.  It's not significant
> in
> all but one case (when tested in 5 different models) and 2.  The one case
> where it is significant, I analyze the two years separately, thus removing
> the interaction from the model.  In my model selection, I establish the
> random effects structure first and then establish the fixed effects
> structure.  If I understand what you and others have responded with, it
> seems that I should leave the possibility of correlations in all models,
> even if a LRT suggests that there is not a significant correlation between
> intercept and slope.  And yes, this amounts to my final random effects
> structure as (Y|B) instead of (Y+0|B)+(1|B).
> 
> The odd thing is, when I allow for correlations in one of my models, the
> R2
> value for my BLUP points (coef(model)) is -1.00, but when I don't allow
> correlations, the R2 value is 0.3857.  When testing for the significance
> of
> correlations with a LRT, the X2 value is 1.68 and p = 0.1946.  How can you
> have an R2 of -1.00, but a non-significant p value?  More importantly, how
> does one get a R2 of -1.00 when working with ecological data?  This seems
> incorrect.
> 
> Thank you again,
> Jacob
> 
> 
> On Tue, Dec 3, 2013 at 7:59 PM, Ben Bolker <bbolker at gmail.com> wrote:
> 
> > Reinhold Kliegl <reinhold.kliegl at ...> writes:
> >
> > >
> > > Does the following relate to the issue you are discussing here?
> > >
> > > # If a and b are intercept and slope, then the
> > > # offset for X to obtain a zero intercept-slope
> > > # correlation parameter is cov(a,b)/var(b).
> > > # (I saw this derivation, which also yields a minimum
> > > # variance estimate for the intercept, in a 2007 GLMM
> > > # handout by Ulrich  Halekoh.)
> > >
> > > # An illustration with the sleepstudy data.
> > > library(lme4)
> > > summary(fm1 <- lmer(Reaction ~ Days + (Days|Subject), sleepstudy))
> > > VarCorr(fm1)
> > >
> > > ## Groups   Name        Std.Dev. Corr
> > > ## Subject  (Intercept) 24.7406
> > > ##          Days         5.9221  0.066
> > > ## Residual             25.5918
> > >
> > > # Compute the offset and add it to Days
> > > # (i.e., a very specific, but linear transformation of Days)
> > > sleepstudy$DaysA <- sleepstudy$Days + (0.066*24.7405*5.9221)/35.072
> > > summary(fm3 <- lmer(Reaction ~ DaysA + (DaysA|Subject), sleepstudy))
> > >
> > > VarCorr(fm3)
> > > ##Groups   Name        Std.Dev. Corr
> > > ##Subject  (Intercept) 24.6874
> > > ##         DaysA        5.9221  0.000
> > > ##Residual             25.5918
> > >
> > > Reinhold Kliegl
> >
> >   Yes, that's a large part of it.  There may have been some other
> > aspects to the example, but I think this illustrates the main point.
> > My interpretation is that unless the zero point of the continuous
> > covariate has strong theoretical meaning, so that a location shift
> > doesn't make sense, the hypothesis (correlation=0) is
> > dubious.  This seems analogous to the old rule that one shouldn't
> > compare between-group (intercept) differences in an ANCOVA model
> > (one with an interaction between a continuous and a categorical
> > predictor), because they can be made to match any arbitrary value
> > by shifting the location of the continuous predictor.
> >
> > _______________________________________________
> > R-sig-mixed-models at r-project.org mailing list
> > https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models
> >
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