On generalized degrees of freedom with application in linear mixed models selection

The concept of degrees of freedom plays an important role in statistical modeling and is commonly used for measuring model complexity. The number of unknown parameters, which is typically used as the degrees of freedom in linear regression models, may fail to work in some modeling procedures, in particular for linear mixed effects models. In this article, we propose a new definition of generalized degrees of freedom in linear mixed effects models. It is derived from using the sum of the sensitivity of the expected fitted values with respect to their underlying true means. We explore and compare data perturbation and the residual bootstrap to empirically estimate model complexity. We also show that this empirical generalized degrees of freedom measure satisfies some desirable properties and is useful for the selection of linear mixed effects models.