[R-sig-ME] [R] coef se in lme

dave fournier otter at otter-rsch.com
Thu Oct 18 09:57:51 CEST 2007

Here is one approach to this problem.
In the AD Model Builder Random Effects package we provide estimated
standard deviations for any function of the fixed and random effects, 
(here I include the parameters which detemine the covarince matrices if 
present) and the random effects. This is for general nonlinear random 
effects models, but the calculations can be used for linear models as 
well. We calculates these estimates as follows. Let L(x,u)
be the log-likelihood function for the parameters x and u given the 
observed data,
where u is the vector of random effects and x is the vector of the other 
parameters. Let F(x) be the log-likelihood for x after the u have been 
integrated out. This integration might be exact or more commonly via the 
Laplace approximation or something else.
For any x let uhat(x) be the value of u which maximizes L(x,u),
and let xhat be the value of x which maximizes
F(x). The the estimate for the covaraince matrix for the x is then
S_xx = inv(F_xx) and the estimated full covariance matrix Sigma for the 
x and u is given by

S_xx                 S_xx * uhat_x
(S_xx * uhat_x)' uhat' * S_xx * uhat_x + inv(L_uu)

where ' denotes transpose _x denotes first derivative wrt x (note that 
uhat is a function of x so that uhat_x makes sense)and _xx _uu denote 
the second derivatives wrt x and u. we then use Sigma and the delta 
method to estimate the standard deviation of any (differentiable) 
function of x and u.
David A. Fournier
P.O. Box 2040,
Sidney, B.C. V8l 3S3
Phone/FAX 250-655-3364

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