# [R] lme parameterization question

John Fieberg John.Fieberg at dnr.state.mn.us
Wed Apr 2 22:36:02 CEST 2003

```Hi,

I am trying to parameterize the following mixed model (following Piepho
and Ogutu 2002), to test for a trend over time, using multiple sites:

y[ij]=mu+b[j]+a[i]+w[j]*(beta +t[i])+c[ij]

where:
y[ij]= a response variable at site i and year j
mu = fixed intercept
Beta=fixed slope
w[j]=constant representing the jth year (covariate)
b[j]=random effect of jth year, iid N(0,sigma2[b])
a[i]=random effect of the ith site, iid N(0, sigma2[a])
t[i]=random effect of ith site, iid N(0, sigma2[t])
c[ij]=random error associated with ith site and jth year

I would like to assume that an unstructured relationship applies to
a[i] and t[i] (i.e., I would like to assume that the random effects a[i]
and t[i] are drawn from a multivariate normal distribution with non-zero
covariance parameter).  These random effects are assumed to be
independent from the b[j]'s and from the c[ij]'s.  I have tried several
approaches, but cannot seem to duplicate the results presented in Piepho
and Ogutu using R's lme function (but I can reproduce the results using
SAS proc mixed).

In SAS, the model is fit using:

proc mixed method=REML nobound;
class year site;
model y=w site/ddfm=satterth s;
random int/sub=year;
random int w/sub=site type=un;
run;

Any help would be greatly appreciated!

Reference:
Piepho, H-P. and J.O.Ogutu. 2002.  A simple mixed model for trend
analysis in wildlife populations.  Journal of Agricultural, Biological,
and Environmental Statistics, 7(3):350-360.

Thanks,

John

```