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Comparing two or more models fit to a common data set. It can be part of the process of "model selection".
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Compare two sets of linear mixed effects models
Predictive power... as in mean squared error, $||Y - \hat{Y}||^2$?
I would just calculate the difference of the two MSE's, and then
cross-validating this (with leave-one-out or so) to get approxima …