  Comparisons are made for the amount of agreement of the composite likelihood
information criteria and their full likelihood counterparts when making
decisions among the fits of different models, and some properties of penalty
term for composite likelihood information criteria are obtained. Asymptotic
theory is given for the case when a simpler model is nested within a bigger
model, and the bigger model approaches the simpler model under a sequence of
local alternatives. Composite likelihood can more or less frequently choose the
bigger model, depending on the direction of local alternatives; in the former
case, composite likelihood has more "power" to choose the bigger model. The
behaviors of the information criteria are illustrated via theory and simulation
examples of the Gaussian linear mixed-effects model.
