The function calculates either log-likelihoods or traces of (residual)
covariance matrices, plus parameter
penalties, to assist in comparative model evaluation or selection.
Because high-dimensional data often
produce singular or ill-conditioned residual covariance matrices,
this function does one of two things: 1) uses
the trace of a covariance matrix rather than its determinant; or 2)
provides a ridge-regularization (Warton, 2008)
of the covariance matrix, only if it is determined that it is
ill-conditioned. Regardless of implementation,
covariance matrices are projected into a principal component (PC)
space of appropriate dimensions.
The parameter penalty is based on that proposed by Bedrick and Tsai
(1994), equal to 2(pk + p(p + 1)/2), where
p is the appropriate dimension (not number of variables) of the
covariance matrix. The parameter, k,
is the rank of the model design matrix.
In the case that "logLik" is chosen for the argument, type,
AIC scores are calculated. These scores
may not perfectly match other packages or software that
calculate AIC for multivariate data, if ridge regularization
was used (and if other packages require p = the number of
data variables). When choosing logLik as the type of comparison,
it might be a good idea to adjust the tolerance or number of data
principal components. The default (NULL) values will
use all data dimensions to calculate log-likelihoods, which
might cause problems if the number of variables exceeds the number
of observations (producing singular residual covariance matrices).
However, one should not reduce data dimensions haphazardly,
as this can lead to poor estimates of log-likelihood. Furthermore,
using the tolerance argument could result in different
numbers of principal components used for each model to calculate
log-likelihoods, which might be a concern for comparing models.
If both tol and pc.no arguments are used, the solution will use
the fewest PCs produced by either argument. Because the trace
of a covariance matrix is not sensitive to matrix singularity,
no PC adjustment is used for the cov.trace argument.
Users can construct their own tables
from the results but this function does not attempt to
summarize results, as interpreting results requires
some arbitrary decisions. The anova function
explicitly tests multiple models and can be used for nested
model comparisons.
Results can also be plotted using the generic plot
function.
Caution: For models with GLS estimation, the number of
parameters used to estimate the covariance matrix
is not taken into consideration. A generalized information
criterion is currently in development.