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@gmail.com> wrote:

> Reinhold Kliegl <reinhold.kliegl@...> 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.
>
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