# [R-sig-ME] Modelling growth data: time polynomials as random effects

Julie Marsh marshj02 at student.uwa.edu.au
Sun Apr 6 12:18:14 CEST 2008

```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

```