Dear List:
My question is more statistical than R oriented (although it originates
from my work with nlme). I know statistical questions are occasionally
posted, so I hope my question is relevant to the list as I cannot turn
up a solution anywhere else. I will frame it in the context of an R
related issue.
To illustrate the problem, consider student achievement test score data
with multiple observations available for each student. One way of
modeling these data might be
Y_{ti} = (\mu + \mu_{i} ) + (\beta_0 + \beta_{i} )*(time) +
\epsilon_{ti} ; t indexes time and i indexes student
The nlme code is
tt<-lme(reponse~time, data, random=~time|ID)
With this, I can extract the growth rate for each individual in the data
set. Conceptually this is the sum of the main effect for time plus the
empirical bayes estimate for each individual:
\beta_0 + \beta_{i}
I can use the coef(tt, ...) to extract these coefficients.
Now, assume that I do not want to include random effects associated with
the slope and intercept, but instead use a gls to account for the
variances and covariances through an unstructured covariance matrix.
For example, assume the following model fit to the same data
Y_{ti} = \mu + \beta_0 * (time) + \epsilon_{ti}; where e~N(0, \Sigma)
With Sigma forming a more complex covariance matrix. We can use the gls
option as follows for example,
tt1<-gls(response~time, data, correlation=corSymm(form=~1|ID),
weights=varIdent(form=~1|time))
On p. 254 of P&B, they note that the mixed model "gives as a by-product,
estimates for the random effects, which may be of interest in
themselves". And in my situation they are. Specifically, I want to
estimate the growth rate for each individual student.
My questions boils down to:
1) Is there any way possible to extract or to compute (estimate) the
growth rate of individual i when the data have been modeled using gls?
2) Can anyone point me to an example or reference where this has been
done? I have searched but have really turned up empty handed.
It seems that there must be a methodology for doing so as we are
accomplishing a similar task. Would there be information in the new
covariance matrix, Sigma, that would help play this role?
These only illustrate the issue, the actual model I am dealing with is
more complex, but the issue generalizes. Fitting random effects in the
current model I am dealing with is not a particularly attractive
solution. I actually have the issue layed out in more detail in a paper
I am working on and would be happy to share if requested.
I would appreciate any thoughts you might have on this problem.
Harold
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