coef se in lme
The AD Model Builder method I posted earlier takes into account the
uncertainty in the mean and both std deviations in Harold's simple model.
Y_ij - mu + u_i +eps_ij
To illustrate this
I built a little model and simulated a data set with 1<=i<=10, 1<=j<=5
observations. Below are the parameter estiamtes tigether with their
estimated std devs. The true values were mu=3.0 sigma_u=2.0 and sigma=3.0
1 mu 2.4091e+00 7.4307e-01
2 log_sigma_u 6.3908e-01 3.4913e-01
3 log_sigma 1.1354e+00 1.1180e-01
4 u -3.1695e-01 6.4561e-01
5 u 2.0441e-01 6.4496e-01
6 u 5.1996e-01 6.4748e-01
7 u 4.3754e-01 6.4661e-01
8 u 4.8381e-01 6.4708e-01
9 u 1.2214e+00 6.6075e-01
10 u -1.7631e+00 6.7793e-01
11 u 3.3987e-01 6.4578e-01
12 u -1.7663e-01 6.4485e-01
13 u -9.5030e-01 6.5439e-01
then I fixed log_sigma_u and log_sigma at their estimated values and obtained.
# fixed sd's
index name value std dev
1 mu 2.4093e+00 7.4307e-01
2 u -3.1736e-01 6.4407e-01
3 u 2.0456e-01 6.4407e-01
4 u 5.2045e-01 6.4407e-01
5 u 4.3794e-01 6.4407e-01
6 u 4.8426e-01 6.4407e-01
7 u 1.2226e+00 6.4407e-01
8 u -1.7650e+00 6.4407e-01
9 u 3.4016e-01 6.4407e-01
10 u -1.7689e-01 6.4407e-01
11 u -9.5138e-01 6.4407e-01
and finally I almost fixed mu (can't fix it completley because then there would be no "fixed" effects and the model thinks there is nothing to do.) and obtained
# all fixed
index name value std dev
1 mu 2.4093e+00 2.2361e-03
2 u -3.1736e-01 5.9145e-01
3 u 2.0456e-01 5.9145e-01
4 u 5.2045e-01 5.9145e-01
5 u 4.3794e-01 5.9145e-01
6 u 4.8426e-01 5.9145e-01
7 u 1.2226e+00 5.9145e-01
8 u -1.7650e+00 5.9145e-01
9 u 3.4016e-01 5.9145e-01
10 u -1.7689e-01 5.9145e-01
11 u -9.5138e-01 5.9145e-01
So for example u(1) the first random effect has estimated std devs 6.4561e-01, 6.4407e-01, and 5.9145e-01 under the three models. It appears that most of the "extra" uncertainty in u(1) comes from the uncertainty in mu.
David A. Fournier P.O. Box 2040, Sidney, B.C. V8l 3S3 Canada Phone/FAX 250-655-3364 http://otter-rsch.com