[R] problem with intervals in mixed model

[Douglas Bates]

Bill Shipley wrote:
> Hello.  I am analysing data from a mixed model perspective using the
> lme() function.  The fixed effects model is a quadratic (Y~X+X2) where
> X2 is the square of X and the data have a 3-level structure.  I fitted a
> series of three models with the same fixed effects but differing in the
> random effects (only intercept, intercept + X, intercept +X +X2).  The
> anova shows that all three parameters vary significantly (p<0.001)
> across groups.  I have therefore chosen the third model, in which all
> three parameters vary.
> 
> When I attempted to obtain the confidence intervals for the correlations
> between the random components, using:
> 
>  
> 
> intervals(fit3,which="var-cov")
> 
>  
> 
> I get the following error message:
> 
>  
> 
> Problem in intervals.lme(fit3, which = "var..: Cannot get confi
> 
> dence intervals on var-cov components: Non-positive definite ap
> 
> proximate variance-covariance
> 
>  
> 
>  
> 
> I assume that this arises because the correlation between two of the
> parameters at the 2nd lowest level is -0.998.  Can anyone tell me how to
> deal with this problem?  Specifically,
> 
> 1) how should I interpret such a strong correlation?
> 
> 2) how can I obtain confidence intervals for these correlations between
> the random components?

It is quite possible that the ML or REML estimate of the
variance-covariance matrix of the random effects is singular.

If this matrix approaches singularity because one of the diagonal terms
is going to zero then it simply means that the model should be reduced
by removing the corresponding random effect.  However, approaching
singularity by getting correlations close to -1 or to +1 takes you out
of the space of linear mixed models.

My only suggestion is possibly to change the origin on the X axis if
that would make sense in the context of your data.