Modelling growth data: time polynomials as random effects

Dear LMER Experts,

I am trying to model some growth data based on multiple ultrasound  
measurements taken across the time course (in days) of each pregnancy.  
The vast majority of subjects have 4+ measurements.  Everything seems  
good up to:

model.ac1 <- lmer(y.ac ~ TIME + I(TIME^2) + I(TIME^3) +  
(TIME|STUDYNO), data=s.workdat, method="ML", na.action=na.omit)

summary(model.ac1)

Linear mixed-effects model fit by maximum likelihood
Formula: y.ac ~ TIME + I(TIME^2) + I(TIME^3) + (TIME | STUDYNO)
    Data: s.workdat
    AIC   BIC logLik MLdeviance REMLdeviance
  -5051 -5004   2532      -5065        -4978
Random effects:
  Groups   Name        Variance   Std.Dev.  Corr
  STUDYNO  (Intercept) 6.3577e-02 0.2521444
           TIME        1.7435e-06 0.0013204 -0.828
  Residual             1.3597e-02 0.1166055
number of obs: 6066, groups: STUDYNO, 1289

Fixed effects:
               Estimate Std. Error t value
(Intercept)  1.309e+00  1.313e-01   9.970
TIME         5.825e-02  2.160e-03  26.963
I(TIME^2)   -1.113e-04  1.144e-05  -9.723
I(TIME^3)    7.197e-08  1.957e-08   3.678

Correlation of Fixed Effects:
           (Intr) TIME   I(TIME^2
TIME      -0.996
I(TIME^2)  0.988 -0.997
I(TIME^3) -0.976  0.991 -0.998


However as the slope consists of the polynomial combination (TIME +  
TIME^2 + TIME^3) intuitively it would seem correct to include these  
polynomial terms as  random effects. However, everything goes  
pair-shaped when I try:

test.ac3 <- lmer(y.ac ~ TIME + I(TIME^2) + I(TIME^3) +  
(TIME+I(TIME^2)+I(TIME^3)|STUDYNO), data=s.workdat, method="ML",  
na.action=na.omit)

Warning messages:
1: In .local(x, ..., value) :
   Estimated variance-covariance for factor ?STUDYNO? is singular

2: In .local(x, ..., value) :
   nlminb returned message false convergence (8)


I am assuming that this is due to the strong correlations between the  
polynomials of TIME. I also performed this analysis on the subset of  
subjects with 5 or greater ultrasound measurements (in desperation!  
n=1024) and obtained the same error message.

My dilemma is explaining why TIME has both a fixed and random  
component, whereas TIME^2 and TIME^3 only have a fixed component.  I  
suspect I am missing something fundamental !!!

I have also had some fun fitting different AR structures but decided  
to strip back the model to the basic correlation structure for this  
question. Any help would be very much appreciated.


kindest regards,  julie marsh

PhD Student
Centre for Genetic Epidemiology
University of Western Autsralia
email: marshj02 at student.uwa.edu.au