[R] polynomials REML and ML in nlme

[Peter Dalgaard]

Spencer Graves <spencer.graves at pdf.com> writes:

>   Am I correct that changing the parameterization should NOT change
> the estimates of the variance components, because you are minimizing
> essentially the same objective function over the same subspace? The
> only thing that changes is the logdet(X'WX) term mentioned by I White?
> Moreover, letting X = QR, and using the fact that det(AB) =
> det(A)*det(B) if they are both square, we get det(R'Q'WQR) =
> det(Q'WQ)*det(R)^2. Thus, the change in parameterization affects only
> R, not Q, which means that it can't affect det(Q'WQ).
> 
> Is this accurate?
> 
> As a simple sanity check, I ran a very simple model with the same
> fixed effects in different parameterizations, and got the same
> estimates for the variance components under REEL but different
> "log-restricted-likelihood" (see below).
> 
> Comments?

Yes, that is accurate. Basically, you are optimizing two functions
that differ by a constant, namely logdet(M). And the expected values
are also identical; if the parameter estimates were betahat, after
reparametrization they become inv(M)betahat.

> >The REML loglikelihood includes a term -(1/2)logdet(X'WX) where X is the
> >design matrix for the fixed effects and W is the inverse covariance matrix
> >for the observations. Under reparametrisation, X becomes XM with M a
> >non-singular matrix, and the REML loglikelihood changes by logdet(M).

-- 
   O__  ---- Peter Dalgaard             Blegdamsvej 3  
  c/ /'_ --- Dept. of Biostatistics     2200 Cph. N   
 (*) \(*) -- University of Copenhagen   Denmark      Ph: (+45) 35327918
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