Confidence intervals on correlation parameter estimates

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Hi Harlan,
From ?intervals.lme it sounds as though normal approximations are
used, with standard errors coming from the inverse Hessian of the log
(RE)ML function evaluated at the
estimates.  I don't have P&B with me, but IIRC Ch 5 begins with a
discussion of how the likelihood functions are adjusted to accommodate
error covariance parameters.

hth,

Kingsford Jones
Hello all,
When incorporating correlation structures into a lme model with the nlme()
library, one can produce confidence intervals for the correlation parameter
estimates. ?The Pinheiro and Bates book has a simple example:

"We can assess the precision of the correlation parameter estimate in
fm20var.lme with the 'intervals' ?method.

fm1 <- lme(distance ~ age, data =
Orthodont,correlation=corAR1(0.5,fixed=FALSE)) # random is ~ age
intervals(fm1)
Approximate 95% confidence intervals
...
?Correlation structure:
? ? ? ? lower ? ? ? est. ? ? ? upper
Phi -0.7615846 -0.4898957 -0.07174523
attr(,"label")
[1] "Correlation structure:"

?Within-group standard error:
? ?lower ? ? ?est. ? ? upper
0.9002138 1.0900639 1.3199522
"

Any thoughts on how these C.I.s are obtained?
As far as I cant tell, the Pinheiro and Bates book does not go into details,
but perhaps I am wrong.

Any thoughts or references would be greatly appreciated,

Harlan Campbell
McGill University
Montreal, Canada

? ? ? ?[[alternative HTML version deleted]]

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Hi,
Thanks for the reply and advice.  unfortunately when incorporating
correlation structures(e.g. AR(1)) into a mixed model, lme() uses numerical
methods for the calculation of the related parameters(e.g. Phi).  (the
numerical methods used are in the nlminb()).  In any case, it is because no
closed form of the derivative of the log-likelihood with respect to these
parameters is available.  This suggests to me that the inverse Hessian of
the log ML function is also unavailable.

If anyone knows of a source that can confirm or refute any of these
thoughts, that would be GREAT!

Harlan

Don't know of a source, but it's extremely standard in this situations
for the system to use the Hessian estimated by (second) finite differences.
  If you need to know badly enough, why not look at the code ... ?

  good luck,
    Ben Bolker