Another case of -1.0 correlation of random effects

Kevin E. Thorpe wrote:

Hello.

I know this has come up a couple times recently, but I'm still not sure 
what to do about it in my data.  Note that my sessionInfo() will be at 
the bottom.

My data come from a crossover trial and are balanced.

 > str(gluc)

'data.frame':	96 obs. of  4 variables:
  $ Subject  : int  1 2 3 5 6 7 10 11 12 13 ...
  $ Treatment: Factor w/ 2 levels "Barley","Oat": 1 1 1 1 1 1 1 1 1 1 ...
  $ Dose     : int  8 8 8 8 8 8 8 8 8 8 ...
  $ iAUC     : num  110 256 129 207 244 ...

 > xtabs(~Treatment+Dose,data=gluc)

          Dose
Treatment  0  2  4  8
    Barley 12 12 12 12
    Oat    12 12 12 12

I plot the data (attached as gluc.pdf, if it comes through).

 From the plot, I think I want to fit the model as:

lmer(iAUC~Treatment+Dose+(Treatment|Subject)+(Dose|Subject),data=gluc)

It could possibly be argued that the (Treatment|Subject) part is not 
needed.  When I fit this, I got -1.0 correlation within the Dose random 
effects.  To simplify, I will fit a simpler model, since the issue persists.

 > lmer(iAUC~Dose+(Dose|Subject),data=gluc,subset=Treatment=="Oat")

Linear mixed model fit by REML
Formula: iAUC ~ Dose + (Dose | Subject)
    Data: gluc
  Subset: Treatment == "Oat"
    AIC   BIC logLik deviance REMLdev
  562.6 573.9 -275.3    563.1   550.6
Random effects:
  Groups   Name        Variance Std.Dev. Corr
  Subject  (Intercept) 8274.324 90.9633
           Dose          16.214  4.0266  -1.000
  Residual             4862.319 69.7303
Number of obs: 48, groups: Subject, 12

Fixed effects:
             Estimate Std. Error t value
(Intercept)  309.352     30.539  10.130
Dose         -14.424      3.596  -4.012

Correlation of Fixed Effects:
      (Intr)
Dose -0.647

Now, a plot created by (and attached as lmlist.pdf):

plot(confint(lmList(iAUC~Dose|Subject,data=gluc,subset=Treatment=="Oat"),pooled=TRUE),order=1)

shows (I think) a strong negative correlation between the intercept and 
slope random effects for Dose.

So, I would appreciate some advice on how I might specify these random 
effects correctly.

  It's not entirely clear whether Subjects are unique within Treatments
(OK, maybe we should say "nested") or whether the same Subject can have
both Treatments  -- what would xtabs(~Subject+Dose+Treatment) look like?

  *If* each Subject appears in only one Treatment (i.e., 5 doses per
Subject, each Subject is either Oat or Barley, 12 Subjects per
Treatment), which is what I initially took for granted, then it makes
perfect sense to me that you can't specify Treatment|Subject, because
Treatment does not vary within Subject -- there doesn't seem to be any
way you can recover information about how the effect of Treatment varies
among Subject.

  Since your plots do look reasonably linear within Subject,

iAUC~Treatment*Dose+(Dose|Subject)

 seems best to me.
For what it's worth, this example closely parallels the Orthodont
example in the nlme package ...