For large data sest, fitme
selects methods that exploits the sparsity of the precision matrix of the random effects.
A call to HLCor
uses the spectral decomposition of the adjacency matrix as further detailed below. This is fast for small datasets but fitme may be preferable otherwise.
A call to corrHLfit
with the additional argument
init.HLfit=list(rho=0)
should be equivalent in speed and result to the HLCor
call.
A call to corrHLfit
without this argument does not use the spectral decomposition. It performs a generic numerical maximization of the likelihood (or restricted likelihood) as function of the correlation parameter \(\rho\). The ML fits by corrHLfit
and HLCor
should be practically equivalent. The REML fits should slightly differ from each other, due to the fact that the REML approximation for GLMMs does not maximize a single likelihood function.
In the adjacency model, the covariance matrix of random effects u can be described as \(\lambda\)(I\(-\rho\) W\()^{-1}\) where W is the (symmetric) adjacency matrix. HLCor
uses the spectral decomposition of the adjacency matrix, written as boldW=VDV' where D is a diagonal matrix of eigenvalues \(d_i\). The covariance of V'u is
\(\lambda\)(I\(-\rho\) D\()^{-1}\), which is a diagonal matrix with elements
\(\lambda_i\)=\(\lambda\)/(1\(-\rho d_i\)). Hence \(1/\lambda_i\) is in the linear predictor form \(\alpha\)+\(\beta d_i\) This can be used to fit \(\lambda\) and \(\rho\) efficiently. If HLCor
is used, the results are reported as the coefficients \(\alpha\) ((Intercept)
) and \(\beta\) (adjd
) of the predictor for \(1/\lambda_i\), in addition to the resulting values of \(\rho\) and of the common \(\lambda\) factor.