[R-sig-ME] Numerical methods used to compute correlation coefficients

[Douglas Bates]

On Tue, Apr 14, 2009 at 12:02 PM, H c <harlancampbell at gmail.com> wrote:
> Thank you for the quick response.
> when I refer to "correlation parameters", I mean the "generally small set of
> parameters \lambda" that parametrize the \Lambda_{i} Variance-Covariance
> matrix.
> For example, one has time series data such that every subject has been
> observed at 4 time points.  One wishes to model this using an AR(1)
> correlation structure within the mixed model.
> the AR(1) is parametrized by a fixed parameter, \phi :
> lme(y~X, random=~1|ID, method="ML", data=data,
> correlation=corAR1(0.5,form=~X,fixed=FALSE))

> Since there is no closed form solution for the maximum-likelihood estimate
> of \phi.  what numerical methods are used to arrive at the given estimate?
> Hopefully this has clarified my question.

In the sense that I know what you mean by the correlation parameters.
I'm not sure how to characterize the numerical methods other than to
say that the deviance (negative twice the log-likelihood) or the
corresponding version of the REML criterion is expressed with respect
to an unconstrained parameter and the resulting function optimized
using optimization software within R (nlminb).  You can check the
definition of the corAR family to determine exactly how the
parameterization is defined.

> Thanks again,
> Harlan
>
> On Tue, Apr 14, 2009 at 12:37 PM, Douglas Bates <bates at stat.wisc.edu> wrote:
>>
>> On Tue, Apr 14, 2009 at 9:20 AM, H c <harlancampbell at gmail.com> wrote:
>> > I have have already posted the following question:  What numerical
>> > methods
>> > are used in nlme to estimate correlation parameters?
>> > I was referred to the Pinheiro and Bates book.  Unfortunately, on p.
>> > 202,
>> > section 5.1.1, under the title "Estimation and Computational Methods",
>> > no
>> > description on a numerical method is provided.  (When the data is
>> > transformed to work with the profiled likelihood(y->ystar), one needs
>> > the
>> > parameters that define Lambda.  How are these parameters estimated?)
>>
>> I'm not sure what you mean by "correlation parameters".  If you mean
>> the correlation parameters in the unconditional distribution of the
>> random effects then those are estimated by maximum likelihood (ML) or
>> residual maximum likelihood (REML).  The profiled deviance or the
>> profiled REML criterion is evaluated with respect to a transformed set
>> of parameters and this value is optimized.
>
>