Description

Function calculates either log-likelihoods or traces of covariance matrices for comparison with respect to parameter penalties.

Usage

Arguments

Details

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.

Value

An object of class model.comparison is a data frame with either log-likelihoods or covariance traces, plus parameter penalties. AIC scores might be include, if applicable

Author(s)

Michael Collyer

References

Bedrick, E.J., and C.L. Tsai. 1994. Model selection for multivariate regression in small samples. Biometrics, 226-231.

Warton, D.I., 2008. Penalized normal likelihood and ridge regularization of correlation and covariance matrices. Journal of the American Statistical Association. 103: 340-349.

Examples